statistika socialur
mecnierebebSi
mesame gamocema
__________________
harold kisi
Tbilisi
2007
socialur mecnierebaTa seria
Authorized translation from the English language edition, entitled STATISTICAL CONCEPTS FOR THE BEHAVIORAL SCIENCES, 3rd Edition, ISBN 0205332919, by KIESS, HAROLD O., published by Pearson Education, Inc, publishing as Allyn & Bacon, Copyright © 2002, 1996, 1989 by Allyn & Bacon, A Pearson Education Company. 75 Arlington St. Boston, MA 02116, Internet : www.ablongman.com. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage retrieval system, without permission from Pearson Education, Inc. GEORGIAN language edition published by the UNION – CENTER FOR SOCIAL SCIENCES, Copyright Georgian Translation © 2007 Center for Social Sciences 10, Chovelidze St. Tbilisi 0108 Georgia, E-mail: [email protected] Internet : www.ucss.ge
harold kisis wignis, «statistika socialur mecnierebebSi», inglisuri gamocemis avtorizebuli
Targmani (mesame gamocema, ISBN 0205332919), gamomcemeli Pearson Education, Inc, Allyn & Bacon. saavtoro ufleba inglisur gamocemaze © 2002, 1996, 1989 by Allyn & Bacon, A Pearson Education Company. bostoni, arlingtonis q. 75. indeqsi MA 02116, internet gverdi : www.ablongman.com. yvela saavtoro ufleba daculia. Pearson Education, Inc.–is Tanxmobis gareSe ar SeiZleba wignis romelime nawilis aslis gadaReba, gamravleba an gavrceleba raime formiT, eleqtronuli an meqanikuri saSualebiT, fotokopirebiT an informaciuli Senaxvis raime sxva saSualebiT. wignis qarTuli Targmani gamocemulia kavSir «socialur mecnierebaTa centris» mier. qarTuli Targmani © socialur mecnierebaTa centri, 2007
q. Tbilisi, 0108, T. WoveliZis q. # 10 el. fosta: [email protected] internet gverdi: www.ucss.ge mTargmnelTa jgufi: Teona aslaniSvili maia robaqiZe lili xeCuaSvili nino wereTeli mTavari samecniero redaqtori: elizbar nadaraia samecniero redaqtorebi: nino javaxiSvili qeTevan manjgalaZe saredaqcio jgufis wevrebi: zaza xeCinaSvili TinaTin zurabiSvili enobrivi redaqtori: lia kaWarava dakabadoneba, ydis dizaini: giorgi bagrationi
ISBN: 978-99940-873-1-0 EAN : 9789994087310 ISSN 1512-4576
wignis qarTuli Targmani momzadda da gamoica “socialur
mecnierebaTa centris” (Center for Social Sciences) mier, ×ÏÍÃÉÓ OSI – Zug da budapeStis Ria sazogadoebis institutis umaRlesi ganaTlebis mxardaWeris programis (HESP) finansuri xelSewyobiT.
Georgian translation of the book has been published by the Center for
Social Sciences, sponsored by the OSI-Zug Foundation and the Higher Education Support Program (HESP) of the Open Society Institute Budapest (OSI).
winasityvaoba saqarTveloSi socialuri mecnierebebis ganviTarebis bunebrivi gza meoce saukuneSi sabWoeTis oficialuri ideologiuri wnexis arsebobis gamo mniSvnelovnad Seferxda, ramac seriozuli ziani miayena Tanamedrove standartebis kvaldakval socialur mecnierebebSi kvlevis ganviTarebas. saqarTvelos axali kanoni umaRlesi ganaTlebis Sesaxeb aqtualurad miiCnevs umaRles saswavleblebSi swavlebisa da kvlevis reintegracias, xolo samagistro da sadoqtoro programebis dafinansebis axali modeli mimarTulia kvlevaze dafuZnebuli swavlebis danergvisaken. umaRlesi ganaTlebis axali kanonis Sesabamisad umaRlesi saswavlebelebi da, gansakuTrebiT, universitetebi cdiloben fexi auwyon reformebis process da Secvalon saswavlo programebi axali moTxovnebis Sesabamisad. cvlilebebi TvalsaCinos xdis saswavlo masalebis deficits da gansakuTrebiT ki kvlevis meTodebis mimarTulebiT. harold kisis wigni “statistika socialur mecnierebebSi” erTiandeba saswavlo literaturis im aucilebel CamonaTvalSi, romlis gareSec sruliad SeuZlebelia romelime mkvlevars hqondes seriozuli pretenzia socialur mecnierebebSi mopovebuli monacemebi warmoadginos, rogorc kvlevis sando Sedegi. ufro metic, dResdReobiT tradiciulad ara raodenobriv meTodebze orientirebuli disciplinebic ki, rogoricaa politikuri mecniereba da saerTaSoriso urTierTobebi sadoqtoro programebis kvleviTi seminarebis aucilebel komponentad miiCneven raodenobriv meTodebSi seriozul trenings (amis magaliTad, mxolod eseqsis universitetis gamocdilebac ki sakmarisia, romelic 1967 wlidan yovelwliurad marTavs zafxulis skolas socialur mecnierebaTa monacemebis Segrovebasa da analizSi). saqarTvelos ganaTlebis ministris 2005 wlis #633 brZanebis safuZvelze ganisazRvra bakalavriatis specialobebis CamonaTvali, romelSic socialuri menierebebis grifiT gaerTianebulia Semdegi specialobebebi – “ekonomika, geografia, fsiqologia, sociologia, politikuri mecnierebani, saerTaSoriso urTierTobebi, sazogadoebrivi urTierTobebi, Jurnalistika, sabiblioTeko, saarqivo da sainformacio saqme da sxva” (http://www.mes.gov.ge). TiTqmis yvela es dargi iswavleba saqarTvelos sxvadasxva umaRles saswavlebelsa da universitetSi da harold kisis qarTul enaze Targmnili wigni udavod SenaZenia socialur mecnierebebSi dakavebuli pedagogebisa da studentebisaTvis. socialur mecnierebaTa centri 2002 wlidan gamoscems socialur mecnierebaTa serias, romlis mizania socialur mecnierebebSi qarTul enaze saswavlo da samecniero literaturis danaklisis Sevseba da samecniero enisa da terminologiis ganviTareba. 2004 wels pirvelad qarTul enaze gamoica politikur da socialur mecnierebaTa terminTa
leqsikoni-cnobari. socialur mecnierebaTa seriiT ukve gamocemulia 16 damxmare saxelmZRvanelo da 23 gafarToebuli saswavlo saleqcio kursi. statistikis es saxelmZRvaneloc am seriis nawilia. harold kisis wignis es Targmani mravali adamianis Zalisxmevis Sedegia. TiToeuli maTganis gansakuTrebuli wvlilis gareSe Zneli iqneboda samuSaos bolomde miyvana. socialur mecnierebaTa centri gansakuTrebul madlobas moaxsenebs mTavar samecniero redaqtors baton elizbar nadaraias; samecniero redaqtorebs qalbatonebs nino javaxiSvilsa da qeTevan manjgalaZes, baton zaza xeCinaSvils; enobriv redaqtors qalbaton lia kaWaravas; mTargmnelebs qalbatonebs Teona aslaniSvils, maia robaqiZes, nino wereTelsa da lili xeCuaSvils; socialur mecnierebaTa centris mTargmnelobiTi programis xelmZRvanels qalbaton manana miqaberiZes. Targmnis procesSi TiToeuli terminis Sesatyvisis moZebnasa da SeTanxmebas didi dro daeTmo. Tu socialuri mecnierebebisaTvis terminologia aqilevsis quslia statistikis terminologiuri aparati nadvilad arsebobs, magram rig SemTxvevebSi is an rusuli kliSea. magaliTad, stiudentis kriteriumi; an erTi da igive termini sxvadasxva konotaciiT gvxvdeba (erTi mxriv, uwyveti da wyvetili cvladi, da meore mxriv, uwyveti da diskretuli cvladi), an sxvadasxva sityviT aRiniSneba erTi da igive Sinaarsi (magaliTad sandooba da mniSvnelovneba). samecniero redaqtorebis mier mizanSewonilad iqna miCneuli, rom teqstSi moyvanili yofiliyo terminis meore variantic gaugebrobebisa da daumsaxurebeli SeniSvnebis Tavidan asacileblad. vimedovnebT, rom proeqtis ganxorcielebaSi Cadebuli dro da energia, samecniero redaqtorebis, mTargmnelebisa da organizatorebis mondomeba saTanado nayofs gamoiRebs da Sedegad qarTvel studentebs statistikis Sesaswavlad SesaniSnavi saxelmZRvanelo eqnebaT da uaxloes momavalSi socialur mecnierebebSi axali Taobis mkvlevarTa mier ganxorcilebuli kvlevebi maqsimalurad iqneba dazRveuli zedapiruli spekulaciebisa da artefaqtuli daskvnebisagan. marine CitaSvili, Tsu profesori socialur mecnierebaTa centris direqtori Tbilisi, 2006
1 w i n a s i t y v a o b a
winamdebare saxelmZRvanelos pirveli da meore gamocemis mizani ZiriTadi
statistikuri meTodebis konceptualuri ganviTarebis aRwera da bihevioristul
mecnierebebSi maTi gamoyeneba iyo. am miznis misaRwevad mkveTri aqcenti davsviT
Tanamedrove kvlevaSi arsebuli problemebis bihevioristul mecnierebebSi
gamoyenebaze, radgan migvaCnia, rom realurad arsebuli problemebi ufro
adekvaturad axdenen mecnierul kvlevaSi statistikuri analizis aucileblobis
ilustrirebas. saxelmZRvanelos winamdebare, mesame gamocema wina gamocemebis
gadasinjvaa, Tumca, SevecadeT, rom saxelmZRvanelos mizani ucvleli darCeniliyo.
teqstis organizacia
winamdebare gamocemis struqtura warmodgenili cnebebis Tanmimdevrulad
mowodebis principiTaa agebuli. statistikis gamoyeneba warmodgenilia
konceptualurad, logikuri Tanmimdevrobis dacviT. pirvel xuT TavSi aRwerilia
statistikis, qcevis gazomvis, qulebis sixSireTa ganawilebis saxiT, grafikulad
warmodgenis saWiroeba da centraluri tendenciisa da cvalebadobis sazomebi. me-6
da me-7 TavebSi warmodgenilia normaluri ganawileba, albaToba, SerCeviTi
ganawilebebi da statistikis gamoyeneba parametrebis SefasebisTvis. statistikuri
hipoTezebis Semowmebis safuZvlebi warmodgenilia me-8 TavSi .z kriteriumiTa da
erTi SerCevis t kriteriumiT. me-9-12 Tavebi eZRvneba saSualoebs Soris gansxvavebis
dadgenis mizniT hipoTezebis Semowmebas, parametruli kriteriumis gamoyenebiT.
detalurad aris warmodgenili dispersieuli analizi, vinaidan dReisaTvis is
yvelaze farTod gamoyenebadi statistikuri kriteriumia bihevioristul
mecnierebebSi da studentebma aucileblad unda icodnen misi gamoyeneba. srulad
aris aRwerili cdis pirTa Soris erTfaqtoriani, cdis pirebs SigniT
erTfaqtoriani da cdis pirTa Soris faqtoruli dispersiuli analizi; aseve,
aRwerilia mravaljeradi Sedarebis kriteriumebis gamoyeneba da gazomvis efeqtis
siZliere. garda amisa, mniSvnelovani yuradReba eTmoba urTierTqmedebis
interpretacias faqtorul analizSi. me-13 da me-14 Tavebi korelaciasa da regresias
eZRvneba. me-13 TavSi ganxilulia pirsonisa da spirmenis korelaciis koeficientebi
da TiToeul maTganTan dakavSirebuli statistikuri hipoTezebis Semowmeba. me-14
TavSi mocemulia martivi wrfivi regresiis safuZvlebis mimoxilva. am ori Tavis
saxelmZRvanelos bolosken moTavseba SeiZleba ucnaurad mogeCvenoT, vinaidan
mraval sxva saxelmZRvaneloSi am TavebSi mocemuli sakiTxebis ganxilva aRweriTi
statistikis Semdeg aris xolme mocemuli; Tumca, migvaCnia, rom es Cveuli
Tanmimdevroba studentebs garkveul sirTuleebs uqmnis. iseTi aRweriTi
statistikebis ganxilvas, rogoricaa saSualo da standartuli gadaxra, bunebrivad
2 mivyavarT SerCevis saSualoebs Soris arsebuli gansxvavebis statistikuri
Semowmebis ganxilvisken; aRweriT statistikebsa da statistikuri hipoTezebis
Semowmebas Soris korelaciisa da regresiis sakiTxebis Casma logikur jaWvs wyvets.
amrigad, vinaidan saxelmZRvanelo cnebebis Tanmimdevrulad mowodebis principiTaa
agebuli, korelaciisa da regresiis sakiTxebi parametruli statistikur
kriteriumebis ganxilvis Semdeg movaTavseT. saxelmZRvanelo me-15 TaviT sruldeba,
sadac warmodgenilia sami araparametruli statistikuri kriteriumi: _ xi-kvadratis
kriteriumi, man-uitnis U kriteriumi da uilkoqsonis niSniani rangebis kriteriumi.
saxelmZRvanelos maxasiaTeblebi
studentebi statistikis rTuli Teoriuli safuZvlebis Seswavlisas mraval
problemas awydebian. winamdebare saxelmZRvanelos aqvs ramdenime meToduri midgoma,
romelic am problemebis daZlevaSi dagexmarebaT da saSualebas mogcemT, rom
warmodgenili masala ukeT gaigoT da realur kvlevebs daukavSiroT.
realuri problemebi
yvela statistikuri meTodi bihevioristuli kvlevis konteqstSia warmodgenili.
mravali problema Tanamedrove publikaciebidan aris amoRebuli; amasTan, wyaro
imdenad srulad aris dasaxelebuli, rom dainteresebul students Tavisuflad
SeuZlia misi moZieba.
statistikebis konceptualuri ganviTareba
mravali statistika cnebiT doneze, ganmartebiTi formulis gamoyenebiT aris
warmodgenili. mniSvnelovania, rom studentebma TvalsaCinod dainaxon, Tu rogor
moqmedebs mocemuli statistika monacemebze. amis gaTvaliswinebiT, SevecadeT, rom
ama Tu im statistikis aRwerisas studentebi ar gadagvetvirTa maTematikuri
formulebiT. saxelmZRvanelos pirvel TavebSi aRwerili statistikebi mxolod
martiv maTematikas saWiroeben, Tumca, studentebs mainc aSinebT xolme pirvel TavSi
warmodgenili maTematikis sawyisebi. amitom winamdebare saxelmZRvaneloSi
maTematikuri cnebebi mxolod saWiroebis SemTxvevaSi aris mocemuli da axsnili;
xolo maTTvis, vinc ufro metad dainteresdeba, maTematikuri operaciebi srulad
aris mocemuli b danarTSi.
„SeamowmeT Tqveni codna“ da „kiTxvebi“
TiToeul TavSi gafantulia qveTavebi _ „SeamowmeT Tqveni codna“, Tavis bolos ki
warmodgenilia qveTavi _ „kiTxvebi“. am qveTavebSi mocemuli kiTxvebi studentebs
saSualebas aZlevs, gaimeoron masala da Seamowmon, Tu rogor gaiges da aiTvises.
bevr kiTxvaSi realuri kvlevebidan aRebuli problemebia warmodgenili. amrigad,
3 maT ara mxolod mimoxilvis funqcia aqvT, aramed sxvadasxva statistikuri meTodis
gamoyenebis magaliTebsac iZlevian. studentebs vurCevT, rom axal masalaze
gadasvlamde srulad upasuxon kiTxvebs da Seasrulon mocemuli savarjiSoebi.
prezentacia samecniero statiebSi
bevri statistikuri kriteriumis ganxilvisas mocemulia amonaridebi samecniero
Jurnalebidan da axsnilia maTSi warmodgenili informacia; amas Tan axlavs
samecniero JurnalSi gamoqveynebuli kvlevis Sedegebis wakiTxvis savarjiSoebi.
saxelmZRvanelos winamdebare gamocemaSi es maxasiaTebeli ufro farTod aris
warmodgenili da Seesabameba amerikis fsiqologiuri asociaciis publikaciebis
saxelmZRvanelos me-4 gamocemaSi (Publication Manual of the American Psychological Association.
American Psychological Association, 1994) mocemul saredaqcio stils.
teqstis Tanmxlebi ganmartebebi da terminebis leqsikoni
winamdebare gamocemaSi Sevida mniSvnelovani terminebis ganmartebebi, romlebic
Tan axlavs teqsts, da am terminTa leqsikoni. mniSvnelovani terminebis Srifti
gamuqebulia da teqstSia ganmartebuli. saswavlo masalis gansamtkiceblad es
ganmartebebi kidev erTxel aris mocemuli teqstis Semdeg. saxelmZRvaneloSi
gamoyenebuli yvela termini ganmartebulia g danarTSi.
statistikuri simboloebi
statistikaSi gamoyenebuli simboloebis mravalferovnebam SeiZleba iolad
dagabnioT. statistikis enis aTvisebisas daxmarebas gagiwevT d danarTi, romelSic
Tavmoyrilia saxelmZRvaneloSi warmodgenili yvela simbolo.
Tanmimdevroba
teqsti Tanmimdevrulad ikiTxeba, mas ar wyvets CarCoSi Casmuli CanarTebi.
TiToeuli qveTavi wina masalidan gamomdinareobs. am uwyvetobis gaumjobesebis
mizniT, dispersiuli analizis nedli qulebis gamosaTvleli formulebi Tavis
boloSi, statistikur damatebaSia mocemuli. amrigad, maTTvis, vinc yuradRebis
miRma tovebs gamosaTvlel formulebs (kompiuteruli statistikuri programebis
gamoyenebis SemTxvevebSi), ar irRveva teqstis Tanmimdevruloba.
vimedovnebT, rom am saxelmZRvaneloTi statistikis Seswavla saintereso da
gamosadegi iqneba TqvenTvis.
madloba1
am saxelmZRvalos SeqmnaSi mraval adamians miuZRvis wvlili. pirveli gamocemis
idea alen uorkmens (Allen Workmen, gamomcemlobidan „Allyn and Bacon“) ekuTvnis,
1 ibeWdeba mcireodeni SemoklebiT (mTargmnelis SeniSvna).
4 romelmac gamoTqva mosazreba, rom fsiqologiuri kvlevis meTodebis
saxelmZRvanelodan _ „fsiqologiuri kvlevis meTodebi, konceptualuri midgoma“ _
aRebul midgomas viyenebdi; es saxelmZRvanelo Cemi da doqtor duglas u.
blumquisTis (Dr. Douglas W. Bloomquist) TanaavtorobiT gamoica. aqedan gamomdinare,
winamdebare saxelmZRvaneloSi gamoyenebuli mravali konceptualuri mosazreba
xsenebul saxelmZRvaneloSi warmodgenilis msgavsia. me madlobas movaxseneb duglas
blumquisTs am mosazrebaTa CamoyalibebaSi Setanili wvlilisTvis da masalisTvis;
aseve, imisTvis, rom man am masalis winamdebare saxelmZRvaneloSi gamoyenebis neba
damrTo. gamomcemlobis _ „Allyn and Bacon“ vice-prezidentma da gamomcemelma suzan
bedgerma (Susan Badger) daamtkica es gamocema da amiT proeqtis dasrulebis impulsi
momca. „Allyn and Bacon“-is aRmasrulebelma redaqtorma fsiqologiis dargSi laura
pirsonma (Laura Pearson) mniSvnelovani roli Seasrula wignis saboloo saxiT
CamoyalibebaSi. saxelmZRvanelos gadasinjvisas misi ideebi da winadadebebi yvelaze
sasargeblo aRmoCnda. redaqtoris asistenti jenifer normandini (Jennifer Normandin)
xalisiT da moxerxebulad ukmlavdeboda xelnaweris momzadebasTan dakavSirebul
Cem mier dasmul mraval sakiTxs. madlobas movaxseneb Allyn and Bacon“-is prezidents
bil barks (Bill Barke), romelmac 12-ze meti wlis win damarwmuna da wamaxalisa, rom
kolejis saxelmZRvanelos dawera mecada.
warmoebis ZiriTad process moxerxebulad gaumklavdnen sintia niubai (Cynthia
Newby) Chestnut Hill Enterprises, Inc. da raia maisi (Raeia Maes) Maes Associates. uiliam o.
tomasi (William O. Thomas) yvelaze aqtiurad CarTuli da gulmodgine redaqtori iyo,
romelmac daxvewa Cemi nawerebi, madlierebiT minda movixsenio.
garda amisa, winamdebare saxelmZRvanelos SeqmnaSi Cemma mravalma profesionalma
kolegam miiRo monawileoba. zogierTma maTganma pasuxebi gasca detalur kiTxvars,
romelic Sinaarssa da midgomebis sxvadasxva variants Seexeboda. sxvebma naxes
xelnaweris sruli varianti an misi fragmentebi. maT mier nawerebis guldasmiT
gadamowmebam fasdaudebeli samsaxuri gamiwia azris naTlad gadmocemaSi da Tavidan
amarida konceptualuri da maTematikuri Secdomebi. cxadia, saxelmZRvaneloSi
darCenil nebismier uzustobaze mxolod me vageb pasuxs. gaweuli daxmarebisTvis
madlobas vuxdi: pol amrheins niu meqsikos universitetidan (Paul Amrhein, University of
New Mexico), denis kogans texasis teqnikuri universitetidan (Dennis Cogan, Texas Tech
University), mark fainmens, samxreT koneqtikutis saxelmwifo universiteti (Mark Fineman,
Southern Connecticut State University), maik greles, centraluri misuris saxelmwifo
universiteti (Mike Grelle, Central Missouri State University), rik hoils, kentukis
universitetidan (Rick Hoyle, University of Kentucky), artur d. kemfs centraluri misuris
saxelmwifo universiteti (Arthur D. Kemp, Central Missouri State University), melani m. kerCers,
sam hiustonis saxelmwifo universiteti (Melanie M. Kercher, Sam Houston State University),
robert langs, centraluri floridis universiteti (Robert Lange, University of Central
5 Florida), j. u. mur umcross, marSalis universiteti (J.W.Moore, Jr., Marshall University),
jon riCs, fulertonis kaliforniis universiteti (Jon Rich, California State University at
Fullerton), riCard serks, tulsas koleji (Richard Serkes, Tusla Junior College), n. s. silvers,
centraluri floridis universiteti (N. C. Silver, University of Central Florida), david d.
simfsons, karolis koleji (David D. Simpson, Carroll College), maikl terels, stounhilis
koleji (Michael Tirrell, Stonehill College) da pol iarnolds, Crdilo-dasavleTis
universitetis samedicino skola (Paul Yarnold, Northwestern University Medical School).
aseve, madloba minda gadavuxado CRC Press, Inc.-s danarTSi mravali statistikuri
cxrilis gamoyenebis nebarTvis mocemisTvis.
da bolos, madlobas vuxdi maT, viszec yvelaze metad imoqmeda winamdebare
saxelmZRvaneloze muSaobam _ Cems Svilebs, qimberlisa da jefris, da Cems meuRles
sandras. maTi mxardaWeris, moTminebis, waxalisebisa da siyvarulis gareSe es wigni
verasodes dasruldeboda da ver ixilavda dRis sinaTles.
h.o.k.
Tavi I. statistikis Sesavali 6
Tavi I. statistikis Sesavali
statistikis Seswavla: rogor gamoviyenoT winamdebare
saxelmZRvanelo
mniSvnelovani terminebi
SeamowmeT Tqveni codna
kiTxvebi
maTematikuri cnebebi
ra aris statistika
statistikis gamoyeneba: sami magaliTi
aRwera da daskvna
eqsperimenti da statistikuri hipoTezis Semowmeba
korelacia da regresia
ratom aris saWiro statistika
Tavi I. statistikis Sesavali 7
„o, es saSineli maRviZara“, _ buzRuniT aCumebT saaTs da cdilobT, kidev cota
xans daiZinoT. ra mokle Rame iyo! rogorRac dgebiT da acnobierebT, rom Tu
xanmokle SxapiTa da swrafi sauzmiT SemoifarglebiT, ar daigvianebT leqciaze,
romelic 8 saaTzea daniSnuli. manqanaSi Cajdomamde amCnevT, rom adreuli
gazafxulis sasiamovno, Tbili dila gaTenebula; gacilebiT ukeTesi, vidre
guSindeli civi, wvimiani dRe iyo. amasobaSi gzatkecilze gadixarT _ moZraoba
Cveulebrivze metadaa gadatvirTuli da ise swrafad ver moZraobT, rogorc
moelodiT. daigvianebT mecadineobaze? sabednierod, ar dagagviandaT. dRes erT-erTi
saukeTeso leqcia gqondaT mTeli semestris ganmavlobaSi, magram gaSinebT
laboratoriuli samuSao, romelic 8 saaTzea daniSnuli; is xSirad imdenad didxans
grZeldeba, rom waxemsebisTvis dro aRar grCebaT xolme; Tumca, dRes yvelaferi
sxvagvarad, SesaniSnavad aris da droulad gamoxvediT mecadineobidan, amitom
sakmarisi dro dagrCaT da or saukeTeso megobarTan erTad auCqareblad da
mxiarulad waixemseT. Semdeg samuSaod midixarT did savaWro centrSi.
guSindelisgan gansxvavebiT, momxmareblebi dRes Zalze Tavazianebi arian, albaT,
kargi amindis gamo. sasiamovno momxmareblebTan urTierTobaSi SuadRec male modis,
Tqven isev gzatkecilze xarT da saxlisken miiCqariT. dRes Cveulebrivze gacilebiT
naklebad datvirTulia moZraoba, albaT, yvela adre wavida samsaxuridan. samecadino
bevri gaqvT, amitom sadilis Semdeg im imediT sxdebiT magidasTan, rom nayofierad
imecadinebT mTeli saRamo. vinaidan (yovel SemTxvevaSi, sadilis Semdeg mecadineobis
sxva mraval SemTxvevasTan SedarebiT) asec xdeba, grZnobT, rom namdvilad mzad
xarT kviris bolos daniSnuli statistikis gamocdisTvis. ajildovebT sakuTar Tavs
_ erT saaTs televizorTan atarebT da dasaZineblad midixarT. cota adre Tu
daiZinebT, iqneb Rame ufro grZeli geCvenoT.
es mokle epizodi, albaT, Tqveni cxovrebis erT Cveulebriv dRes asaxavs an ar
asaxavs. nebismier SemTxvevaSi, albaT, dagainteresebT, ra kavSirSia es epizodi
statistikasTan. is gviCvenebs, rom Tqven cxovrebaSi amCnevT garkveul
kanonzmierebas: _ drois Cveuli odenoba, romelsac uTmobT Sxapsa Tu sauzmes,
adreuli gazafxulisTvis Cveuli amindi, kolejisken mimavali gzisTvis
damaxasiaTebeli garkveulwilad gadatvirTuli moZraoba da dro, romelic
gWirdebaT kolejamde misvlisTvis, mecadineobis sainteresoobis zogadi done,
laboratoriuli samuSaos saSualo xangrZlivoba, mxiarulebis Cveuli zoma
waxemsebisas, savaWro centris momxmarebelTa tipuri qceva da saRamos
mecadineobisas Cveuli nayofiereba. amCnevT, rom Tqvens cxovrebaSi aris
kanonzomiereba da sistematuroba da es faqti imazec miuTiTebs, rom
cvalebadobasac ganicdiT. zogierTi Rame, danarCenTan SedarebiT, ufro grZeli
geCvenebaT, dRidan dRemde varirebs im drois xangrZlivoba, romelsac Sxapis
miRebasa da sauzmes uTmobT, weliwadis am dros iSviaTad aris kargi amindi,
Tavi I. statistikis Sesavali 8
gzatkecilze moZraoba met-naklebad gadatvirTulia xolme, sxvadasxva dRes (da
sxvadasxva kursze) mecadineoba met-naklebad saintereso SeiZleba iyos,
laboratoriuli samuSao zogjer saSualoze didxans grZeldeba, zogjer ki _ ufro
mcire xans, myidvelTa qceva sakmaod cvalebadia da zogjer ufro nayofierad
mecadineobT, vidre sxva dros.
movlenebis es cvalebadoba garemoSi cvladebis arsebobas gulisxmobs. cvladi
aris garemos nebismieri mdgomareoba an movlena, nebismieri stimuli, pirovnuli
maxasiaTebeli an qceva, romelmac SeiZleba miiRos sxvadasxva mniSvneloba
sxvadasxva dros an sxvadasxva
adamianTan. cvladia im drois xangrZlivoba,
ramden xansac geZinaT, Sxapis miRebis
xangrZlivoba, gzatkecilze moZraobis
intensivoba, mecadineobis sainteresoobis done,
laboratoriuli samuSaos xangrZlivoba,
myidvelTa guneba-ganwyobileba da mecadineobis
nayofiereba. statistika aris mecniereba, romelic poulobs raRac mudmivs da am
mudmivis mimarT cvladebadobas zomavs. miuxedavad imisa, rom statistikis aseTi
gansazRvreba araformaluri da arasrulia, am etapze sruliad sakmarisia.
cxadia, Tqven ar xarT erTaderTi adamiani, romelic amCnevs movlenaTa
kanonzomierebebs. farTo interesebis mqone belgiel profesor lambert adolf Jak
ketles (Lambert Adomphe Jacques Quetelet, 1796-1874) miaCnda, rom cvalebadobis aseT
kanonzomierebebs Tavisi mizezebi aqvs da Tu is daadgenda am kanonzmierebebs, Semdeg
maTi mizezebis povnasac SeZlebda. amrigad, man ganizraxa am kanonzomierebebis
gamovlena da muSaoba daiwyo imis daxasiaTebiT, rasac Tavad „saSualo adamians“
uwodebda. amisaTvis man didi raodenobis frangi jariskacebisgan Sekriba iseTi
cvladebis anTropologiuri monacemebi (anu sxeulebrivi gazomvebis Sedegad
miRebuli monacemebi), rogoricaa wona da simaRle (Stigler, 1986). mas imedi hqonda, rom
saSualo individis daxasiaTebiT is SeZlebda adamianis maxasiaTeblebis
cvladi (variable): _ garemos
nebismieri mdgomareoba, nebismieri
stimuli, pirovnuli maxasiaTebeli
an qceva, romelmac SeiZleba miiRos
sxvadasxva mniSvneloba sxvadasxva
dros an sxvadasxva adamianTan.
cxrili N1.1. gasamarTlebaTa sixSire safrangeTis sasamarTloSi, rogorc wlisa da
braldebulis wera-kiTxvis unaris funqcia
braldebulis wera-kiTxvis unari weli
saSualo 1828 1829 1830
ar SeuZlia wera an kiTxva
SeuZlia wera-kiTxva, magram arasrulyofilad
kargad wers da kiTxulobs
aqvs umaRlesi ganaTleba
0.63
0.62
0.56
0.35
0.63
0.60
0.55
0.48
0.62
0.58
0.52
0.37
0.627
0.600
0.543
0.400
ketles monacemebis mixedviT. adaptirebulia s. m. stigleris (S.M.Stigler, 1986) Sromidan statistikis istoria: ganusazRvrelobis gazomva 1900 wlamde, cxrili N5.2.
Tavi I. statistikis Sesavali 9
kanonzomierebebis povnas da Seadarebda mamakacebisa da qalebis saSualoebs, an
moaxdenda Sedarebas sacxovrebeli adgilis, asakisa da rasis mixedviT. ketles
saSualo adamianiT daintereseba vrceldeboda imazec, rasac igi „moralis
statistikas“ uwodebda anu kanonzomierebebs iseT cvladebSi, rogoricaa suicidis
an danaSaulis sixSire da gasamarTlebis sixSire. magaliTad, is wlebis manZilze
Seiswavlida safrangeTis sasamarTloSi gasamarTlebaTa sixSires da moaxdina maTi
kategorizacia imis mixedviT, Tu ramdenad SeeZlo braldebuls wera-kiTxva. Sedegebi
mocemulia cxrilSi N1.1, romelSic ricxviTi mniSvnelobebi albaTobebs gamoxatavs.
magaliTad, Tu Tqven safrangeTis sasamarTloSi 1829 wels gqondaT saqme da mxolod
arasrulyofilad SegeZloT wera-kiTxva, albaToba imisa, rom Tqven damnaSaved
gcnobdnen, iqneboda 0.60 (anu 6 Sansi 10-dan). yuradReba miaqcieT, rom aq Cans aSkara
kavSiri (anu Sesabamisoba) braldebulis wera-kiTxvis unarebsa da damnaSaved cnobis
albaTobas Soris. am statistikuri kvlevebis dawyebisTvis ketles xSirad
socialuri mecnierebis mamas uwodeben (Dudycha & Dudycha, 1972).
statistikis Seswavla: rogor gamoviyenoT winamdebare
saxelmZRvanelo
rogorc SesavalSi vnaxeT, statistika Seexeba movlenaTa kanonzmierebebs da
cvalebadobas, romelsac yvela Cvengani amCnevs cxovrebaSi. amrigad, garkveuli
azriT, yvelani damwyebi statistikosebi varT: miuxedavad amisa, bevr adamians
afrTxobs statistikis srul kursze fiqri, Tumca, es ase ar unda iyos. statistikis
Seswavla axali enis aTvisebas hgavs. statistikis ena sityvebisgan, simboloebisgan
da formulebisgan Sedgeba. nebismieri axali enis msgavsad, is seriozul
mecadineobas moiTxovs am komponentebis dasaZlevad da asaTviseblad, magram
praqtikasTan da gamocdilebasTan erTad is iseTive bunebrivi unda gaxdes TqvenTvis,
rogorc mSobliuri ena. winamdebare saxelmZRvanelos zogierTi Tavisebureba
daxmarebas gagiwevT statistikis Seswavlisas.
mniSvnelovani terminebi da simboloebi
teqstSi warmodgenili mniSvnelovani terminebi gamuqebulia da Semdeg calkea
gamotanili maTi ganmartebebi. rodesac teqstSi SegxvdebaT gamuqebuli termini,
unda mixvdeT, rom is mniSvnelovania statistikis gagebisaTvis. nu gaagrZelebT
teqstis kiTxvas, sanam ar SeiswavliT termins. amasTanave, terminTa leqsikoni
mocemulia g danarTSi. Tu teqstSi SegxvdebaT termini, romlis mniSvneloba
dagaviwydaT, CaixedeT am leqsikonSi. statistikuri simboloebis leqsikoni
mocemulia d danarTSi.
Tavi I. statistikis Sesavali 10
qveTavebi „SeamowmeT Tqveni codna“
TiToeul TavSi mocemulia qveTavebi _ „SeamowmeT Tqveni codna“, romlebic
SesaZleblobas gaZlevT, rom Semdeg qveTavze gadasvlamde gaimeoroT masala da
naxoT, Tu ramdenad gaigeT is. am kiTxvaTa didi nawili mimarTulia kvlevis realur
problemebze. Sesabamisad, maTSi mocemulia Tqvens mier ukve naswavli statistikuri
meTodebis gamoyenebis magaliTebi. am qveTavebis kiTxvebze pasuxebis didi nawili
mocemulia saxelmZRvanelos bolo e danarTSi _ amosaxsneli amocanebis pasuxebi.
kiTxvebi
TiToeuli Tavis bolos mocemulia ganvlili masalis gasameorebeli kiTxvebi.
rogorc qveTavebis „SeamowmeT Tqveni codna“ SemTxvevaSi, aqac SegiZliaT upasuxoT
ganvlili masalis gasameorebel kiTxvebs da SeadaroT Tqveni pasuxebi e danarTSi
mocemul pasuxebs.
maTematikuri cnebebi
winamdebare saxelmZRvaneloSi gamoyenebulia elementaruli maTematika, romelic
mxolod Sekrebis, gamoklebis, gamravlebisa da gayofis operaciebs saWiroebs.
maTematikuri simboloebi da operaciebi imdenad aris axsnili, ramdenadac saWiroa
isini. Tu raime problema SegeqmnebaT romelime maTematikur operaciasTan
dakavSirebiT, naxeT danarTi b _ maTematikuri cnebebi, sadac ganxilulia ZiriTadi
maTematikuri simboloebi da operaciebi.
ra aris statistika
ketles Semdeg
bihevioristuli mecnierebis
warmomadgenlebi cdiloben
adamianisa da cxovelis
qcevis gagebasa da axsnas.
isini amas monacemebis
Segrovebis saSualebiT
akeTeben. monacemebi aris is
qulebi an qcevis Tu
maxasiaTeblebis gazomvis
Sedegebi, romlebic miiReba
adamianebze an cxovelebze
dakvirvebis Sedegad.
procedurebs, romlebic gamoiyeneba monacemebis mimarT maTi sistematurobis da
cvalebadobis dasadgenad, statistika ewodeba. amrigad, statistika niSnavs im
meTodebsa Tu procedurebs, romlebic gamoiyeneba monacemebis mowesrigebis,
analizisa da maTgan daskvnebis gamosatanad. statistika (statistikuri sidide) aris
monacemebi (Data _ mravlobiTi, datum _ mxolobiTi): _ is
qulebi an qcevis Tu maxasiaTeblebis gazomvis Sedegebi,
romelic miiReba adamianebze an cxovelebze dakvirvebis
Sedegad.
statistika (mecniereba) (Statistics): _ is meTodebi Tu
procedurebi, romlebic gamoiyeneba monacemebis
sistematizaciisTvis, gasaanalizeblad da maTgan daskvnebis
gamosatanad.
statistika (statistikuri sidide) (Statistic): _ monacemebze
dayrdnobiT gamoTvlili calkeuli ricxvi, romelic
SeiZleba gamoyenebul iqnes SerCevis monacemTa
erTobliobis aRsawerad, an am monacemTa ufro sruli
gaanalizebisTvis.
Tavi I. statistikis Sesavali 11
monacemebze dayrdnobiT gamoTvlili calkeuli ricxvi, romelic SeiZleba
gamoyenebul iqnes SerCevis monacemTa erTobliobis aRsawerad an am monacemTa ufro
sruli gaanalizebisaTvis.
statistikis gamoyeneba: sami magaliTi
bihevioristul mecnierebebSi statistikur procedurebs sxvadasxvagvarad iyeneben.
qvemoT mocemulia bihevioristul mecnierebaSi statistikuri meTodebis gamoyenebis
sami magaliTi. aseve, mocemulia mTeli rigi terminebi, romlebic Semdeg SegxvdebaT
teqstSi; amitom unda iswavloT isini. ar aReldeT, Tu ver gaigebT warmodgenili
magaliTebis yvela detals. TiToeuli midgomis Sesaxeb ufro dawvrilebiT da
srulad Semdeg TavebSi visaubrebT.
aRwera da daskvna
yoveldRiurad presidan didi raodenobiT monacemTa statistikur aRwerebs
vecnobiT. magaliTad, gazeTi USA Today gvTavazobs 1980-iani wlebis tipuri
individebis saqmianobaTa aRweras („The Way“, 1985). erT-erTi SemoTavazebuli
daxasiaTebis mixedviT, saSualo ojaxSi yoveldRiurad televizori CarTuli iyo 7-
12 saaTis ganmavlobaSi. es ricxobrivi maCvenebeli gviCvenebs statistikuri
meTodebis gamoyenebis or mniSvnelovan gzas: _ aRwerasa da daskvnas. miuxedavad
imisa, rom ricxobrivi maCvenebeli _ 7-12 saaTi mowodebulia mTlianad aSS-is
ojaxebis populaciis dasaxasiaTeblad im drois xangrZlivobis mixedviT, romlis
manZilzec yoveldRiurad CarTulia televizori, is ar aris miRebuli qveyanaSi
mcxovrebi yvela ojaxze dakvirvebisa da gazomvis Sedegad. metic, yvela ojaxis
mxolod nawils, anu SerCevas daakvirdnen da gazomes. monacemTa arsebuli
erToblioba miiRes im saaTebis raodenobis Caweris Sedegad, ramden xansac hqondaT
yoveldRiurad CarTuli televizori SerCevaSi Sesul ojaxebs. amis Semdeg SerCevaSi
Semavali sxvadasxva ojaxidan miRebuli monacemebi Seajames anu gamoTvales erTi
ricxvi, aRweriTi statistika _ 7-12 saaTi. es statistika mowodebulia im saaTebis
yovedRiuri raodenobis aRsawerad, romlis ganmavlobaSic SerCevaSi Semaval tipur
ojaxs CarTuli aqvs televizori. Semdeg am statistikis safuZvelze askvnian, anu
zomaven im saaTebis raodenobas, romlis ganmavlobaSic yoveldRiurad CarTuli aqvs
televizori aSS-Si mcxovrebi yvela ojaxis populaciaSi Semaval tipur ojaxs.
Cven gamoviyeneT terminebi populacia da SerCeva. populacia aris saerTo
maxasiaTeblis mqone adamianTa, cxovelTa an
movlenaTa sruli erToblioba. magaliTad,
aSS-is kolejis yvela studenti Seadgens
populacias. am populaciis saerTo
maxasiaTebelia is, rom masSi Sesuli yvela individi swavlobs kolejSi. aseve,
populaciis magaliTia aSS-is yvela ojaxi, CrdiloeT karolinis yvela mozrdili
populacia (Population): _ aris saerTo
maxasiaTeblis mqone adamianTa, cxovelTa
an movlenaTa sruli erToblioba.
Tavi I. statistikis Sesavali 12
qali, nebraskis fermerebi an yvela adamiani, visac erTi an meti sabanko sakredito
baraTi aqvs; TiToeuli dasaxelebuli jgufi Seadgens populacias. individTa
nebismieri jgufi SeiZleba ganvixiloT rogorc populacia mas Semdeg, rac
mocemuli saerTo maxasiaTeblis mqone yvela individi Sesuli iqneba jgufSi.
bihevioristuli mecnierebis warmomadgenlebs xSirad populaciis daxasiaTeba
iseTi debulebebiT surT, rogoricaa, magaliTad, „tipur amerikul ojaxs
yoveldRiurad 7-12 saaTis ganmavlobaSi aqvs CarTuli televizori“, an „aSS-Si
mcxovrebi saSualo mamakacis Semosavali weliwadSi $19,438-s Seadgens maSin, roca
qalis _ mxolod $9,584-s“ („The Way“, 1985); Tumca, naTelia, rom SemTxvevaTa
umravlesobaSi SeuZlebelia aseTi maxasiaTeblebis misaRebad mTeli populaciis
gazomva. Sesabamisad, mecnierebi, rogorc wesi, mxolod populaciis SerCevas iyeneben.
SerCeva aris populaciidan mTeli
rigi wesebis dacviT (raTa SerCeva
populaciis reprezentaciuli iyos)
arCeuli qvesimravle anu qvejgufi.
SerCevis arCevis erT-erTi
gavrcelebuli formaa SemTxveviTi
SerCeva. martiv SemTxveviT SerCevaSi
individebi SerCeulni arian ise, rom
populaciis TiToeul wevrs SerCevaSi
moxvedris toli Sansi aqvs da erTi
wevris arCeva ar aris damokidebuli
populaciis sxva wevrebis arCevaze.
Semdeg SerCevis wevrebisgan miiReba
monacemebi da maT aRsawerad
gamoiTvleba aRweriTi statistika,
romelic gamoiyeneba populaciis
maxasiaTeblis Sesaxeb daskvnis
gasakeTeblad. sityva „daskvna“
gulisxmobs raRac cnobilze
dayrdnobiT raRac ucnobis Sesaxeb msjelobas. amrigad, statistikuri daskvna aris
SerCevidan miRebul statistikebze dayrdnobiT populaciis ucnobi maxasiaTeblis
(magaliTad, im saaTebis raodenoba, romlis ganmavlobaSic yoveldRiurad rTavs
televizors tipuri amerikuli ojaxi) anu parametrebis Sesaxeb daskvnebis gamotanis
procesi.
aRwerisa da daskvnisaTvis statistikis gamoyenebis mravali magaliTia cnobili.
magaliTad, aSS-is mosaxleobis aRweris biuro agrovebs sxvadasxvagvar informacias
amerikaSi cxovrebis Sesaxeb da zogierT maTgans yovelwliurad aqveynebs
SerCeva (Sample): _ populaciidan mTeli rigi
wesebis dacviT (raTa SerCeva populaciis
reprezentaciuli iyos) arCeuli qvesimravle anu
qvejgufi.
SemTxveveiTi SerCeva (Random sample): _ SerCeva,
romelSic individebi SerCeulni arian ise, rom
populaciis TiToeuls wevrs SerCevaSi
moxvedris toli Sansi aqvs da erTi wevris
arCeva ar aris damokidebuli populaciis sxva
wevrebis arCevaze.
aRweriTi statistika (Descriptive statistics): _
statistikuri procedurebi, romlebic
gamoiyeneba SerCevidan miRebuli monacemebis
Sesajameblad da aRsawerad.
statistikuri daskvna (Statistical inference): _
populaciis maxasiaTeblis Sefaseba SerCevidan
miRebul statistikebze dayrdnobiT.
parametri (Parameter): _ ricxvi, romelic aRwers
populaciis maxasiaTebels.
Tavi I. statistikis Sesavali 13
SeerTebuli Statebis statistikuri cnobaris saxiT. 1999 wlis aseTi gamocema
gvamcnobs, rom
• aSS-Si mcxovreb ojaxTa 98%-s aqvs televizori, maTgan 84.6%-s ki _ video
magnitofoni;
• 1997 wels saSualo statistikurma moqalaqem tansacmelSi $353.30, xolo
$957.30 samedicino momsaxurebaSi daxarja;
• mamakacebis 58% da qalebis 54.9% daojaxebulni arian.
TiToeuli es mniSvneloba aRweriTi statistikaa _ ricxvi, romelic aRwers
SerCevis wevrebis tipur pasuxs. cxadia, praqtikulad SeuZlebelia aSS-Si arsebuli
TiToeuli televizorisa Tu video magnitofonis daTvla, anda TiToeuli
moqalaqisTvis tansacmelSi daxarjuli Tanxebis Sesaxeb gamokiTxva. amrigad,
TiToeuli am mniSvnelobaTagani statistikaa, romelic populaciaSi amave
maCveneblis mniSvnelobas afasebs. is, Tu ramdenad kargad afasebs calkeuli
aRweriTi statistika populaciis mniSvnelobas, damokidebulia rogorc SerCevis
Seqmnis meTodze, ise _ mis reprezentaciulobaze. SerCevis Sedgenisa da aRweris da
populaciaze daskvnebis gamotanis procedurebs ufro dawvrilebiT me-3-7 TavebSi
ganvixilavT.
SeamowmeT Tqveni codna I.1
1. ganmarteT: _ monacemebi, parametri, populacia, SemTxveviTi SerCeva,
SerCeva, statistika (mecniereba), statistikuri daskvna, statistika
(statistikuri sidide), cvladi.
2. bolo dros Catarebul arCevnebSi winaswar gamokiTxes saarCevno olqis
600 potenciuri amomrCeveli. gamokiTxvis mizani iyo imis garkveva, Tu
visTvis apirebdnen amomrCevlebi xmis micemas. Sedegebma aCvena, rom 312
amomrCeveli, anu SerCevis 52% A kandidatisTvis apirebda xmis micemas, 240
amomrCeveli anu 40% _ B kandidatisTvis, 48 amomrCevelma, anu 8%-ma
miuTiTa, rom ar apirebda arCevnebSi monawileobis miRebas an ar aniWebda
upiratesobas arc erT kandidats. gamokiTxvis am Sedegebze dayrdnobiT
mkvlevrebma daadgines, rom kandidati A miiRebda arCevnebSi Semosuli
yvela xmis 52%-s.
a) axseniT, ratom aris aucilebeli aRweriTi statistikebi am kvlevis
Sedegebis SejamebisTvis.
b) miRebul statistikebze dayrdnobiT, ra daskvnebi gakeTda populaciis
Sesaxeb?
Tavi I. statistikis Sesavali 14
eqsperimenti da statistikuri hipoTezis Semowmeba
bihevioristuli mecnierebis warmomadgenlebi dainteresebulni arian ara marto
SerCevis aRweriTa da daxasiaTebiT, aramed calkeuli individebisa Tu jgufebis
qcevis mizezebis povniTac. am sakiTxisadmi erT-erTi midgomaa eqsperimentis Catareba.
eqsperimentis dros mkvlevari gamoyofs cvlads, romelsac damoukidebeli cvladi
ewodeba da romelic, misi azriT, gavlenas axdens pirovnebis qcevaze. qcevas,
romelzec, mkvlevris molodiniT, gavlenas moaxdens damoukidebeli cvladi,
damokidebuli cvladi ewodeba.
magaliTad, SeiZleba movelodeT, rom alkoholis moxmareba gavlenas axdens
pirovnebisagan Zaladobis reaqciaze: _ mosalodnelia, rom alkoholis zemoqmedebis
qveS myofi adamianebisTvis ufro misaRebi iyos Zaladoba, vidre im adamianebisTvis,
romlebic ar arian alkoholis zemoqmedebis qveS (Gustafson, 1987a). am SemTxvevaSi
alkoholis moxmareba aris damoukidebeli cvladi. eqsperimentatori akontrolebs
_ miiRebs individi alkohols Tu ara da moelis, rom is (alkoholi) gavlenas
damoukidebeli cvladi (Independent variable): _ cvladi, romliTac manipulireben eqsperimentSi,
damokidebul cvladze misi gavlenis gansazRvris mizniT.
damokidebuli cvladi (Dependent variable): _ cvladi eqsperimentSi, romelic damokidebulia
damoukidebel cvladze. SemTxvevaTa umravlesobaSi damokidebuli cvladi qcevis garkveuli
sazomia.
kvlevis hipoTeza (Research hypothesis): _ debuleba or an met cvlads Soris mosalodneli an
nawinaswarmetyvelebi kavSiris Sesaxeb. eqsperimentSi kvlevis hipoTeza aris
nawinaswarmetyvelebi kavSiri damoukidebel da damokidebul cvladebs Soris.
cdis piri (Subject): _ pirovneba, romelic monawileobs eqsperimentSi.
ekvivalenturi jgufebi (Equivalent groups): _ cdis pirTa jgufebi, romlebic, savaraudod, ar
gansxvavdebian erTmaneTisgan raime mudmivi maxasiaTebliT eqsperimentis damoukidebeli
cvladis miwodebamde.
cdis pirTa Soris sqema (Between-subject design): _ kvlevis sqema, romelSic monawileobs ori an
meti jgufi.
qula (Score): _ cdis piris mier davalebis Sesrulebis gazomviT miRebuli ricxviTi
mniSvneloba.
nedli monacemebi (Raw data): _ yvela cdis pirisgan miRebuli qula, romelTac jer ar
gauvliaT statistikuri analizi.
statistikuri hipoTezis Semowmeba (Statistical hypothesis testing) _ procedurebi, romlebic
jgufebs Soris SemTxveviTi gansxvavebis zomis winaswarmetyvelebisTvis gamoiyeneba.
SemTxveviTi gansxvaveba (Chance difference): _ gansxvaveba ekvivalentur jgufebs Soris,
romelic miiReba SemTxveviT an gauTvaliswinebeli mizeziT.
Tavi I. statistikis Sesavali 15
moaxdens Zaladobis mimReblobaze. individis mier Zaladobis mimReblobis sazomi
aris damokidebuli cvladi. varaudoben, rom Zaladobis mimRebloba damokidebulia
imaze, aqvs Tu ara individs miRebuli alkoholi. debulebas, romelic asaxavs
mkvlevris molodins imis Taobaze, rom alkoholis moxmareba gavlenas moaxdens
Zaladobis mimReblobaze, kvlevis hipoTeza ewodeba. kvlevis hipoTeza aris debuleba
or an met cvlads Soris mosalodneli an nawinaswarmetyvelebi kavSiris Sesaxeb.
eqsperimentSi kvlevis hipoTeza aris nawinaswarmetyvelebi kavSiri damoukidebel da
damokidebul cvladebs Soris.
kvlevis im hipoTezis Sesamowmebeli umartivesi eqsperimenti, romelic alkoholis
moxmarebas akavSirebs Zaladobis mimReblobasTan, adamianTa ori ekvivalenturi
jgufis SedgeniT iwyeba. im adamianebs, romlebic monawileoben samecniero kvlevebSi,
mecnierebi cdis pirebs an monawileebs uwodeben. ekvivalenturi jgufebi, romlebic,
savaraudod, ar gansxvavdebian erTmaneTisgan raime mudmivi maxasiaTebliT
eqsperimentis damoukidebeli cvladis miwodebamde. eqsperiments, romelSic
monawileobs ori an meti jgufi, cdis pirTa Soris sqemis mqone eqsperimenti
ewodeba. ekvivalenturi jgufebis Seqmnis Semdeg mkvlevari manipulirebs
damoukidebeli cvladiT anu cvlis (axdens varirebas) mas da zomavs damokidebul
cvlads; amasTan, akontrolebs yvela im sxva cvlads, romelmac SeiZleba gavlena
moaxdinos damokidebul cvladze. Cvens magaliTSi eqsperimentatorma SeiZleba cdis
pirebis pirvel jgufs misces 0.8 mililitri alkoholi amave odenobis forToxlis
wvenTan erTad sxeulis 1 kilogramze, meore jgufs ki _ 1.6 mililitri sufTa,
ualkoholo forToxlis wveni. erTaderTi, riTac es ori jgufi SeiZleba
gansxvavdebodes erTmaneTisgan, aris miRebuli sasmelis raodenoba. damokidebuli
cvladi SeiZleba gaizomos imiT, rom cdis pirs ayurebinon Zaladobis Semcveli
filmi da Semdeg Seafasebinon is nanaxi Zaladobis mimReblobis mixedviT. Sefasebis
Sedegad TiToeuli cdis pirisgan miiReba qula. qula aris cdis piris mier
davalebis Sesrulebis gazomviT miRebuli ricxviTi mniSvneloba. qula aris
damokidebuli cvladi, romelic, Cveni molodinis Tanaxmad, damokidebulia imaze,
miiRo cdis pirma filmis naxvamde alkoholi Tu ara. yvela cdis pirisgan miRebuli
qula Seadgens eqsperimentis nedl monacemebs. am nedli monacemebis mixedviT unda
gadawyvitos mkvlevarma, axdens Tu ara damoukidebeli cvladi _ alkoholis miReba
gavlenas cdis piris mier filmSi nanaxi Zaladobis Sefasebaze.
procesi, romlis saSualebiTac askvnis eqsperimentatori, axdens Tu ara
damoukidebeli cvladi gavlenas damokidebul cvladze, ori etapisgan Sedgeba: _
pirveli etapia TiToeuli jgufis mier mocemuli Sefasebis aRwera; amisaTvis
iTvleba aRweriTi statistikebi, romlebic warmoadgenen TiToeul jgufSi Semavali
cdis pirebis mier mocemul tipur Sefasebebs. es etapi monacemTa aRweris zemoT
Tavi I. statistikis Sesavali 16
ganxiluli procesis identuria. Tqven, albaT, ukve icnobT iseT zogierT cnobil
aRweriT statistikas, rogoricaa moda, mediana an saSualo.
meore etapze irkveva: _ qulaTa ori jgufis aRweriT statistikebs Soris
miRebuli gansxvaveba aris Tu ara sakmarisad didi imisaTvis, rom is damoukidebeli
cvladis gavlenis Sedegad iqnes miCneuli. am etapze saWiroa statistikuri
hipoTezis Semowmebis gamoyeneba. statistikuri hipoTezis Semowmebis mizania,
gansazRvros or jgufs Soris SemTxveviTi gansxvavebis mosalodneli zoma.
eqsperimentSi ekvivalenturi jgufebi yovelTvis gansxvavdebian erTmaneTisgan
raime gauTvaliswinebeli niSniT, rasac SemTxveviT gansxvavebas vuwodebT.
SemTxveviTi gansxvaveba eqsperimentSi maSinac ki miiReba, rodesac damoukidebeli
cvladi ar axdens gavlenas damokidebul cvladze. statistikuri hipoTezis
Semowmebisas or jgufs Soris eqsperimentSi realurad miRebuli gansxvaveba
Sedardeba maT Soris mosalodnel SemTxveviT gansxvavebasTan. Tu iseTive didi
SemTxveviTi gansxvavebis miRebis albaToba sakmarisad mcirea, rogoricaa realurad
miRebuli gansxvaveba, maSin mkvlevari gadawyvets, rom miRebuli gansxvaveba ar aris
SemTxveviTi. amasTan, miRebuli gansxvaveba damoukidebeli cvladis gavlenas
miewereba.
eqsperimentirebisa da statistikuri hipoTezebis Semowmebis magaliTad Semdeg
kiTxvaze pasuxis gacemis procesi ganvixiloT: _ gqoniaT SemTxvevebi, rom xandaxan
dRis manZilze gasakeTebeli zogierTi saqme dagviwyebiaT? cxadia, TiToeul Cvengans
gvqonia aseTi SemTxveva, Tumca, sainteresoa, Tu ramdenad aumjobesebs dRis ganrigis
damaxsovrebas mnemonikuri teqnikebis gamoyeneba. harisma da blaiserma (Harris and
Blaiser, 1997) eqsperimentulad ikvlies es sakiTxi, risTvisac isini adamianebs dRis
manZilze Sesasrulebel saqmeTa damaxsovrebas sTxovdnen. kvlevis monawileebi
SemTxveviTad gayves or ekvivalentul jgufad da TiToeul maTgans 10 iseTi saqmis
damaxsovreba daavales, rogoricaa gaqiravebis punqtSi videokasetis dabruneba,
telefonze darekva da sursaTis maRaziaSi Sesvla. damaxsovrebidan 24 saaTis Semdeg
Seamowmes, Tu risi damaxsovreba SeZles maT. damoukidebel cvlads monawilisTvis
micemuli instruqciis forma, damokidebul cvlads ki _ TiToeul monawilis mier
gaxsenebuli saqmeebis raodenoba warmoadgenda. sakontrolo jgufSi Semaval
adamianebs ar eZleodaT instruqcia imis Taobaze, Tu rogor daemaxsovrebinaT
masala, eqsperimentul jgufs ki sityvebis erTmaneTTan dakavSirebis teqnikas
aswavlidnen, rac niSnavs imas, rom sityvas ricxvi eriTmeba, magaliTad, „erTi aris
merxi, ori aris doli, sami aris skami, oTxi _ noxi, xuTi _ yuTi, eqvsi _ nemsi,
Svidi _ nivTi, rva _ Tma, cxra _ ca, aTi _ Tafli“. riTmis daswavlis Semdeg
individi dasamaxsovrebeli saqmis dakavSirebas cdilobs vizualuri warmodgenis
xatTan; magaliTad, Tu I saqme gaqiravebis punqtSi video kasetis dabrunebaa, man
Tavi I. statistikis Sesavali 17
SeiZleba warmoidginos, rom kaseta merxze devs, xolo Tu II saqme telefonze
darekvaa, warmoidgonos, rom telefoni dolze dgas.
harisma da blaiserma ivaraudes, rom eqsperimentul jgufSi Semavali adamianebi
am teqnikis gamoyenebiT ufro met saqmes gaixsenebdnen, vidre sakonrolo jgufi.
eqsperimentuli jgufis mier gaxsenebul saqmeTa saSualo raodenoba 7.1, xolo
sakontrolo jgufis mier gaxsnebul saqmeTa saSualo raodenoba _ 4.1 iyo. imis
gasarkvevad, eqsperimentuli da sakontrolo jgufebis mier gaxsenebul saqmeTa
raodenobebs Soris sxvaoba ubralo SemTxveviTobaze metia Tu ara, harisma da
blaiserma dispersiuli analizi gamoiyenes, romelic aCvenebs ori jgufis mier
gaxsenebuli saqmeebis raodenobebis gansxvaveba mosalodnel SemTxveviT
gansxvavebaze metia Tu ara. gairkva, rom eqsperimentul jgufSi Semavali individebis
mier gaxsenebul saqmeTa raodenoba sandod aRemateboda sakontrolo jgufisas. es
Sedegi adasturebda kvlevis hipoTezas: _ Cven ukeT vimaxsovrebT gasakeTebel
saqmeebs sityvebis mibmis mnemonikuri teqnikis gamoyenebis SemTxvevaSi.
statistika, rogorc mecniereba, farTod gamoiyeneba bihevioristul mecnierebaSi,
Tumca, masSi Semavali cnebebis gasagebad saWiroa aRweriTi statistikis,
statistikuri daskvnisa da Sefasebis Teoriis codna, romelsac me-3-7 TavebSi
ganvixilavT. statistikur kriteriumebze me-8-13 TavebSi visaubrebT.
SeamowmeT Tqveni codna I.2
1. ganmarteT: _ cdis pirTa Soris sqema, SemTxveviTi gansxvaveba, damokidebuli
cvladi, ekvivalenturi jgufebi, damoukidebeli cvladi, nedli monacemebi,
kvlevis hipoTeza, qula, cdis piri.
2. baletSi ZiriTadi poziciebis swavlisas studentebs xSirad uWirT fexebis
swori poziciis miReba. aq erT-erTi problema isaa, rom fexebi TavisTavad
ixreba muxlebSi, rodesac quslebi mityupebulia da cerebi gareT iyureba.
am problemis dasaZlevad klarksonma, jemsma, uotkinsma da foleim (Clarkson,
James, Watkins, Foley, 1986) Caatares eqsperimenti, romelSic monawileobda
qalTa ori ekvivalenturi jgufi, romelTac ar hqondaT baletSi
mecadineobis gamocdileba. orive jgufs aswavles baletis ZiriTadi
poziciebi da daTvales fexebis moxris raodenoba. erT jgufs, cdis pirebis
gasakvirad, misces fexebis muxlebSi moxris ufleba, meore jgufs ki
araferi uTxres amis Taobaze. mkvlevrebma aRmoaCines, rom sacdeli
periodis manZilze pirvel jgufSi cdis pirebi mxolod 3,1 wamis manZilze
xridnen muxlebSi fexebs maSin, roca meore jgufSi (romelsac araferi
uTxres) es dro 21,8 wams Seadgenda. statistikurma Semowmebam daadastura,
rom muxlebSi fexebis moxris droSi gansxvaveba jgufebs Soris realuri
iyo da ara _ SemTxveviTi.
Tavi I. statistikis Sesavali 18
a) daasaxeleT damoukidebeli cvladi, romlis manipulirebasac
axdendnen klarksoni da sxvebi.
b) daasaxeleT damokidebuli cvladi, romelsac zomavdnen.
g) axseniT, ratom aris es kvleva eqsperimenti.
d) axseniT, ratom aris am eqsperimentSi statistikuri hipoTezis
Semowmeba aucilebeli.
korelacia da regresia
kvlevisas eqsperimentators xSirad surs
ara damoukidebeli cvladiT manipulireba,
aramed mxolod is ainteresebs,
ukavSirdeba Tu ara erTmaneTs anu
korelirebs Tu ara erTmaneTTan
individebis ori an meti maxasiaTebeli. amboben, rom ori cvladi korelirebs, Tu
erTi cvladis cvlileba ukavSirdeba meore cvladis SeTanxmebul sistematur
cvlilebas. magaliTad, Tu pirovnebis SfoTvis tipuri done dakavSirebulia mis mier
alkoholis miRebasTan ise, rom SfoTvis maRali done ukavSirdeba didi raodenobiT
alkoholis miRebas, xolo SfoTvis dabali done _ naklebi raodenobis alkoholis
miRebas, SegviZlia vTqvaT, rom SfoTva da alkoholis miReba korelaciaSia
erTmaneTTan.
imis dasadgenad, ori cvladi
korelaciaSia Tu ara, vzomavT adamianTa
SerCevas da TiToeuli individisgan
viRebT or qulas: _ SfoTvis qulas da
miRebuli alkoholis qulas. Semdeg
viTvliT korelaciis koeficients. korelaciis koeficienti aris statistika,
romelic ricxobrivad gviCvenebs maCvenebelTa ori jgufis dakavSirebulobis
xarisxs da am kavSiris mimarTulebas. am koeficientis mniSvneloba SeiZleba
icvlebodes -1.00-dan +1.00-mde.
statistikuri hipoTezis Semowmeba korelaciis koeficientTan mimarTebaSic
gamoiyeneba. yovelTvis arsebobs ori gansxvavebuli cvladis mniSvnelobebs Soris
SemTxveviTi kavSiris gamovlenis albaToba. amrigad, ibadeba kiTxva: _ aris Tu ara
korelaciis koeficientiT gamoxatuli, realurad miRebuli kavSiri mosalodnel
SemTxveviT kavSirze meti? am kiTxvaze pasuxs korelaciis koeficientis
statistikuri Semowmeba iZleva.
Tu davadgineT, rom
maCvenebelTa or jgufs
Soris kavSiri
korelacia (Covary): _ ori cvladi
korelaciaSi imyofeba erTmaneTTan,
rodesac erTi cvladis cvlileba
ukavSirdeba meore cvladis SeTanxmebul
sistematur cvlilebas.
korelaciis koeficienti (Correlation coefficient):
_ statistika, romelic ricxobrivad
gviCvenebs maCvenebelTa ori jgufis
dakavSirebulobis xarisxs da am kavSiris
mimarTulebas.
regresiuli analizi (Regression analysis): _ statistikuri
meTodebis gamoyeneba qulaTa erTi erTobliobis mixedviT
qulaTa meore erTobliobis Sesaxeb winaswarmetyvelebisTvis.
Tavi I. statistikis Sesavali 19
SemTxveviT kavSirze metia, SeiZleba SevecadoT erTi maCvenebelis mixedviT meoris
Sesaxeb winaswarmetyvelebas. Tu viciT pirovnebis SfoTvis maCvenebeli, SevZlebT
miRebuli alkoholis maCveneblis winaswarmetyvelebas? da Tu SegviZlia miRebuli
alkoholis maCveneblis winaswarmetyveleba, ramdenad zusti iqneba es
winaswarmetyveleba? erTi cvladis mniSvnelobis mixedviT meore cvladis
mniSvnelobis winaswarmetyveleba gulisxmobs regresiuli analizis gamoyenebas.
korelaciuri da regresiuli analizis teqnikebi farTod gamoiyeneba
bihevioristuli mecnierebis mraval sferoSi. sailustraciod ganvixiloT uoerisa
da qavelis (Woehr and Cavel, 1993) mier Catarebuli kvleva, romelic im faqtorebis
gamovlenas Seexeboda, rac, SesaZloa, fsiqologiis safuZvlebSi pirvel Sualedur
testSi naCveneb Sedegs ukavSirdebodes. mkvlevrebma 325 studentis iseTi
maCveneblebi Seiswavles, rogoricaa skolaSi miRebuli niSnebis saSualo, sagamocdo
masalis wakiTxvis raodenoba da kviris manZilze televizoris yurebaze daxarjuli
dro. Semdeg daiTvales TiToeuli dasaxelebuli maCveneblisa da pirvel Sualedur
testSi miRebul qulas Soris korelacia. skolaSi miRebuli niSnebis saSualosa da
Sualedur gamocdaSi miRebul qulas Soris korelaciis koeficienti +.31 gamovida
(anu rac ufro maRali iyo skolaSi miRebuli niSnebis saSualo, miT ufro maRali
iyo Sualedur SefasebaSi miRebuli qula); sagamocdo masalis wakiTxvaTa
raodenobasa da Sualedur gamocdaSi miRebul qulas Soris korelaciis
koeficienti _ +.22 (anu rac ufro metjer waikiTxa studentma masala, miT ufro
maRali qula miiRo), kviris manZilze televizoris yurebaze daxarjuli drosa da
Sualedur gamocdaSi miRebul qulas Soris korelaciis koeficienti ki _ _.13 (anu
rac ufro met dros xarjavda studenti kviris manZilze televizoris yurebaSi, miT
ufro naklebi qula miiRo). statistikuri Semowmeba aCvenebs, rom korelacia araa
ubralo SemTxveviTi kavSiri. avtorebi Tvlian, rom es Sedegebi, sxva monacemebTan
erTad, SesaZloa, sasargeblo aRmoCndes studentebisTvis sakuTari Sedegebis
gasaumjobeseblad. korelaciasa da regresias me-13 da me-14 TavebSi ganvixilavT.
SeamowmeT Tqveni codna I.3
1. ganmarteT: _ korelaciis koeficienti, korelacia, regresiuli analizi.
2. bevr adamians sjera iseTi paranormaluri fenomenebis arsebobis, rogoricaa
naTelxilva, eqstrasensoruli aRqma da fsiqokinezi. ukavSirdeba Tu ara aseTi
fenomenebis rwmenis xarisxi imas, Tu ra tipis ganaTleba miiRo individma
skolaSi? tobasikma, milerma da jonsma (Tobacyk, Miller & Jones) 193 me-11
klasels gauzomes paranormaluri fenomenebis arsebobis rwmena. maT, aseve,
aRnusxes TiToeuli moswavlis mier gavlili kursebis saerTo raodenoba.
rodesac gavlili samecniero kursebis raodenoba paranormaluri fenomenebis
arsebobis rwmenas daukavSires, maT Soris Sebrunebuli kavSiri aRmoCnda. rac
Tavi I. statistikis Sesavali 20
ufro meti samecniero kursi hqonda moswavles gavlili, miTYufro susti iyo
misgan paranormaluri movlenebis arsebobis rwmena.
a) axseniT, ratom aris saWiro am kvlevaSi korelaciuri statistika.
b) romeli ori sazomi gamoiyenes am kvlevaSi?
ratom aris saWiro statistika
zemoT warmogidgineT bihevioristul mecnierul kvlevaSi statistikuri analizis
gamoyenebis sami ZiriTadi meTodi, Tumca, Tqven SeiZleba ibuzRunoT da SegawuxoT
kiTxvam imis Taobaze, Tu risTvis aris statistikuri meTodebi ase aucilebeli da
saWiro bihevioristul mecnierebaSi. pasuxi sakmaod pirdapiria da ucvleli
TiToeuli warmodgenili statistikuri meTodisTvis: _ cxovrebis sasiamovno da
aRmafrTovanebeli faqtia, rom adamianebi mravali maxasiaTeblis xarisxiT
gansxvavdebian erTmaneTisgan. samyaro mosabezrebeli iqneboda TiToeuli adamiani
meoris identuri rom yofiliyo, Tumca, aseTi garemoebis upiratesoba statistikuri
meTodebis saWiroebis ararseboba iqneboda. ratom? ganvixiloT statistikis zemoT
warmodgenili sxvadasxvagvari gamoyeneba.
pirveli magaliTi individebisgan miRebuli raodenobrivi Sefasebis Sedegebis
aRweras Seexeba. yvela adamiani rom erTnairi yofiliyo, nebismieri erTi pirovnebis
SefasebiT miRebuli monacemebi aRwerda yvela danarCen adamians. magaliTad, yvela
maTganis Semosavali iqneboda $19.438 weliwadSi. populaciis Sesaxeb daskvnis
gasakeTeblad sakmarisi iqneboda mxolod erTi adamianis Sefaseba. Sesabamisad, ar
iqneboda aRweriTi statistikis saWiroeba, romelic warmoadgenda SerCevis wevrTa
tipur maCvenebels.
meore magaliTi Seexeboda statistikuri hipoTezis Semowmebas da, Sesabamisad,
pasuxis gacemas kiTxvaze: _ axdens Tu ara damoukidebeli cvladi gavlenas qcevaze.
warmoidgineT, ra advili iqneboda amis gansazRvra, TiToeuli individis qceva
nebismieri sxva individis qcevis identuri rom yofiliyo. davuSvaT, gvinda
gavarkvioT, amcirebs Tu ara konkretuli medikamenti guliscemis sixSires.
medikamentis gareSe yvela adamianis guliscemis sixSirea 60 dartyma wuTSi,
medikamentis miRebis SemTxvevaSi ki _ 55 dartyma wuTSi. kidev eWvqveSaa, amcirebs Tu
ara mocemuli medikamentis miReba guliscemis sixSires? vinaidan ar arsebobs
individidan individze cvalebadobis Sansi, ueWveli iqneba, rom medikamenti amcirebs
guliscemis sixSires, statistikuri hipoTezis Semowmeba ki aRar dagvWirdeba.
mesame magaliTi korelaciisa da regresiuli analizis problemas Seexeba. aris Tu
ara ori qceva erTmaneTTan kavSirSi da Tu aris, ramdenad SegviZlia erTi qcevis
ganxorcielebis prognozireba meore qcevis ganxorcielebis mixedviT? es kiTxva
mxolod imitom Cndeba, rom adamianebi gansxvavdebian erTmaneTisgan da
sxvadasxvagvarad iqcevian erTsa da imave situaciaSi. TiToeuli individi yvela sxva
Tavi I. statistikis Sesavali 21
individis identuri rom yofiliyo, yvela qceva srulad prognozirebadi iqneboda. am
SemTxvevaSi ar iqneboda cvalebadoba individidan individze da srulyofilad
SevZlebdiT yvela qcevis winaswarmetyvelebas, korelaciuri da regresiuli
analizis gamoyenebis gareSe.
erTi individidan meoreze cvalebadobis ararsebobis pirobebSi sruliad
moxsnili iqneboda statistikuri analizis saWiroeba, magram ramdenad mosawyeni
iqneboda es samyaro. sabednierod, Cveni samyaro ar aris aseTi; Tumca,
kanonzimierebebiTa da cvalebadobiT savse cxovreba statistikur meTodebsac
saWiroebs, raTa aRiweros da aixsnas qceva.
Sejameba
• statistika efuZneba meTodebsa da procedurebs, romlebic gamoiyeneba
monacemebis Sejamebis, analizisa da maTze dayrdnobiT daskvnebis
gamosatanad.
• statistika (statistikuri sidide) aris monacemebze dayrdnobiT gamoTvlili
ricxvi, romelic SeiZleba gamoyenebul iqnes SerCevidan miRebuli qulebis
aRsawerad.
• statistika, aseve, SeiZleba gamoyenebul iqnes populaciis parametris Sesaxeb
daskvnis gasakeTeblad.
• statistikuri hipoTezis Semowmeba gamoiyeneba imis gansasazRvrad,
eqsperimentSi or jgufs Soris miRebuli gansxvaveba aris Tu ara sakmarisad
didi imisaTvis, rom miCneul iqnes damoukidebeli cvladis moqmedebis
Sedegad da ara _ SemTxveviTad miRebulad.
• korelaciuri statistika gamoiyeneba imis gansasazRvrad, arsebobs Tu ara
kavSiri qulaTa or jgufs Soris.
• regresiuli analizi gamoiyeneba qulaTa erTi jgufis mixedviT qulaTa
meore jgufis Sesaxeb prognozirebisTvis.
ZiriTadi terminebi da simboloebi
aRweriTi statistika
damokidebuli cvladi
damoukidebeli cvladi
ekvivalenturi jgufebi
kvlevis hipoTeza
korelacia
korelaciis koeficienti
monacemebi
nedli monacemebi
parametri
populacia
regresiuli analizi
statistika (mecniereba)
statistika (statistikuri sidide)
statistikuri daskvna
statistikuri hipoTezis Semowmeba
qula
SemTxveviTi gansxvaveba
SerCeva
cdis pirebi
cdis pirTa Soris sqema
cvladi
kiTxvebi
Tavi I. statistikis Sesavali 22
1. daasaxeleT oTxi movlena an aqtivoba Tqvens cxovrebaSi, romelic
kanonzmierebiT xasiaTdeba (magaliTad, drois Cveuli raodenoba, romelsac
uTmobT Zils an yoveldRiuri samuSao saaTebis raodenoba).
2. moifiqreT da daasaxeleT sami sxvadasxva populaciis magaliTi. ra saerTo
maxasiaTebliT erTiandebian TiToeul maTganSi wevrebi?
3. qriusma, Sirefsma, tomasma, krahenbalma da helfriCma (Crews, Shirreffs, Thomas,
Krahenbuhl & Helfrich, 1986) gazomes 23 profesionali golfis moTamaSe qalis
sxeulSi cximis Semcveloba. miRebuli Sedegebi maT daukavSires golfis
moTamaSeTa mier erTi wlis manZilze SejibrebSi miRebul saSualo qulebs
da aRmoaCines, rom maT SeeZloT golfSi miRebuli qulebis prognozireba
sxeulSi cximis Semcvelobis odenobis mixedviT. sxeulis cximis nakleb
procentul Semcvelobas ukavSirdeboda golfSi dabali saSualo qula
maSin, roca sxeulis cximis Semcvelobis ufro maRali maCvenebeli
dakavSirebuli iyo golfSi maRal saSualo qulasTan.
a) am TavSi ganxiluli romeli statistikuri meTodi gamoiyenes am
kvlevaSi?
b) axseniT, risTvis iyo saWiro am kvlevaSi korelaciuri statistika.
g) romeli ori sazomi iyo gamoyenebuli am kvlevaSi?
4. kavasakis sindromi bavSvebis davadebaa, romlis simptomebia maRali sicxe,
damskdari tuCebi, dawiTlebuli ena, xelisgulebisa da fexisgulebis
siwiTle da gamonayari. rogorc wesi, am daavadebas aspiriniT mkurnaloben,
Tumca, ukanasknelma kvlevebma aCvena, rom aspirinze gemaglobulinis
damateba SeiZleba ufro efeqturi iyos, vidre mxolod aspirini. kvlevis am
hipoTezis Sesamowmeblad niuburgerma da sxvebma (Newburger et al, 1986) 168
bavSvi SemTxveviTad gaanawiles or jgufSi: _ erTi jgufi iRebda mxolod
aspirins, meore ki _ aspirinze damatebul gemaglobulins. maT aRmoaCines,
rom mkurnalobis pirveli dRis Sedegad im bavSvebis temperatura,
romlebic aspirinze damatebul gemaglobulins iRebdnen, saSualod 2.3°F-iT
(0.26°C) Semcirda maSin, roca meore jgufSi (anu im bavSvebTan, vinc mxolod
aspirins iRebda) es cvlileba mxolod 0.8°F (0.09°C) iyo. statistikurma
Semowmebam aCvena, rom am or jgufs Soris temperaturis Semcirebis
mixedviT gansxvaveba ar iyo SemTxveviTi; aspirinze damatebuli
gemaglobulinis miRebis Sedegad ufro metad Semcirda temperatura, vidre
mxolod aspirinis miRebisas.
a) am TavSi ganxiluli romeli statistikuri meTodi gamoiyenes am
kvlevaSi?
b) daasaxeleT, romeli damoukidebeli cvladis manipulirebas axdendnen
niubergeri da sxvebi?
Tavi I. statistikis Sesavali 23
g) daasaxeT damokidebuli cvladi.
d) axseniT, ratom aris es kvleva eqsperimenti.
e) axseniT, risTvis aris saWiro statistikuri hipoTezis Semowmeba am
eqsperimentSi.
5. jakobi (Jacoby, 2000) aRniSnavs, rom amerikelTa erovnuli satelefono
gamokiTxvis Sedegebis mixedviT, mamakacebis 72%-s da qalebis 58%-s surs,
materialurad uzrunvelyofili iyos. respondentTa 19% miuTiTebs, rom
fuliT SesaZlebelia bednierebis yidva, 56%-ki sjera, rom rac ufro meti
fuli gaqvs, miT naklebad stresulia cxovreba.
a) am TavSi ganxiluli romeli statistikuri meTodi gamoiyenes am
kvlevaSi?
b) axseniT, risTvis aris saWiro aRweriTi statistikis gamoyeneba am
kvlevis Sedegebis Sesajameblad.
g) romeli aRweriTi statistika gamoiyenes am kvlevaSi?
6. miuTiTeT, ras aRwers qvemoT mocemuli TiToeuli debuleba: _ nedl
monacemebs, statistikas Tu daskvnas populaciis parametris Sesaxeb.
a) linda samsaxurSi misvlas yovel dRe 36 wuTs andomebs.
b) saSualo amerikeli mamakaci mgzavri yoveldRiurad 42 wuTs
transportSi atarebs.
g) 75 mgzavrisgan Semdgari SerCeva yoveldRiurad saSualod 46 wuTs
mgzavrobs.
d) benJamin da anita hiustonebs sabanko angariSze $5126 aqvT Senaxuli.
e) tipur amerikel mozrdils sabanko angariSze $6024 aqvs Senaxuli.
v) samxreTis patara qalaqis SemTxveviTad SerCeuli 50 moqalaqes sabanko
angariSze saSualod $5692 aqvs Senaxuli.
7. moifiqreT magaliTi, romelSic erTi cvladis quliT SeZlebT meore
cvladis qulis winaswarmetyvelebas (magaliTad, qolesterinis done,
romliTac SesaZlebelia gulis daavadebis Sansis prognozireba).
Tavi II. mecnieruli kvleva
ra aris mecniereba
mecnieruli kvlevis kiTxvebi
kvlevis meTodebi
operaciuli gansazRvrebebi
gazomva
gazomvis skalebi
Tvisebrivi da raodenobrivi monacemebi
diskretuli (wyvetili) da uwyveti cvladebi
თავი II. მეცნიერული კვლევა 2
statistikuri meTodebi aTaswleulebis manZilze viTardeboda praqtikuli
problemebis gadasaWrelad. sityva „statistika“ laTinuri status-idan anu
sazogadoebrivi mdgomareobidan (State) momdinareobs; statistikis adreuli gamoyeneba
saxelmwifos Tu imperiis Sesaxeb codnis miRebas isaxavda miznad. romaeli
imperatorebi yovelTvis awarmoebdnen axali dapyrobili teritoriebis aRweras.
aRweriT miRebul informaciaze dayrdnobiT, romelic warmoadgenda mosaxleobisa
da maT mflobelobaSi arsebuli sakuTrebis statistikas, wesdeboda gadasaxadebi.
miuxedavad imisa, rom statistika am miznisTvis dResac gamoiyeneba, misi farglebi
gafarTovda da moicavs meTodebs, romlebic mowodebulia adamianebisa da
cxovelebis qcevis SeswavlisTvis.
ra aris mecniereba
mecniereba codnis SeZenis saSualebaa, romelic kiTxvis dasmas, am kiTxvaze
savaraudo pasuxis gacemas kvlevis hipoTezis formiT, hipoTezis empiriulad
SemowmebisTvis kvlevis sqemis Camoyalibebas, monacemebis Segrovebas, monacemTa
analizs da monacemTa analizze dayrdnobiT hipoTezis Sesaxeb daskvnis gamotanas
moicavs.
mecnieruli kvlevis kiTxvebi
mecnieruli kiTxva empiriuli monacemebis
mogrovebis gziT pasuxis miRebis saSualebas
iZleva. sityva „empiriuli“ sensorul
gamocdilebas an dakvirvebas gulisxmobs.
amrigad, empiruli monacemebi aris sistematur dakvirvebasa da sensorul
gamocdilebaze dafuZnebuli gazomvis Sedegad miRebuli qulebi. I TavSi ganxiluli
TiToeuli magaliTi mecnierul kiTxvebs efuZneboda da empiriuli monacemebis
mogrovebis gziT pasuxis miRebis SesaZleblobas iZleoda.
dasmul kiTxvaze eqsperimenti xSirad savaraudo pasuxs kvlevis hipoTezis saxiT
gvTavazobs. I TavSi kvlevis hipoTeza ganvsazRvreT, rogorc debuleba or an met
cvlads Soris mosalodneli an savaraudo kavSiris Sesaxeb. uoletma da ioSim
(Ouellet and Joshi, 1986) gamoTqves hipoTeza, rom martooba da depresia
urTierTdakavSirebuli maxasiaTeblebia. vinaidan kvlevis hipoTezaSi mocemulia
cvladebi da miTiTebulia maT Soris mosalodneli kavSiris ragvaroba, is
mimarTulebas aZlevs mkvlevars kvlevis sqemis SerCevisas da monacemTa
mogrovebisas.
mecnieruli kvlevis kiTxvebi
kvlevis hipoTezis Sesamowmeblad empiriuli
monacmebis mogroveba saWiroebs kvlevis meTods.
empiriuli monacemebi (Empirical data): _
dakvirvebasa da sensorul
gamocdilebaze dafuZnebuli gazomvis
Sedegad miRebuli qulebi.
kvlevis meTodi (Research method): _
midgoma monacemebis mogrovebisadmi.
თავი II. მეცნიერული კვლევა 3
kvlevis meTodi aris midgoma, romelsac mecnieri iyenebs monacemTa mosagroveblad.
bihevioristul mecnierebebSi mravali meTodi gamoiyeneba. aq ZiriTadi meTodebidan
mokled mimovixilavT mxolod oTxs. TiToeulis SemTxvevaSi mogrovili monacemebis
sistematizaciisa da analizisTvis gamoiyeneba statistika.
bunebrivi dakvirveba
bunebrivi dakvirveba gulisxmobs bunebriv
pirobebSi ganxorcielebul qcevebze, situaciaSi
Carevis gareSe, dakvirvebas. bunebrivi dakvirvebis
magaliTad SeiZleba movitanoT trinkosis (Trinkaus,
1982, 1983, 1988, 1993) kvleva, romelic 10 wlis
manZilze sistematurad akvirdeboda mZRolebis qcevas erT gzajvaredinze. erT-erTi
misi aRmoCenaa, rom dakvirvebis periodSi Semcirda SuqniSnis wiTel Suqze
damorCileba. am dakvirvebisas eqsperimentatori praqtikulad ar ereoda procesSi;
mZRolebma arc ki icodnen, rom vinme akvirdeboda wiTeli Suqisadmi maT
damorCilebas. maT rom scodnodaT amis Sesaxeb, mosalodneli iqneboda maTi qcevis
Secvla.
saarqivo Canawerebi
saarqivo Canawerebi gulisxmobs kiTxvebze
pasuxis gacemas arsebuli Canawerebis gamoyenebiT.
sazogadoebebSi, rogorc wesi, grovdeba
mravalferovani informacia maTi wevrebis Sesaxeb:
_ dabadebis, qorwinebisa da gardacvalebis Canawerebi; kriminaluri angariSebi,
mosaxleobis aRweris Canawerebi da a.S. es arsebuli Canawerebi SeiZleba mecnierul
kiTxvaze pasuxisTvis saWiro monacemebs iZleodnen. magaliTad, londonelma
maRaziis mflobelma jon grantma (John Grant, 1620-1674) mamakacebisa da qalebis
sikvdili mizezebSi gansxvavebebis dasadgenad dabadebisa da gardacvalebis
Canawerebi gamoiyena. uaxloes warsulSi zusne (Zusne, 1986-87) dainteresebuli iyo
Semdegi sakiTxiT: _ SeuZliaT Tu ara sikvdilis piras myof individebs sikvdilis
gadavadeba. man Seiswavla 3000-ze meti adamianis dabadebisa da gardacvalebis
Canawerebi, romelic aCvenebda, rom es SesaZlebelia, Tumca, is gansxvavdeba qalebTan
da mamakacebTan.
meore magaliTi: tomsonma (Thomson, 1993) gaaanaliza aSS-is janmrTelobis
statistikis nacionaluri centris monacemebi mkvlelobaTa sixSiris Sesaxeb. am
kvlevam aCvena, rom mkvlelobaTa ricxvi yvelaze metia kolumbiis olqSi; aq
gardacvalebis yoveli 1000 SemTxvevidan daaxloebiT 39 SemTxveva mkvlelobaa. es
ricxvi yvelaze naklebia aiovaSi, sadac 1000 SemTxvevidan mxolod 2 mkvlelobaa.
bunebrivi dakvirveba (Naturalistic
observation): _ kvleva, romelic
gulisxmobs bunebriv pirobebSi
ganxorcielebul qcevebze
dakvirvebas.
saarqivo Canawerebi (Archival recodrs):
_ kvleva, romelSic gamoiyeneba
arsebuli Canawerebi.
თავი II. მეცნიერული კვლევა 4
gamokiTxva
gamokiTxva gulisxmobs monacemebis
miRebas adamianebis zepiri an werilobiTi
interviuebiT miiReba. gamokiTxva farTod
aris gavrcelebuli sazogadoebaSi. Tu Tqven erTxel mainc SegivsiaT restoranSi
baraTi, romelSic mocemuli iyo kiTxvebi sakvebisa Tu momsaxurebis xarisxis
Sesaxeb an Tu erTxel mainc SegivsiaT produqtis sagarantio furceli, sadac
mocemuli iyo kiTxvebi Tqveni Semosavlis, ojaxuri mdgomareobisa da hobis Sesaxeb,
Tqven monawileoba migiRiaT gamokiTxvaSi. gamokiTxvis kidev erTi cnobili
magaliTia arCevnebze xmis micemis procedura.
eqsperimenti
eqsperimenti aris zedmiwevniT
gakontrolebuli situacia, sadac mecnieri
axdens erTi an meti damoukidebeli cvladis
manipulirebas da akvirdeba, Tu ra gavlenas
axdens es damokidebul cvladze. magaliTad,
harisma da blaiserma (1997, ix. gv. XX)
operacionalurad gansazRvres mnemonikuri teqnika, rogorc cdis pirebisTvis
sityvis da misi Sesatyvisi riTmis mqone ricxvebis Sewyvilebis swavlebaa. Amis
mixedviT, „erTi aris merxi, ori aris doli, sami aris skami, oTxi _ noxi, xuTi _
yuTi, eqvsi _ nemsi, Svidi _ nivTi, rva _ Tma, cxra _ ca, aTi _ Tafli“. amrigad, Tu
sxva mecnieri moisurvebs am kvlevis gagrZelebas, mas ukve ecodineba, Tu rogor
gansazRvres harisma da blaiserma mnemonikuri teqnika.
operaciuli gansazRvrebebi
sxvadasxva kvlevis sqemebis gamoyenebiT
miRebuli mecnieruli codna ganmeorebadi
unda iyos da mis Sesaxeb unda
Tanxmdebodnen sxvadasxva damkvirveblebi.
mecnieruli codna qcevis Sesaxeb maSin iZens
Zalas da xdeba yvelasTvis misaRebi,
rodesac sxva mecnierebs SeuZliaT am qcevis gameoreba sxvadasxva monawileebTan
sxvadasxva garemoSi da sxvadasxva adgilze. codna rom mecnieruli gaxdes, srulad
unda iyos cnobili masTan dakavSirebuli mosazrebebi da Sedegebi ise, rom sxvas
SeeZlos Sedegebis gameoreba, gakritikeba da gavrcoba. kvlevis Sesaxeb informaciis
gadacemis procesSi gadamwyveti nabijia im operaciebis zustad gansazRvra, romelic
gamoyenebul iqna dakvirvebis, damoukidebeli cvladebiT manipulirebisa Tu
damokidebuli cvladis gazomvis dros. am process mecnierebi kvlevaSi gamoyenebuli
procedurebis operaciulad gansazRvras uwodeben. erT-erT kvlevaSi uoletma da
gamokiTxva (Survey research): _ kvleva,
romelSic monacemebi adamianebis zepiri
an werilobiTi interviuebiT miiReba.
eqsperimenti (Experiment): _ zedmiwevniT
gakontrolebuli situacia, sadac
mecnieri axdens erTi an meti
damoukidebeli cvladis manipulirebas
da akvirdeba, Tu ra gavlenas axdens es
damokidebul cvladze.
operaciuli gansazRvreba (Operational
definition): _ dakvirvebisas, damoukidebeli
cvladiT manipulirebisas an
damokidebuli cvladis gazomvisas
gamoyenebuli operaciebis detaluri
aRwera.
თავი II. მეცნიერული კვლევა 5
ioSim operaciulad gansazRvres martooba, rogorc martoobis skalaze cdis piris
mier miRebuli qula (Russell, Peplau & Cutrona, 1980). amrigad, Tu romelime mecniers
moesurveba am kvlevis gagrZeleba, man unda icodes, Tu rogor gazomes martooba
uoletma da ioSim.
SeamowmeT Tqveni codna II.1
1. ganmarteT: _ saarqivo Canawerebi, empiriuli monacemebi, eqsperimenti,
bunebrivi dakvirveba, operaciuli gansazRvreba, kvlevis meTodi, kvlevis
hipoTeza, gamokiTxva.
2. qvemoT mocemuli TiToeuli kiTxvisTvis daasaxeleT kvlevis meTodi,
romelic, Tqveni azriT, unda gamoiyenon masze pasuxis gasacemad.
a) ras fiqrobs mocemuli dasaxlebuli punqtis mosaxleoba dasaxlebis
teritoriaze narCenebis gadamamuSavebeli qarxnis funqcionirebis
Sesaxeb, romelmac SeiZleba safrTxe Seuqmnas maT janmrTelobas?
b) rogor Seicvleba cdis pirebis damokidebuleba imis mixedviT, Tu ra
tipis _ SiSis gamomwvev Tu informacis matarebel _ Setyobinebas
gadavcemT?
g) saSualod ra asakSi uCndebaT arizonel mamakacebs pirveli Svili?
d) ramdenad xSirad TamaSoben 4 wlis bavSvebi sami bavSvisgan Semdgar
jgufebSi sabavSvo baRebSi?
gazomva
mecnierul kiTxvaze pasuxis
gasacemad saWiroa gazomva. gazomva
aris garkveuli wesebis mixedviT
cvladebisTvis ricxviTi mniSvnelobebis miniWebis procesi. rogorc wesi, gazomvis
Sedegi miekuTvneba oTxidan erT-erT skalas. es skalebia: _ saxeldebis, rigis,
intervalebisa da Sefardebis skalebi.
gazomvis skalebi
saxeldebis skala
saxeldebis skala aris gazomili
cvladis sxvadasxva kategoriebSi
gaerTianeba. magaliTad, iseTi cvladi
rogoricaa individis eTnikuri warmomavloba, saxeldebis skalaze izomeba; am
cvladis mixedviT aSS-Si mcxovrebi adamianebi SeiZleba gavaerTianoT kategoriebSi:
_ aborigeni, azieli amerikeli, afroamerikeli, TeTrkaniani an espanuri warmoSobis
amerikeli da Semdeg TiToeul kategorias mivaniWoT pirobiTi nomeri (aqedan
momdinareobs dasaxeleba saxeldebis, romelic niSnavs „saxelis darqmevas“),
gazomva (Measurement): _ garkveuli wesebis
mixedviT cvladebisTvis ricxviTi
mniSvnelobebis miniWeba.
saxeldebis skala (Nominal measurement): _
gazomili cvladis sxvadasxva kategoriebSi
gaerTianeba.
თავი II. მეცნიერული კვლევა 6
magaliTad, 1 = aborigeni, 2 = azieli amerikeli, 3 = afroamerikeli, 4 = TeTrkaniani,
5 = espaneli warmoSobis amerikeli. erTsa da imave kategoriaSi gaerTianebul yovel
individs mieniWeba erTi da igive nomeri. cxadia, es nomeri ar aris ricxobrivi
mniSvnelobis matarebeli; igive warmatebiT kategoriebis aRsaniSnad cifrebis
nacvlad SesaZlebelia asoebis gamoyeneba. bihevioristuli mecnierebis mravali
warmomadgeneli saxeldebis skalas ar miiCnevs realur sazomad, radgan
kategoriebisTvis miniWebuli ricxvebi mxolod identifikaciis funqcias asruleben;
isini ar iZlevian raodenobriv informacias gazomili cvladis Sesaxeb.
rigis skala
rodesac cvladis sidide
mniSvnelobebis mixedviT aris
dalagebuli skalaze, es niSnavs, rom
gamoyenebulia rigis skala. magaliTad, adamianebs SeiZleba vTxovoT
multiplikaciuri filmis seriebis dalageba skalaze ufro sasacilodan naklebad
sasacilomde. xSirad ricxviTi mniSvneloba 1 eniWeba stimuls an individs,
romelsac gasazomi maxasiaTebeli yvelaze didi odenobiT aqvs, mniSvneloba 2 _ im
stimuls an individs, romelsac aReniSneba mocemuli maxasiaTeblis Semdegi yvelaze
didi odenoba da a.S. magaliTad, Tu SfoTva rigis skalaze izomeba, individs,
romelic yvelaze met SfoTvas avlens, mieniWeba 1; individs, romelic avlens Semdeg
yvelaze met SfoTvas, eniWeba 2, Semdegs _ 3 da a.S. ase dalagdebian individebi ufro
mSfoTvaredan naklebad mSfoTvaremde.
rigis skala, faqtobrivad, ranJirebas axdens, sadac viRaca an raRaca, romelsac
mieniWa rangi 1, ufro metad xasiaTdeba mocemuli cvladiT, vidre is, visac mieniWa
rangi 2; Tumca, rigis skala ar iZleva imis gansazRvris saSualebas, Tu ramdenad
gansxvavebulia gazomili cvladi rangebs Soris. gazomili cvladis mixedviT
SeiZleba mciredi gansxvaveba iyos 1 da 2 rangebs Soris, magram es gansxvaveba
SeiZleba gacilebiT didi iyos 2 da 3 rangebs Soris. magaliTad, maraTonSi
monawile pirveli sami morbenalis dro SeiZleba iyos 3 sT da 19 wT, 3 sT da 20 wT
da 3 sT da 36 sT, Sesabamisad. finiSze am morbenalTa rangebi iqneba 1, 2 da 3;
Tumca, droSi gansxvaveba pirvel da meore morbenals Soris iqneba 1 wT maSin, roca
meore da mesame morbenals Soris gansxvaveba 16 wT-s Seadgens.
rigis skala ar aCvenebs gazomili cvladis sidides; is iZleva informacias
mxolod am cvladis mixedviT individebis Tu sagnebis Tanmimdevrobis (rigis)
Sesaxeb. ver vityviT, rom rangi 1 aucileblad ukavSirdeba gazomili cvladis met
odenobas; verc imas vityviT, rom umciresi rangis mqone individs an sagans yvelaze
naklebad axasiaTebs gazomili maxasiaTebeli. individma, romelsac moeTxoveboda
multiplikaciuri filmis seriebis dalageba ufro sasacilodan naklebad
sasacilomde, SeiZleba verc erTi maTgani ver aRiqvas sasacilod; amasTan, am
rigis skala (Ordinal measurement): _ skalaze
cvladis sidide dalagebulia mniSvnelobebis
mixedviT.
თავი II. მეცნიერული კვლევა 7
individis mier multiplikaciuri filmisTvis miniWebuli rangi 1 imave informaciis
matarebelia, romelic gaaCnia rangs 1, romelic mianiWa individma, romelmac yvela
seria Zalze sasacilod aRiqva. miuxedavad damaxasiaTebeli naklovanebebisa, rigis
skala pirveli nabijia ufro zusti sazomebis ganviTarebisken. magaliTad, autizmi
ganviTarebis darRvevaa, romelic gavlenas axdens bavSvis metyvelebis ganviTarebaze,
azrovnebis procesze da socialur da fizikur gamRizianeblebze reagirebis unarze.
naklebad aris nakvlevi, Tu rogor SeiZleba gansxvavdebodnen erTmaneTisgan
autizmis diagnozis mqone bavSvebi. am mimarTulebiT pirveli nabiji iyo bavSvis mier
ganxorcielebeuli autisturi qcevis xarisxis sazomi rigis skala, romelic
dihofma, heznekerma brosviCma, karpenterma da hofmanma (Dihoff, Hetznecker, Brosvic,
Carpenter & Hoffman, 1993) Seqmnes. am skalaze izomeba qcevis eqvsi komponenti: _
metyvelebis unari, garemoze reagirebis unari, socialur gamRizianeblebze
reagirebis unari, fizikur gamRizianeblebze reagirebis unari, masalebis
gamoyenebisadmi adaptaciis unari da kognituri funqcionireba. TiToeuli
komponentis farglebSi gamoiyo funqcionirebis xuTi done. es doneebi rigis
skalaze gansxvavdeba erTmaneTisgan da dalagebulia funqcionirebis umdablesi
donidan umaRles donemde. magaliTad, metyvelebis ganviTarebis komponentis
farglebSi es doneebia:
• araferi
• Jestebi
• komunikaciis survili
• komunikaciis survili da sityvebi
• sxva mimarTuli komunikacia
Jestebi miuTiTeben ufro metad ganviTarebul metyvelebaze, vidre metyvelebis
saerTod ganuviTarebloba, komunikaciis survili da sityvebi ki _ ufro metad
ganviTarebul metyvelebaze, vidre mxolod Jestebi, magram Cven ar viciT, ramdenad
didia gansxvaveba kategoriebs Soris; amitom es rigis skalaa.
SeamowmeT Tqveni codna II.2
1. ganmarteT: _ skala, saxeldebis skala, rigis skala.
2. CamoTvaleT gazomvis oTxi skala.
3. ra nakli aqvs saxeldebis skalas?
4. ra informacias gvawvdis individisTvis miniWebuli rangi 2 miRwevis
motivaciis skalaze?
5. depresiisken midrekilebis sklaze (Zlierad midrekilidan naklebad
midrekilisken) 13 individisgan Semdgar jgufSi jenifers mieniWa rangi 1,
debis _ 2, anas _ 12 da martis _ 13.
თავი II. მეცნიერული კვლევა 8
a) axseniT, ratom ver daaskvniT, rom jeniferi midrekilia depresiisken
da rom marti _ saerTod ara.
b) SeiZleba davaskvnaT, rom depresiisken midrekilebiT jeniferi da debi
iseve gansxvavdebian erTmaneTisgan, rogorc ana da marti? axseniT
Tqveni pasuxi.
intervalebis skala
Tu dakmayofilebulia rigis skalis
moTxovnebi da, amasTan miniWebul
ricxvebs Soris gansxvaveba asaxavs
cvladis mniSnelobebs Soris igive
sididis sxvaobas, maSin saqme gvaqvs
intervalebis skalasTan, Tumca,
intervalTa skalas ar aqvs absoluturi nuli, sadac saerTod ar iqneba mocemuli
gasazomi cvladi. intervalTa skalaze nulovani mniSvneloba pirobiTad aris
arCeuli sawyis wertilad. is SeiZleba Canacvldes nebismieri sxva mniSvnelobiT da
am mniSvnelobidan daviwyoT aTvla.
intervalebis skalis magaliTebia farenhaitisa da celsiusis temperaturis
skalebi. nulovani maCvenebeli arc erTze ar niSnavs temperaturis saerTod arqonas.
arc erT am skalaTagans ar aqvs absoluturi nuli, radgan temperatura SeiZleba
nul gradusze qvemoTac daeces. swori iqneba, Tu vityviT, rom temperaturis
mixedviT iseTive sxvaobaa 20°C-sa da 40°C-s Soris, rogoric 40°C-sa da 60°C-s Soris,
magram ar iqneba swori, Tu vityviT, rom 40°C orjer ufro maRalia, vidre _ 20°C.
naTeli gaxdeba, Tu ratom ar SegviZlia amis Tqma, rodesac SevadarebT celsiusisa
da farengeitis skalebze gazomil Tanabar temperaturebs:
Tanabari temperaturebi
°C °F
20
40
60
68
104
140
yuradReba miaqcieT, rom 20°C-iT sxvaoba yovelTvis tolia 36°F-iT sxvaobisa,
magram vinaidan TiToeuli skala pirobiTi nulovani wertilidan iwyeba, erT
skalaze erTmaneTis gverdigverd mdebare qulebis fardoba ar udris meore skalis
erTmaneTis gverdigverd mdebare qulebis fardobas. magaliTad, Sefardeba
40°C/20°C=2.0, xolo Sesabamisi temperaturebis fardoba 104°F/68°F=1.5.
bihevioristul mecnierebSi skalebi intervalebis dones zustad iSviaTad
Seesabameba, Tumca, Sefasebis skalebi, romelTac xSirad iyeneben adamianebis iseTi
intervalebis skala (Interval measurement): _
cvladis mniSvnelobebi dalagebulia skalaze
da miniWebul ricxvebs Soris gansxvaveba
asaxavs cvladis mniSnelobebs Soris erTi da
igive sididis sxvaobas. intervalTa skalaze
nuli pirobiTad sawyisi wertilia.
თავი II. მეცნიერული კვლევა 9
maxasiaTeblebis Sesafaseblad, rogoricaa pirovnuli maxasiaTeblebi,
damokidebulebebi, ojaxuri bedniereba, sagnebis SiSi, miRwevis motivacia an
lideroba, xSirad gamoiyeneba, rogorc intervalebis skala. rogorc wesi, am
skalebSi warmodgenilia debuleba da pasuxebis skala ori ukiduresi polusiTa da
maT Soris moTavsebuli ramdenime wertiliT. magaliTad, damokidebulebebis skalaze
individs SeiZleba moeTxovebodes, rom upasuxos debulebas: _ „iaraRis tarebis
makontrolebeli Zlieri kanoni amcirebs danaSauls“ da aRniSnos Tavisi pasuxi aseT
skalaze:
sruliad
veTanxmebi
veTanxmebi ver gadamiwyvetia ar veTanxmebi sruliad ar
veTanxmebi
sxva formis Sefasebis skalaze SeiZleba mogeTxovebodeT pirovnebis Sefaseba
bipolaruli zedsarTavi saxelebiT:
aragulwrfeli _____ : ______ : ______ : ______ : ______ : ______ : ______ gulwrfeli
mimzidveli _____ : ______ : ______ : ______ : ______ : ______ : ______ aramimzidveli
civi ______ : ______ : ______ : ______ : ______ : ______ : ______ cxeli
Sefasebis skalebis or ukidures poluss Soris SeiZleba nebismieri raodenobis
kategoria moTavsdes; Sesabamisad, am skalebs ewodebaT 5-quliani, 7-quliani, 9-
quliani da a.S. imisda mixedviT, Tu ramdeni kategoriaa masze mocemuli. xSirad
skalis polusebze moTavsebuli zedsarTavi saxelebis prioritetuloba inversiulia
debulebaTa naxevrisTvis. amrigad, prioritetul zedsarTav saxelTa naxevari
skalis marcxena nawilSi Tavsdeba, xolo meore naxevari _ marjvena nawilSi, rac
Tavidan gvaridebs pasuxebis gacemisas wamoWril iseT problemebs, rogoricaa
mxolod skalis marjvena mxares moTavsebul debulebaTa gadasinjva.
mas Semdeg, rac pirovneba gascems pasuxebs, skalis TiToeul kategorias eniWeba
ricxobrivi mniSvneloba:
aragulwrfeli _____ : ______ : ______ : ______ : ______ : ______ : ______ gulwrfeli
1 2 3 4 5 6 7
mimzidveli _____ : ______ : ______ : ______ : ______ : ______ : ______ aramimzidveli
1 2 3 4 5 6 7
civi ______ : ______ : ______ : ______ : ______ : ______ : ______ cxeli
1 2 3 4 5 6 7
xSirad mocemulia ramdenime skala da cdis piris saboloo, erTi qulis
misaRebad saWiroa calkeul skalebze miRebul qulaTa dajameba. aseT skalebs
თავი II. მეცნიერული კვლევა 10
jamuri Sefasebis skalebs uwodeben. magaliTad, rotonma da kelim (Rotton & Kelly, 1985)
Seqmnes e.w. mTvaris efeqtis rwmenis skala, romliTac fasdeboda qcevaze savse
mTvaris gavlenis rwmena. skala cxra debulebisgan Sedgeba. respondenti irCevs
ricxvs 1-dan (sruliad ar veTanxmebi) 9-mde (sruliad veTanxmebi) da amiT miuTiTebs,
Tu ramdenad eTanxmeba TiToeul debulebas. magaliTad, skalis erT-erTi debuleba1
aseTia:
kargi azria savse mTvaris dros saxlSi darCena.
1 2 3 4 5 6 7 8 9
sruliad ar
veTanxmebi
arc
veTanxmebi,
arc ar
veTanxmebi
sruliad
veTanxmebi
Sefasebis skalisTvis saWiro saboloo qula miiReba cxrave debulebaze miRebuli
qulebis dajamebiT. qulaTa SesaZlebeli diapazonia 9-dan (Tu yoveli debuleba
Sefasebulia 1 quliT) 81 qulamde (Tu yoveli debuleba Sefasebulia 9 quliT).
yuradReba miaqcieT, rom am skalaze qulaTa mocemuli rigi pirobiTia; Tu
TiToeuli debulebis Sesafasebeli sayrdeni punqtebi (romelTac raodenoba am
skalaze aris 9) dalagebuli iqneboda 10-dan 90-mde (da ara 1-dan 9-mde), maSin jamuri
Sefasebis skalis diapazoni iqneboda 90-dan 810 qulamde.
gazomvis romel skalas miekuTvneba mTvaris efeqtis rwmenis skalaze da Sefasebis
sxva msgavs skalebze miRebuli qulebi? cxadia, rom TiToeuli debulebis Sefaseba
moTavsebulia, sul mcire, rigis skalaze. qula 4 miuTiTebs debulebaSi aRwerili
konkretuli maxasiaTeblis ufro metad mocemulobaze, vidre _ qula 3, xolo qula
3 _ ufro metad mocemulobaze, vidre qula 2; Tumca, ver vityviT gadaWriT, rom
gansxvaveba 4-sa da 3-s Soris maxasiaTebelSi imdenive gansxvavebas gamoxatavs,
ramdensac gansxvaveba 3-sa da 2-s Soris. Tumca, aseTi Sefasebebi ufro mets unda
iZleodes, vidre mxolod rigis skalis informaciaa. Cven SeiZleba movelodeT, rom
respondenti subieqturad dayofs kategoriebs daaxloebiT tol intervalebad
skalis polusebs Soris. Tu davajamebT respondentisgan ramdenime calkeul
skalaze miRebul Sefasebebs, samarTliani iqneba, Tu davuSvebT, rom saboloo qula
ufro mets gvaZlevs, vidre mxolod rigis Sesaxeb informacias. gardneris (Gardner,
1975) mixedviT, jamur skalebze miRebuli qulebi rigisa da intervalebis skalebs
Soris xvdeba. amrigad, ar aris aucilebeli, rom romelime konkertuli skala
zustad ergebodes gazomvis oTxi skalidan erT-erTs; Tumca, rodesac statistikuri
1 debuleba ibeWdeba avtorebis _ j. rotonisa da i. u. kelis da gamomcemlis nebarTviT, „A
Scale for assessing belief in lunar effects: reliability and concurrent validity“, Psychological Reports, 1985, 57, 239-245,
© Psychological Reports, 1985.
თავი II. მეცნიერული კვლევა 11
analizis saWiroeba dgeba, Sefasebis skalebs mkvlevrebi xSirad intervalebis
skalebad ganixilaven.
Sefardebis skala
intervalebis skalis pirobiTi
nulovani wertili Sefardebis
skalaze Canacvlebulia
absoluturi nuliT, romelic
Seesabameba gasazomi cvladis
srul ararsebobas. Sefardebis
skalaze SesaZlebelia vTqvaT, rom erTi obieqti (magaliTad, gamRizianebeli,
movlena an individi) orjer, naxevarjer an samjer metad xasiaTdeba gasazomi
cvladiT, vidre _ meore. Sefardebis skala maSin gamoiyeneba bihevioristul
mecnierebebSi, rodesac izomeba iseTi sidideebi, rogoricaa reaqciis dro an
amocanis Sesrulebaze daxarjuli dro. am sidideebs aqvT absoluturi nuli, skalis
erTeulebi ki Tanabari intervalebiT arian erTmaneTisgan daSorebuli. magaliTad,
Tu vzomavT dros, romelsac individi Tavsatexis amoxsnas andomebs, 15 wamsa da 20
wams Soris (anu 10 wami) sxvaoba tolia 50 wamsa da 60 wams Soris (isev 10 wami)
sxvaobisa; amasTanave, is, vinc 60 wamSi xsnis Tavsatexs, orjer met dros andomebs
amas, vidre is, vinc Tavsatexis amoxsnas 30 wamSi axerxebs.
perceptuli iluziis sididis gazomva xSirad Sefardebis skalazea
SesaZlebeli. iluzia Cndeba maSin, rodesac gamRizineblis Cveneuli aRqma ar
Seesabameba mis fizikur maxasiaTeblebs; Cven arasworad aRviqvamT imas, rac
arsebobs. amis cnobili magaliTia miuler-laieris iluzia, romelic N2.1 suraTzea
mocemulia. aq Tqven xedavT (an metad albaTuria, rom dainaxoT), rom (a) xazi (b)
xazze grZelia, Tumca, fizikurad es xazebi tolia. Tqveni aRqma mcdaria. am iluziis
sidide SeiZleba gavzomoT imis gansazRvriT, Tu ramdenad gansxvavdeba stimulis
aRqmuli sigrZe misi realuri sigrZisgan. milimetrebiT (mm) gazomili perceptuli
Secdoma ganekuTvneba Sefardebis skalas. sruliad SesaZlebelia, rom cdis piris
Secdoma iyos 0 mm (anu ar iyos Secdoma), rodesac stimulis aRqmuli sigrZe ar
gansxvavdeba misi realuri sigrZisgan. 5 mm-iT Secdomasa da 8 mm-iT Secdomas Soris
gansxvaveba igivea, rac 10 mm-iT Secdomasa da 13 mm-iT Secdomas Soris. garda amisa,
10 mm-iT Secdoma gamoxatavs sididis iluzias, romelic orjer aRemateba 5 mm-iT
Secdomas.
Sefardebis skala (Ratio measurement): _ cvladis sidide
dalagebulia skalaze da miniWebul ricxvebs Soris
gansxvaveba asaxavs cvladis mniSnelobebs Soris igive
sididis gansxvavebas. Sefardebis skalaze arsebobs
WeSmaritad nulovani wertili, romelic Seesabameba
gasazomi maxasiaTeblis srul ararsebobas.
თავი II. მეცნიერული კვლევა 12
Tvisebrivi da raodenobrivi monacemebi
adamianebisa Tu sagnebis sxvadasxva
kategoriebSi gasaerTianeblad
saxeldebis skalaze gamoiyeneba
ricxvebi, romlebic ar iZlevian
raodenobriv informacias im cvladis
Sesaxeb, romlis mixedviTac moxda
maTi kategoriebSi gaerTianeba.
amrigad, saxeldebis skala gvaZlevs
Tvisebriv monacemebs. kategoriebad gaerTianebuli adamianebi an sagnebi raRac
TvisebiT arian gansxvavebulni. rigis, intervalebisa da Sefardebis skalebi
gvaZleven raodenobriv monacemebs. cvladisTvis miniWebuli ricxvebi gamoxataven am
cvladis sidides anu raodenobas. am sami skalidan bihevioristuli mecnierebebis
wamomadgenlebi upiratesobas aniWeben intervalebisa da Sefardebis skalebs, radgan
isini gasazomi cladis ZiriTadi Tvisebebisa Tu maxasiaTeblebis raodenobis ufro
zustad gansazRvris saSualebas iZlevian, vidre rigis skala. garda amisa, vinaidan
intervalebisa da Sefardebis skalebi iseTi maTematikuri operaciebis gamoyenebis
saSualebas iZlevian, rogoricaa Sekreba, gamokleba, gamravleba da gayofa, am
skalebze SesaZlebelia metad mravalferovani statistikuri teqnikis gamoyenebac.
aqedan gamomdinare, winamdebare saxelmZRvaneloSi warmodgenili mravali
statistikuri meTodi mowodebulia intervalebis an Sefardebis skalebze miRebuli
monacemebisTvis.
SeamowmeT Tqveni codna II.3
1. ganmarteT: _ intervalebis skala, Tvisebrivi monacemebi, raodenobrivi
monacemebi, Sefardebis skala.
2. liderobis unaris Sesafasebel rigis skalaze cdis pirebma miiRes
Semdegi qulebi: rozana _ 84, eva _ 78, pirsi _ 71, viqtoria _ 65, nensi _ 42
da uiliami _ 0.
a) marTebuli iqneba, Tu davaskvniT, rom rozanasa da evas liderobis
unarebs Soris iseTive gansxvavebaa, rogoric pirsisa da viqtorias
unarebs Soris? axseniT Tqveni pasuxi.
b) marTebuli iqneba, Tu davaskvniT, rom rozanas liderobis orjer meti
unari aqvs, vidre nensis? axseniTYTqveni pasuxi.
g) marTebuli iqneba, Tu davaskvniT, rom radgan uiliamma 0 qula miiRo,
mas saerTod ar aqvs liderobis unari? axseniTYTqveni pasuxi.
3. kolejis ramdenime studentma monawiloba miiRo Zilis kvlevaSi,
romelSic Tvalebis swrafi moZraobis fazaSi (REM Zili; faza, rodesac
Tvisebrivi monacemebi (Qualitative data): _
saxeldebis skalaze miRebuli monacemebi,
romlebic miuTiTeben, rom cvladebi
Tvisebrivad gansxvavdebian erTmaneTisgan.
raodenobrivi monacemebi (Quantitative data): _
rigis, intervalebisa an Sefardebis skalaze
miRebuli monacemebi, romlebic miuTiTeben, Tu
ra sididis aris cvladi.
თავი II. მეცნიერული კვლევა 13
adamiani sizmars xedavs) gatarebuli dro (wuTebiT)
eleqtroencefalogramis gamoyenebiT izomeboda. erT Rames REM Zilis
xangrZlivoba iyo: dominikTan _ 204 wT, gretasTan _ 194 wT, andresTan _
175 wT, judisTan _ 165 da karmenTan _ 102 wT.
a) marTebuli iqneba, Tu davaskvniT, rom sxvaoba dominikisa da gretas
REM Zilis xangrZlivobas Soris imdenivea, ramdenic andresa da judis
REM Zilis xangrZlivobas Soris? axseniTYTqveni pasuxi.
b) marTebuli iqneba, Tu davaskvniT, rom dominikma orjer meti dro
gaatara REM Zilis fazaSi, vidre karmenma? axseniTYTqveni pasuxi.
g) davuSvaT, erT-erTma studentma, martinma REM Zilis fazaSi gaatara 0
wT. marTebuli iqneba, Tu davaskvniT, rom im Rames mas ar hqonia REM
Zilis faza? axseniTYTqveni pasuxi.
diskretuli (wyvetili) da uwyveti cvladebi
cvladebi SeiZleba iyos diskretuli (wyvetili) da uwyveti
diskretuli (wyvetili) cvladebi
diskretulia (wyvetilia) cvladi,
romelmac SeiZleba miiRos
mniSvnelobebi mxolod sasruli an
potenciurad Tvladi simravlidan. sityva „sasruli“ niSnavs SesaZlo mniSvnelobebis
gansazRvrul simravles. mocemuli kolejis studentebis raodenoba diskretuli
cvladia: am kolejSi SeiZleba swavlobdes 5652 studenti an 5653 studenti, magram
ara _ 5652½ studenti. studentebis raodenoba SeiZleba gaizardos an Semcirdes
erTi mTliani, magram ara _ ½ an ⅓ studentis mimatebiT an gamoklebiT. amrigad,
studentebis raodenoba iRebs mxolod gansazRvrul Tvlad mniSvnelobebs.
Cveulebisamebr, cvladebs, romelTa SesaZlo mniSvnelobebi SeiZleba daviTvaloT,
diskretul (wyvetil) cvladebs uwodeben: qalaqSi mcxovreb adamianTa raodenoba,
ojaxSi bavSvebis raodenoba, qurdobebis raodenoba erTi wlis manZilze, StatSi
regisgrirebuli avtomanqanebis raodenoba da a.S.
uwyveti cvladebi
uwyvetia cvladi, romelmac SeiZleba miiRos
mniSvnelobebi cvladis nebismier or dones
Soris moTavsebuli usasrulo simravlidan.
amrigad, uwyvet cvlads aqvs mniSvnelobaTa
SemousazRvreli simravle. aseTi cvladis magaliTia Tqveni wonis gazomva: is
SeiZleba iyos 55.6 kg an 55.64 kg. Tqveni wona ar SemoisazRvreba iseTi diskretuli
mniSvnelobebiT, rogoricaa 55 kg an 56 kg. Tumca, Tqveni wona mainc ar iqneboda
diskretuli (wyvetili) cvladi (Discrete variable):
_ cvladi, romelmac SeiZleba miiRos
sasruli an Tvladi mniSvneloba.
uwyveti cvladi (Continuous variable): _
cvladi, romelmac SeiZleba miiRos
usasrulo mniSvneloba cvladis
nebismier or dones Soris.
თავი II. მეცნიერული კვლევა 14
SemosazRvruli am konkretuli mniSvnelobebiT; man SeiZleba miiRos nebismieri
mniSvneloba 55 kg-sa da 56 kg-s Soris. analogiurad, mravali fsiqologiuri cvladi,
rogoricaa, magaliTad, SfoTva an inteleqti, arsebobs uwyveti cvladis saxiT,
Tumca, maTi gazomva SesaZlebelia skalebiT, romlebzec cdis pirs diskretuli
qulebi eniWeba.
gazomvis realuri sazRvrebi. rodesac vzomavT uwyvet cvlads, cvladis arsebul
sidideebs eniWeba konkretuli ricxviTi mniSvnelobebi, magram es aris cvladis
realur sidideebTan mxolod miaxloebuli mniSvnelobebi. aqedan gamomdinare,
ganurCevlad imisa, gazomvis romel skalas avirCevT, unda vicodeT, ra sizustiT
vzomavT. uwyveti cvladis gazomvis sizuste moTavsebulia miniWebuli ricxvis
realur sazRvrebSi.
gazomvis realuri sazRvrebis cnebis
gageba gaioldeba, Tu ganvixilavT drois
cvladis gazomvis magaliTs. davuSvaT,
gvainteresebs, ramden xans SeuZlia
adamians 40°F (5°C) wyalSi xelis gaCereba.
am davalebas xSirad iyeneben tkivilis
usafrTxod gamosawvevad eqsperimentebSi,
sadac ikvleven tkivilisadmi
tolerantobis gazrdaSi relaqsaciis
strategiebis rols. davuSvaT, erTma cdis
pirma, Cveni gazomvis mixedviT
(vakvirdebodiT mas da vuyurebdiT wamzoms), wyalSi xelis gaCereba SeZlo 37 wm.
ramdenad zustia gazomvis es Sedegi? vinaidan Cven vzomavdiT uaxloesi mTeli wamis
sizustiT, dafiqsirebuli dro ver iqneboda 36 an 38 wm, magram SesaZlebelia, rom
realuri dro 37,2 wm an 36,7 wm yofiliyo? pasuxia: _ „diax“; Cven SegveZlo 37 wm-mde
dagvemrgvalebina 36,7 wamic da 37,2 wamic. am gazomvis (37 wm) sizuste mocemulia
ricxvis realuri sazRvrebiT. ricxvis realuri sazRvrebi aris ricxvisa da misi
wina ricxvis Suawertili an ricxvisa da misi momdevno ricxvis Suawertili im
skalaze, romelsac viyenebT mocemuli cvladis gasazomad. miRebuli
mniSvnelobisTvis _ 37 wm-isTvis, Sesabamisad, realuri qveda sazRvari iqneba 36,5 wm,
xolo realuri zeda sazRvari _ 37,5 wm. 36,5 wamisTvis realuri qveda zRvari aris
Sualedi 36 wamsa (skalaze momdevno gazomvad mniSvnelobasa) da 37 wams Soris. 37,5
wamisTvis realuri zeda zRvari aris Sualedi 37 wamsa da 38 wams (skalze momdevno
gazomvad mniSvnelobas) Soris.
axla davuSvaT, rom vzomavT eleqtronuli wamzomiT, romelsac SeuZlia gazomos
uaxloesi 100
1 wamis sizustiT da miviReT, magaliTad, 37,21 wm. ra iqneba am gazomvis
ricxvis realuri sazRvrebi (Real limits of a
number): _ ricxvisa da misi wina ricxvis
Suawertili an ricxvisa da misi momdevno
ricxvis Suawertili im skalaze, romelsac
viyenebT mocemuli cvladis gasazomad.
ricxvis realuri qveda sazRvari (Lower real
limit of a number) _ ricxvisa da misi wina
ricxvis Suawertili.
ricxvis realuri zeda sazRvari(Upper real
limit of a number) _ ricxvisa da misi momdevno
ricxvis Suawertili.
თავი II. მეცნიერული კვლევა 15
realuri zeda da qveda zRvari? am SemTxvevaSic viyenebT igive wess: _ vpoulobT
momdevno umdables da umaRles mniSvnelobebs Soris Sua wertils im skalaze,
romelzec vzomavT cvlads. amrigad, realuri qveda zRvari iqneba Suawertili 37,20
wamsa (momdevno umdables mniSvnelobas) da 37,21 wams Soris anu 37,205 wm, xolo
realuri zeda zRvari _ Suawertili 37,21 wamsa da 37,22 wams (momevno umaRles
mniSvnelobas) Soris anu 37,215 wm.
uwyveti cvladis yovel ricxviT mniSvnelobas aqvs masTan dakavSirebuli
realuri zeda da qveda zRvrebi. davuSvaT, viyenebT Sefasebis 7-qulian skalas,
romelic mocemulia N2-2 suraTze. vTqvaT, cdis pirma am skalaze aRniSna pasuxi 3.
zemoT aRwerili wesebis mixedviT, am qulis realuri qveda da zeda zRvari,
Sesabamisad, aris 2,5 da 3,3-ia.
gazomvis realuri sazRvrebi mniSvnelovania qulebis sixSireTa ganawilebis
agebisas da statistikur gamoTvlebSi mniSvnelobebis damrgvalebisas. sixSireTa
ganawilebis Sesaxeb me-3 TavSi visaubrebT.
suraTi N2.2. 7-qulian skalaze gacemuli pasuxis (aRniSnulia -iT) realuri zeda da
qveda zRvrebi.
1 2 3 4 5 6 7
arasodes
Zalian
xSirad
2,5
realuri
qveda
zRvari
3,5
realuri
zeda
zRvari
damrgvaleba: gamoTvlis saboloo pasuxi. davuSvaT, sam adamians vTxoveT N2-2
suraTze mocemuli skalis Sevseba da miviReT Semdegi qulebi: _ 4, 4 da 3. tipuri
pasuxis aRsawerad unda SevkriboT es ricxvebi (miviRebT 11-s) da miRebuli jami
gavyoT 3-ze, Sedegad gveqneba 6666,33
11= . aRniSvna (‾) bolo 6-ianis Tavze niSnavs, rom
6-ianebi usasrulod meordeba. mZimis Semdeg ramdeni TanrigiT unda warmoadginoT
miRebuli pasuxi?
erTi SeTanxmebaa, rom gamoTvlis saboloo mniSvneloba unda damrgvaldes mZimis
Semdeg or Tanrigamde realurad miRebuli mniSvnelobis Semdeg. warmodgenil
magaliTSi qulebi mocemulia erTeulebSi: cdis pirma unda airCios 3 an 4 da ara _
3,2 an 4,6. am wesis mixedviT, gamoTvlis saboloo Sedegi unda damrgvaldes arCeuli
pasuxis Semdeg or Tanrigamde anu 3,67-mde.
meore SeTanxmebiT, gamoTvlis saboloo Sedegi mrgvaldeba mZimis Semdeg erT
Tanrigamde realurad miRebuli qulis Semdeg. am midgomis mixedviT, mocemul
თავი II. მეცნიერული კვლევა 16
magaliTSi gamoTvlis saboloo Sedegi damrgvaldeba miRebuli qulis mZimis Semdeg
erT Tanrigamde anu iqneba 3,7. qvemoT ganxilulia orive midgomis sailustracio
magaliTi.
damrgvaleba: gamoTvlis Sualeduri nabijebi. Tu saboloo pasuxis gamoTvlas
ramdenime nabiji sWirdeba, Sualeduri ricxviTi mniSvnelobebi, sul mcire, ori
TanrigiT metiT unda gadavitanoT saboloo pasuxis gamoTvlis etapamde. magaliTad,
Tu 3
11 -is mniSvneloba Semdgom gamoTvlebSic unda gamoviyenoT, is ar unda
damrgvaldes 3,67-mde; metic, Semdgomi gamoTvlebisTvis is ori damatebiTi TanrigiT
unda gadavitanoT _ 3,6667. am midgomas minimumamde dayavs gamoTvlebSi damrgvalebis
Secdoma.
damrgvalebis wesebi. davuSvaT, gamoTvlis Sedegad miviReT pasuxi 3,6AB, sadac A
da B SesaZlo ricxviTi mniSvnelobebia. magaliTad, Tu A=2 da B=9, maSin 3,6AB=3.629.
vTqvaT, gvsurs am ricxvis mZimis Semdeg or Tanrigamde damrgvaleba. ricxvebis
damrgvaleba, Cvelebriv, ramdenime wesis dacviT xdeba.
• Tu B-Ti aRniSnuli ricxvi 5-ze metia, A izrdeba 1-iT da ugulebeliyofa B
sidide. am wesis mixedviT, 3,629 (aq A=2 da B=9) mrgvaldeba 3,63-mde.
• Tu B 5-ze naklebia, A-s mniSvneloba ucvleli rCeba da ugulebeliyofa B
sidide. am wesis mixedviT, 3,623 (aq A=2 da B=3) mrgvaldeba 3,62-mde.
• Tu B-Ti aRniSnuli ricxvi 5-is tolia, A izrdeba 1-iT, Tu A kentia; xolo A-
s mniSvneloba ucvleli rCeba, Tu A luwia, B sidide ki ugulebeliyofa. am
wesis mixedviT, 3,635 (aq A=3 da B=5) mrgvaldeba 3,64-mde, xolo 3,625 (aq A=2 da
B=5) _ 3,62-mde da ara 3,63-mde.
SeamowmeT Tqveni codna II-4
1. ganmarteT: _ uwyveti cvladi, diskretuli (wyvetili) cvladi, ricxvis
realuri sazRvrebi, realuri qveda zRvari, realuri zeda zRvari.
2. gaarkvieT, qvemoT CamoTvlili cvladebidan romelia diskretuli da
romeli _ uwyveti.
a) individis xelis sigrZe, gazomili santimetrebiT.
b) sasamarTlos darbazSi bralmdebelisa da braldebulis TvalebiT
kontaqtis xangrZlioba, gazomili wamebiT.
g) tomsonis gazelis naxtomebis (vertikaluri naxtomi, rodesac oTxive
fexi erTdroulad scildeba miwas) raodenoba 1 saaTis ganmavlobaSi.
d) mgalobeli Citis simReris xangrZlivoba (wamebSi) Sewyvilebis
periodSi.
თავი II. მეცნიერული კვლევა 17
e) multiplikaciur filmebze sicili, romelic izomeba im drois
xangrZlivobis mixedviT (wamebSi), ramden xansac icinis maTze cdis
piri.
v) sabavSvo baRis bavSvebis altruistul qcevaTa raodenoba erTi dRis
ganmavlobaSi.
z) bavSvis megobrebis raodenoba.
T) statistikis gamocdaze sworad gacemuli pasuxebis raodenoba.
3. ipoveT TiToeuli qulis realuri zeda da qveda zRvrebi.
a) 152,4 sm simaRle.
b) TvalebiT kontaqtis xangrZlioba 17,3 wm.
g) 7-qulian skalaze micemuli Sefaseba 4.
d) reaqciis dro 0,437 wm.
e) 6,9 mm Secdoma miuler-laieris iluziis aRqmisas.
v) xelis sigrZe 67,81 sm.
4. daamrgvaleT Semdegi mniSvnelobebi frCxilebSi miTiTebul Tanrigamde.
a) 27,436 (or Tanrigamde).
b) 119,0278 (sam Tanrigamde).
g) 1,445 (or Tanrigamde).
d) 1263.75915 (oTx Tanrigamde).
e) 0,352 (or Tanrigamde).
v) 13,2659 (or Tanrigamde).
Sejameba
• statistika qcevis mecnieruli Seswavlis saSualebaa.
• mecnieruli kvleva iwyeba kiTxvis dasmiT, romelzec pasuxis gacema
SesaZlebelia empiriuli monacemebis mopovebis saSualebiT.
• kvlevis meTodi aris monacemebis mogrovebisas gamoyenebuli midgoma. aq
ganxilulia kvlevis oTxi meTodi: _ bunebrivi dakvirveba, saarqivo
Canawerebi, gamokiTxva da eqsperimenti.
• bunebrivi dakvirveba aris kvleva, romelic gulisxmobs bunebriv pirobebSi
ganxorcielebul qcevebze dakvirvebas.
• saarqivo CanawerebiT kvlevisas gamoiyeneba arsebuli Canawerebi.
• gamokiTxva aris kvleva, romelSic monacemebi adamianebis zepiri an
werilobiTi interviuebiT miiReba.
• eqsperimenti aris zedmiwevniT gakontrolebuli situacia, sadac mecnieri
axdens erTi an meti damoukidebeli cvladis manipulirebas da akvirdeba, Tu
ra gavlenas axdens es damokidebul cvladze.
თავი II. მეცნიერული კვლევა 18
• operaciuli gansazRvreba aRwers dakvirvebisas an damoukidebeli cvladiT
manipulirebisas gamoyenebul operaciebs Tu etapebs.
• gazomva aris garkveuli wesebis mixedviT cvladebisTvis ricxviTi
mniSvnelobebis miniWeba.
• saxeldebis skala aris gazomili cvladis sxvadasxva kategoriebSi
gaerTianeba. saxeldebis skala iZleva Tvisebriv monacemebs.
• rigis skala mocemuli parametris mixedviT TanmimdevrobiT ganalagebs
cvladis sidideebs.
• intervalebis skalaze Tanabrad izrdeba gasazomi cvladis sidide, Tumca, aq
mas ara aqvs absoluturi nuli.
• Sefardebis skala xasiaTdeba intervalebis skalis TvisebebiT da damatebiT
aqvs absoluturi nulovani wertili.
• rigis, intervalebisa da Sefardebis skalebi raodenobriv monacemebs iZleva.
• diskretulma cvladma SeiZleba miiRos mniSvnelobebi mxolod sasruli an
potenciurad Tvladi simravlidan.
• uwyvetma cvladma SeiZleba miiRos mniSvnelobebi cvladis nebismier or dones
Soris moTavsebuli usasrulo simravlidan.
• ricxvis realuri sazRvrebi aris ricxvsa da mis wina da momdevno umdables
da umaRles ricxvebs Soris Sua wertilebi im skalaze, romelsac viyenebT
mocemuli cvladis gasazomad.
ZiriTadi terminebi da simboloebi
bunebrivi dakvirveba
gamokiTxva
diskretuli cvladi
empiriuli monacemi
eqsperimenti
Tvisebrivi monacemebi
intervalebis skala
kvlevis sqema
kvlevis hipoTeza
operaciuli gansazRvreba
raodenobrivi monacemebi
rigis skala
ricxvis realuri zeda
zRvari
ricxvis realuri zRvari
ricxvis realuri qveda
zRvari
saarqivo Canawerebi
saxeldebis skala
skala
uwyveti cvladi
Sefardebebis skala
cvladi
kiTxvebi
1. davuSvaT, gTxoves qvemoT CamoTvlil kiTxvebze pasuxis gasacemad
monacemebis Segroveba. kvlevis romel sqemas airCevdiT TiToeul SemTxvevaSi
da ratom?
a) statistikis kursze Tqveni jgufelebi, Cveulebriv, ramden dros uTmoben
saSinao davalebis Sesrulebas?
თავი II. მეცნიერული კვლევა 19
b) ramdenad xSirad isvris kalaTburTeli burTs kalaTSi maSin, rodesac
msaji uSualod srolis win drois amowurvis niSans iZleva?
g) zrdis Tu ara gamyidvelTa daxelovnebis programa im adamianTa navaWris
raodenobas, romlebmac gaiares es programa?
d) ra tipur qcevebs axorcielebs patara bavSvi maRaziidan gamosvlisas?
2. davuSvaT, gTxoves TanamSromelTa jgufSi samsaxureobrivi stresis gazomva.
rogor gansazRvravT operaciulad samsaxureobrivi stresis zomas?
3. gazomes policiis oficerTa orientacia profesiul miRwevaze. amisaTvis
gamoiyenes oTxi teqsti. TiToeul maTganSi aRwerili iyo profesiul
miRwevaze sxvadasxvagvari oriantacia, romlis Sinaarsi icvleboda 1-dan
(invidividis profesiul miRwevasTan yvelaze miaxloebulidan) 4-mde
(individis profesiul miRwevisgan yvelaze daSorebulamde).
a) ra ZiriTadi cvladi gazomes am kvlevaSi?
b) gazomvis romel skalas miekuTvneboda gamoyenebuli sazomi?
4. profesionali golfis moTamaSe qalebi awones, raTa daedginaT,
ukavSirdeboda Tu ara maTi sxeulis wona maTi TamaSis xarisxs.
a) ra ZiriTadi cvladi gazomes am kvlevaSi?
b) gazomvis romel skalas miekuTvneboda gamoyenebuli sazomi?
g) golfis erTi moTamaSis wona iyo 54,6 kg. ra iqneba am gazomvis realuri
zeda da qveda zRvari.
d) 6 golfis moTamaSe awones kilogramis sizustiT. maTi saSualo wona
iyo 57,648 kg. daamrgvaleT es ricxvi.
5. pirvelklaseli bavSvebis mSoblebi Seafases emociuri stabilobis
pirovnuli maxasiaTeblis mixedviT. amisaTvis gamoiyenes Sefasebis skala.
a) ra ZiriTadi cvladi gazomes am kvlevaSi?
b) gazomvis romel skalas miekuTvneboda gamoyenebuli sazomi?
g) vTqvaT, cdis pirma daTanxmeba _ ardaTanxmebis 9-qulian skalaze
debulebas mianiWa 8. ra iqneba am gazomvis realuri qveda da zeda
zRvari?
d) vTqvaT, am skalis mixedviT SeafaseT eqvsi adamiani da mocemuli
debulebis maTi tipuri Sefaseba iyo 5,833. ramdenad sworad aris
mocemuli es Sedegi mZimis Semdeg warmodgenili Tanrigebis raodenobis
TvalsazrisiT? axseniT Tqveni pasuxi.
6. Tqven ikvlevT, Tu ramden dros uTmoben studentebi yoveldRiurad saSinao
davalebis Sesrulebas. erTma studentma aRniSna, rom qimias 97 wuTs uTmobs.
a) gazomvis romel skalas miekuTvneboda gamoyenebuli sazomi?
b) ra iqneba studentis mier dasaxelebuli 97 wuTis realuri zeda da
qveda zRvari?
თავი II. მეცნიერული კვლევა 20
g) samma studentma aRniSna, rom saSinao davalebis Sesrulebas uTmobs 97,
68 da 83 wuTs, Sesabamisad. maTi qulebis saSualoa 82,666666.
daamrgvaleT es ricxvi.
7. pirovnebaTSorisi Rirebulebebis Sesafaseblad cdis pirs moeTxoveboda
Rirebulebebis amsaxveli 12 debuleba daelagebina yvelaze
mniSvnelovanidan yvelaze naklebad mniSvnelovanamde. gazomvis romel
skalas miekuTvneboda gamoyenebuli sazomi?
8. sabavSo baRSi bavSvebis urTierTqmedeba gaarTianes kategoriebSi:
agonisturi (anu kamaTi an Cxubi), neitraluri (ar iyo interaqcia) da
altruistuli (anu daxmarebis gamwevi). gazomvis romel skalas
miekuTvneboda gamoyenebuli sazomi?
9. gansazRvreT, qvemoT CamoTvlili cvladebi diskretulia Tu uwyveti.
a) Tqveni jgufelebis raodenoba statistikis kursze.
b) im adamianebis mier daklebuli wona kilogramebSi, romlebic dietas
icaven da varjiSoben.
g) caciebis ricxvi 100 kacian jgufSi.
d) individis sisxlis wneva, romelic izomeba vercxlis wylis svetis
simaRliT.
10. gansazRvreT qvemoT CamoTvlili maCveneblebis realuri zeda da qveda
zRvari.
a) wona 61,3 kg.
b) 34 wuTi, romelic eTmoba video TamaSs.
g) manqaniT gavlili 175.6 km.
d) sxeuli temperatura 36,6°C.
e) sistoluri sisxlis wneva vercxlis wylis simaRlis 109 mm.
11. daamrgvaleT Semdegi ricxvebi mZimis Semdeg or Tanrigamde.
a) 42,255
b) 1,165
g) 163,651
d) 27,426
e) 13,895
v) 24,6250
z) 19,2751
12. sadilis dasrulebis Semdeg restornis mepatroneebi avseben baraTebs. mas
sTxoven, rom TiToeul baraTSi 5-qulian skalze (cudidad saukeTesomde)
Seafasos sakvebis xarisxi, momsaxurebisas keTilganwyobis xarisxi,
momsaxurebis siswrafe da sisufTave. TiToeuli baraTidan miRebuli
saboloo qulaa 4-dan 20-mde. gazomvis romeli skalaa gamoyenebuli aq?
axseniT Tqveni pasuxi.
13. fsiqologi ikvlevda, Tu ramdenad SeeZloT adamainebs tkivilis atana;
amisaTvis cdis pirebi xels yofdnen 5°C-ian wyalSi; izomeboda drois
თავი II. მეცნიერული კვლევა 21
xangrZlivoba, ramden xansac SeeZloT maT xelis gaCereba wyalSi. man
Caatara ori kvleva.
a) pirvel kvlevaSi cdis pirebi dalagda imis mixedviT, Tu ramden xans
SeeZloT xelis gaCereba wyalSi da miiRes Semdegi Tanmimdevroba: liza
_ 1, stacia _ 2, marita _ 3, robini _ 4 da judi _ 5. gazomvis romeli
skala gamoiyena mkvlevarma?
b) meore kvlevaSi tkivilis amtanobis xangrZlivoba wamebiT gaizoma da
miiRes Semdegi Sedegebi: liza _ 111,3 wm, stacia _ 97,8 wm, marita _ 71,3
wm, robini _ 18,7 wm da judi _ 13,6 wm. gazomvis romeli skala gamoiyena
mkvlevarma?
14. jandacvis msoflio organizaciis mier janmrTelobis dacvis sistemebis
ranJirebisas (Mishra, 2000) aSS-is janmrTelobis dacvis sistemam 37-e rangi
miiRo. pirveli da meore adgilebi, Sesabamisad, safrangeTisa da italiis
janmrTelobis dacvis sistemebma daikaves. iaponias mieniWa me-10, xolo
Sveicarias _ me-20 rangi.
a) ramdenad ukeTesia safrangeTis janmrTelobis dacvis sistema
italiasTan SedarebiT?
b) aris Tu ara aSS-is janmrTelobis dacvis sistema orjer ukeTesi,
vidre Sveicariis? axseniT Tqveni pasuxi.
g) ras geubnebaT aSS-is janmrTelobis dacvis sistemis xarisxis Sesaxeb
is faqti, rom mas 37-e rangi mieniWa?
Tavi III. monacemTa aRwera: sixSireTa ganawileba da grafikebi
sixSireTa ganawileba daujgufebeli monacemebisTvis
martiv sixSireTa ganawileba
fardobiT sixSireTa ganawileba
sixSireTa ganawileba procentebSi
informaciis miReba sixSireTa ganawilebidan
dajgufebul sixSireTa ganawileba
dajgufebul sixSireTa ganawilebis ageba
procentili da procentuli rangebi
monacemis procentuli rangis povna
ganawilebis procentilis povna
sixSireTa ganawilebis grafikulad gamosaxva
histograma
sixSireTa poligoni
fardobiTi sixSiris histograma da poligoni
informaciis miReba histogramidan an sixSireTa
poligonidan
foTlebiani Reroebis diagrama
Tvisebrivi monacemebis grafikulad gamosaxva
sixSireTa ganawilebis formebi
sixSireTa simetriuli ganawileba
sixSireTa asimetriuli ganawileba
modaloba
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 2
gifiqriaT rodesme: _ „SfoTvis kontroli rom SemZleboda, ukeTesad
Cavabarebdi gamocdebs“? Tu ki, getyviT, rom marto Tqven ar gawuxebT es
problema. ukanesknel wlebSi konsultantebma da fsiqologebma daadgines, rom
gamocdebTan dakavSirebuli SfoTva zemoqmedebas axdens studentTa akademiur
warmatebaze. gamocdebTan dakavSirebuli SfoTvis gazomvis mizniT spilbergerma
(Spielberger, 1980) SeimuSava gamocdebTan dakavSirebuli SfoTvis kiTxvari (TAI). es
kiTxvari Sedgeba 20 iseTi debulebisgan, rogoricaa, magaliTad, „aforiaqebuli da
SeSinebuli var didi gamocdis win.“ yovel debulebas aqvs oTxi savaraudo pasuxi:
_ TiTqmis arasdros, xandaxan, xSirad da TiTqmis yovelTvis. kiTxvaris Sevsebisas
studentma TiToeuli debulebisTvis unda airCios erT-erTi pasuxi, romelic ukeT
miesadageba mas. mas Semdeg, rac kiTxvari Sevsebulia, TiToeul variants eniWeba
ricxviTi mniSvneloba: _ TiTqmis arasdros = 1; xandaxan = 2; xSirad = 3; TiTqmis
yovelTvis = 4. saboloo qula miiReba arCeuli variantebisTvis miniWebuli
ricxviTi mniSvnelobebis SekrebiT. amrgad, es kiTxvari jamuri Sefasebis skalaa,
romlis qulebi 20-dan 80-mde meryeobs. 20 qula gviCvenebs SfoTvis minimalur
dones (am kiTxvaris mixedviT), 80 qula ki _ maqsimalurs. rogorc me-2 TavSi
vnaxeT, jamuri Sefasebis skala, savaraudod, skalaTa Soris Tavsdeba; Tumca,
statistikuri miznebisTvis Sefasebis skalebze miRebuli qulebi intervalTa
skalis monacemebad ganixileba. qvemoT ganxilul magaliTSic am tradicias
mivsdevT.
davuSvaT, 20 studentisgan
(romelsac gamocdebTan dakavSirebuli
SfoTvis kontrolSi daxmareba
esaWiroeboda) Semdgar SerCevas
kolejis konsultantma gamocdebTan
dakavSirebuli kiTxvari misca.
miRebuli Sedegebi mocemulia cxrilSi
N3.1. aseT qulebs nedl monacemebs an
nedl qulebs uwodeben, radgan isini
uSualod kvlevis dros arian Segrovili
da ar arian statistikurad
damuSavebulni. laTinuri aso N
gamoiyeneba miRebuli qulebis saerTo
raodenobis aRsaniSnad; amrigad, am
magaliTSi N aris 20.
ganvixiloT studentis SemTxveva,
romelmac am kiTxvaris mixedviT 60 qula
miiRo. ras gveubneba es qula am studentis gamocdasTan dakavSirebuli SfoTvis
N: _ jgufSi miRebuli qulebis saerTo
raodenoba.
nedli monacemebi an nedli qulebi (raw data or
raw scores) _ kvlevis monawileebisgan
miRebuli qulebi, romelTac jer ar
gauvliaT statistikuri analizi.
cxrili N3.1. gamocdebTan
dakavSirebuli SfoTvis kiTxvarSi 20
studentis mier miRebuli hipoTeturi
qula.
61 67 63 60 62 67 59 60 68 64
57 55 59 59 59 64 65 62 59 60
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 3
donis Sesaxeb? is sxva studentebis mier miRebul qulebze metia Tu naklebi?
SegviZlia ki am kiTxvaze pasuxis gacema mxolod am erTi qulis codnis
pirobebSi? cxadia, ara da aseTive uaryofiTia pasuxi bihevioristul mecnierebaSi
gamoyenebuli mravali sazomis SemTxvevaSi, radgan erTi calkeuli qula mxolod
sxva qulebTan Sedarebis Semdeg iZens mniSvnelobas. amrigad, miRebuli qulebis
analizis da maTgan informaciis miRebis pirveli nabijia monacemebis warmodgena
sixSireTa ganawilebis saxiT. dasawyisisTvis ganvixilavT intervalebisa da
Sefardebis skalebze miRebuli raodenobrivi monacemebis warmodgenas.
sixSireTa ganawileba daujgufebeli monacemebisTvis
martiv sixSireTa ganawileba
sixSireTa ganawilebas uwodeben
cxrils, romelSic mocemulia
TiToeul qula da is sixSire, ra
sixSiriTac gvxvdeba es qula. cxrilSi N3.1 mocemuli gamocdebTan dakavSirebul
kiTxvarSi miRebuli qulebis daujgufebel sixSireTa ganawileba warmodgenilia
cxrilSi N3.2.
cxrili N3.2. daujgufebel sixSireTa ganawileba N3-1 cxrilSi
mocemuli (gamocdebTan dakavSirebuli SfoTvis kiTxvarSi
miRebuli) qulebisTvis.
(a) (b) (g) (d) (e)
qula Tvlis
Sedegi
sixSire
(ƒ)
fardobiTi
sixSire
(rƒ)
sixSire
procentebSi
(%ƒ)
68 67 66 65 64 63 62 61 60 59 58 57 56 55
/ // / // / // /
/// /////
/ /
1 2 0 1 2 1 2 1 3 5 0 1 0 1
.05
.10 .00 .05 .10 .05 .10 .05 .15 .25 .00 .05 .00 .05
5 10 0 5 10 5 10 5 15 25 0 5 0 5
sixSireTa ganawileba (Frequency distribution): _
cxrili, romelSic mocemulia TiToeuli qula da
is sixSire, ra sixSiriTac gvxvdeba es qula.
daujgufebel sixSireTa ganawileba (Ungroup frequency distribution): _ sixSireTa ganawileba,
romelic agebulia miRebul minimalur da maqsimalur qulaTa Soris yvela SesaZlo
mniSvnelobis dalagebiT da Semdeg qulis gaswvriv Tvlis Sedegis aRniSvnis CxiriT (/)
imdenjer moTavsebiT, ramdenjerac iqna miRebuli es qula.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 4
es daujgufebel sixSireTa ganawileba agebulia miRebul umdables da umaRles
qulaTa Soris yvela SesaZlo mniSvnelobis (cxrilis N3.2 (a) sveti) dalagebiT da
Semdeg qulis gaswvriv Tvlis Sedegis aRniSvnis Cxiris (/) imdenjer moTavsebiT,
ramdenjerac iqna miRebuli es qula (amave cxrilis (b) sveti). yuradReba miaqcieT,
rom (a) svetSi mocemulia yvela SesaZlo qula ganurCevlad imisa, miRebul iqna
is realurad Tu ara. im qulebis gaswvriv, romlebic ar iqna miRebuli qulaTa
mocemul simravleSi, ar iwereba Tvlis Sedegis aRmniSvneli Cxiri. TiToeuli
qulis gaswvriv moTavsebuli Cxirebis raodenoba daiTvleba da Seva (g) svetSi,
rogorc TiToeuli qulis Sexvedris martivi sixSire (aRniSneba ƒ-iT). yuradReba
miaqcieT, rom qula 60 Cvens mier ganxiluli ganawilebis daaxloebiT Sua nawils
warmoadgens da qulaTa daaxloebiT naxevari masze naklebi, meore naxevari ki _
masze metia.
fardobiT sixSireTa ganawileba
daujgufebel sixSireTa ganawilebaSi
qulebis Sexvedris sixSire SeiZleba
fardobiTi sixSireebis saxiTac
warmovadginoT. fardobiT sixSireTa
ganawileba miiReba TiToeuli qulis sixSiris gayofiT ganawilebaSi qulaTa
saerTo raodenobaze. amrigad, fardobiTi sixSire (aRiniSneba rƒ-iT) ase
gamoisaxeba:
qulis rƒ = qulis ƒ/N
magaliTad, cxrilidan N3.2 59-is Sexvedris fardobiTi sixSiris
gamosaTvlelad misi sixSire (anu ƒ=5) unda gavyoT qulaTa saerTo raodenobaze
(N=20) da miviRebT .25-s1. gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli
qulebis fardobiTi sixSireebi mocemulia cxrilis N3.2 (d) svetSi.
sixSireTa ganawileba procentebSi
sixSireTa ganawileba procentebSi gamoxatavs
qulis Sexvedris sixSires miRebuli qulebis
saerTo raodenobis procentis saxiT. qulis
procentebSi gamoxatuli sixSiris (aRiniSneba %ƒ-
1 teqstSi meaTedi wiladebi mocemulia amerikis fsiqologTa asociaciis poblikaciebis
saxelmZRvanelos (Publication Manual of the American Psychological Association, 1994) mixedviT, rac
gulisxmobs, rom rodesac wiladi 1-ze naklebia, mZimis win nuli ar iwereba. amitom
fardobiTi sixSireebi mocemulia mZimis win nulis gareSe.
ƒ: _ sixSireTa ganawilebaSi TiToeuli qulis Sexvedris martivi sixSire.
fardobiTi sixSire (rƒ) (Relative frequency):
_ qulis sixSire gayofili miRebul
qulaTa saerTo raodenobaze.
sixSiris procenti (%ƒ) (Relative
frequency): _ qulis fardobiTi
sixSire gamravlebuli 100-ze.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 5
iT) gamosaTvlelad qulis fardobiTi sixSire mravldeba 100-ze anu
qulis %ƒ = (qulis rƒ)x100
cxrilSi N3.2 qulis sixSiris procenti mocemulia (e) svetSi. yuradReba
miaqcieT, rom am cxrilis (g), (d) da (e) svetebSi mocemulia erTi da igive
informacia cxrilSi N3.1 warmodgenili qulebis Sesaxeb. fardobiTi sixSireebis
an procentebSi gamoxatuli sixSiris saxiT sixSireebis gamosaxvis upiratesoba
isaa, rom maTi mniSvnelobebi ar aris damokidebuli miRebuli qulebis saerTo
raodenobaze. amrigad, Tu gvaqvs studentTa sxva jgufisgan miRebuli 35 qulisgan
Semdgari SerCeva (N=35), ufro martivi iqneba sxvadasxva zomis ori SerCevis
sixSireTa ganawilebis urTierTSedareba, rodesac sixSireTa ganawileba
warmodgenilia fardobiTi sixSireebiT an sixSireTa procentebis saxiT, vidre
mxolod martivi sixSireebis saxiT warmodgenis SemTxvevaSi.
informaciis miReba sixSireTa ganawilebidan
cxrilSi N3.2 mocemuli martivi sixSireebi, fardobiTi sixSireebi an sixSireebi
procentebSi garkveul warmodgenas gviqmnis gamocdebTan dakavSirebuli SfoTvis
kiTxvarSi miRebul qulebze, rac arc ise naTlad Cans cxrilSi N3.1 mocemuli
nedli monacemebidan. magaliTad, TiToeuli ganawilebis SemTxvevaSi SeiZleba
SeamCnioT, rom:
• yvelaze dabali qulaa 55;
• yvelaze maRali qulaa 68;
• yvelaze xSirad gvxvdeba qula 59;
• 59-sa da 65-s Soris moTavsebulia TxuTmeti qula.
naTelia, rom nedli monacemebis warmodgena sixSireTa ganawilebis saxiT
aiolebs am monacemebis mniSvnelovani Tvisebebis danaxvas.
magaliTi N3.1
sixSireTa ganawilebis ageba
amocana
fsiqologebi dainteresdnen orazrovani figurebis aRqmiT. maT miaCndaT, rom
aseTi figurebis aRqmis Taviseburebebis SeswavliT samyaros Cveulebrivi aRqmis
Sesaxeb moipovebdnen codnas. suraTze N3.1 mocemuli naxati, romelic meCis
piramidis (The Mach Pyramid) saxeliT aris cnobili, orazrovani figuris magaliTia,
vinaidan sxvadasxvanairad aris SesaZlebeli misi siRrmeSi aRqma. is SeiZleba
aRviqvaT rogorc piramida, romlis mcire kvadrati Cvens mxares aris moqceuli,
fuZe ki furcelzea ganlagebuli; an rogorc oTaxi, romelSic igive mcire
kvadrati Cvens pirdapir mdebare, daSorebul kedlad aRiqmeba. vTqvaT, gyavdaT 25
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 6
cdis piri, romelTagan TiToeuli 1 wuTis ganmavlobaSi uyurebda am figuras da
afiqsirebdiT, ramden xans SeeZlo TiToeul maTgans piramidis danaxva. Sedegad
miiReT Semdegi droebi (1 wamis sizustiT):
2, 7, 10, 5, 5, 3, 9, 11, 5, 8, 2, 4, 7, 6, 10, 7, 3, 9, 7, 11, 4, 6, 3, 1 da 8.
es qulebi gamosaxeT martiv sixSireTa ganawilebis, fardobiT sixSireTa
ganawilebisa da sixSireTa procentebSi ganawilebis saxiT. ra daskvnebamde
SeiZleba mixvideT am ganawilebebze dayrdnobiT?
amoxsna
qvemoT mocemul cxrilSi warmodgenilia
sixSireTa dasaxelebuli ganawilebebi.
daskvnebi
qulebi icvleba 1-dan 11 wamamde da yvelaze xSirad gvxvdeba 7 wami. yvelaze
xSirad dafiqsirebuli 7 wamis garda, yvela danarCeni qula daaxloebiT Tanabari
sixSiriT gvxvdeba.
SeamowmeT Tqveni codna III.1
1. ganmarteT: _ sixSireTa ganawileba, sixSireTa ganawileba procentebSi,
%ƒ, nedli monacemebi, nedli qulebi, fardobiT sixSireTa ganawileba, rƒ,
sixSireTa ganawileba daujgufebeli monacemebisTvis.
2. gudma da gudma (Good & Good, 1973) Seqmnes warmatebis SiSis Sefasebis
skala. iTvleba, rom am skalaze maRali maCveneblis mqone individebi Tavs
arideben dawyebul saqmeebSi didi warmatebis miRwevas sxvebis
arakeTilganwyobis gamowvevis SiSis gamo. warmatebis SiSis skalaze
miRebuli qulebi SeiZleba icvlebodes 0-dan (rac miuTiTebs, rom ar
aRiniSneba warmatebis SiSi am skalaze) 29-mde (rac miuTiTebs warmatebis
maqsimalur SiSze am skalaze). davuSvaT, Sefasebis es skala mieciT 20
cdis pirs da miiReT Semdegi qulebi: _ 6, 13, 4, 19, 3, 12, 9, 8, 5, 11, 6, 8, 15,
5, 8, 10, 9, 14, 4 da 8.
qula Tvlis Sedegi
ƒ rƒ %ƒ
11 10 9 8 7 6 5 4 3 2 1
// // // // //// // /// // /// // /
2 2 2 2 4 2 3 2 3 2 1
.08
.08
.08
.08
.16
.08
.12
.08
.12
.08
.04
8 8 8 8 16 8 12 8 12 8 4
suraTi N3.1. meCis piramida
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 7
a) es qulebi gamosaxeT martiv sixSireTa ganawilebis, fardobiT
sixSireTa ganawilebisa da procentebSi gamoxatuli sixSiris saxiT.
b) miRebul qulebs Soris romelia maqsimaluri da minimaluri qula?
g) romeli qula gvxvdeba yvelaze xSirad?
sixSireTa ganawileba dajgufebuli monacemebisTvis
Tu miRebulia didi raodenobiT qulebi, an
Tu monacemebSi qulaTa mniSvnelobebis farTo
diapazonia, cxrilSi N3.2 mocemuli
daujgufebel sixSireTa ganawileba SeiZleba
Zalze gafantuli gamovides da amiT
garTuldes monacemTa naTeli suraTis danaxva.
am SemTxvevaSi sasargebloa dajgufebul
sixSireTa ganawilebis gamoyeneba, sadac qulebi dajgufebulia intervalebSi.
intervalebi lagdeba yvelaze maRlidan yvelaze dablisken, iTvleba TiToeuli
intervalis qulebis sixSireebi da gamoisaxeba cxrilis saxiT. dajgufebul
sixSireTa ganawilebis sailustraciod, davuSvaT, rom konsultantma gadawyvita,
gaezoma gamocdebTan dakavSirebuli SfoTva 100 studentisgan Semdgari
SerCevisTvis, romelTac ar miumarTavT daxmarebisTvis am problemis gamo.
miRebuli qulebi mocemulia cxrilSi N3.3.
cxrili N3.3. gamocdebTan dakavSirebuli SfoTvis
kiTxvarSi 100 studentisgan Semdgari SerCevis mier
miRebuli qulebi.
42 28 37 45 58 41 39 52 40 34 25 43 49 45 48 42 44
44 38 41 41 38 37 50 40 31 50 41 46 37 40 38 26 41
41 44 22 32 35 41 38 47 52 48 55 41 28 26 41 44 55
36 34 48 43 46 43 42 28 41 38 41 65 42 46 34 38 40
39 40 39 36 25 31 40 39 38 45 70 36 44 39 42 44
50 46 37 60 60 47 36 73 43 46 37 46 34 29 40 37
dajgufebul sixSireTa ganawilebis ageba
dajgufebul sixSireTa ganawileba
(Grouped frequency distribution): _
ganawileba, sadac qulebi
dajgufebulia intervalebSi da
TiToeuli maTganSi moxvedrili
qulebis sixSireebi gamoisaxeba
cxrilis saxiT.
intervali (Class interval): _ intervalis sidide, romelic gamoiyeneba dajgufebul sixSireTa
ganawilebaSi nedli qulebis dasajgufeblad. intervalis sidide aRiniSneba i-iT.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 8
dajgufebul sixSireTa ganawilebaSi nedli qulebi nawildeba intervalebSi.
intervali aris qulaTa diapazoni, romelSic jgufdeba nedli qulebi. cxrilSi
N3.3 mocemuli qulebis dajgufebul sixSireTa ganawileba warmodgenilia
cxrilSi N3.4. am SemTxvevaSi intervalis zoma aris 5. dajgufebul sixSireTa
iseTi ganawilebis agebas, rogoric mocemulia cxrilSi N3.4, sWirdeba ori ram: _
(1) gamoyenebul intervalTa raodenoba da (2) intervalis zoma.
cxrili N3.4. dajgufebul sixSireTa ganawileba cxrilSi N3.3 mocemuli gamocdebTan
dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebisTvis.
(a) (b) (g) (d) (e) (v) (z) (T) (i) (k)
intervali
realuri zeda da qveda
sazRvari
intervalis Suawertili
Tvlis Sedegi ƒ rƒ %ƒ cƒ crƒ c%ƒ
70-74 65-69 60-64 55-59 50-54 45-49 40-44 35-39 30-34 25-29 20-24
69.5-75.5 64.5-69.5 59.5-64.5 54.5-59.5 49.5-54.5 44.5-49.5 39.5-44.5 34.5-39.5 29.5-34.5 24.5-29.5 19.5-24.5
72 67 62 57 52 47 42 37 32 27 22
// / // /// ///// //////////////// ///////////////////////////////// //////////////////////// /////// /////// /
2 1 2 3 5 15 33 24 7 7 1
.02 .01 .02 .03 .05 .15 .33 .24 .07 .07 .01
2 1 2 3 5 15 33 24 7 7 1
100 98 97 95 92 87 72 39 15 8 1
1.00 .98 .97 .95 .92 .87 .72 .39 .15 .08 .01
100 98 97 95 92 87 72 39 15 8 1
intervalTa raodenoba
imas, Tu ramden intervals gamoviyenebT, ganapirobebs qulaTa diapazoni;
Cveulebisamebr, gamoiyeneba 10-dan 20-mde intervali. rogorc wesi, intervalebis
raodenobas irCeven ise, rom intervalTa sidide iyos 1, 2, 3, 5 an 5-is jeradi.
cxrilSi N3.3 mocemul qulaTa diapazonia 51 [22-dan (yvelaze dabali qula) 73-mde
(yvelaze maRali qula)]. qulaTa aseTi diapazonis SemTxvevaSi monacemebis
dasajgufeblad sakmarisi iqneba 10 intervali.
intervalis zoma
intervalTa raodenobis gansazRvris Semdeg gamoiTvleba intervalis zoma.
amisaTvis maqsimaluri da minimaluri qulebis sxvaoba iyofa intervalTa
raodenobaze.
statistikaSi simbolo X xSirad gamoiyeneba qulis aRsaniSnad. aqedan
gamomdinare, intervalis sidide, romelic aRiniSneba i-iT, ase gamoiTvleba:
raodenoba intervalTa
minimaluriimaqsimalur XXi
−=
X _ simbolo, romelic calkeuli, individualuri qulis aRsaniSnad gamoiyeneba.
i _ intervalis zoma dajgufebul sixSireTa ganawilebaSi.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 9
sadac Xmaqsimaluri aris yvelaze maRali qula, xolo Xminimaluri _ yvelaze dabali
qula ganawilebaSi. cxrilSi N3.3 mocemuli monacemebisTvis, sadac gvaqvs 10
intervali, xolo Xmaqsimaluri = 73 da Xminimaluri = 22, intervalis sidide tolia:
10.51051
102273
==−
=i
Tu am formulaSi mniSvnelobebis Setanis Semdeg miviRebT aTwilads (rogorc
ganxilul magaliTSi _ 5.10), is mrgvaldeba intervalTa klasis uaxloes
rekomendebul sigrZemde (anu 1, 2, 3, 5 an 5-is jeradamde). magaliTad, am
SemTxvevaSi i mrgvaldeba 5-mde. Tu qulebi gamosaxulia 1-ze naklebi aTwiladiT,
intervalis sidide ar mrgvaldeba mTel ricxvamde; am dros gamoiyeneba iseTi
intervalebi, rogoricaa 0.3, 0.5, 0.03 an 0.05.
intervalis ageba
intervalis asagebad unda sruldebodes ori ZiriTadi piroba:
• qveda intervali unda moicavdes yvelaze dabal qulas;
• pirveli intervalis qveda sazRvari (anu pirveli intervalis yvelaze
dabali SesaZlo qula) unaSTod unda iyofodes intervalis sigrZeze.
cxrilSi N3.3 mocemul qulaTa Soris yvelaze dabalia 22. Sesabamisad,
cxrilis N3.4 (a) svetSi yvelaze dabali intervali iwyeba qveda sazRvriT 20,
radgan 20 aris pirveli qula, romelic naklebia 22-ze da romelic unaSTod iyofa
intervalis sidideze anu 5-ze. Semdegi intervalia 25-29. damatebiTi intervali
gamoiTvleba im wertilis zeviT, romelic saWiroa bolo intervali aigeba ise,
rom masSi CarTuli iyos yvelaze maRali qula, mocemul magaliTSi, es aris
intervali 70-74.
intervalis dadgenili da realuri
sazRvrebi. cxrilis N3.4 (a) svetSi mocemul
TiToeul intervals aqvs zeda da qveda
dadgenili sazRvari (romelsac nominalur
sazRvars an qulis sazRvarsac uwodeben).
intervalis zeda da qveda sazRvari aris maqsimaluri da minimaluri qulebi,
romelic SeiZleba moxvdes am intervalSi. magaliTad, 40-44 intervalisTvis qveda
dadgenili sazRvaria 40, xolo zeda dadgenili sazRvari _ 44. 40 aris minimaluri
qula, xolo 44 _ maqsimaluri qula, romelic SeiZleba Sevides intervalSi.
suraTze N3-2 warmodgenilia 40-44 intervalis dadgenili zeda da qveda sazRvari.
ricxvis realuri sazRvrebi
me-2 TavSi ganmartebulia,
rogorc ricxvisa da mis wina
ricxvis da ricxvisa da mis
intervalis dadgenili sazRvari
(Stated limits of a class interval): _
maqsimaluri da minimaluri qulebi,
romelic SeiZleba moxvdes am
intervalSi.
intervalis realuri sazRvari (Real limits of a class
interval): _ intervalis dadgenil sazRvrebsa da am
intervalis wina an momdevno intervalebis dadgenil
sazRvrebs Soris Suawertili.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 10
momdevno ricxvis Suawertili im skalaze, romelsac viyenebT mocemuli cvladis
gasazomad. realuri sazRvrebi dajgufebuli sixSireebis ganawilebis
intervalebsac aqvs da isini moiZebneba iseve, rogorc ricxvis realuri sazRvrebi.
intervalis qveda realuri sazRvari aris intervalis qveda dadgenil sazRvarsa
da winamdebare intervalis zeda dadgenil sazRvars Soris Suawertili.
magaliTad, vipovoT intervalis 40-44 (cxrilidan N3.4) qveda realuri sazRvari. am
intervalis qveda miRebuli sazRvaria 40. Semdegi dabali intervalis (anu
intervalis 35-39) zeda miRebuli sazRvaria 39. Sesabamisad, intervalis 40-44 qveda
realuri sazRvaria 39-sa da 40-s Soris Suawertili anu 39.5. es qveda realuri
sazRvaric mocemulia suraTze N3.2.
intervalis zeda realuri sazRvari aris intervalis zeda dadgenil sazRvarsa
da Semdegi maRali intervalis qveda dadgenil sazRvars Soris Suawertili.
intervalis 40-44 zeda realuri sazRvari miiReba Semdegi intervalis 40-44 zeda
dadgenil sazRvarsa (anu 44) da Semdegi maRali intervalis 45-49 qveda dadgenil
sazRvars Soris Suawertilis povniT. am intervalis qveda dadgenili sazRvaria
45. amrigad, intervalis 40-44 zeda realuri sazRvaria Suawertili 44-sa da 45-s
Soris anu 44.5. es zeda realuri sazRvaric naCvenebia suraTze N3.2. yuradReba
miaqcieT, rom intervalis 40-44 zeda realuri sazRvari Semdegi 45-49 intervalis
intervalis Suawertili (Midpoint of a class interval): _ intervalis realur sazRvrebs Soris
Suawertili.
suraTi N3.2. qveda da zeda dadgenili sazRvrebis, qveda da zeda realuri sazRvrebisa da
dajgufebuli sixSireebis ganawilebis intervalis Suawertilis ilustracia.
intervali
35-39
intervali
40-44
intervali
45-49
35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
zeda dadgenili
zRvari
zeda dadgenili
zRvari
qveda dadgenili
zRvari
qveda dadgenili
zRvari
intervalis Suawertili
39.5 intervali 40-44 qveda realuri sazRvari
intervali 35-39 zeda realuri sazRvari
44.5 intervali 40-44 zeda realuri sazRvari
intervali 45-49 qveda realuri sazRvari
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 11
qveda realuri sazRvaricaa. intervalis zeda da qveda realuri sazRvrebi
warmodgenilia cxrilis N3.4 (b) svetSi.
intervalis Suawertili. TiToeul intervals Suawertilic aqvs. intervalis
Suawertili aris intervalis realur sazRvrebs Soris Suawertili. 40-44
intervalis Suawertili aris 42, radgan 42 aris 39.5-sa da 44.5-s Soris
Suawertili. intervalis Suawertilis povna SesaZlebelia intervalis realuri
sazRvrebis mniSvnelobaTa jamis 2-ze gayofiT. 40-44 intervalis Suawertili
gamoiTvleba Semdegnairad:
422
842
5.445.39==
+
TiToeuli intervalis Suawertili warmodgenilia N3.4 cxrilis (g) svetSi.
miuxedavad imisa, rom am cxrilSi erTad aris mocemuli intervalis zeda da qveda
realuri sazRvrebi da Suawertili, dajgufebul sixSireTa ganawileba xSirad
mxolod (a) svetSi mocemuli intervalebiT aris xolme warmodgenili.
dajgufebul martiv sixSireTa ganawileba
dajgufebul martiv sixSireTa ganawileba miiReba im intervalSi qulebis
daTvlis Sedegad, romelSic es qulebi xvdeba (ix. cxrili N3.4, (d) sveti).
magaliTad, qula 35 daTvlilia intervalSi 35-39, qula 52 _ intervalSi 50-54.
mocemuli intervalis martivi sixSiris (ƒ) misaRebad jamdeba daTvlis Sedegebi
intervalSi.
dajgufebul fardobiT sixSireTa ganawileba da sixSireTa ganawileba
procentebSi
dajgufebul sixSireTa ganawilebisTvis SesaZlebelia fardobiT sixSireTa
ganawilebisa da procentebSi sixSireTa ganawilebis miRebac, rac iseve
xorcieldeba, rogorc daujgufebel sixSireTa ganawilebis SemTxvevaSi.
dajgufebul fardobiT sixSireTa
ganawileba. dajgufebul fardobiT
sixSireTa ganawilebis gamosaTvlelad
intervalSi qulaTa raodenoba iyofa qulaTa
saerTo raodenobaze ganawilebaSi. amgvarad,
Nf
rf qulaTa intervalSi
qulaTa intervalSi =
gamocdebTan dakavSirebul SfoTvis kiTxvarSi miRebuli qulebis dajgufebul
fardobiT sixSireTa ganawileba mocemulia cxrilis N3.4 (v) svetSi.
dajgufebul fardobiT sixSireTa
ganawileba (Grouped relative frequancy
distribution): _ dajgufebul sixSireTa
ganawileba miiReba qulaTa
fardobiTi sixSiris 100-ze
gamravlebiT.
dajgufebul sixSireTa ganawileba procentebSi (Grouped
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 12
dajgufebul sixSireTa
ganawileba procentebSi.
dajgufebul sixSireTa
ganawileba procentebSi
miiReba formuliT
100% ×=N
ff
qulaTa i intervalS qulaTa intervalSi
gamocdebTan dakavSirebul SfoTvis kiTxvarSi miRebuli qulebis dajguebul
sixSireTa ganawileba procentebSi mocemulia cxrilis N3.4 (z) svetSi.
dagrovil sixSireTa ganawileba
dagrovili intervaluri
sixSire. yvela ganxiluli
ganawileba: _ martivi sixSire,
fardobiTi sixSire da sixSire
procentebSi _ dagrovil
sixSireTa saxiTac SeiZleba
warmovadginoT. qulis
dagrovili sixSire (aRiniSneba
cƒ-iT an cum ƒ-iT) aris am
qulis Sexvedris sixSires
damatebuli yvela ufro
naklebi qulis sixSireTa jami.
dagrovili intervaluri
sixSire aris am intervalSi
qulebis Sexvedris raodenobas
damatebuli yvela ufro dabal
intervalSi qulebis sixSireTa
jami. magaliTad, 30-34 intervalis (ix. cxrili N3.4) dagrovili sixSire aris 15,
radgan mTlianobaSi 15 qula gvxvdeba intervalSi 20-24 (1 qula), intervalSi 25-29
(7 qula) da intervalSi 30-34 (7 qula). TiToeuli dagrovili intervaluri sixSire
mocemulia cxrilis N3.4 (T) svetSi.
dagrovili intervaluri fardobiTi sixSiris ganawileba da sixSiris
ganawileba procentebSi. fardobiTi sixSireebi da sixSireebi procentebSi
SeiZleba warmodgenil iqnes dagrovili sixSireebis saxiTac. intervalis
dagrovili intervaluri fardobiTi sixSire (aRiniSneba crƒ-iT an cum rƒ-iT) aris
intervalSi qulebis fardobiT sixSires damatebuli yvela ufro dabal
intervalSi qulaTa fardobiT sixSireTa jami.
percentage frecuancy distribution): _ dajgufebul sixSireTa
ganawileba, sadac intervalSi qulaTa sixSire
Sefardebulia ganawilebaSi qulaTa saerTo
raodenobasTan.
qulis dagrovili sixSire (Cumulative frecuency of a score):
_ qulis Sexvedris sixSires damatebuli yvela
ufro naklebi qulis sixSireTa jami.
dagrovili intervaluri sixSire (Cumulative frecuency of
a class interval): _ intervalSi qulebis Sexvedris
raodenobas damatebuli yvela ufro dabal
intervalSi qulebis sixSireTa jami.
dagrovili intervaluri fardobiTi sixSire
(Cumulative grouped relative frecuency of a class interval): _
intervalSi qulebis fardobiT sixSires damatebuli
yvela ufro dabal intervalSi qulaTa fardobiT
sixSireTa jami.
dagrovili intervaluri sixSiris ganawileba
procentebSi ( Cumulative grouped percentage frecuency of a
class interval): _ intervalSi qulebis sixSires
procentebSi damatebuli yvela ufro dabal
intervalSi qulaTa sixSireTa jami procentebSi.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 13
procentebSi gamoxatuli dagrovili intervaluri sixSiris ganawileba
(aRiniSneba c%ƒ-iT an cum %ƒ-iT) aris intervalSi qulebis procentebSi gamoxatul
sixSires damatebuli yvela ufro dabal intervalSi procentebSi gamoxatul
qulaTa sixSireTa jami. dagrovili intervaluri fardobiTi sixSireebi da
sixSireebi procentebSi TiToeuli intervalisTvisA mocemulia cxrilis N3.4 (i)
da (k) svetebSi, Sesabamisad.
informaciis miReba dajgufebul sixSireTa ganawilebidan
cxrilSi N3.4 mocemuli dajgufebul sixSireTa sxvadasxva ganawileba
gviCvenebs, rom qulaTa umravlesoba moTavsebulia 30-sa da 54-s Soris. garda
amisa, ganawilebebi gviCvenebs, rom SfoTvis qulaTa daaxloebiT erTi mesamedi
(100-dan 33) moTavsebulia intervalSi 40-44, Tumca, am cxrilSi ar Cans qulaTa
zusti ricxviTi mniSvnelobebi. Cven ver davasaxelebT konkretul maqsimalur an
minimalur, anda yvelaze xSir qulas, magram informaciis es danakargi im
simartivis sazRauria, romliTac SegviZlia ganawilebis mTliani, zogadi formis
aRqma.
procentili da procentuli rangebi
dagrovil sixSireTa ganawilebebi
xSirad gamoiyeneba procentilebisa da
procentiluri rangebis gamosaTvlelad.
procentili aris qula, romelzec qulaTa
garkveuli procenti ganawilebaSi tolia
an naklebia am qulaze. magaliTad, Tu 35-e
procentili gamocdaze aris 70, gamocdaze
miRebul qulaTa 35% tolia an naklebia
70-ze. qulis procentuli rangi gviCvenebs ganawilebaSi im qulaTa procents,
romlebic tolia an naklebia am qulaze. amrigad, ganxilul magaliTSi testSi
miRebuli 70 qulis procentuli rangi aris 35. qulaTa 35% tolia an naklebia 70-
ze.
qulis procentilis an procentuli rangis gamoTvliT viRebT informacias
ganawilebaSi sxva qulebis mimarT am qulis adgilmdebareobis Sesaxeb. davuSvaT,
Tqven im 100 studentTa Soris xarT, vinc Seavso gamocdebTan dakavSirebuli
SfoTvis kiTxvari da Tqveni qulaa 41. es 41 qula maRalia Tu dabali sxva
studentebis qulebTan SedarebiT? Tu Tqven getyvian, rom 41 aris 49-e procentili,
an 41 qulis procentuli rangia 49, Tqven gecodinebaT, rom qulaTa 49% tolia an
naklebia 41-ze.
qulis procentuli rangis povna
procentili (Percentile): _ qula, romelzec
qulaTa garkveuli procenti
ganawilebaSi tolia an naklebia am
qulaze.
qulis procentuli rangi (Percentile rank of
a score): _ im qulaTa procenti
ganawilebaSi, romlebic tolia an
naklebia am qulaze.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 14
davuSvaT, gvsurs gamovTvaloT cxrilSi N3.4 mocemuli ganawilebidan 41 qulis
procentuli rangi. es qula Sedis intervalSi 40-44. am intervalis qveda realur
sazRvramde yvela qulis dagrovili sixSire procentebSi aris 39 (ix. sveti (k) anu
c%ƒ), xolo am intervalis zeda realur sazRvramde qulaTa dagrovili sixSire
procentebSi _ 72 (ix. (k) sveti). amrigad, 41-is procentuli rangi iqneba 39-sa da 72-
s Soris. zusti procentuli rangis gamosaTvlelad unda movaxdinoT 41-is
interpolireba 40-44 intervalSi. qvemoT mocemulia formula, romliTac xdeba
interpolireba da gamoiTvleba qulis procentiluri rangi:
( )[ ] 100/×
−+==
NfiXXcfP iLL
Xrangi riprocentilu
sadac PX = X qulis procentuli rangi.
cƒL = X-is Semcveli intervalis qveda realur sazRvramde qulebis
dagrovili sixSire.
X = qula, romlisTvisac gamoiTvleba procentuli rangi.
XL = X-is Semcveli intervalis qveda realuri sazRvari.
i = intervalis sidide.
ƒ i = X-is Semcveli intervalis qulaTa sixSire.
N = qulaTa saerTo raodenoba ganawilebaSi.
cxrilSi N3.4 mocemuli 41 qulisTvis es mniSvnelobebi iqneba:
PX = P41.
cƒL = 39 (ix. cxrili N3.4, (T) sveti).
X = 41.
XL = 39.5 (ix. cxrili N3.4, (b) sveti).
i = 5.
ƒ i = 33 (ix. cxrili N3.4, (e) sveti).
N = 100.
Tu am mniSvnelobebs CavsvamT formulaSi, miviRebT:
( )[ ]
( )( )[ ]
( )( )[ ]
9.48
100100
9.48
100100
9.939
100100
333.39
100100
335/5.139
100100
335/5.39413941
=
×=
×+
=
×+
=
×+
=
×−+
=P
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 15
procentuli rangi, rogorc wesi, mrgvaldeba mTel ricxvamde, amitom
P41 = 49
41 qulis procentuli rangi aris 49 anu cxrilSi N3.4 qulaTa 49% tolia an
naklebia 41-ze. es formula daujgufebel sixSireTa ganawilebisTvisac SeiZleba
gamoviyenoT; am SemTxvevaSi i = 1.
ganawilebis procentilis povna
Semdegi amocanaa mocemuli procentis Sesabamisi qulis povna. magaliTad,
davuSvaT, gvsurs cxrilSi N3.4 warmodgenili ganawilebis 49-e procentilis povna.
yuradReba miaqcieT, rom es procentili aris qula, romelic Sedis intervalSi
40-44, qulaTa 39% xvdeba am intervalis qveda realur sazRvars qvemoT, xolo
qulaTa 72% _ zeda realur sazRvars qvemoT. im konkretuli qulis sapovnelad,
romelic aris 49-e procentili, unda movaxdinoT interpolireba. qvemoT
mocemulia formula, romliTac xdeba interpolireba da procentilis gamoTvla:
if
cfPNXXi
LLP
−+==procentili
sadac XP = qula konkretuli procentilis toli (anu qula, romlis povnac
gvsurs).
XL = konkretuli procentilis Semcveli intervalis qveda realuri
sazRvari.
P = mocemuli procenti, mocemuli 0-dan 1.00-mde wiladis saxiT.
N = qulaTa saerTo raodenoba ganawilebaSi.
cƒL = XP-is Semcveli intervalis qveda realur sazRvramde qulebis
dagrovili sixSire.
ƒ i = XP-is Semcveli intervalSi qulaTa sixSire.
i = intervalis sidide.
49-e procentilisTvis es mniSvnelobebi iqneba:
XP = X49.
XL = 39.5 (ix. cxrili N3.4, (b) sveti).
P = .49
N = 100.
cƒL = 39 (ix. cxrili N3.4, (T) sveti).
ƒ i = 33 (ix. cxrili N3.4, (e) sveti).
i = 5.
Tu am mniSvnelobebs CavsvamT formulaSi, miviRebT:
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 16
02.4152.15.39
533105.39
533
39495.39
533
39)100(49.5.3949
=+=
+=
−
+=
−
+=X
rodesac am mniSvnelobas mTel ricxvamde davamrgvalebT, miviRebT: X49 = 41. 49-e
procentili aris 41 qula. ganawilebaSi qulaTa 49% tolia an naklebia 41-ze. es
formula daujgufebel sixSireTa ganawilebisTvisac SeiZleba gamoviyenoT; am
SemTxvevaSi i = 1.
magaliTi N3-2
dajgufebul sixSireTa ganawileba
amocana
davuSvaT, vikvlieT N3-1 magaliTSi ganxiluli orazrovani naxatebis aRqmis
problema da gavzomeT drois xangrZlioba, romlis manZilzec cdis pirebs
SeeZloT piramidis danaxva meCis piramidaSi. gvyavda 50 cdis piri da miviReT
Semdegi monacemebi (wamis sizustiT): _ 18, 2, 7, 23, 16, 10, 5, 30, 5, 15, 17, 3, 31, 9, 11, 13,
5, 13, 8, 19, 16, 2, 14, 4, 7, 7, 6, 2, 10, 6, 7, 24, 3, 8, 9, 7, 19, 7, 11, 9, 4, 7, 6, 1, 3, 10, 1, 26, 8
da 5.
a) am qulebisTvis aageT dajgufebul sixSireTa Semdegi ganawilebebi: _ martiv
sixSireTa ganawileba, fardobiT sixSireTa ganawileba, sixSireTa
ganawileba procentebSi, dagrovili intervaluri sixSire, dagrovili
fardobiTi sixSire da dagrovil sixSireTa ganawileba procentebSi.
gansazRvreT intervalebis qveda da zeda realuri sazRvrebi da TiToeuli
intervalis Suawertilebi.
b) ra daskvnebis gakeTeba SegiZliaT am monacemebis Sesaxeb am ganawilebebze
dayrdnobiT?
g) ipoveT 16 wamis procentiluri rangi.
d) am ganawilebaSi romelia 25-e procentili?
amoxsna
a) saWiroa intervalTa raodenobis gansazRvra, intervalTa sididis gamoTvla da
Semdeg dajgufebul sixSireTa ganawilebis ageba am intervalebis gamoyenebiT.
intervalTa raodenoba: qulaTa diapazonia 1-dan 31 wamamde anu 30 wami.
monacemebis dasajgufeblad sakmarisi iqneba 10 intervali.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 17
intervalis sidide: i= (31-1)/10=3.0.
ganawilebis ageba: miRebul qulaTa Soris yvelaze dabali qulaa 1 wami,
intervalis sidide _ 3 wami. amitom pirveli intervali iqneba 0-2. es intervali
Seicavs yvelaze dabal qulas da misi qveda dadgenili sazRvari (anu 0)
unaSTod iyofa intervalis sidideze (anu 3-ze). dajgufebul sixSireTa
intervalebi warmodgenilia qvemoT mocemuli cxrilis (a) svetSi. TiToeuli
intervalis zeda da qveda sazRvrebi miRebulia momijnave intervalebs Soris
Suawertilebis povnis Sedegad. magaliTad, zeda da qveda realuri sazRvrebi
intervalisTvis 24-26 aris Suawertili 23-sa da 24-s (qveda realuri sazRvari)
Soris da Suawertili 26-sa da 27-s (zeda realuri sazRvari) Soris,
Sesabamisad. cxrilis (b) svetSi mocemulia TiToeuli intervalis realuri
sazRvrebi, (g) svetSi _ TiToeuli intervalis Suawertilebi, (d) svetSi _
qulebis Tvlis Sedegi, xolo (e), (v), (z), (T), (i) da (k) svetebSi _ dajgufebul
sixSireTa ganawilebebi.
(a) (b) (g) (d) (e) (v) (z) (T) (i) (k)
intervali
realuri zeda da qveda
sazRvari
intervalis Suawertili
Tvlis Sedegi ƒ rƒ %ƒ cƒ crƒ c%ƒ
30-32 27-29 24-26 21-23 18-20 15-17 12-14 9-11 6-8 3-5 0-2
29.5-32.5 26.5-29.5 23.5-26.5 20.5-23.5 17.5-20.5 14.5-17.5 11.5-14.5 8.5-11.5 5.5-8.5 2.5-5.5 -0.5-2.5
31 28 25 22 19 16 13 10 7 4 1
// // / /// //// /// //////// ///////////// ///////// /////
2 0 2 1 3 4 3 8 13 9 5
.04
.00
.04
.02
.06
.08
.06 .16 .26 .18 .10
4 0 4 2 6 8 6 16 26 18 10
50 48 48 46 45 42 38 35 27 14 5
1.00 .96 .96 .92 .90 .84 .76 .70 .54 .28 .10
100 96 9 92 90 84 76 70 54 28 10
b) daskvnebi: martivi sixSireebidan (sveti (e)), fardobiTi sixSireebidan (sveti (v))
anda sixSireebidan procentebSi (sveti (z)) vxedavT, rom qulebi yvelaze
xSirad gvxvdeba intervalSi 6-8. dagrovil sixSireTa ganawileba gviCvenebs,
rom 0.70 anu qulaTa 70% aris 11 wami an masze naklebi.
g) qulis procentuli rangi gamoiTvleba formuliT
( )[ ] 100/×
−+=
NfiXXcfP iLL
X
16 wamis procentuli rangis gamosaTvlelad saWiro mniSvnelobebia:
PX = P16
cƒL = 38
X = 16
XL = 14.5
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 18
i = 3
ƒ i = 4
N = 50
CavsvaT es mniSvnelobebi formulaSi. miviRebT:
( )[ ]
[ ]
8010080.
1005040
10050
238
10050
4)5(.38
10050
43/5.138
10050
43/5.14163816
=×=
×=
×+
=
×+
=
×+
=
×−+
=P
16 wamis procentuli rangi aris 80. es niSnavs, rom drois miRebuli
xangrZliobebis 80% aris 16 wami an 16 wamze naklebi.
d) 25-e procentili gamoiTvleba formuliT
if
cfNPXXi
LLP
−+=
)(
25-e procentilisTvis es mniSvnelobebi iqneba:
XP = X25.
XL = 2.5
P = .25
N = 50
cƒL = 5
ƒ i = 9
i = 3
CavsvaT es mniSvnelobebi formulaSi. miviRebT:
0.55.25.2
395.75.2
39
55.125.2
39
5)50(25.5.216
=+=
+=
−
+=
−
+=X
25-e procentilia qula 5.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 19
SeamowmeT Tqveni codna III.2
1. ganmarteT: _ intervali, intervalis dagrovili sixSire, qulis
dagrovili sixSire, intervalis dagrovili sixSire procentebSi,
intervalis dagrovili fardobiTi sixSire, cƒ, c%ƒ, crƒ, cum ƒ, cum %ƒ, cum
rƒ, dagrovil sixSireTa ganawileba, intervalis qveda realuri sazRvari,
intervalis qveda dadgenili sazRvari, intervalis Suawertili,
procentili, procentuli rangi, intervalis zeda realuri sazRvri,
intervalis zeda dadgenili sazRvari.
2. dajgufebul sixSireTa ganawilebis agebisas ra raodenobis
intervalebis gamoyenebaa rekomendebuli?
3. ganawilebaSi maqsimaluri qulaa 8 da minimaluri _ 14. Tqven apirebT
dajguebul sixSireTa ganawilebis agebas 15 intervaliT. ra sigrZis
iqneba Tqveni intervali?
4. ganawilebaSi yvelaze didi qulaa 95 ada yvelaze mcire _ 62. Tqven
gsurT, rom gqondeT intervali sididiT 3. ramdeni intervali geqnebaT?
5. ra ori ZiriTadi piroba unda sruldebodes dagjufebul sixSireTa
ganawilebis agebisas?
6. dajgufebul sixSireTa ganawilebis intervalebia 120-122, 123-125, 126-128,
129-131 da 132-134.
a) ipoveT 123-125 intervalis qveda da zeda dadgenili sazRvrebi;
b) ipoveT 129-131 intervalis qveda da zeda dadgenili sazRvrebi;
g) ipoveT 123-125 intervalis qveda da zeda realuri sazRvrebi;
d) ipoveT 129-131 intervalis qveda da zeda realuri sazRvrebi;
e) ipoveT 120-122 intervalis Suawertili;
v) ras udris i am ganawilebisTvis?
7. mtaceblis miaxloebisas tompsonis qurcikebi xtian ise, rom oTxive fexs
erTdroulad aSoreben miwas. es qceva, romelsac maRali naxtomi
ewodeba, rogorc Cans, mniSvnelovania qurcikisTvis, radgan is
saSualebas aZlevs mas, Tavi aaridos mtacebelTan Setakebas. am naxtomis
Seswavlis mizniT karom (Caro, 1986) bunebrivi dakvirveba gamoiyena. is
ikvlevda qurcikebs serengetis erovnuli parkSi, tanzaniaSi. erT-erTi
maCvenebeli iyo qurciksa da mtacebels (magaliTad, gepards) Soris
manZili, rodesac qurciki pirvel maRali naxtoms axorcielebda. vTqvaT,
aRinusxa gepardis qurcikTan miaxloebis 50 SemTxveva da dafiqsirda
Semdegi manZilebi (metrebSi) qurciksa da gepards Soris pirveli maRali
naxtomis ganxorcielebamde: 47, 61, 72, 63, 56, 54, 67, 75, 91, 72, 83, 140, 31, 37,
25, 38, 57, 62, 49, 205, 66, 39, 43, 81, 66, 72, 42, 29, 190, 64, 76, 54, 38, 67, 61, 33,
48, 55, 59, 68, 18, 49, 37, 83, 70, 154, 91, 50, 58 da 66.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 20
a) am qulebisTvis aageT Semdegi dajgufebuli ganawilebebi: _ martiv
sixSireTa ganawileba, fardobiT sixSireTa ganawileba, sixSireTa
ganawileba procentebSi, dagrovil martiv sixSireTa ganawileba,
dagrovil sixSireTa fardobiTi ganawileba da dagrovil sixSireTa
ganawileba procentebSi. ipoveT intervalTa qveda da zeda realuri
sazRvrebi da TiToeuli intervalis Suawertili. gamoiyeneT 20
intervali.
b) ra daskvnebis gakeTeba SegiZliaT qurcikis maRali naxtomis sigrZis
Sesaxeb dajgufebul sixSireTa ganawilebebSi mocemul
informaciaze dayrdnobiT?
g) ipoveT 68 metriani manZilis procentuli rangi.
d) am ganawilebaSi romelia 50-e procentili?
sixSireTa ganawilebis grafikulad gamosaxva
sixSireTa ganawileba iZleva monacemebis sistematuri gziT aRqmis saSualebas.
monacemebis ukeT aRsaqmelad sixSireTa ganawilebiT mocemuli informacia xSirad
gamoisaxeba iseTi grafikebis saxiT, rogoricaa histograma, sixSireTa poligoni
an foTlebiani Reroebis diagrama. Tavdapirvelad gavecnoT grafikebis agebis
zogad wesebs.
• grafikis ori RerZi erTmaneTis mimarT marTi (90°-iani) kuTxiT aris
ganlagebuli. horizontalur RerZs ewodeba x RerZi anu abscisa,
vertikalurs ki _ y RerZi anu ordinata.
• SesaZlo qulebi an dajgufebul sixSireTa SesaZlo qulebis intervalebi
moTavsebulia x RerZis gaswvriv. sixSiris sazomi (magaliTad, martivi
sixSire, fardobiTi sixSire, sixSire procentebSi, dagrovili sixSire da a.S.)
moTavsebulia y RerZze.
• TiToeul RerZs erqmeva iseTi saxeli, rom mkiTxveli mixvdes, Tu ra
informaciaa mocemuli grafikze.
• TiToeuli RerZis sawyisi wertili, Cveulebisamebr, aris nuli. Tu skalis
nawili sawyis nulovan wertilsa da pirvel mocemul qulas an qulis
sixSires Soris carielia RerZis marcxena mxares, RerZze mocemulia amis
maniSnebeli niSani -/ /-.
• y RerZis skala ise unda iqnes SerCeuli, rom masze gadazomili sixSiris
maqsimaluri simaRle x RerZis daaxloebiT mde dan −−43
32
siganes Seadgendes.
histograma
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 21
histogramaSi vertikaluri
svetebis simaRle gamoiyeneba
intervalSi qulis
Sexvedris sixSiris
saCveneblad. TiToeuli intervalis sidide warmodgenilia abscisaze svetis
siganiT. histogramebs, Cveulebriv, mxolod dajgufebuli sixSireebisTvis ageben,
sixSiris sazomad ki iyeneben martiv intervalur sixSires, an fardobiT
intervalur sixSires (ix. cxrili N3.4, (e) da (v) svetebi, Sesabamisad). gamocdebTan
dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebis dajgufebul martiv
sixSireTa histograma mocemulia suraTze N3.3. rogorc wesi, abscisaze qulebi
mieTiTeba TiToeuli intervalis Suawertilebis ricxviTi mniSvnelobebiT,
rogorc es naCvenebia suraTebze N3.3 da N3.5. qulaTa moxvedris sixSire
gamosaxulia ordinataze da TiToeuli svetis simaRle Seesabameba im intervalTa
sixSires, romelsac is warmoadgens.
suraTi N3.3. martiv sixSireTa histograma gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebisTvis
(cxrilidan N3.3)
0
10
20
30
40
0 22 27 32 37 42 47 52 57 62 67 72
gamocdebTan dakavSirebuli SfoTvis kiTxvaris qula
six
Sir
e
sixSireTa poligoni
sixSireTa poligonis asagebad saWiroa,
rom abscisaTa RerZze ganlagdes
intervalis Suawertilebi, intervalis
sixSire ki aRiniSneba Suawertilebis
Tavze wertilebiT. es wertilebi Semdeg erTdeba swori xazebiT. suraTze N3.4
mocemulia gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli
dajgufebuli qulebis (ix. cxrili N3.4) sixSireTa poligoni. aq gamoyenebulia
histograma (Histogram): _ svetebiani grafikis nairsaxeoba,
sadac intervalebSi qulaTa Sexvedris sixSire
gamosaxulia svetis simaRliT, TiToeuli intervalis
sigrZe ki _ abscisaze svetis siganiT.
sixSireTa poligoni (Frequency polygon): _
grafiki, romlis abscisebia intervalTa
Suawertilebi, ordinatebi ki _ am
intervalTa Sesabamisi sixSireebi.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 22
martiv sixSireTa ganawileba, romelic mocemulia am cxrilis (e) svetSi.
yuradReba miaqcieT, rom suraTze N3.4 ganawilebis orive boloSi mocemulia
TiTo-TiTo damatebiTi intervali, romelic ar Seicavs arc erT qulas da rom
sixSireTa poligoni grafikis orive boloSi miyvanilia nulamde
(Suawertilebamde 17 da 77). sixSireTa poligonis agebisas es yovelTvis ase xdeba.
suraTi N 3.4. fardobiT sixSireTa poligoni gamocdebTan dakavSirebuli SfoTvis
kiTxvaris qulebisTvis (cxrilidan N3.3)
0
10
20
30
40
17 22 27 32 37 42 47 52 57 62 67 72 77
gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebi
far
dobiT
i six
Sir
e
fardobiTi sixSiris histograma da poligoni
histogramisa da sixSireTa poligonis asagebad, martivi sixSireebis (suraTebi
N3.3 da N3.4) nacvlad, fardobiTi sixSireebic SeiZleba gamoviyenoT. gamocdebTan
dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebis fardobiT sixSireTa (ix.
cxrilis N3.4 (v) sveti) histograma da sixSireTa poligoni warmodgenilia
suraTebze N3.5 da N3.6, Sesabamisad.
histogramis an sixSireTa poligonisTvis fardobiTi sixSireebis gamoyenebis
upiratesoba isaa, rom fardobiTi sixSireebis ricxviTi mniSvnelobebi gviCveneben
qulaTa wils, romelic sxvadasxva intervalSi xvdeba. amrigad, Tu SevadarebT
qulaTa sxvadasxva raodenobaze agebul sixSireTa or gansxvavebul ganawilebas,
fardobiTi sixSireebis SemTxvevaSi pirdapiri Sedareba ufro iolia, vidre
martivi sixSireebisas.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 23
suraTi N3.5. fardobiT sixSireTa histograma gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebisTvis
(cxrilidan N3.3)
0
10
20
30
40
0 22 27 32 37 42 47 52 57 62 67 72
gamocdebTan dakavSirebuli SfoTvis kiTxvaris qula
far
dobiT
i six
Sir
e
informaciis miReba histogramidan an sixSireTa poligonidan
histograma da sixSireTa poligoni, sixSireTa ganawilebis msgavsad, erTsa da
imave informacias gvawvdis. histogramisa da sixSireTa poligonis upiratesoba
isaa, rom xSirad gacilebiT iolia grafikulad warmodgenili informaciis aRqma.
nebismier suraTze, iqneba es suraTi N3.3, N3.4, N3.5 Tu N3.6, SegviZlia davinaxoT,
rom SfoTvis qulebis ganawilebis piki mdebareobs qula 40-is midamoebSi da rom
qulaTa umravlesoba SfoTvis skalis qulaTa qveda sazRvarTan (anu 20-Tan) ufro
axlosaa ganlagebuli, vidre _ zeda sazRvarTan (anu 80-Tan), Tumca, qulaTa
raodenoba mcirdeba maRali SfoTvis qulebis mimarTulebiT.
suraTi N3.6. fardobiT sixSireTa poligoni gamocdebTan dakavSirebuli SfoTvis
kiTxvaris qulebisTvis (cxrilidan N3.3)
0
10
20
30
40
17 22 27 32 37 42 47 52 57 62 67 72 77
gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebi
far
dobiT
i six
Sir
e
.
.
.
.
.
.
.
.
.
.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 24
foTlebiani Reroebis diagrama
foTlebiani Reros diagrama, romelic
tukis (Tukey, 1977) ekuTvnis, xasiaTdeba
sixSireTa ganawilebisa da monacemTa
grafikulad gamosaxvis Tvisebebis
kombinaciiT. gamocdebTan dakavSirebuli SfoTvis
kiTxvarSi miRebuli qulebis (ix. cxrili N3.3)
foTlebiani Reroebis diagarama warmodgenilia
cxrilSi N3.5. am grafikze marcxena svetSi mocemulia
Reroebi, romelic qulis pirvel cifrs aRniSnavs.
magaliTad, qulisTvis 3.5 Reroa 3. Reroebi,
Cveulebisamebr, dalagebulia ise, rom yvelaze dabali
Rero (am magaliTSi, 2-dan) moTavsebulia svetis bolos,
yvelaze didi (am magaliTSi, 7) ki _ svetis TavSi.
marjvena svetSi mocemuli ricxvebi warmoadgenen
foTlebs da Seesabamebian qulis meore cifrs; 35-is
SemTxvevaSi, es iqneba 5. magaliTad, pirvel rigSi, sadac
mocemulia Rero 7 da foTlebi 0 da 3, warmodgenilia
qulebi 70 da 73, Sesabamisad. Tu gvqvs samniSna qula,
rogoricaa, magaliTad, 321 wami, pirveli ori cifri (anu
32) gamoiyeneba Reros formirebisTvis, xolo bolo
cifri (anu 1) iqneba foToli.
miuxedavad amisa, mainc rTulia cxrilSi N3.5
mocemuli informaciis wakiTxva, gansakuTrebiT me-3 da
me-4 grZel Reroebze. am sirTulis dasaZlevad martivi
foTlebiani Reros diagrama SeiZleba ise
modificirdes, rom TiTo striqonze mxolod ramdenime
foToli darCes. cxrilSi N3.6 swored aseTi
modificirebuli variantia mocemuli, sadac TiToeuli
Rero xuTi striqoniTaa warmodgenili. yuradReba
miaqcieT ·, t, ƒ, s da * svetebs, romelic ricxviT aRniSnul svetebs mosdevs. aseT
ganlagebaSi ·-is gaswvriv mocemulia 0-isa da 1-is toli mniSvnelobebi, t-is
gaswvriv _ 2 da 3, ƒ-is gaswvriv _ 4 da 5, s-is gaswvriv _ 6 da 7, xolo *-is
foTlebiani Reroebis diagrama (Stem-and-leaf
display): _ monacemTa gamosaxvis forma, sadac
qulis pirveli (an pirveli da meore) cifri
aris Rero, meore (an bolo) ki _ foToli.
cxrili N3.5. gamocdebTan dakavSirebuli
SfoTvis kiTxvarSi miRebuli qulebis (ix.
cxrili N3.3) foTlebiani Reroebis diagrama.
Rero foTlebi
7 6 5 4 3 2
03 005 00022558 000000011111111111222223333444444555666666778889 1124444566666777777888888899999 25568889
cxrili N3.6. gamocdebTan
dakavSirebuli SfoTvis
kiTxvarSi miRebuli
qulebis (ix. cxrili 3.3)
foTlebiani Reroebis
modificirebuli diagrama.
Rero foTlebi
t 3
7 · 0
6 *
s
ƒ 5
t
6 · 00
5 * 8
s
ƒ 55
t 22
5 · 000
4 * 8889
s 66666677
ƒ 444444555
t 222223333
4 · 000000011111111111
3 * 888888899999
s 66666777777
ƒ 44445
t 2
3 · 11
2 * 8889
s 6
ƒ 55
t 2
2 ·
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 25
gaswvriv _ 8 da 9. aseTi modifikacia cxrils ufro advilad wasakiTxs xdis da
ganawilebis formis ukeT da iolad danaxvis saSualebas iZleva.
foTlebiani Reros diagrama martivad aigeba nedl monacemebze dayrdnobiT; am
dros ar ikargeba informacia, rogorc es xdeba dajgufebuli sixSireebis
ganawilebisas. am diagramidan yvela nedli qulis aRdgenaa SesaZlebeli. cxrilis
N3-6 ganxilviTa da qulebis ubralod daTvliT swrafad SegviZlia ganvsazRvroT,
rom qulebi icvleba 22-dan 73-mde da rom qula 41 yvelaze xSirad gvxvdeba.
foTlebiani Reroebis kidev erTi magaliTis saxiT ganvixiloT mTvaris efeqtis
rwmenis skala, romelic rotonsa da kelis (Rotton & Kelly, 1985) ekuTvniT da qcevaze
savse mTvaris gavlenis rwmenas afasebs. SegaxsenebT, rom am skalaze minimaluri
qulaa 9, rac mTvaris efeqtis minimalur rwmenaze an am rwmenis arqonaze
miuTiTebs, xolo maqsimaluri qulaa 81, rac mTvaris efeqtis maqsimaluri rwmenis
arsebobas gulisxmobs. rotonma da kelim es skala kolejis 157 students
miawodes. miRebuli qulebi mocemulia foTlebiani Reroebis diagramis saxiT2.
Rero warmoadgens qulis aTeuls, xolo
foToli _ erTeuls. amrigad, Rero 3 da
foToli 7 aris qula 37. rotonma da kelim
daxvewes Reroebiani foTlebis TavianTi
diagrama. yuradReba miaqcieT wertilebiTa da
varskvlavebiT Sedgenil svets. wertili
foTlebis 0-dan 4-is CaTvliT mniSvnelobis
aRsaniSnad gamoiyeneba, varskvlavi ki foTlis 5-
dan 9-is CaTvliT mniSvnelobebze miuTiTebs.
magaliTad, 1 Reros mqone qulebi cxrilSi
Semdegnairad aris mocemuli:
1 * .
556666778888899 122334
foTlis im mniSvnelobaTa diapazoni, romelTac win uZRvis wertili, 1-dan 4-is
CaTvliTaa. isini gamosaxaven qulebs 11, 12, 12, 13, 13 da, Sesabamisad. foTlis im
mniSvnelobebsac, romelTac win uZRvis varskvlavi (*), aqvT Rero 1, Tumca, maTi
diapazoni 5-dan 9-is CaTvliTaa. Rero 1-is SemTxvevaSi es mniSvnelobebi qmnis
qulebs 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 19 da 19, Sesabamisad.
foTlebiani Reroebis am diagramidan SegviZlia davinaxoT, rom miRebul
qulaTa diapazonia 11-dan 72-mde; yvelaze xSirad _ 9-jer gvxvdeba qula 45. am
2 ibeWdeba avtorebis _ j. rotonisa da i. u. kelis da gamomcemlis nebarTviT, „A Scale for assessing belief in lunar effects: reliability and concurrent validity“, Psychological Reports, 1985, 57, 239-245, © Psychological Reports, 1985.
Reroebi foTlebi
7 . 12
6 * .
59 0011114
5 * .
5666668999 000111111222334
4 * .
555555555666678888899 001111223333344444
3 * .
566666677788899999 0000112223344444
2 * .
55667777899999 00001122234444
1 * .
556666778888899 122334
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 26
diagramaze ar ikargeba informacia, SesaZlebelia 157-dan TiToeuli qulis
aRdgena.
Tvisebrivi monacemebis grafikulad gamosaxva
zemoT ganxiluli sakiTxebi Seexeboda raodenobrivi monacemebis grafikulad
gamosaxvas. Tvisebrivi monacemebi mxolod imas gviCvenebs, rom gasazomi cvladi
Tvisebrivad gansxvavebulia. magaliTad, trinkusma (Trinkaus, 1982, 1983, 1988, 1993)
bunebrivi dakvirvebis Sedegad mZRolebis qceva _ gzatkecilze Sesabamis niSanze
gaCereba _ miakuTvna samidan erT-erT kategorias: _ (1) srulad gaCereba, (2) svlis
Seneleba, (3) ar gaCereba. es kategoriebi Tvisebrivad gansxvavdeba erTmaneTisgan.
isini aRniSnaven qcevas, magram arafers gveubnebian imis Sesaxeb, Tu ramdenad
gansxvavdeba es qcevebi erTmaneTisgan raodenobrivad. trinkusma daiTvala
TiToeuli qcevis Sexvedris raodenoba da TiToeuli kategoriisTvis gamoTvala
SexvedraTa sixSiris procenti. procentebiT gamosaxuli es sixSireebi SeiZleba
warmodgenil iqnes grafikulad iseTi svetebiani diagramis saxiT, rogoric
mocemulia suraTze N3.7.
svetebiani diagrama (Bar graph) _ raodenobrivi monacemebis sixSireTa ganawilebis
grafikulad gamoxatvis saSualeba. mocemul kategoriaSi qulaTa gamoCenis sixSires
svetis simaRle Seesabameba.
svetebiani diagrama histogramas
hgavs. gansxvaveba isaa, rom
svetebi gancalkevebuladaa
ganlagebuli X RerZze, rac imaze
miuTiTebs, rom kategoriebi ar
warmoadgenen uwyvet cvlads.
svetebs Soris manZili SeiZleba
iyos nebismieri, magram
erTmaneTis toli. TiToeuli
kategoriis Sexvedris sixSiris
gamosaxatad SeiZleba gamoviyenoT
martivi sixSireebi, fardobiTi
sixSireebi anda sixSireebi
procentebSi (rogorc mocemulia
suraTze N3.7). am diagramidan advilad davinaxavT, rom gzajvaredinebze yvelaze
xSirad dafiqsirebuli qceva ekuTvnis kategoria 3-s da es niSnavs saerTod ar
gaCerebas.
SeamowmeT Tqveni codna III.3
suraTi N3.7. gaCerebis qcevis sami kategoriis Sexvedris sixSire
procentebSi (Trinkaus, 1982, 1983, 1988, 1993).
0
10
20
30
40
50
60
70
80
90
100
srulad
gaCereba
svlis
Seneleba
ar gaCereba
six
Sir
e pr
ocenteb
Si
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 27
1. „SeamowmeT Tqveni codna III.2“-is 7a punqtSi mogeTxovebodaT qurcikis
mier pirvelad Sesrulebuli maRali naxtomis dros mtacebelsa da
qurciks Soris arsebuli manZilis sixSireTa ganawilebis ageba.
gamoiyeneT sixSireTa es ganawileba da upasuxeT (a) da (b) kiTxvebs.
a) dajgufebuli ganawilebis martivi sixSireebis gamoyenebiT
(intervalTa klasis sogrZe 10 metri) miRebuli manZilebisTvis aageT
histograma da sixSireTa poligoni.
b) aageT fardobiT sixSireTa histograma da fardobiT sixSireTa
poligoni 1a kiTxvaSi aRwerili dajgufebuli ganawilebisTvis.
g) ra daskvnebis gakeTeba SegiZliaT qurcikis maRali naxtomis sigrZis
Sesaxeb Tqvens mier agebul grafikebze dayrdnobiT?
2. aageT foTlebiani Reroebis diagrama „SeamowmeT Tqveni codna III.2-is“ me-
7 kiTxvaSi mocemuli 50 manZilisTvis.
a) romelia minimaluri dafiqsirebuli manZili?
b) romelia maqsimaluri dafiqsirebuli manZili?
g) romelia yvelaze xSirad dafiqsirebuli manZili?
sixSireTa ganawilebis formebi
qulaTa ganawilebam SeiZleba miiRos nebismieri forma arsebuli mTeli rigi
sxvadasxvagvari formebidan. amrigad, rogorc wesi, ar arsebobs sixSireTa
ganawilebis formis ramdenime sityviT zustad aRweris martivi gza; Tumca,
arsebobs ramdenime zogadi aRweriTi termini, romelic gamoiyeneba ganawilebis
formis mimarT da aiolebs ganawilebis sityvebiT aRweras. suraTze N3.8
warmodgenilia sxvadasxva formis sixSireTa poligonis magaliTebi.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 28
sixSireTa simetriuli ganawileba
Tu sixSireTa simetriul
ganawilebas orad
gadavkecavT Suawertilis
gaswvriv, miviRebT erTnairi
formis or naxevars. ganawilebis erTi naxevari warmoadgens meoris sarkisebur
asaxvas. suraTze N3.8 ganawilebebi (a), (b) da (g) simertiulia, (d) da (e) ki _ ara;
isini asimetriuli ganawilebebia. simetriuli zaris formis anu normalur
ganawilebas, romelic naCvenebia suraTze N3.8a, gansakuTrebuli mniSvneloba aqvs
bihevioristul mecnierebaSi. amaze ufro dawvrilebiT me-6 TavSi visaubrebT.
asimetriuli ganawileba
nedli monacemebis srulyofilad
simetriuli ganawileba iSviaTad miiReba.
ufro mosalodnelia, rom sixSireTa
ganawileba iyos asimetriuli anu
gadaxrili. asimetriul ganawilebaSi
qulebi Tavmoyrilia ganawilebis erT
boloSi; ganawilebis meore boloSi (anu kudSi) ki ganlagebulia iSviaTad
dafiqsirebuli qulebi. ganawilebas ewodeba dadebiTad asimetriuli, Tu maRali
qulebis Semcveli bolo (anu kudi) ganawilebis marjvena mxares mdebareobs
(suraTi N3.8d), xolo ganawilebas ewodeba uaryofiTad asimetriuli, Tu dabali
sixSireTa simetriuli ganawileba (Symmetrical frecuency
distribution): _ sixSireTa ganawileba, romlis
Suawertilze orad gadakecvis SemTxvevaSi miiReba
erTnairi formis ori naxevari.
asimetriuli ganawileba (Skewed
distribution): _ sixSireTa ganawileba, sadac
qulebi Tavmoyrilia ganawilebis erT
boloSi; ganawilebis meore boloSi (anu
kudSi) ki ganlagebulia iSviaTad
dafiqsirebuli qulebi.
suraTi N3.8.sixSireTa hipoTeturi ganawilebebis formebi.
(g)
marTkuTxedi
simetriuli
(d)
asimetriuli
dadebiTad asimetriuli uaryofiTad asimetriuli
(e)
(a) (b)
bimodaluri
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 29
qulebis Semcveli bolo (anu kudi) mdebareobs ganawilebis marcxena mxares
(suraTi N3.8e).
modaloba
ganawileba aRiwereba imiTac, Tu
ramdeni modiT xasiaTdeba is. moda
qulaTa ganawilebaSi yvelaze xSirad dafiqsirebuli qulaa. unimodalur
ganawilebas mxolod erTi yvelaze xSirad dafiqsirebuli qula aqvs (ix. suraTi
N3.8(a), (d) da (e)). ganawilebas, romelsac ori moda aqvs, bimodaluri ewodeba (ix.
suraTi N3.8b). bihevioraluri mecnierebis mravali warmomadgeneli ganawilebas
bimodalurad miiCnevs, Tu is ori gancalkevebuli pikiT xasiaTdeba; im
SemTxvevaSic ki, rodesac pikebi ar gamoxataven toli sixSiriT dafiqsirebul
qulebs. multimodalur ganawilebaSi warmodgenilia sami an meti qula, romelic
yvelaze xSirad fiqsirdeba. ganawileba maSinac ki SeiZleba CaiTvalos
multimodalurad, rodesac masSi mocemuli sami an meti pikis sixSire ar aris
toli. ar gaaCnia moda marTkuTxedis formis iseT ganawilebas, rogoric suraTze
N3.8g-zea naCvenebi. aseT ganawilebaSi yvela qula Tanabrad an TiTqmis Tanabrad
fiqsirdeba.
SeamowmeT Tqveni codna III.4
1. ganmarteT: _ asimetriuli ganawileba, bimodaluri ganawileba, moda,
multimodaluri ganawileba, uaryofiTad asimetriuli ganawileba,
dadebiTad asimetriuli ganawileba, simetriuli ganawileba,
unimodaluri ganawileba.
2. aRwereT suraTze N3.9 mocemuli sixSireTa ganawilebebi simetriis,
asimetriulobisa da modalobis mixedviT.
moda (Mode): _ qulaTa ganawilebaSi
yvelaze xSirad dafiqsirebuli qula.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 30
Sejameba
• sixSireTa ganawileba aris cxrili, romelic gviCvenebs individTa jgufis
mier miRebul TiToeul qulas da am qulis miRebis sixSires.
• sixSireTa ganawileba gamoiyeneba monacemebis mowesrigebisTvis.
• daujgufebel sixSireTa ganawileba Seileba gamoviyenoT mcire raodenobis
monacemebis SemTxvevaSi.
• dajgufebul sixSireTa ganawileba gamoiyeneba didi raodenobiT qulebis,
an qulaTa farTo diapazonis SemTxvevaSi.
• daujgufebel an dajgufebul sixSireTa ganawileba SeiZleba gamoisaxos
martivi, fardobiTi an procentebSi sixSireebis saxiT.
• procentili aris qula, romlis qvemoTac Tavsdeba ganawilebaSi qulaTa
garkveuli procenti.
• qulis procentuli rangi gviCvenebs ganawilebaSi im qulaTa procents,
romelic tolia an naklebia am qulaze.
• sixSireTa ganawileba SeiZleba gamoisaxos grafikulad histogramis,
sixSireTa poligoniT an foTlebiani Reroebis diagramis saxiT.
• Tvisebrivi monacemebi SeiZleba gamoisaxos grafikulad svetebiani
diagramis saxiT.
(a) (b)
(d) (g)
suraTi N3-.9. sixSireTa ganawileba me-2 kiTxvisTvis.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 31
• sixSireTa ganawilebis forma aRiwereba simetriulobis, asimetriulobisa
da modalobis terminebiT.
ZiriTadi terminebi da simboloebi
asimetriuli ganawileba
bimodaluri ganawileba
dagrovili sixSire
procentebSi
dadebiTad asimetriuli
ganawileba
daujgufebel sixSireTa
ganawileba
dajgufebul sixSireTa
ganawileba
intervali
intervalis dagrovili
sixSire
intervalis zeda
dadgenili sazRvari
intervalis zeda
realuri sazRvari
intervalis dadgenili
sazRvrebi
intervalis realuri
sazRvrebi
intervalis qveda realuri
sazRvari
intervalis qveda dadgenili
sazRvari
intervalis Suawertili
moda
multimodaluri ganawileba
nedli monacemebi
nedli qulebi
procentili
procentuli rangi
svetebiani diagrama
simetriuli ganawileba
sixSireTa ganawileba
sixSireTa ganawileba
procentebSi
sixSireTa poligoni
uaryofiTad asimetriuli
ganawileba
unimodaluri ganawileba
fardobiT sixSireTa
ganawileba
foTlebiani Reroebis
diagrama
qulis dagrovili sixSire
histograma
cƒ anu cum ƒ
crƒ anu cum rƒ
c%ƒ anu cum %ƒ
%ƒ
rƒ
kiTxvebi
1. hiksi, misTri, luCero, meriqali da pelegrini (Hicks, Mistry, Lucero, Marical and
pellegrini, 1990) aRniSnaven, rom kolejis studentebis gamokiTxvis Sedegebis
mixedviT, maT dRe-RameSi saSualod 6 saaTi da 87 wuTi (anu 412 wuTi)
ZinavT. Tqven dainteresdiT, Tu ramdenad SeiZleba am monacemis
gavrceleba kolejis mamrobiTi sqesis studentebze. amitom sTxoveT
kolejis 50 mamrobiTi sqesis students, rom CaeweraT, Tu ramden xans
eZinaT erTi Ramis ganmavlobaSi. Tqven miiReT wuTebSi Cawerili Semdegi
droebi: _ 440, 223, 309, 427, 463, 2371, 75, 315, 290, 356, 407, 399, 263, 417, 328, 377,
406, 275, 316, 391, 350, 422, 431, 382, 350, 371, 410, 295, 305, 327, 400, 227, 275, 340,
364, 279, 300, 412, 388, 296, 320, 342, 371, 258, 470, 496, 384, 439, 284, 351, 417.
a) am monacemebiT aageT dajgufebul sixSireTa ganawileba
procentebSi da dajgufebul dagrovil sixSireTa ganawileba
procentebSi. gamoiyenebT 14 intervali.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 32
b) romeli inervali Seicavs yvelaze met qulas? Tanxvdeba Tu ara am
intervalis Suawertili kvlevaSi dasaxelebul 412 wuTs?
g) dajgufebul dagrovil sixSireTa ganawilebaze dayrdnobiT
SeafaseT drois is raodenoba, romelic ganawilebas or naxevrad
yofs anu romlis qvemoTac da zemoTac moTavsebulia 25-25 cdis piri.
d) ipoveT am ganawilebis 50-e procentili. Tanxvdeba Tu ara is (g)
punqtSi miRebul pasuxs?
e) ra iqneba 431 wuTis procentuli rangi?
v) aageT histograma da sixSireTa poligoni (a) punqtSi agebuli
ganawilebis martivi sixSireebis gamoyenebiT.
z) aRwereT ganawilebis forma asimetriulobisa da modalobis
terminebiT.
2. mamri beRura mondomebiT aSenebs budes da mas naWuWebiT, feradi
foTlebiTa da TuTiyuSis cisferi bumbulebiT rTavs, radgan
SewyvilebaSi warmateba budis morTulobis odenobasTan aris
dakavSirebuli. bevrjer iqna SemCneuli, rom mamri beRurebi sxva
frinvelebis budeebidan iparaven morTulobebs da maT TavianTi
budeebisTvis iyeneben. borgia da gori (Borgia & Gore, 1986) am qcevas axal
samxreT uelsSi, avstraliaSi akvirdebodnen. maTi dakvirvebis obieqti iyo
59 beRura. maT daniSnes budeSi arsebuli bumbulebi da iniSnavdnen, Tu
romeli bumbulebi iyo moparuli sxva budeebidan. davuSvaT, dakvirvebis
periodis ganmavlobaSi maT CainiSnes TiToeuli frinvelis mier moparuli
bumbulebis Semdegi raodenobebi: _ 2, 6, 14, 1, 10, 3, 0, 26, 37, 47, 4, 0, 6, 21, 3,
6, 6, 51, 8, 11, 0, 0, 4, 2, 5, 9, 4, 6, 24, 4, 1, 18, 63, 1, 5, 6, 4, 7, 12, 5, 3, 4, 3, 15, 46,
0, 5, 0, 26, 5, 17, 1, 17, 1, 58, 9.
a) am monacemebiT aageT dajgufebul fardobiT sixSireTa ganawileba
da dagrovil dajgufebul fardobiT sixSireTa ganawileba.
gamoiyeneT 13 intervali.
b) romeli intervali Seicavs yvelaze bevr qulas?
g) ipoveT am ganawilebis 50-e procentili.
d) ra iqneba 24 qulis procentuli rangi?
e) am ganawilebisTvis aageT foTlebiani Reroebis grafiki. romelia
minimaluri da maqsimaluri qula am ganawilebaSi? romelia yvelaze
xSirad dafiqsirebuli qula?
v) aageT histograma da sixSireTa poligoni 2a kiTxvaSi agebuli
ganawilebis martivi sixSireebis gamoyenebiT.
v) aRwereT ganawilebis forma asimetriulobisa da modalobis
terminebiT.
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 33
3. davuSvaT, daakvirdiT mamrobiTi da mdedrobiTi sqesis mZRolebs
avtosadgomze da daiTvaleT, Tu ra sixSiriT aCerebdnen isini
avtomanqanebs iseT adgilebSi, romlebic ar ekuTvnoda avtosadgoms.
samdRiani dakvirvebis manZilze Tqven SeamCnieT, rom mamrobiTi sqesis 100
mZRolidan 12 da mdedrobiTi sqesis 100 mZRolidan 7 arasaTanado
adgilze aCerebda avtomanqanas. aageT arasaTanado adgilze avtomanqanis
gaCerebis fardobiTi sixSireebis svetebiani diagrama mamrobiTi da
mdedrobiTi sqesis mZRolebisTvis.
4. stargenma da birma (Sturgeon & Beer, 1990) aCvenes, rom im studentebis
waxaliseba, romlebic regularulad dadian leqciebze, semestris bolos
gamocdidan ganTavisuflebiT, zrdis leqciebze maT daswrebas. davuSvaT,
erTmaneTs SeadareT ori skolis daswrebis monacemebi; aqedan erT
skolaSi xorcieldeboda aseTi waxaliseba, meoreSi ki _ ara. Tqven miiReT
gacdenebis martiv sixSireTa Semdegi saxis ganawileba:
a) TiToeuli skolisTvis aageT
fardobiTi sixSireebis ganawileba.
b) ra iqneba 50-e procentili TiToeuli
skolisTvis?
g) ra iqneba TiToeul skolaSi oTxi
gacdenis procentuli rangi?
d) gansxvavdebian Tu ara skolebi
erTmaneTisgan gacdenebis sixSiris
mixedviT? aRwereT ganawilebebi.
5. vinaidan cxovrebis tempi Zalze
Cqaria, gviWirs erTi calkeuli
dRis gaxseneba. bunebriv
pirobebSi mexsierebis kvlevis farglebSi fsiqologma kolejis 90
students sTxova, rom CamoeweraT maTTvis Cveuli saqmianobebi dasvenebis
dReebSi. davuSvaT, miRebuli Sedegebi fsiqologma qvemoT mocemuli
foTlebiani Reros diagramis saxiT warmoadgina.
Rero foToli
2 * s ƒ t 2 · 1 * s ƒ t 1 · 0 *
888899 66667 44555 2222333 0000000111111 888999999 6666677777 44455555 2233333 0001111
gacdenili dReebis
raodenoba
waxalisebis mqone skola
waxalisebis ar mqone skola
25-ze meti 24-25 22-23 20-21 18-19 16-17 14-15 12-13 10-11 8-9 6-7 4-5 2-3 0-1
2 0 2 0 1 0 2 0 3 8 4 13 23 42
6 6 1 8 0 4 22 10 18 20 3 14 34 54
jami 100 200
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 34
s ƒ t 0 ·
888999 6677 555
a) aageT nedli qulebis cxrili im studentebisTvis, romelTac 14-dan
19 saqmianoba gaixsenes.
b) romelia dasaxelebuli yvelaze maRali da dabali qulebi?
g) romelia yvelaze xSirad dasaxelebuli qula?
d) gamosaxeT qulebi histogramiT. gamoiyeneT intervali, romlis
sididea i=2.
e) foTlebiani Reros diagramasTan SedarebiT ra informacia ikargeba
qulebis histogramis saxiT gamosaxvis SemTxvevaSi?
6. pirvel TavSi iTqva, rom 25-54 wlis amerikeli qalebi, rogorc wesi, dReSi
4.4 saaTs (264 wT) televizorTan atareben. socialuri fsiqologiis
studentebis saklaso proeqtis farglebSi am asakis 100 qali gamoikiTxa,
romelTac sTxoves maT mier dRis manZilze televizorTan gatarebuli
saaTebis raodenobis Sefaseba. dasaxelebuli dro (wuTebSi) dajgufebul
sixSireTa Semdegi ganawilebis saxiTaa mocemuli:
a) ra sididis (i) intervalia gamoyenebuli am
cxrilSi?
b) ra iqneba 210-239 intervalis qveda da zeda
realuri zRvari?
g) gansazRvreT 210-239 intervalis Suawertili.
d) am qulebisTvis aageT martiv sixSireTa
ganawileba da dagrovil sixSireTa
ganawileba.
e) ipoveT am ganawilebis 75-e procentili.
v) ipoveT procentiluri rangi televizorTan
gatarebuli 264 wuTisTvis
7. gamajansaRebeli klubis 25 mepatronem
Seafasa, Tu dRis ganmavlobaSi ra dros
(wuTebiT) uTmobda TiToeuli maTgani
varjiSs. Sedegad dasaxelda Semdegi ricxvebi: _ 67, 73, 79, 78, 50, 80, 98, 88,
79, 71, 57, 81, 41, 65, 50, 71, 91, 75, 60, 79, 79, 71, 65, 85 da 64.
a) am qulebisTvis i=5 intervalis gamoyenebiT aageT fardobiTi
sixSireebisa da dagrovili fardobiTi sixSireebis ganawilebebi.
ra iqneba pirveli intervalis qveda dadgenili sazRvari? ratom?
b) ra formisaa es ganawileba?
g) ipoveT 78 wuTis procentiluri rangi.
intervali Tvlis Sedegi
510-539 480-509 450-479 420-449 390-419 360-389 330-359 300-329 270-299 240-269 210-239 180-209 150-179 120-149 90-119 60-89 30-59 0-29
/ // ///// ////// ///// ///////// //////// ///////// ////////// ///////// /// /////// // //////// ///// /////// ////
თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 35
d) davuSvaT, pirovnebas surs, rom sakmarisi raodenobiT ivarjiSos
ise, rom am ganawilebaSi pirvel 25%-Si Soris moxvdes. sul mcire
ramdeni wuTi unda ivarjiSos man yoveldRiurad, rom man es
SeZlos?
Tavi IV. monacemTa aRwera: centraluri tendenciis
sazomebi
moda
mediana
medianis gamoTvla kenti raodenobis monacemebiT:
monacemebi ar aris dajgufebuli
medianis gamoTvla luwi raodenobis monacemebiT:
monacemebi ar aris dajgufebuli
medianis gamoTvla procentilis formuliT
medianis Tvisebebi
SerCevis saSualo
SerCevis saSualos gamoTvla
SerCevis saSualos Tvisebebi
SerCevis saSualos warmodgena samecniero statiebSi
populaciis saSualo
centraluri tendenciis sazomebis Sedareba
centraluri tendenciis sazomebis aRweriTi gamoyeneba
centraluri tendenciis sazomebis daskvniTi gamoyeneba
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 2
sixSireTa ganawileba da grafikebi did informacias gvaZlevs qulaTa
ganawilebis formisa da miRebuli qulebis diapazonis Sesaxeb; Tumca, xSirad
mecnierebs sWirdebaT, rom qulaTa ganawileba mxolod erTi ricxviT aRiweros. am
ricxvebs aRweriTi statistikebi ewodebaT, radgan isini nedl monacemebs erTi
ricxviT aRweren. Cveulebisamebr, gamoiyeneba ori tipis aRweriTi statistikebi: _
erTni, romelTac centraluri tendenciis sazomebs uwodeben, aRweren miRebul
tipur qulas; meoreni ki, romelTac gafantvis sazomebs uwodeben, miuTiTeben
tipuri qulis irgvliv cvalebadobisa da gafantvis zomaze.
aRweriT statistikebs ori funqcia aqvT: _ (1) SerCevaSi miRebuli qulebis
aRwera; (2) SerCevis monacemebze dayrdnobiT populaciis maxasiaTeblebis Sefaseba
anu maT Sesaxeb daskvnebis gakeTeba. magaliTad, me-3 TavSi xsenebuli mkvlevari
SeiZleba dainteresebuli iyos Tavisi kolejis studentTa populaciaSi
gamocdebTan dakavSirebuli SfoTvis tipuri donis SefasebiT. misi Sefaseba
dafuZnebuli iqneba 100 gogonasgan Semdgari SerCevis qulebisgan miRebul
aRweriT statistikaze.
winamdebare Tavi Seexeba centraluri tendenciis sazomebis aRweriT
gamoyenebas. cvalebadobis sazomebi da aRweriTi statistikebis gamoyeneba
populaciis maxasiaTeblebis Sesaxeb daskvnebis gakeTebisas ganxilulia me-5 da me-
7 TavebSi.
centraluri tendenciis sazomebi aris ricxvebi, romlebic warmoadgenen
SerCevidan miRebul saSualo an tipur qulas. Cveulebisamebr, gamoiyeneba
centraluri tendenciis sami sazomi: _ moda, mediana da saSualo ariTmetikuli.
moda
moda aris qula, romelic yvelaze xSirad gvxvdeba qulaTa ganawilebaSi. moda
moiZebneba martivi, daujgufebeli sixSireebis ganawilebis gadasinjviT da imis
dafiqsirebiT, Tu romeli qula gvxvdeba yvelaze xSirad. magaliTad, cxrilSi N3-
2 yvelaze xSirad (xuTjer) gvxvdeba qula 59. Sesabamisad, am ganawilebis modaa 59.
centraluri tendenciis sazomebi (Measures of central tendency): _ ricxvebi, romlebic
warmoadgenen SerCevidan miRebul saSualo an tipur qulas.
cvalebadobis sazomi (measures of central tendency) _ ricxvi, romelic miuTiTebs, Tu ramdenad
gansxvavdebian qulebi erTmaneTisgan da ramdenad gansxvavdeba maTgan scentraluri
tendenciis sazomi.
moda (Mode): _ qula, romelic yvelaze xSirad gvxvdeba ganawilebaSi.
unimodaluri (Unimodal): _ ganawileba, romelsac aqvs erTi moda.
bimodaluri (Bimodal): _ ganawileba, romelsac aqvs ori moda.
multimodaluri (Multimodal): _ ganawileba, romelsac aqvs orze meti moda.
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 3
vinaidan qulaTa am erTobliobaSi mxolod erTi modaa, am ganawilebas
unimodaluri ewodeba. qulaTa ganawileba SeiZleba iyos bimodaluri (anu hqondes
ori moda), an multimodaluri (anu hqondes orze meti moda). SegaxsenebT, rom
sixSireTa ganawilebis formis aRweris erT-erTi gza modalobas efuZneba.
unimodaluri da bimodaluri ganawilebebi mocemulia suraTze N3.8.
miuxedavad imisa, rom moda advilad gamoiTvleba martivi, daujgufebeli
sixSireebis ganawilebidan, bihevioristul mecnierul kvlevaSi is iSviaTad
gamoiyeneba centraluri tendenciis sazomad. Tu ganawilebaSi erTze meti modaa,
romeli maTgani iqneba yvelaze tipuri qula? amrigad, modam SeiZleba ver
uzrunvelyos ganawilebis aRsawerad sakmarisi erTi tipuri qula.
modis iSviaTad gamoyenebis meore mizezi isaa, rom erTi qulis cvlilebamac ki
SeiZleba mniSvnelovnad Secvalos ganawilebis moda. magaliTad, davuSvaT, rom
cdis pirebs mieciT xanmokle mexsierebis testi da sTxoveT 10-cifriani
mwkrividan SeZlebisdagvarad meti cifris gameoreba. vTqvaT, 6 cdis pirisgan
miiReT cifrebis Semdegi raodenoba:
cdis
pirebi
cifrebis
raodenoba
1 2 3 4 5 6
5 6 7 7 8 9
am ganawilebis modaa 7, radgan orma cdis pirma gaimeora 7 cifri. davuSvaT,
meoTxe cdis pirma 7-is nacvlad gaimeora 5 cifri. axla ganawilebis moda 7-is
nacvlad aris 5. mxolod erTi qulis cvlilebam mniSvnelovnad Secvala modac.
axla davuSvaT, rom mesame cdis pirma 7-is nacvlad gaixsena 10 cifri. am
SemTxvevaSi ganawilebas ar eqneba moda, TiToeuli qula gveqneba toli
raodenobiT. vinaidan moda ase Zalian aris damokidebuli mxolod ramdenime
qulaze, is ar aris centraluri tendenciis stabiluri sazomi.
mediana
mediana (SemoklebiT Mdn) ukavSirdeba ganawilebaSi
im qulas, romlis zemoT da qvemoT ganlagebulia
qulaTa toli raodenoba. amrigad, mediana 50-e
procentilia ganawilebaSi. qulaTa ganawilebis
umravlesobisTvis martivad SeiZleba medianis povna.
mediana (Median, Mdn): _ qula
ganawilebaSi, romlis zemoT
da qvemoT ganlagebulia
qulaTa toli raodenoba,
mediana aris 50-e procentili
ganawilebaSi.
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 4
medianis gamoTvla kenti raodenobis monacemebiT: monacemebi ar aris
dajgufebuli
ganawilebaSi Semavali kenti raodenobis monacemebisTvis, rodesac monacemebi
ar aris dajgufebuli, rogorc wesi, mediana sixSireTa ganawilebaSi SuaTana
qulis povniT ganisazRvreba. magaliTad, gamocdebTan dakavSirebuli SfoTvis
kiTxvaris qulebis _ 22, 34, 36, 37, 38, 44 da 72 _ medianaa 37. esaa meoTxe qula da
Svidquliani ganawilebis SuaSi xvdeba. sami qula naklebia 37-ze da sami _ metia
masze. yuradReba miaqcieT, rom medianis gansasazRvrad daTvlis procedura
gamoiyeneba, xolo Tavad qulaTa ricxviTi mniSvnelobebi mxedvelobaSi ar miiReba.
medianis gamoTvla luwi raodenobis monacemebiT: monacemebi ar aris
dajgufebuli
ganawilebaSi Semavali luwi raodenobis monacemebisTvis, rodesac monacemebi
ar aris dajgufebuli, medianad pirobiTad aiReba mniSvneloba, romelic
sixSireTa ganawilebaSi or SuaTana qulas Soris mdebareobs. magaliTad,
ganvixiloT gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebi: _ 34, 35, 37, 38,
44, 72. am qulebis medianaa 37.5 anu mniSvneloba, romelic xvdeba 37-isa (mesame
qula) da 38-is (meoTxe qula) SuaSi. eqvsi qulidan erTi naxevari mdebareobs 37.5-is
qvemoT da meore naxevari ki _ mis zemoT.
magaliTi N4-1
medianis moZebna
amocana
realuri samyaros vizualuri aRqmis zogierT kvlevaSi cdis pirs eZleva
suraTi, romelzec gamosaxulia raime scena, da Semdeg moeTxoveba, rom gaixsenos
am suraTis detalebi. davuSvaT, aseTi Tanmimdevrobis cda CaatareT 18 cdis
pirTan da TiToeulma maTganma gaixsena Semdegi raodenobis detalebi: _ 7, 3, 7, 9,
6, 10, 14, 3, 6, 9, 14, 9, 8, 5, 4, 6, 9 da 13. ipoveT mediana.
amoxsna
ganawileba Sedgeba Tvrameti qulisgan, amitom mediana iqneba qula, romlis
zemoT da qvemoT moTavsebulia cxra qula. Sesabamisad, rodesac qulebi
dalagebulia sixSireTa ganawilebaSi (rogorc mocemulia qvemoT), mediana
mdebareobs mecxre da meaTe qulebs Soris.
mecxre qulaa 7, xolo meaTe _ 8, amitom mediana iqneba 7-sa da 8-s Soris anu
7.5.
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 5
medianis gamoTvla procentilis formuliT
dajgufebuli monacemebisTvis aRar gamogvadgeba medianis gamoTvlis martivi
gzebi. am SemTxvevaSi medianis misaRebad SesaZlebelia formulis gamoyeneba.
mediana aris ganawilebis 50-e procentili. gavisxenoT me-3 Tavidan, rom
procentili aris mniSvneloba, romlis qvemoTac moTavsebulia ganawilebis
qulaTa garkveuli procenti. medianis SemTxvevaSi misi mniSvnelobis qvemoT xvdeba
qulaTa 50%. Sesabamisad, medianis miReba SesaZlebelia dajgufebul sixSireTa
ganawilebidan, me-3 TavSi mocemuli procentilis formulis gamoyenebiT:
if
cfNXXMdni
LL
−+===
)(50.50.mediana
sadac X.50 = mediana, romlis gamoTvlac gvsurs.
XL = 50-e procentilis Semcveli intervalis qveda realuri sazRvari.
N = ganawilebaSi qulaTa saerTo raodenoba.
cƒL = medianis Semcveli intervalis qveda realur sazRvramde qulebis
dagrovili sixSire.
ƒi = medianis Semcvel intervalSi Semaval qulaTa sixSire.
i = intervalis sigrZe.
medianis Tvisebebi
mediana, faqtobrivad, miiReba qulebis daTvliT da yuradRebis miRma rCeba
yvela qulis ricxviTi mniSvneloba, garda erTisa; gamonaklisia mxolod im qulis
mniSvneloba, romelic medianas warmoadgens. garda amisa, medianaze ar moqmedebs
ganawilebaSi Semaval qulaTa ukiduresi ricxviTi mniSvnelobebi. magaliTad,
qvemoT mocemuli TiToeuli ganawilebis medianaa 37 (gamuqebulia).
9 qula
9 qula
7.5 = mediana
qula ƒ
14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
2 1 0 0 1 4 1 2 3 1 1 2 0 0 0
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 6
mediana ar iTvaliswinebs b da g ganawilebis
boloebSi ganlagebul ukidures qulebs. es Tvisebebi
medianas sasurvel sazomad aqcevs, rodesac
asimetriuli ganawilebis centraluri tendenciis
gazomva xdeba saWiro. magaliTad, mediana xSirad
gamoiyeneba, rodesac saubaria populaciis tipur
Semosavalze, radgan Semosavlis ganawileba mkveTrad
dadebiTad asimetriulia. mxolod mcire raodenobis adamianebs aqvT
mravalmilioniani Semosavali. Tu am Semosavlebis ricxviTi mniSvnelobebi
gaTvaliswinebuli iqneba tipuri Semosavlis gazomvaSi, is Zalze maRali iqneba,
raTa saSualo adamianis Semosavlis reprezentaciuli iyos. vinaidan medianaze ar
zemoqmedebs es mcire raodenobiT didi Semosavali, is warmoadgens tipuri
Semosavlis reprezentaciul centraluri tendenciis sazoms.
SeamowmeT Tqveni codna IV.1
1. ganmarteT: _ bimodaluri ganawileba, aRweriTi statistikebi, centraluri
tendenciis sazomebi, mediana, moda, multimodaluri ganawileba,
unimodaluri ganawileba.
2. daasaxeleT aRweriTi statistikebis ori funqcia.
3. davuSvaT, mkvlevari zomavs 25 meSvide klaseli gogonas TviTSefasebas.
gogonebma TavianTi TviTSefaseba ganalages Sefasebis skalaze 10-dan
(yvelaze dabali TviTSefaseba) 70-mde (yvelaze maRali TviTSefaseba).
miRebul iqna Semdegi qulebi: _ 49, 53, 67, 20, 27, 36, 49, 27, 61, 17, 49, 50, 29,
56, 65, 27, 48, 24, 49, 15, 32, 66, 46, 24 da 38.
a) ipoveT am ganawilebis moda.
b) rogoria es ganawileba: _ unimodaluri, bimodaluri Tu
multimodaluri?
4. davuSvaT, erT-erTma gogonam, romelmac miiRo zemoT (me-3 kiTxvaSi)
dasaxelebuli 49 qula, 49-is nacvlad miiRo 27 qula.
a) ipoveT am ganawilebis moda.
b) rogoria es ganawileba: _ unimodaluri, bimodaluri Tu
multimodaluri?
5. davuSvaT, erT-erTma gogonam, romelmac miiRo zemoT (me-3 kiTxvaSi)
dasaxelebuli 49 qula, 49-is nacvlad miiRo 24 qula.
a) ipoveT am ganawilebis moda.
b) rogoria es ganawileba: _ unimodaluri, bimodaluri Tu
multimodaluri?
ganawileba
a b g
22 34 35 37 39 44 44
34 34 35 37 39 44 109
22 34 35 37 39 44 117
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 7
6. modis, rogorc aRweriTi statistikis, romeli SezRudvis magaliTs
warmoadgens me-4 da me-5 kiTxvebze Tqvens mier miRebuli pasuxebi?
7. ipoveT me-3 kiTxvaSi mocemuli ganawilebis mediana.
8. davuSvaT, eqsperimentatorma TviTSefaseba kidev erT me-7 klasel gogonas
gauzoma da miiRo 28 qula. CarTeT es qula ganawilebaSi da ipoveT am
ganawilebis mediana.
9. aviRoT me-3 kiTxvaSi mocemuli sawyisi ganawileba (26-e cdis piris
monacemis damatebis gareSe). davuSvaT, rom qulebi 61, 65, 66 da 67
Seicvala qulebiT 51, 55, 56 da 57-iT, Sesabamisad. ra iqneba qulaTa axali
ganawilebis mediana?
10. SeadareT erTmaneTs: _ (a) me-7 da me-8 kiTxvebze miRebuli pasuxebi; (b) me-7
da me-9 kiTxvebze miRebuli pasuxebi. medianis, rogorc aRweriTi
statistikis, ra SezRudvis magaliTia Tqvens mier miRebuli pasuxebi?
SerCevis saSualo
centraluri tendenciis yvelaze cnobili
sazomia SerCevis saSualo, romelsac xSirad
saSualo ariTmetikuls, an ubralod
saSualos uwodeben. SerCevis saSualo aris
miRebul qulaTa jami gayofili qulaTa mTlian raodenobaze. statistikuri
simboloebiT es ase gamoiTqmis:
N
XX
N
ii∑
=== 1saSualo SerCevis
sadac X aris saSualo.
∑ (berZnuli didi sigma) aris Sekrebis niSani. is miuTiTebs ricxvebis
erTobliobis dajamebaze anu Sekrebaze. i=1 da N aRniSvnebi niSnavs
imas, rom Sekrebilia yvela miRebuli qula, pirveli (i=1) cdis
piridan dawyebuli, bolo (me-N) cdis piriT damTavrebuli.
X i aris nebismieri cdis piris mier miRebuli qula, indeqsi i miuTiTebs
konkretul cdis pirze (magaliTad, X1 miuTiTebs pirvel cdis pirze,
X2 _ meore cdis pirze X10 _ meaTe cdis pirze da a.S).
N aris SerCevaSi cdis pirTa raodenoba.
rodesac cnobilia, rom xdeba SerCevaSi miRebuli yvela N qulis dajameba,
formulaSi aRar iwereba Sekrebis niSnis sazRvrebi da ugulebelyofilia indeqsi
i; aqedan gamomdinare, saSualos formula aseTi saxisaa:
SerCevis saSualo ( X ) (Sample mean):
_ miRebul qulaTa jami gayofili
qulaTa mTlian raodenobaze.
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 8
NX
X ∑=
umravles SemTxvevaSi am gamartivebul formulas gamoviyenebT.
SerCevis saSualos gamoTvla
davuSvaT, gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miviReT Semdegi
qulebi: _ 34, 35, 37, 38, 44 da 50. qulaTa am erTobliobisTvis SerCevis saSualo
Semdegnairad gamoiTvleba:
pirvel rigSi, vkrebT qulebs anu
∑ =+++++= 228404438373534X
qulaTa raodenoba N udris 6-s. qulaTa saSualos gamosaTvlelad miRebul
jams anu 228-s vyofT 6-ze, anu
0.386
228=== ∑
NX
X
gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli 100 qulisTvis,
romelic mocemulia cxrilSi N3-3, saSualo gamoiTvleba 100 qulis SekrebiTa da
miRebuli jamis 100-ze gayofiT. formulis saxiT:
6.4164.411004164
100374029...372842
===++++++
=X
magaliTi N4-2
saSualos gamoTvla
amocana
davuSvaT, magaliTSi N4-1 ganxilul suraTis gaxsenebis davalebaSi miiReT
Semdegi 18 qula: _ 7, 3, 7, 9, 6, 10, 14, 3, 6, 9, 14, 9, 8, 5, 4, 6, 9 da 13. gamoTvaleT am
qulebis saSualo.
saWiro statistika: N
XX ∑=
amoxsna
pirveli nabiji iqneba ∑ X -is miReba, romelic udris:
1421396458914963141069737 =+++++++++++++++++
Sekrebil qulaTa raodenobaa N, Cvens magaliTSi N udris 18-s. Sesabamisad,
gveqneba:
9.789.718
142==== ∑
NX
X
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 9
SerCevis saSualos Tvisebebi
saSualos gamoTvlisas gamoiyeneba ganawilebaSi Semavali yvela qulis
ricxviTi mniSvneloba, amitom ganawilebaSi Semaval qulaTa nebismieri cvlileba
aucileblad Secvlis am ganawilebis saSualos mniSvnelobas. magaliTad,
ganvixiloT gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli Semdegi
qulebi: _ 26, 26, 26, 30, 34, 40 da 42. am ganawilebis modaa 26, mediana _ 30, xolo
saSualo _ 32. Tu SevcvliT mxolod erT qulas, magaliTad, 40-is nacvlad
aviRebT 43-s, moda isev iqneba 26 da mediana _ 30, magram saSualo axla iqneba 32.4.
vinaidan saSualo damokidebulia TiToeuli qulis sidideze, ganawilebis
ukiduresi qulebi gavlenas axdenen saSualos sidideze da cvlian (amaxinjeben)
mis, rogorc centraluri tendenciis sazomis, mniSvnelobas. magaliTad, davuSvaT,
ganxilul ganawilebaSi qula 40 SevcvaleT ara 43-iT, aramed _ 78-iT. ganawilebis
moda da mediana isev 26 da 30 iqneba, Sesabamisad, magram gaizrdeba saSualo da
iqneba 37.4. aqedan gamomdinare, Tu ganawilebaSi Sedis ramdenime ukiduresi qula,
saSualo SeiZleba veRar warmoadgendes tipur qulas.
saSualos meore Tviseba isaa, rom ganawilebaSi Semaval TiToeul qulasa da
ganawilebaSi Semaval qulaTa saSualos Soris gansxvavebaTa jami nulis tolia,
anu
( )∑ =− 0XX
es gamosaxuleba gviCvenebs, rom ganawilebaSi Semaval TiToeul qulas
gamovakeliT saSualo, Semdeg SevkribeT miRebuli sxvaobebi da Sedegad miviReT
nuli. cxrilSi N4-4 mocemulia es mimarTeba: am cxrilis (a) svetSi mocemulia
gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli Svidi qula da maTi
saSualo 35.0; (b) svetSi mocemulia Semdegi moqmedeba: _ saSualos ricxviTi
mniSvneloba akldeba (a) svetSi mocemul TiToeul qulas da am svetSi miRebuli
mniSvnelobebis jami ( )∑ − XX aris nuli.
qulaTa nebismieri erTobliobisTvis
( )∑ − XX jamis mniSvneloba nulis
toli iqneba, magram ( )2∑ − XX mxolod
im SemTxvevaSi iqneba nulis toli,
rodesac ganawilebaSi Semavali
TiToeuli qula saSualos tolia; sxva
SemTxvevaSi ( )2∑ − XX ar iqneba nulis
toli. es sidide gamoTvlilia (g)
svetSi, sadac TiToeul qulasa da
kvadratebis jami (SS) (Sum of squares): _
ricxvi, romelic miiReba ganawilebaSi
Semavali TiTouli qulidan ganawilebis
saSulos gamoklebiT, TiToeuli sxvaobis
kvadratSi ayvaniTa da miRebuli sidideebis
SejamebiT.
M _ simbolo, romelic amerikis
fsiqologiuri asociaciis publikaciebis
saxelmZRvaneloSi (1994) miTiTebuli
normebiT Sesrulebul naSromSi
gamoiyeneba SerCevis saSualos aRsaniSnad.
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 10
saSualos Soris sxvaoba (romelsac xSirad gadaxras uwodeben) kvadratSia
ayvanili (anu gamravlebulia Tavis Tavze). magaliTad, qula 22 X -sgan
gansxvavdeba -13-iT, xolo (-13)2 = 169. (g) svetSi mocemul sxvaobaTa kvadratebis
jamia 264. ( )2∑ − XX sidide xSirad gamoiyeneba statistikur gamoTvlebSi da
kvadratebis jami (SemoklebiT SS) ewodeba. amrigad,
( )2∑ −== XXSSjami skvadratebi
maTematikurad SesaZlebelia imis Cveneba, rom saSualodan qulaTa gadaxrebis
kvadratebis jami (anu ( )2∑ − XX ) ufro naklebia, vidre nebismieri sxva
statistikidan, magaliTad, medianidan an modidan, qulaTa gadaxrebis kvadratebis
jami. es Tviseba SesaZleblobas iZleva, rom saSualo gamoyenebul iqnes ramdenime
Zalze mniSvnelovan statistikur meTodSi, romelzec mogvianebiT, Semdeg TavebSi
visaubrebT.
cxrili N4.1. sveti (a): Svidi qula da maTi
saSualo 35.0; sveti (b): XX − sxvaobis
mniSvneloba TiToeuli qulisTvis; sveti (g)
XX − gadaxra TiToeuli qulisTvis
ayvanilia kvadratSi.
(a)
X
(b)
XX −
(g)
2)( XX −
22 34 35 35 37 38 44
22-35 = -13 34-35 = -1 35-35 = 0 35-35 = 0 37-35 = +2 38-35 = +3 44-35 = +9
169 1 0 0 4 9 81
X 0.35 0 jami 264
SerCevis saSualos warmodgena samecniero statiaSi
statistikosebi, rogorc wesi, SerCevis saSualos aRsaniSnad iyeneben X
simbolos, Tumca, samecniero JurnalebSi xSirad gamoiyeneba statistikaSi
miRebulisgan gansxvavebuli simboloebi. bihevioristuli mecnierebis mravali
Jurnali icavs amerikis fsiqologiuri asociaciis publikaciebis
saxelmZRvaneloSi (Publication Manual of the American Psychological Association. American
Psychological Association, 1994) mocemul saredaqcio stils. am stilis mixedviT,
SerCevis saSualo aRiniSneba laTinuri didi daxrili M-iT. amrigad, samecniero
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 11
JurnalSi gamoqveynebul statiaSi SeiZleba SexvdeT Semdegnairad warmodgenil
saSualos:
qali cdis pirebis mier qali politikosebis kompetenturobis saSualo Sefaseba (M =
6.13) ufro maRali iyo, vidre mamakaci cdis pirebis saSualo Sefaseba (M = 5.42).
sabeWd manqanaze akrefil an xelnawer teqstSi statistikuri simbolos
gamosayofad gamoiyeneba xazgasma1. amrigad, laboratoriul angariSs aseTi saxe
SeiZleba hqondes:
qali cdis pirebis mier qali politikosebis kompetenturobis saSualo Sefaseba (M =
6.13) ufro maRali iyo, vidre mamakaci cdis pirebis saSualo Sefaseba (M = 5.42).
populaciis saSualo
populaciisTvis qulaTa
saSualo SerCevis saSualos
msgavsad gamoiTvleba, im
gansxvavebiT, rom N populaciaSi
qulaTa raodenobas aRniSnavs, saSualos mniSvneloba ki berZnuli µ-iTaa
(berZnuli anbania me-12 aso, warmoiTqmis, rogorc „miu“) warmodgenili da ara _
X -iT. amrigad, populaciis saSualo gamoisaxeba Semdegi formuliT:
populacia
saSualo populaciisN
X∑== µ
umravles SemTxvevaSi µ-s Sesaxeb daskvna X -dan gamomdinare keTdeba, radgan
SeuZlebelia populaciaSi yvela qulis gazomva. SerCevis saSualo, rogorc
populaciis saSualos Sefaseba, ufro detalurad me-7 TavSia ganxiluli.
SeamowmeT Tqveni codna IV.2
1. ganmarteT: _ M, µ, N, populaciis saSualo, ∑ , SerCevis saSualo,
kvadratebis jami, ∑ X , X .
2. davuSvaT, miiReT Semdegi qulebi: _ 97, 90, 86, 93, 76 da 86.
a) ipoveT maTi saSualo.
b) aCveneT, rom am qulebisTvis ( )∑ =− 0XX .
1 es meTodi gamoiyeneba laTinuri asoebiT (magaliTad, inglisurad an germanulad)
dawerili teqstis SemTxvevaSi; qarTulSi statistikuri simbolos xazgasma ar aris
saWiro (mTargmnelis SeniSvna).
populaciis saSualo (Population mean – µ) _
populaciaSi yvela SemTxvevis jamis fardoba
qulebis saerTo raodenobasTan.
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 12
g) am qulebisTvis gamoTvaleT ( )2∑ − XX .
3. davuSvaT, mkvlevarma gazoma me-7 klaseli gogonebis TviTSefaseba da
miiRo Semdegi qulebi: _ 49, 53, 67, 20, 27, 36, 49, 27, 61 da 28.
a) gamoTvaleT am qulebis saSualo.
b) davuSvaT, eqsperimentatorma kidev erTi me-7 klaseli gogonas
TviTSefaseba gazoma da miiRo 17 qula. CarTeT es qula zemoT
mocemul ganawilebaSi da ipoveT miRebuli ganawilebis saSualo.
g) davuSvaT, qula 67 Seicvala 27-iT da miRebuli qulebi gaxda 49, 53,
27, 20, 27, 36, 49, 27, 61 da 28. ra iqneba qulaTa am axali simravlis
saSualo?
d) SeadareT erTmaneTs am kiTxvis (a), (b) da (g) punqtebSi miRebuli
pasuxebi. saSualos romel Tvisebebs aCvenebs Tqvens mier miRebuli
pasuxebi?
centraluri tendenciis sazomebis Sedareba
romelia „saukeTeso“ saSualo qula? am kiTxvaze pasuxis gacema ar aris ioli.
centraluri tendenciis saukeTeso sazomi damokidebulia imaze, Tu risTvis aris
mowodebuli mocemuli statistika, anu risTvis viyenebT mas. centraluri
tendenciis sazomi gamoiyeneba rogorc aRsawerad, ise _ daskvnebis gasakeTeblad.
centraluri tendenciis sazomebis aRweriTi gamoyeneba
arCeva imisa, centraluri tendenciis romeli sazomi gamoviyenoT, rogorc
aRweriTi statistika, damokidebulia gazomvis skalaze, romelzec imyofeba
miRebuli qulebi da am qulebis sixSireTa ganawilebis formaze.
qulebis gazomvis skalebi
saxeldebis skala. saxeldebis skalaze pasuxebi ubralod nawildeba or
kategoriaSi: _ pasuxi aris, an ar aris. centraluri tendenciis erTaderTi sazomi,
romelic SeiZleba gamoviyenoT saxeldebis skalaze, aris moda anu ganvsazRvroT
kategoria, romelic yvelaze xSirad gvxdeba.
rigis skala. rigis skalaze qulebi lagdeba sididis mixedviT, magram ver
viqnebiT darwmunebulni, rom cvladis gverdigverd ganlagebuli mniSvnelobebi
skalaze toli intervalebiT aris daSorebuli erTmaneTisgan. Sesabamisad,
SegviZlia im qulis gansazRvra, romelic iqneba mediana. mediana qulaTa
ganawilebas or tol nawilad yofs. SesaZlebelia modaluri qulis moZebnac.
intervalebisa da Sefardebis skalebi. orive skalis SemTxvevaSi miRebulia, rom
am skalaze gverdigverd ganlagebul qulebs Soris intervali Seesabameba im
skalis intervalebs, romelzec gaizoma cvladi. amitom SesaZlebelia
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 13
intervalebisa Tu Sefardebis skalaze ganlagebul qulaTa saSualos gamoTvla.
am doneze myofi qulebisTvis medianisa da modis povnac SesaZlebelia.
qulebis ganawilebis forma
is, Tu ramdenad kargad axasiaTebs centraluri tendenciis sazomi ganawilebas,
ganawilebis formazec aris damokidebuli. suraTze N3.8 warmodgenilia qulaTa
ramdenime tipuri ganawileba. Tu ganawileba unimodaluria da simetriuli,
rogoricaa suraTze N3.8a mocemuli normaluri ganawileba, maSin moda, mediana da
saSualo erTi da igive ricxvi iqneba. TiToeuli statistika erTnairad kargad
axasiaTebs ganawilebas; Tumca, miRebuli ganawilebebi xSirad ar aris
unimodaluri da simetriuli; isini dadebiTad an uaryofiTad asimetriulia.
asimetriul ganawilebaSi modas, medianasa da saSualos mniSvnelobebi
gansxvavdeba erTmaneTisgan. vinaidan saSualo mgrZnobiarea asimetriuli
ganawilebis ukiduresi qulebis mimarT, is ufro metad gadainacvlebs ukiduresi
qulebis mimarTulebiT, vidre moda an mediana. amrigad, saSualosa da medianas
Soris sxvaobis niSani gviCvenebs ganawileba dadebiTad asimetriulia Tu
uaryofiTad. uaryofiTad asimetriul ganawilebaSi saSualo ufro naklebi iqneba,
vidre mediana. analogiurad, ramdenime ukiduresad maRali qula dadebiTad
asimetriul ganawilebaSi gazrdis saSualos mniSvnelobas da is medianaze meti
iqneba. suraTze N4.1 naCvenebia medianisa da saSualos fardobiTi adgilmdebareoba
dadebiTad da uaryofiTad asimetriul ganawilebebSi. rogorc suraTze xedavT,
Tu ganawileba sakmaod asimetriulia da didi sxvaobaa saSualosa da medianas
mniSvnelobebs Soris, mediana, rogorc wesi, centraluri tendenciis ukeTes
sazoms warmoadgens, vidre saSualo. amis magaliTia beisbolis ZiriTadi ligis
moTamaSeTa Semosavlebis ganawileba. erT-erT amerikul gundSi wliuri
Semosavalia $109.000-dan $5.155.250-mde (Cafardo, 1994). am ganawilebis saSualoa
$1.357.163, magram medianaa $500.000. ganawileba mkveTrad dadebiTad asimetriulia;
gundis 25 moTamaSidan mxolod 8 maTgani iRebs saSualoze mets. amrigad, am
gundis moTamaSeTa tipuri anazRaurebis saSualoTi daxasiaTeba araswor suraTs
gvaZlevs, radgan moTamaSeTa naxevari $500.000-s an naklebs iRebs (Fudin & Levinson,
1994).
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 14
yovelTvis, rodesac mniSvnelovania aRwera, umjobesia qulaTa ganawilebis
centraluri tendenciis samive sazomis warmodgena; Tumca, rogorc naxavT
bihevioristuli mecnierebis literaturaSi, centraluri tendenciis yvelaze
farTod gamoyenebad sazoms saSualo warmoadgens, xolo moda iSviaTad
gamoiyeneba.
SeiZleba, rom zogierT ganawilebas ar hqondes centraluri tendencia?
bimodaluri ganawileba ver aRiwereba saTanadod centraluri tendenciis
mxolod erTi sazomiT, gansakuTrebiT maSin, rodesac modaluri qulebi didi
manZiliT arian daSorebuli erTmaneTisgan. magaliTad, davuSvaT, rom mkvlevarma
miiRo gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebis iseTi ganawileba,
rogoric mocemulia cxrilSi N4.2. es ganawileba aSkarad bimodaluria; misi erTi
modaa 34, xolo meore _ 58. ganawilebis medianaa 45.5, saSualo ki _ 45.8. cxrilidan
Cans, rom arc erTma cdis pirma ar miiRo medianis an saSualos axlos mdebare
qula. am SerCevaSi arc mediana da arc saSualo ar warmoadgens tipuri pirovnebis
gamocdebTan dakavSirebul SfoTvas. moda 34 da moda 58 am SemTxvevaSi tipuri
qulebis saukeTeso aRweraa.
dadebiTad asimetriuli
mediana
saSualo
mediana
saSualo
suraTi N4.1. medianisa da saSualos fardobiTi adgilmdebareoba dadebiTad
da uaryofiTad asimetriul ganawilebaSi.
uaryofiTad asimetriuli
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 15
ganvixiloT kidev erTi magaliTi.
soflis meurneobis departamentis
bolo wlis monacemebis mixedviT,
aSS-Si yovelwliurad erT sul
mosaxleze modioda 1.33 funti
moxmarebuli saReWi Tambaqo.
Sesabamisia es aRwera TqvenTvis? Tu
Tqven ar ReWavT Tambaqos, es ricxvi
Zalze maRali iqneba, magram Tu Tqven
ReWavT Tambaqos, es sidide SeiZleba
Zalze mcirec ki iyos TqvenTvis.
saReWi Tambaqos gamoyenebis
ganawileba bimodaluria. populaciis
mcire nawili iyenebs saReW Tambaqos
da misi moxmareba mniSvnelovnad
aRemateba 1.33 funts weliwadSi.
populaciis didi nawili ar moixmars
saReW Tambaqos da maTi modaluri
gamoyeneba iqneba nuli funti
weliwadSi.
centraluri tendenciis sazomebis daskvniTi gamoyeneba
centraluri tendenciis sazomis arCevas, nawilobriv, ganapirobebs is, Tu
ramdenad gamoiyeneba mocemuli statistika SemdgomSi daskvniTi miznebisTvis.
pirvel TavSi vTqviT, rom populaciis parametrebi xSirad fasdeba SerCevisgan
miRebul statistikebze dayrdnobiT. SerCevis saSualo X populaciis saSualos
(romelic aRiniSneba berZnuli asoTi μ) saukeTeso Sefasebaa. aqedan gamomdinare,
SerCevis saSualo Sedis mniSvnelovan statistikur SefasebaTa ricxvSi. vinaidan
saSualo aseTi gamosadegia daskvnisTvis, bihevioristuli mecnierebis
warmomadgenlebi xSirad iyeneben mas monacemTa aRsawerad. SerCevis saSualos,
rogorc populaciis saSualos Sefasebis Tvisebebs ufro srulad me-7 TavSi
ganvixilavT.
cxrili N4.2. gamocdebTan dakavSirebuli
SfoTvis kiTxvaris 26 qulis bimodaluri
ganawileba.
qula Tvlis
Sedegi
ƒ
61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31
/ // ///// /// / / / /// ///// // / /
1 2 5 3 1 1 1 3 5 2 1 1
moda
8.45=X , Md = 45.5 moda
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 16
SeamowmeT Tqveni codna IV.3
1. centraluri tendenciis romel sazoms airCevT qvemoT aRweril
TiToeul SemTxvevaSi?
a) ipoveT mniSvneloba, romelic warmoadgens ganawilebis 50-e
procentils.
b) romeli sazomisTvis aris minimaluri ganawilebaSi Semaval
TiToeul qulasa da centraluri tendenciis sazoms Soris
sxvaobebis kvadratebis jami.
g) aRwereT qula, romelic yvelaze xSirad gxvdebaT ganawilebaSi.
d) mis gamosaTvlelad gamoiyeneT TiToeuli qulis ricxviTi
mniSvneloba.
2. ganawileba mkveTrad uaryofiTad asimetriulia. rogor fiqrobT,
ganawilebis saSualo iqneba medianaze naklebi, medianis toli Tu masze
meti? axseniT Tqveni pasuxi.
3. ganawilebis saSualo mniSvnelovnad aRemateba medianas. rogor fiqrobT,
ganawileba simetriulia, uaryofiTad asimetriulia Tu dadebiTad
asimetriuli? ratom?
Sejameba
• centraluri tendenciis sazomebis gamoyeneba SesaZlebelia aRsawerad
da daskvnebis gasakeTeblad.
• centraluri tendenciis sami yvelaze xSirad gamoyenebadi sazomia: _
moda, mediana da SerCevis saSualo.
• moda aris qula, romelic yvelaze xSirad gvxvdeba qulaTa
ganawilebaSi.
• mediana aris qula ganawilebaSi, romlis zemoT da qvemoT
ganlagebulia qulaTa toli raodenoba.
• SerCevis saSulo gamoiTvleba ∑ NX / SefardebiT.
• saSualo centraluri tendenciis yvelaze xSirad gamoyenebadi sazomia
daskvniTi miznebisTvis.
• TiToeuli statistikis saTanado gamoyeneba aRweriTi miznebisTvis
damokidebulia gazomvis skalaze da qulebis sixSireTa ganawilebis
formaze.
ZiriTadi terminebi da simboloebi
aRweriTi statistika
bimodaluri ganawileba
mediana (Mdn)
moda
multimodaluri
ganawileba
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 17
populaciis saSualo
saSualo
unimodaluri ganawileba
SerCevis saSualo
cvalebadobis sazomebi
centraluri tendenciis
sazomebi
M
μ (miu)
∑ (dajameba)
X
∑ X
( )2∑ − XX
kiTxvebi
1. rotonma da kelim (Rotton & Kelly, 1985) Seqmnes skala, romelsac mTvaris
efeqtis rwmenis skala uwodes. am skaliT xdeba imis Sefaseba, Tu
ramdenad swams adamians, rom savse mTvare gavlenas axdens qcevaze.
skalaze qulaTa diapazonia 9-dan 81-is CaTvliT. am skalaze individTa
erTi SerCevis saSualoa 37.9, xolo mediana _ 39. SeZlebT Tu ara am
informaciaze dayrdnobiT daaskvnaT qulaTa ganawilebis Sesaxeb: _
daaxloebiT rogoria is, simetriuli Tu asimetriuli? ratom?
2. aSS-is Census Bureau-is monacemebis mixedviT, 1992 wels mamrobiTi sqesis
advokatebisa da mosamarTleebis saSualo Semosavalma $94.757 Seadgina
maSin, roca maTi Semosavlis mediana $78,052 iyo. asimetriulia es
ganawileba? Tu fiqrobT, rom asimetriulia, maSin ra mimarTulebiT aris
asimetriuli? ratom?
3. eglim da meiersma (Egli & Meyers, 1984) video TamaSebis pavilionSi
gamokiTxes 151 mozardi. gamokiTxvis Sedegebma aCvena, rom individebis
mier kviris manZilze TamaSSi gatarebuli saSualo dro iyo 5.2 saaTi,
xolo mediana _ 2.5 saaTi kviraSi. asimetriulia Tu ara kviris manZilze
TamaSis saaTebis miRebuli ganawileba? Tu fiqrobT, rom asimetriulia,
maSin ra mimarTulebiT aris asimetriuli? ratom?
4. bihevioristul medicinaSi momuSave mkvlevrebs ainteresebdaT tkivilis
kontrolis fsiqologiuri meqanizmebi. cdis pirebSi tkivilis
usafrTxod gamosawvevad maT SeeZloT civi presoris amocanis gamoyeneba.
am amocanis erT-erTi forma gulisxmobs civ wyalSi xelis Cayofasa da
im drois dafiqsirebas, ramden xansac SeZlebs cdis piri xelis
gaCerebas wyalSi. davuSvaT, eqsperimetSi monawileobda 17 cdis piri,
romelTac xeli Cayofili hqondaT 40°F-ian (4°C) wyalSi da fiqsirdeboda
dro, ramden xansac aCerebda cdis piri xels wyalSi manam, sanam tkivili
gausaZlisi gaxdeboda. eqsperimentatorma cda oTxjer gaimeora da miiRo
qulaTa oTxi jgufi, romelic qvemoT aris mocemuli (qulebi
warmoadgens wamebiT gazomil dros):
5.
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 18
jgufi qulebi
1 35, 39, 38, 30, 37, 43, 30, 38, 41, 52, 25, 44, 33, 41, 38, 56, 37.
2 20, 28, 26, 29, 60, 21, 37, 30, 28, 57, 29, 25, 63, 26, 32, 30, 53.
3 45, 23, 57, 50, 47, 52, 20, 53, 50, 30, 52, 60, 52, 49, 52, 49, 53.
4 52, 23, 40, 60, 52, 21, 20, 52, 50, 23, 55, 26, 28, 53, 26, 49, 26.
qulaTa TiToeuli jgufisTvis:
a) aageT daujgufebel martiv sixSireTa ganawileba. gamoiyeneT es
sixSireTa ganawileba qulaTa ganawilebis formis Sesaxeb daskvnis
gasakeTeblad.
b) gamoTvaleT moda, mediana da saSualo.
g) SeadareT centraluri tendenciis sazomebi erTmaneTs. asimetriulia
ganawileba? axseniT, rogor mixvediT am daskvnamde. am Sedarebis
Sedegad miRebuli pasuxi Tanxvdeba Tu ara (a) punqtSi agebul
sixSireTa ganawilebas?
d) rogor fiqrobT, centraluri tendenciis romelime sazomi aRwers
qulebs ukeTesad? Tu ki, romeli sazomi da ratom?
5. Tqveni statistikis profesori apirebs, rom xuTi gamocdis safuZvelze
dagiweroT saboloo niSani. TiToeul gamocdaSi maqsimaluri qulaa 100.
man giTxraT, rom Tqveni niSani gamoiTvleba qulebis centralur
tendenciaze dayrdnobiT da rom Tqven unda airCioT, romel qulaze
dayrdnobiT gamoTvlili Sefaseba girCevniaT: _ modalur, medianur Tu
saSualo qulaze. davuSvaT, sami studentis niSnebia:
gamocdebi
1 2 3 4 5
roberto
dimitriosi
kareni
35
10
56
40
50
56
65
65
67
90
80
78
90
85
98
Tu Tqven erT-erTi am studentTagani xarT, centraluri tendenciis romel
sazoms airCevdiT Tqveni saboloo niSnis gansasazRvrad? axseniT
TiToeuli studentis SemTxvevaSi gakeTebuli arCevanis mizezi.
6. grosma (Gross, 1993) gamokiTxa studentebi did centralur evropul
universitetSi. gamokiTxva Seexeboda kolejis mdedrobiTi da mamrobiTi
sqesis studentebSi alkoholis moxmarebas. gamokiTxvis erT-erTi kiTxvaSi
respondentebs moeTxovebodaT mieTiTebinaT, Tu ramdenjer iRebdnen
alkohols kviraSi. grosma aRmoaCina, rom 21 wlis da ufrosi asakis
studentebSi mamakacebTan kviraSi dalevis SemTxvevaTa saSualo raodenoba
iyo 23.9, qalebTan _ 7.5.
თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 19
a) gaakritikeT dalevis qcevis centraluri tendenciis sazomad
saSualos gamoyeneba; kerZod, fiqrobT, rom dalevaTa
ganawileba asimetriulia? Tu ki, ra mimarTulebiT aris
asimetriuli?
b) rogor fiqrobT, centraluri tendenciis romeli sazomi iqneba
saukeTeso am SemTxvevaSi?
7. martivi reaqciis dro (dro, romelic sWirdeba reaqcias iseT martiv
gamRizianeblebze, rogoricaa sinaTle an toni) warmoadgens zogadi
anesTeziidan individis gamosvlis sazoms. davuSvaT, zomavT 8 qirurgiuli
pacientis martiv reaqciis dros, mcire qirurgiuli operaciisTvis zogadi
anesTeziis miRebidan 2 saaTis Semdeg da miiReT reaqciis Semdegi droebi
miliwamebSi: _ 782, 793, 793, 817, 832, 847, 866, 893. mecxre pacienti ar iyo
sakmarisad gamosuli anesTeziidan, rom moeca reaqciis dro, amitom am
cdis piris monacemi ganisazRvra, rogorc reaqciis usasrulo dro.
a) Tu gsurT cxrave pacientis monacemis erTad ganxilva,
centraluri tendenciis romel sazoms an sazomebs
gamoTvlidiT?
b) gamoTvaleT am sazomis ricxviTi mniSvneloba.
8. aSS-is Census Bureau-is monacemebis mixedviT, 1998 wels pirveli qorwinebis
asaki mamakacebisTvis 26.7 weli iyo. axseniT, ratom aris ukeTesi am
statistikis aRsawerad medianis gamoyeneba da ara _ saSualo
ariTmetikulis.
Tavi V. monacemTa aRwera: cvalebadobis sazomebi
cvalebadobis sazomebi
diapazoni
kvartilTSorisi gabnevis diapazoni da kvartilTSorisi
gabnevis diapazonis naxevari
cvalebadobis sazomebi SerCevis saSualos mimarT
dispersia
standartuli gadaxra
populaciis dispersiisa da standartuli gadaxris
Sefasebis gamosaTvleli formula
dispersiisa da standartuli gadaxris gamoTvla
kalkulatoris gamoyenebiT
simboloebisa da formulebis mimoxilva
dispersiisa da standartuli gadaxris interpretacia
rodis aris cvalebadoba didi
cvalebadobis sazomis warmodgena samecniero statiebSi
cvalebadobis sazomebis Sedareba
aRweriTi statistikis arCeva
gazomvis skala
sixSireTa ganawilebis forma
monacemebis Semdgomi analizi
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 2
cvalebadoba gulisxmobs imas, Tu ramdenad gansxvavdebian qulebi
erTmaneTisgan da ganawilebis centraluri tendenciis sazomisgan. rac ufro
metad gansxvavdebian qulebi erTmaneTisgan, miT metad gansxavebulni iqnebian isini
centraluri tendenciis sazomisganac da ufro meti cvalebadoba iqneba
ganawilebaSi; da, piriqiT, rac ufro naklebad gansxvavdebian qulebi
erTmaneTisgan ganawilebaSi, miT naklebad gansxvavebulni iqnebian isini
centraluri tendenciis sazomisgan da miT naklebi cvalebadoba iqneba
ganawilebaSi. amrigad, centraluri tendenciis sazomi individTa SerCevisgan
miRebuli qulebis Sesaxeb informaciis mxolod nawils gvatyobinebs. mxolod am
sazomze dayrdnobiT ver vityviT, qulebi
farTod arian gafantulni, Tu mWirdod
arian Sejgufebulni centraluri tendenciis
sazomis irgvliv. mxolod moda, mediana anda
saSualo ver gviCvenebs, Tu ramdenad didia
ganawilebaSi Semavali qulebis gafantuloba an cvalebadoba.
cvalebadobis cnebis sailustraciod, davuSvaT, rom mkvlevarma gazoma cxra
cdis pirisgan Semdgari sami SerCeva imis mixedviT, Tu ramden xans icinodnen
isini komiqsebis kiTxvisas. wamebiT gazomili qulebi mocemulia cxrilSi N5.1.
moda, mediana da saSualo TiToeuli jgufisTvis aris 41.0; magram cxadia, rom es
ganawilebebi Zalze gansxvavdeba erTmaneTisgan da centraluri tendenciis
sazomebi erTnairad kargad ver aRweren qulaTa
TiToeul jgufs. pirvel jgufSi TiToeuli qula
41-is tolia; amrigad, am jgufSi ar aris
cvalebadoba. aqedan gamomdinare, centraluri
tendenciis TiToeuli sazomi srulyofilad
warmoadgens ganawilebaSi Semaval qulebs; magram
am jgufSi miRebuli Sedegi Zalze iSviaTi iqneba,
Tu sicilis xangrZliobas realurad gavzomavT
cxra cdis pirTan. meore jgufis qulebi mcire
cvalebadobiT xasiaTdebian. qulebi Sejgufebulni
arian centraluri tendenciis sazomis irgvliv.
arc erTi qula gansxvavdeba 41 wamisgan 3 wamze
metiT. centraluri tendenciis TiToeuli sazomi
jgufSi Semavali yvela individis
reprezentaciulia, radgan mcire cvalebadobaa calkeul qulebs Soris.
gansxvavebuli situacia gvaqvs mesame jgufSi. aq ufro meti cvalebadobaa
sicilis xangrZliobis TvalsazrisiT. qulebi mniSvnelovnad gansxvavdeba
erTmaneTisgan da, meore jgufisgan gansxvavebiT, zogierTi qula 20 wamze metiTac
cvalebadoba (Variability): _ ramdenad
gansxvavdebian qulebi erTmaneTisgan
da ganawilebis centraluri
tendenciis sazomisgan.
cxrili N5.1. sicilis
xangrZliobis qulebi (wamebiT)
cxra cdis pirisgan Semdgari
sami jgufisTvis.
jgufi
1 2 3
moda Md
X
41 41 41 41 41 41 41 41 41 41 41 41
38 39 40 41 41 41 42 43 44 41 41 41
21 26 33 41 41 41 50 51 65 41 41 41
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 3
gansxvavdeba centraluri tendenciis (41 wm) sazomisgan. naTelia, rom centraluri
tendenciis verc erTi sazomi ver iqneba qulaTa am erTobliobis
reprezentaciuli, rogorc es xdeba pirveli da meore jgufis qulebis
SemTxvevaSi.
centraluri cendenciis sazomebis msgavsad, unda gavzomoT qulebis
cvalebadoba; amitom ganawilebaSi Semavali qulebis cvalebadobis aRweris
ramdenime sazomi arsebobs.
cvalebadobis sazomebi
cvalebadobis erT-erTi sazomi aRwers ganawilebaSi qulebis garkveuli
procentis Semcvel intervals. Cveulebisamebr, gamoiyeneba sami aseTi sazomi;
esenia: _ diapazoni, kvartilTSorisi gabnevis diapazoni da kvartilTSorisi
gabnevis diapazonis naxevari.
diapazoni
qulaTa erTobliobis cvalebadobis umartivesi
sazomia diapazoni, romelic Semdegnairad
gamoiTvleba:
diapazoni = Xudidesi _ Xumciresi
sadac Xudidesi warmoadgens ganawilebis maqsimalur qulas, xolo Xumciresi _
minimalur qulas. cxrilSi N5.1 mocemuli mesame jgufisTvis Xudidesi = 65 da
Xumciresi = 21; amrigad, am jgufis diapazonia 65_21 anu 44. meore jgufis qulebis
diapazonia 44_38 anu 6, xolo pirveli jgufis _ 41_41 anu 0. cxrilSi 3-3
mocemuli gamocdebTan dakavSirebuli SfoTvis kiTxvaris 100 qulis SemTxvevaSi
maqsimaluri qulaa 73, minimaluri _ 22, amitom am ganawilebis diapazoni iqneba
73_22 anu Sefasebis skalis 51 qula.
diapazonis Tvisebebi
diapazoni aRwers ganawilebis maqsimalur da minimalur qulebs. maqsimaluri da
minimaluri qulis nebismieri cvlileba gavlenas axdens diapazonze, maSinac ki,
Tu yvela sxva danarCeni qula ucvleli darCeba. amrigad, diapazoni cvalebadobis
SedarebiT arastabiluri sazomia. garda amisa, rac ufro didia SerCeva, miT metia
albaToba imisa, rom erTma eqstremalurma qulam gazardos diapazonis ricxviTi
mniSvneloba. diapazonis dadebiTi Tviseba isaa, rom martivad gamoiTvleba, magram
vinaidan aklia stabiluroba, xSirad ar gamoiyeneba cvalebadobis aRsawerad.
diapazoni (Range): _ sxvaoba
ganawilebis maqsimalur da
minimalur qulebs Soris.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 4
kvartilTSorisi gabnevis diapazoni da kvartilTSorisi gabnevis
diapazonis naxevari
kvartilTSorisi gabnevis diapazoni (IQR)
aRwers ganawilebis qulebis SuaTana 50%-s da
xSirad gamoiyeneba rigis skalaze. is
warmoadgens sxvaobas 75-e procentilis
Sesabamis qulasa (X75) da 25-e procentilis
(X25) Sesabamis qulas Soris. kvartilTSorisi
diapazoni gamoiTvleba tolobiT:
kvartilTSorisi gabnevis diapazoni =IQR =
X75 _ X25
X75-sa da X25-is mniSvnelobebi gamoiTvleba
me-3 TavSi mocemuli, ganawilebis konkretuli
procentilis gamosaTvleli formuliT.
kvartilTSorisi gabnevis diapazonis naxevari (SIQR) gamoiTvleba formuliT:
kvartilTSorisi gabnevis diapazonis naxevari = SIQR = IQR/2= (X75 _ X25)/2
IQR-isa da SIQR-is Tvisebebi
IQR-iT aRiwereba medianis irgvliv dajgufebuli individebis SuaTana 50%-is
tipur qulaTa diapazoni. vinaidan IQR-Si ar Sedis ganawilebis qulaTa zeda 25%
da qveda 25%, mis mniSvnelobaze gavlenas ar axdens asimetriuli ganawilebis
ukiduresi qulebi. naTelia, rom kvartilTSorisi gabnevis diapazoni ufro
naklebi iqneba, vidre qulaTa erTobliobis diapazoni. magaliTad, cxrilSi N3-3
mocemuli gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebisTvis X75 = 45 da
X75 = 37 (damrgvalebulia uaxloes mTel ricxvamde). amrigad, IQR iqneba
IQR = X75 _ X25 = 45 _ 37 = 8
maSin, rodesac am qulaTa diapazonia 51. miuxedavad imisa, rom cxrilSi N3-3
mocemuli qulebis sruli diapazoni didia, SedarebiT mcirea qulaTa Sua 50%-is
diapazoni.
SIQR-iT aRiwereba medianis zemoT da qvemoT ganlagebul qulaTa 25%-is
diapazoni; Tumca, mxolod simetriuli ganawilebis SemTxvevaSi aRwers SIQR
medianis zemoT da qvemoT ganlagebul qulaTa zustad 25%-is diapazons.
magaliTad, cxrilSi N3.3 mocemuli gamocdebTan dakavSirebuli SfoTvis
kiTxvaris qulebiTvis SIQR aris 8/2 anu 4. es SIQR gviCvenebs, rom ganawilebaSi
qulaTa 50% ar aris medianaze daaxloebiT 4 quliT meti da daaxloebiT 4 quliT
naklebi. yuradReba miaqcieT, rom am winadadebaSi gamoyenebulia sityva
„daaxloebiT“. vinaidan gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebis
kvartilTSorisi gabnevis diapazoni
(Interquartile range, IQR): _ ganawilebaSi
Semaval qulaTa SuaTana 50%.
X75 _ ganawilebis 75-e procentilis
Sesabamisi qula.
X25 _ ganawilebis 25-e procentilis
Sesabamisi qula.
kvartilTSorisi gabnvis diapazonis
naxevari (Semi-interquartile range, SIQR) _
kvartilTSorisi gablevis
diapazonis naxevari.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 5
sixSireTa ganawileba mcired asimetriulia, SIQR-Si ar Sedis medianis TiToeul
mxares moTavsebul qulaTa zustad 25%.
SeamowmeT Tqveni codna IV.1
1. ganmarteT: _ kvartilTSorisi gabnevis diapazoni, diapazoni,
kvartilTSorisi gabnevis diapazonis naxevari, cvalebadoba, X25, X75..
2. klasSi Catarebul testSi miRebuli maqsimaluri qulaa 98, xolo
minimaluri _ 54. miRebuli qulebis 75-e procentilia 92 qula, xolo 25-e
procentili _ 74. gamoTvaleT am qulaTa diapazoni, IQR da SIQR.
3. rTuli reaqciis drois testSi miRebuli yvelaze didi qulaa 929
miliwami, xolo yvelaze mcire _ 642 miliwami. qulaTa 75-e procentilia
809 miliwami, xolo 25-e procentili _ 724 miliwami.
a) gamoTvaleT am qulebis diapazoni, IQR da SIQR.
b) ra informacias iZleva qulebis Sesaxeb IQR?
g) ra informacias iZleva qulebis Sesaxeb SIQR?
4. axseniT, ratom aris diapazoni cvalebadobis SedarebiT aramdgradi
sazomi.
5. axseniT, ratom aris IQR cvalebadobis ufro mdgradi sazomi, vidre
diapazoni.
cvalebadobis sazomebi SerCevis saSualos mimarT
TiToeuli statistika, rogoricaa diapazoni, IQR da SIQR, migviTiTebs
ganawilebis centralur nawilSi ganlagebul qulaTa garkveuli procentis
Semcvel intervalze (magaliTad, diapazoni Seesabameba 100 procents), magram arc
erTi maTganis gamoTvlisas ar xdeba ganawilebaSi Semavali yvela qulis
gamoyeneba; Tumca, arsebobs cvalebadobis ori mniSvnelovani sazomi, romelic
iTvaliswinebs yvela qulis cvalebadobas saSualos mimarT; esenia: _ dispersia da
standartuli gadaxra.
me-4 TavSi gamovTvaleT ganawilebaSi Semavali TiToeuli qulis gansxvaveba anu
gadaxra ganawilebis saSualodan; Cven daviTvaleT XX − TiToeuli qulisTvis.
es gadaxra migviTiTebs imaze, Tu ramdenad gansxvavebulia qula saSualosgan. Tu
qula saSualos tolia, maSin XX − nulis toli iqneba; magram Tu qula ar aris
saSualos toli, maSin XX − miiRebs nulisgan gansxvavebul ricxviT
mniSvnelobas da rac ufro gansxvavebuli iqneba erTmaneTisgan X da X , miT meti
iqneba XX − sxvaobis mniSvneloba. Cven SegviZlia miviRoT cvalebadobis sazomi
imiT, rom SevkriboT ganawilebaSi Semavali yvela qulis gadaxra da miRebuli
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 6
jami gavyoT qulaTa raodenobaze. formulis saxiT es ase Caiwereba: ( )∑ − NXX / ;
Tumca, miRebuli sazomis SemTxvevaSi arsebobs erTi problema. rogorc naCvenebia
cxrilis N4.1 (b) svetSi, ganawilebis qulebis XX − gadaxraTa jami yovelTvis
nulis toli iqneba anu ( )∑ =− 0XX . am problemuri situaciis Tavidan
asacileblad, Sekrebamde da gayofamde Cven kvadratSi unda aviyvanoT TiToeuli
gadaxris mniSvneloba (anu ( )2XX − ). ( )∑ −2
XX gamosaxuleba mxolod maSin iqneba
nulis toli, Tu TiToeuli XX − sxvaobis mniSvneloba nulis toli iqneba. es
procedura gamoiyeneba dispersiisa da, Semdgom, standatruli gadaxris misaRebad.
Cven jer gavecnobiT dispersiisa da standartuli gadaxris formulebs da maT
gamoyenebas, Semdeg ki visaubrebT maT interpretaciaze.
dispersia
SerCevis dispersia
SerCevis dispersia (aRiniSneba S2)
ganisazRvreba, rogorc
( )N
XXS ∑ −
==2
2dispersia SerCevis
SerCevis dispersiis gamoTvlis es
formula migviTiTebs, rom:
1. TiToeul qulas vaklebT
saSualos [anu ( )XX − ];
2. TiToeuli miRebuli sxvaoba
agvyavs kvadratSi [anu ( )2XX − ];
3. SevkrebT kvadratSi ayvanil sxvaobebs [anu ( )∑ −2
XX ];
4. miRebuli jams vyofT N-ze, ganawilebaSi Semaval qulaTa ricxvze.
populaciis dispersia
Tu N populaciis tolia da populaciis saSualo (µ) cnobilia, maSin
formulaSi X-is nacvlad µ-is CasmiT populaciis dispersias miviRebT. populaciis
dispersia σ2-iT (berZnuli aso sigma, kvadratSi) aRiniSneba da miuTiTebs, rom es
dispersia populaciis parametria da ara _ SerCevidan miRebuli statistikuri
sidide. amrigad, populaciis dispersia gamoisaxeba formuliT:
( )populacia
dispersia populaciisN
X∑ −==
22 µ
σ
SerCevis dispersia (Sample variance, S2): _
cvalebadobis sazomi, romelic miiReba
ganawilebaSi Semavali ToToeuli qulisa
da am ganawilebis saSualos sxvaobebis
kvadratSi ayvaniT, miRebuli sidideebis
dajamebiTa da miRebuli jamis gayofiT
ganawilebaSi Semavali qulebis
raodenobaze.
populaciis dispersia (Population variance,
σ2) _ populaciis yvela qulis gazomviT
miRebuli dispersia.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 7
sadac X _ populaciis calkeuli wevris qula;
µ _ populaciis qulaTa saSualo;
Npopulacia _ populaciaSi qulaTa saerTo raodenoba.
populaciis dispersiis Sefaseba
zemoT mocemuli formula SerCevis
dispersias aRwers; Tumca, rogorc pirvel
TavSi aRvniSneT, mkvlevars xSirad
sWirdeba SerCevidan miRebul dispersiaze
dayrdnobiT populaciis dispersiis
Sefaseba. Sesabamisad, S2-iT gamoTvlili
SerCevis dispersia srulad ver afasebs
populaciis dispersiis realur sidides
da iZleva σ2-is (berZnuli aso sigma,
ayvanili kvadratSi) Canacvlebis mqone Sefasebas. es Canacvleba SeiZleba
gaswordes imiT, rom formulis mniSvnelSi N-is nacvlad gamoviyenoT N_1.
amrigad, populaciis dispersiis Sefaseba aRiniSneba s2-iT da ganisazRvreba,
rogorc
( )1
s2
2
−
−== ∑
NXX
Sefaseba dispersiis populaciis .
rogorc me-4 TavSi aRvniSneT, dispersiis formulis mricxvelSi mocemul
kvadratSi ayvanil sxvaobebis jams ( )∑ −2
XX kvadratebis jami ewodeba da
aRiniSneba simboloTi SS. amrigad, populaciis dispersiis Sefaseba, aseve,
SeiZleba ganisazRvros, rogorc
12
−==
NSSs Sefaseba dispersiis populaciis
populaciis dispersiis Sefaseba _ s2 xSirad, mravali miznisTvis ufro
gamosadegia, vidre SerCevis dispersia _ S2 da ufro xSirad gamoiyeneba, vidre S2.
amrigad, am TvalsazrisiT, rodesac dispersiaze vsaubrobT, yovelTvis
igulisxmeba s2, garda im SemTxvevebisa, rodesac konkretulad vuTiTebT, rom
viyenebT S2-s. s2-isa da S2-is Tvisebebze, rogorc σ2-is Sefasebaze, ufro srulad me-
7 TavSi visaubrebT.
standartuli gadaxra
SerCevis standartuli gadaxra
qulaTa SerCevis standartuli gadaxra,
romelsac SerCevis standartul gadaxras
populaciis dispersiis Sefaseba (Estimated
population variance, s2): _ cvalebadobis
sazomi, romelic miiReba ganawilebaSi
Semavali ToToeuli qulisa da am
ganawilebis saSualos sxvaobebis
kvadratSi ayvaniT, miRebuli sidideebis
dajamebiTa da miRebuli jamis gayofiT
ganawilebaSi Semavali qulebis
raodenobaze erTiT nakleb ricxvze.
SerCevis standartuli gadaxra (Sample
standard deviation, S): _ kvadratuli fesvi
SerCevis dispersiidan.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 8
uwodeben, S-iT aRiniSneba da warmoadgens kvadratul fesvs SerCevis dispersiidan
anu
( )N
XXSS ∑ −
===2
2 gadaxra istandartul SerCevis
populaciis standartuli gadaxra
Tu N populaciis individTa
raodenobas gamoxatavs da populaciis
saSualo µ cnobilia, maSin populaciis
standartuli gadaxra, romelic σ-iT
aRiniSneba, gamoisaxeba formuliT
populaciaNX∑ −
===2
2 )( µσσgadaxra istandartul populaciis
Tumca, vinaidan, Cveulebisamebr, µ ar aris xolme cnobili, σ-is mniSvneloba
SerCevidan gamomdinare fasdeba.
populaciis standartuli gadaxris Sefaseba
populaciis standartuli gadaxris Sefaseba _ s, aris kvadratuli fesvi
populaciis dispersiis Sefasebidan, anu
.1
)( 22
−
−=== ∑
NXX
ssSefaseba gadaxris istandartul populaciis
kvadratebis jamis gamosaxulebis formulaSi Setanis SemTxvevaSi gveqneba
1−=
NSSs .
populaciis standartuli gadaxris Sefasebis gamosaTvlelad:
1. TiToeul qulas vaklebT saSualos [anu ( )XX − ];
2. TiToeuli miRebuli sxvaoba agvyavs kvadratSi [anu ( )2XX − ];
3. SevkrebT kvadratSi ayvanil sxvaobebs [anu ( )∑ −2
XX ];
4. miRebuli jams vyofT N_1-ze, ganawilebaSi Semaval qulaTa
raodenobaze erTiT nakleb ricxvze.
5. miRebuli Sefardebidan viRebT kvadratul fesvs.
am SemTxvevaSic, populaciis standartuli gadaxris Sefaseba mravali
miznisTvis ufro gamosadegia, vidre SerCevis standartuli gadaxra da rodesac
viyenebT termins standartuli gadaxra, vgulisxmobT s-s.
populaciis standartuli gadaxra (σ) _ kvadratuli fesvi populaciis dispersiidan.
populaciis standartuli gadaxris Sefaseba
(Estimated populaation standard deviation) _
kvadratuli fesvi populaciis dispersiis
Sefasebidan.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 9
magaliTi N5-1
populaciis dispersiisa da standartuli gadaxris Sefasebis gamoTvla
ganmartebiTi formuliT
amocana
ipoveT N5.1 cxrilidan meore jgufSi gaerTianebuli cxra cdis piris mier
miRebuli sicilis xangrZliobis qulebze dayrdnobiT populaciis dispersiisa da
standartuli gadaxris Sefaseba; gamoiyeneT s2-is ganmartebuli formula.
saWiro statistika: ( ) ( )∑ −−= 1/22 NXXs .
amoxsna
qvemoT mocemul cxrilSi warmodgenilia s2-isa da s-is gamoTvlisTvis
aucilebeli nabijebi. Tavdapirvelad viTvliT s2-s. formulaSi Casasmelad
gvWirdeba kvadratebis jamis, SS-is anu ( )∑ −2
XX gamoTvla. cxra qula mocemulia
(a) svetSi. am qulebis saSualoa 41 da misi mniSvneloba mocemulia (b) svetSi.
pirveli nabiji: _ TiToeul qulas gamovakloT saSualo, rogorc es naCvenebia (g)
svetSi. meore nabiji: _ TiToeul qulasa da saSualos Soris sxvaoba aviyvanoT
kvadratSi, rogorc es naCvenebia (d) svetSi. kvadratSi ayvanili sxvaobebi
Seikribeba da miRebuli jami, romelic udris 28-s, mocemulia (d) svetis bolos.
es jami aris ( )∑ −2
XX anu SS, romelic warmoadgens s2-is formulis mricxvels.
bolo nabiji: _ SS gavyoT (N_1)-ze. vinaidan sul gamoviyeneT cxra qula anu N=9,
amitom N_1=8. Sesabamisad, s2=28/8=3.50. yuradReba miaqcieT, rom vinaidan TiToeuli
qulis gadaxra saSualodan kvadratSia ayvanili, s2 yovelTvis iqneba nuli an
dadebiTi ricxvi. uaryofiTi s2 gviCvenebs, rom gamoTvlaSi Secdoma davuSviT.
populaciis standartuli gadaxris Sefaseba s
warmoadgens s2-idan kvadratul fesvs. amrigad,
87.150.3 ==s wams. yuradReba miaqcieT, rom s
izomeba igive erTeulebiT anu am SemTxvevaSi
wamebiT, romliTac nedli qulebi. mniSvnelovania,
davimaxsovroT, rom s-is gamoTvlisas s2-idan unda
amoviRoT kvadratuli fesvi, radgan s-is
gamoTvlisas gavrcelebuli daSvebuli Secdomaa s2-
idan kvadratuli fesvis aramoReba.
gamoTvlis safexurebis Sejameba
1. ipoveT (a) svetSi mocemuli qulebis saSualo da Semdeg SeitaneT
miRebuli mniSvneloba (b) svetSi, rogorc naCvenebia cxrilSi.
(a)
X
(b)
X
(g)
( )XX −
(d)
( )2XX −
38 39 40 41 41 41 42 43 44
41 41 41 41 41 41 41 41 41
-3 -2 -1 0 0 0 +1 +2 +3
9 4 1 0 0 0 1 4 9
jami 0 28
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 10
2. (b) svetSi mocemuli saSualo gamoakeliT (a) svetSi mocemul TiToeul
qulas. miRebuli sxvaoba mocemulia (g) svetSi.
3. (g) svetSi mocemuli TiToeuli sxvaoba aiyvaneT kvadratSi. kvadratSi
ayvanis Sedegebi mocemulia (d) svetSi.
4. SekribeT (d) svetSi mocemuli sxvaobebis kvadratebi.
5. (d) svetis jami gayaviT (N_1)-ze. miRebuli sidide aris s2.
6. s-is misaRebad amoiReT kvadratuli fesvi s2-dan.
populaciis dispersiisa da standartuli gadaxris Sefasebis
gamosaTvleli formula
dispersiis ganmartebiTi formula saukeTesod gamoxatavs, Tu ras warmoadgens
cnebis doneze s2. formula martivia gamosayeneblad, rodesac mxolod ramdenime
qulaa saWiro; amasTan, qulebi da maTi saSualo mTeli ricxvebia. rodesac qulebi
an maTi saSualo aTwiladebiT gamoixateba an rodesac gvWirdeba didi
raodenobiT qulebi, gamoTvla ufro mouxerxebeli xdeba. amitom xSirad s2-s
gamoTvla ufro martivia formuliT, romelic algebrulad tolfasia
ganmartebiTi formulisa. s2-is gamosaTvleli formulaa
( )[ ]1
/222
−
−= ∑∑
NNXX
s
populaciis dispersiis Sefaseba miiReba s2-idan kvadratuli fesvis amoRebiT,
anu
( )[ ]1
/222
−
−== ∑ ∑
NNXX
ss
magaliTi N5-2
populaciis dispersiis Sefasebisa da standartuli gadaxris gamoTvla
gamosaTvleli formuliT
amocana
ipoveT cxrilidan N5.1 meore jgufis cxra cdis piris mier miRebuli sicilis
xangrZliobis qulebze dayrdnobiT populaciis dispersiisa da standartuli
gadaxris Sefaseba; gamoiyeneT s2-is gamosaTvleli formula.
(a)
X
(b)
2X
38 39
1,444 1,521
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 11
saWiro statistika: ( )[ ]
1/22
2
−
−= ∑∑
NNXX
s
amoxsna
qvemoT mocemul cxrilSi warmodgenilia s2-is gamoTvlis
nabijebi. cxra qula mocemulia (a) svetSi. formulaSi
Casasmelad am qulebidan gvWirdeba ori sididis miReba; esaa yvela qulis jami
(anu ∑ X ) da TiToeuli qulis kvadratebis jami (anu ∑ 2X ). qulaTa jami ∑ X
aris 369 da mocemulia (a) svetis bolos. TiToeuli qulis kvadratebi mocemulia
(b) svetSi (magaliTad, (38)2 = 1,444). kvadratSi ayvanili qulebis jami ∑ 2X aris
15,157 da mocemulia (b) svetis bolos. sul gamoyenebulia cxra qula anu N = 9,
xolo N _1 = 8.
es mniSvnelobebi CavsvaT formulaSi. miviRebT:
( )[ ]( )
50.3828
8129,15157,15
89/161,136157,15
199/369157,15 2
2
==
−=
−=
−−
=s
s2-is am formulis gamoyenebiT igive 3.50 miviReT, rac _ N5-1 magaliTSi s2-is
ganmartebiTi formuliT; da isev, 87.150.32 === ss wami.
dispersiisa da standartuli gadaxris gamoTvla kalkulatoris
gamoyenebiT
mas Semdeg, rac gaecaniT dispersiasa da standartul gadaxras, mosalodnelia,
rom mcire moculobis monacemebisTvis kalkulatoris gamoyenebiT mogiwioT am
statistikebis gamoTvla. kalkulatorTa umravlesobas aqvs rogorc X -is, ise _
dispersiisa da standartuli gadaxris gamosaTvleli programebi, Tumca, maTi
gamoyenebis meTodebi gansxvavdeba sxvadasxva kalkulatorze. aqedan gamomdinare,
Tqveni sakuTari kalkulatoris instruqciis wakiTxva mogiwevT. bevri
kalkulatori ∑ X -is, ∑ 2X -isa da N-is gamoTvlis saSualebasac iZleva, Tumca,
maT aRsaniSnad gamoyenebuli simboloebi SeiZleba gansxvavdebodes winamdebare
teqstSi gamoyenebulisgan. populaciis standartuli gadaxris Sefaseba xSirad
(Tumca, ara yovelTvis) s-iT aRiniSneba. es simboloa gamoyenebuli am teqstSic.
rac Seexeba SerCevis standartul gadaxras, mas xSirad σ-iT an SD-iT aRniSnaven.
40 41 41 41 42 43 44
1,600 1,681 1,681 1,681 1,764 1,849 1,936
jami 369 15,157
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 12
cxadia, mniSvnelovania imis codna, Tu romeli statistikuri sidide rogor aris
aRiniSnuli Tqvens kalkulatorze.
simboloebisa da formulebis mimoxilva
winamdebare nawilSi warmogidgineT mTel rig simboloebsa da formulebs.
qvemoT gTavazobT maT mimoxilvas.
S2 SerCevis dispersia. S2 aris mxolod aRweriTi statistika SerCevisTvis.
S SerCevis standartuli gadaxra. S aris mxolod aRweriTi statistika
SerCevisTvis.
s2 populaciis dispersiis Sefaseba. es statistika gamoiyeneba populaciis
dispersiis Sesafaseblad.
s populaciis standartuli gadaxris Sefaseba. es statistika gamoiyeneba
populaciis standartuli gadaxris Sesafaseblad.
σ2 populaciis dispersia. es parametri ufro srulad aris ganxiluli me-6
TavSi.
σ populaciis standartuli gadaxra, ufro srulad aris ganxiluli me-6
TavSi.
populaciis dispersiis Sefaseba s2 da standartuli gadaxra s is statistikebia,
romlebic, Cveulebisamebr, SerCevis qulebiT gamoiTvleba. cxrilSi N5.2
warmodgenilia ganmartebiTi, gamoTvliTi kvadratebis jamis formulebi s2-isa da s
isTvis. qulaTa erTi da imave erTobliobis mimarT gamoyenebisas TiToeuli
formuliT statistikis erTi da igive ricxviTi mniSvneloba miiReba.
cxrili N5.2. populaciis dispersiisa da standartuli gadaxris Sefasebis ganmartebiTi,
gamoTvliTi da kvadratebis jamis formulebi.
formulebi
ganmarteba gamosaTvleli kvadratebis jami
dispersia, s2 ( )1
2
−
−∑N
XX
( )[ ]1
/22
−
−∑ ∑N
NXX 1−N
SS
strandartuli
gadaxra, 2ss =
( )1
2
−
−∑N
XX
( )[ ]1
/22
−
−∑ ∑N
NXX 1−N
SS
dispersiisa da standartuli gadaxris interpretacia
kvlevis angariSebSi standartuli gadaxra ufro xSirad aris xolme
warmodgenili rogorc aRweriTi statistika, vidre dispersia. amis erT-erTi
mizezi isaa, rom vinaidan standartuli gadaxra warmoadgens kvadratul fesvs
dispersiidan, is gvaZlevs qulaTa cvalebadobis sazoms maT sawyis erTeulebSi.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 13
N5-1 da N5-2 magaliTebSi s2 aris 3.50 wami kvadratSi (wm2); am mniSvnelobis
interpretireba ar aris ioli; Tumca, s-is mniSvnelobaa 1.87 wami; aq gvaqvs igive
erTeuli (anu wami), romliTac gavzomeT nedli qulebi. amrigad, am qveTavSi s-is
interpretaciaze gveqneba saubari. dispersia gamoiyeneba statistikuri hipoTezebis
Semowmebis meTodebSi da ufro srulad ganixileba me-10-12 TavebSi.
standartuli gadaxra zomavs, saSualod ramdenad gansxvavdebian ganawilebaSi
Semavali qulebi saSualosgan. Tu yvela ganawilebaSi Semavali qula saSualos
tolia (aseTi ganawileba mocemulia cxrilSi N5-1, pirveli jgufi), maSin s nulis
toli iqneba. amrigad, saSualo sidide, romliTac pirveli jgufis qulebi
gansxvavdebian saSualosgan, aris nuli. am SemTxvevaSi, Tu ganawilebaSi nebismieri
qulis Sesaxeb winaswarmetyvelebisTvis gamoviyenebT X -s, Cvens
winaswarmetyvelebaSi Secdoma nulis toli iqneba. Tu viciT, rom s-is mniSvneloba
nulis tolia, maSin Cven isic viciT, rom ganawilebaSi Semavali yvela qula
SerCevis saSualos tolia da saSualo srulyofilad winaswarmetyvelebs yvela
qulis Sesaxeb.
meore mxriv, Tu ganawilebaSi, sul mcire, erTi qula mainc ar aris saSualos
toli, maSin s miiRebs nulze met mniSvnelobas da misi mniSvneloba miT ufro
meti iqneba, rac ufro gaizrdeba qulaTa cvalebadoba. s-is mateba, romelic Tan
sdevs ganawilebaSi Semavali qulebis cvalebadobis matebas, SegviZlia vnaxoT
cxrilSi N5-1 warmodgenili meore da mesame jgufebis monacemebis Sedarebisas.
yuradReba miaqcieT, rom mesame jgufis qulebi ufro cvalebadia saSualos (41
qula) mimarT, vidre meore jgufis qulebi. meore jgufis qulebisTvis s = 1.87 wams
maSin, roca mesame jgufis s = 13.44 wams. amrigad, meore jgufisTvis saSualo
sidide, romliTac qulebi gansxvavdebian X -isgan, aris 1.87 wami, xolo mesame
jgufisTvis _ 13.44 wami. Tu 41=X wams gamoviyenebT meore jgufSi Semavali
nebismieri qulis Sesaxeb winaswarmetyvelebisTvis, Cveni winaswarmetyvelebis
Secdoma daaxloebiT iqneba 1.87 wami, xolo mesame jgufisTvis aseTi
winaswarmetyvelebis Secdoma daaxloebiT 13.44 wams Seadgens.
s-is meore interpretacia gulisxobs qulebis aRweras imis mixedviT, Tu ramdeni
standartuli gadaxriT arian isini daSorebulni ganawilebis saSualosgan. bevr
ganawilebaSi qulaTa umravlesoba saSualosgan erTi standartuli gadaxris
farglebSia daSorebuli. magaliTad, cxrilSi N3-3 mocemuli gamocdebTan
dakavSirebuli SfoTvis kiTxvaris 100 qulis saSualoa 41.64, xolo am ganawilebis
s _ 8.66. X -is erTi standartuli gadaxris farglebSi moTavsebuli qulebia 32.98-
dan 50.30-is CaTvliT. es aris mniSvnelobebi, romlebic SemosazRvrulia, erTi
mxriv, erTi standartuli gadaxriT saSualos qvemoT (anu
98.3266.864.411 =−=− sX ) da, meore mxriv, erTi standartuli gadaxriT saSualos
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 14
zemoT (anu 30.5066.864.411 =+=+ sX ). cxrilSi N3-3 mocemuli qulebis daTvliT
vnaxavT, rom am intervalSi 100-dan 79 qula xvdeba. Tu ganawilebas normaluri
forma aqvs (rogorc naCvenebia suraTze N3-7a), maSin standartuli gadaxris codna
ufro met informacias gvaZlevs ganawilebaSi Semavali qulebis Sesaxeb. s-is es
interpretacia ufro srulad aris ganxiluli me-6 TavSi.
garda amisa, yuradReba miaqcieT, rom qulebi SeiZleba aRiweros imaze
miTiTebiTac, Tu saSualodan ramdeni standartuli gadaxriT zemoT an qvemoT
mdebareoben isini. magaliTad, Tu qulaTa ganawilebis saSualoa 80 da
standartuli gadaxraa 6, maSin qula 86 mdebareobs saSualodan erTi
standartuli gadaxriT zemoT, xolo qula 74 _ erTi standartuli gadaxriT
qvemoT. standartuli gadaxris es gamoyeneba ufro srulad aris axsnili me-6
TavSi.
rodis aris cvalebadoba didi
ar arsebobs martivi wesi imis gansasazRvrad, Tu ra SemTxvevaSi gamoxatavs
cvalebadobis nebismieri sazomis mniSvneloba qulaTa ganawilebis did Tu mcire
cvalebadobas. cvalebadobis sidide fardobiTia cvladis SesaZlo mniSvnelobebis
diapazonisa da qulaTa saSualos mimarT. magaliTad, senderi (Sender, 1958)
miuTiTebs, rom 100 metriani eleqtrogadamcemi mavTulisTvis 7.5 sm ar iqneba
uCveulo standartuli gadaxra, radgan misi saSualo sigrZe sakmaod didia;
Tumca, Tu Tqven adamianebis cxvirebis sigrZes zomavT, 7.5 sm-iani standartuli
gadaxra sagangaSo iqneba, vinaidan adamianis cxviris sigrZis SemTxvevaSi
standartuli gadaxra 7.5 sm Zalian didia.
cvalebadobis warmodgena samecniero statiebSi
populaciis standartuli gadaxris Sefaseba s cvalebadobis is sazomia,
romelic xSirad aris warmodgenili samecniero statiebSi ganxilul kvlevebSi.
JurnalebSi, sadac daculia amerikis fsiqologiuri asociaciis publikaciebis
saxelmZRvaneloSi (Publication Manual of the American Psychological Association. American
Psychological Association, 1994) mocemuli saredaqcio stili, standartuli gadaxris
aRsaniSnad iyeneben simbolos SD. amrigad, JurnalSi gamoqveynebul samecniero
statiaSi SegxvdebaT Semdegnairad formulirebuli nawyveti:
agresiuli qmrebis saSualo asaki iyo 34.3 weli (SD = 6.8).
sabeWd manqanaze akrefil an xelnawer teqstSi igive fraza SegxvdebaT aseTi
formiT:
agresiuli qmrebis saSualo asaki iyo 34.3 weli (SD = 6.8).
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 15
SeamowmeT Tqveni codna V.2
1. ganmarteT: _ populaciis standartuli gadaxris Sefaseba, populaciis
dispersiis Sefaseba, SerCevis dispersia, kvadratebis jami, s, s2, SS, σ, σ2.
2. homenma, topingma da holma (Homan, Toppung & Hall, 1986) ikvlies gansxvaveba
studentebis testur qulebs Soris, romelTagan erTi jgufi ismenda
maswavleblis mier wakiTxul tests, meore ki _ Tavad kiTxulobda.
davuSvaT, Tqvenc am problemiT xarT dainteresebuli da ikvlevT
studentTa or jgufs. erTi jgufi TviTon kiTxulobs tests, meores ki
Tqven ukiTxavT xmamaRla. miiReT testis qulaTa cxrilSi mocemuli ori
jgufi (minimaluri qulaa 0 da maqsimaluri _ 100).
a) TiToeuli jgufisTvis ipoveT X , s2 da s, ganmartebiTi formulis
gamoyenebiT, romelic mocemulia magaliTSi N5-1.
b) TiToeuli jgufisTvis ipoveT X , s2 da s, gamoTvliTi formulis
gamoyenebiT, romelic mocemulia magaliTSi N5-2.
g) s2-isa da s-is miRebuli mniSvnelobebis mixedviT gansazRvreT, romel
jgufSi Sedis metad cvalebadi qulebi.
d) Tu meore jgufis (romelsac maswavlebeli ukiTxavda) tipuri
qulis winaswarmetyvelebisTvis X -s gamoiyenebT, ra iqneba am
winaswarmetyvelebis saSualo Secdoma?
3. mravali fsiqologi dainteresebulia pirovnuli sivrcis sakiTxiT anu
mosazrebiT imis Taobaze, rom TiToeul adamians uxilavi sazRvrebi aqvs,
romelic ar surs, rom daurRvion ucxo adamianebma. am fenomenis erT-erT
kvlevaSi pirovnebis piradi sivrcis darRveva xdeboda sazogadoebrivi
TavSeyris iseT adgilebSi, rogoricaa, magaliTad, kolejis biblioTeka
an kafe, imiT, rom eqsperimentatori jdeboda pirovnebis gverdiT
Tavisufal skamze. davuSvaT, mamakaci eqsperimentatori arRvevda kolejis
asakis 10 gogonasa da 10 vaJis pirovnul sivrces, rodesac isini
biblioTekaSi isxdnen. fiqsirdeboda drois xangrZlivoba (wuTebiT),
jgufi
kiTxulobdnen
TviTon
ukiTxavdaT
maswavlebeli
60 50 64 53 61 65 60
51 74 64 63 52 45 50
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 16
romlis manZilzec pirovneba Cerdeboda eqsperimentatoris gverdiT;
miRebul iqna Semdegi Sedegebi:
pirovnebis sqesi
mamrobiTi mdedrobiTi
10.6 19.8 4.3 21.6 15.7 8.1 26.4 17.9
8.6 4.7 9.1 6.3 2.4 7.0 7.8 10.5
a) TiToeuli jgufisTvis ipoveT X , s2 da s, gamoTvliTi formulis
gamoyenebiT, romelic mocemulia magaliTSi N5-2.
b) s2-isa da s-is miRebuli mniSvnelobebis mixedviT gansazRvreT,
romel jgufSi Sedis metad cvalebadi qulebi.
g) ra iqneba saSualo Secdoma, Tu vityviT, rom mamrobiTi sqesis cdis
pirebi pirovnuli sivrcis darRvevis Semdeg kidev 15.55 wuTs
Cerdebian TavianT adgilebze?
d) Tu cdis piris tipuri qcevis aRsawerad gamoiyenebT jgufis
saSualos, romeli jgufisTvis (vaJebis Tu gogonebis) ufro zusti
iqneba winaswarmetyveleba? axseniT Tqveni pasuxi.
cvalebadobis sazomebis Sedareba
cvalebadobis sazomebis Sedarebis gamartivebis mizniT cxrilSi N5.3
warmodgenilia cvalebadobis sazomebi cxrilSi N3.3 mocemuli gamocdebTan
dakavSirebuli SfoTvis kiTxvaris qulebisTvis. naTelia, rom miRebuli sidideebi
mniSvnelovnad gansxvavdeba erTmaneTisgan. vinaidan TiToeuli sazomi qulebis
Sesaxeb gansxvavebuli informaciis matarebelia, sxvadasxva sazomis ricxviT
mniSvnelobebs pirdapir ver SevadarebT erTmaneTs. diapazoni, IQR da SIQR asaxaven
cdis pirTa qulebis garkveuli procentis Semcvel intervals; 100% _ diapazoni,
SuaTana 50% _ kvartilTSorisi gabnevis diapazoni da 25% medianis TiToeul
cxrili N5.3. cvalebadobis sazomebis Sedareba gamocdebTan
dakavSirebuli SfoTvis kiTxvaris 100 qulisTvis (ix. cxrili 3-3).
cvalebadobis sazomebi mniSvneloba
diapazoni
kvartilTSorisi gabnevis diapazoni (IQR)
semikvartilTSorisi gabnevis diapazoni (SIQR)
populaciis standartuli gadaxris Sefaseba (s)
populaciis dispersiis Sefaseba (s2)
51
8
4
8.66
74.92
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 17
mxares _ semikvartilTSorisi gabnevis diapazoni. dispresia da standartuli
gadaxra warmoadgens imis maCvenebels, saSualod ramdenad gadaxrilia qulebi
maTi saSualodan; Tumca, cvalebadobis yvela sazomis SemTxvevaSi, rac ufro
mcirea sazomis mniSvneloba, miT naklebad iqneba qulebi gafantuli. Tu yvela
individs erTi da igive qula aqvs da saerTod ar aris cvalebadoba, TiToeuli
sazomis mniSvneloba nulis toli iqneba.
aRweriTi statistikebis arCeva
Cven ganvixileT mTeli rigi statistikebisa, romelic saSualo qulisa da
SerCevaSi miRebuli qulebis cvalebadobis aRsawerad gamoiyeneba; Tumca,
mkvlevrebi miRebuli monacemebis Sesajameblad irCeven centraluri tendenciisa
da cvalebadobis mxolod TiTo-TiTo sazoms. konkretuli aRweriTi statistikis
arCeva, ZiriTadad, damokidebulia sam pirobaze: _ (1) gazomvis skalaze, romelzec
mocemulia qulebi, (2) qulebis sixSireTa ganawilebis formaze da (3) Semdgom
statistikur analizSi rogor gamoiyeneba mocemuli aRweriTi statistika.
cxrilSi N5.4 mocemulia rekomendebuli sazomebi konkretuli gazomvis skalisa
da ganawilebis formisaTvis.
cxrili N5.4. rekonendebuli sazomebi mocemuli gazomvis skalisa da ganawilebis
formisaTvis.
sixSireTa ganawilebis forma
daaxloebiT simetriuli,
unimodaluri asimetriuli, unimodaluri
gazomvis skala
centraluri
tendenciis
sazomi
cvalebadobis
sazomi
centraluri
tendenciis
sazomi
cvalebadobis
sazomi
rigis
intervalebis an
Sefardebis
Mdn
X
SIQR
s
Mdn
Mdn
SIQR
SIQR
gazomvis skala
saxeldebis skala
saxeldebis skalaze pasuxebi dalagebulia kategoriebSi da kategoriis
farglebSi yvela pasuxi gaigivebuli erTmaneTTan. Sesabamisad, Tvisebrivi
monacemebi, Cveulebriv, mxolod sixSireebis (anu jgufidan ramdeni individi
Sedis erT, an meore kategoriaSi) an procentebis terminebiT aRiwereba. amitom
saxeldebis skala ar aris mocemuli cxrilSi N5.4.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 18
rigis skala
rigis skalis monacemebisTvis mediana warmoadgens centraluri tendenciis
saukeTeso sazoms, ganurCevlad sixSireTa ganawilebis formisa, an imisa,
SemdgomSi Tu ra mizans isaxavs statistikuri analizi. rodesac centraluri
tendenciis sazomad gamoiyeneba mediana, maSin cvalebadobis sazomad, rogorc
wesi, gamoiyeneba diapazoni, kvartilTSorisi gabnevis diapazoni an
semikvartilTSorisi gabnevis diapazoni.
intervalebisa da Sefardebis skalebi
rodesac intervalebisa da Sefardebis skalebze qulebis sixSireTa ganawileba
daaxloebiT simetriulia, umjobesia, rom centraluri tendenciis sazomad
saSualo gamoviyenoT. saSualosTan erTad cvalebadobis sazomad, Cveulebisamebr,
gamoiyeneba populaciis standartuli gadaxris Sefaseba.
sixSireTa ganawilebis forma
Cveulebriv, ganawilebis forma mxolod maSin gansazRvravs aRweriTi
statistikis arCevas, rodesac damokidebuli cvladi intervalebis an Sefardebis
skalaze izomeba. Tu qulaTa ganawileba daaxloebiT simetriulia, maSin
upiratesoba eniWeba SerCevis saSualosa da standartul gadaxras; magram Tu
qulaTa ganawileba mniSvnelovnad asimetriulia, maSin SeiZleba mediana da SIQR
airCion.
monacemebis Semdgomi analizi
centraluri tendenciisa da cvalebadobis konkretuli sazomebis arCeva imazec
aris damokidebuli, Tu ramdenad gamoiyeneba mocemuli aRweriTi statistikebi
statistikuri daskvnebis gakeTebisas. daskvnisTvis iyeneben saSualos, populaciis
standartuli gadaxris Sefasebasa da populaciis dispersiis Sefasebas, vinaidan
isini ukavSirdebian normaluri ganawilebis cnobil maxasiaTeblebs. am kavSiris
Sesaxeb me-6 TavSi visaubrebT. aqedan gamomdinare, bihevioristuli mecnierebis
warmomadgenlebma monacemebis sistematizaciisTvis maSinac ki SeiZleba gamoiyenon
SerCevis saSualo da standartuli gadaxra, rodesac gazomvis skalis done da
sixSireTa ganawilebis forma sawinaaRmdegoze metyveleben. xandaxan aris
SemTxvevebi, rodesac monacemebze dayrdnobiT daksvnebis gamotanas eniWeba
upiratesoba da ar xdeba am pirobebis (gazomvis skalisa da sixSireTa
ganawilebis formis) gaTvaliswineba. Tqven naxavT, rom bihevioristul kvlevebSi
yvelaze xSirad gamoyenebad aRweriT statistikebs saSualo da standartuli
gadaxra warmoadgens.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 19
SeamowmeT Tqveni codna V.3
1. qulaTa TiToeuli qvemoT mocemuli ganawilebisTvis daasaxeleT
centraluri tendenciisa da cvalebadobis is sazomebi, romlebic
saukeTesod aRweren am qulebs.
a) intervalebis skalaze mocemuli qulebi, romelTa ganawileba
mkveTrad uaryofiTad asimetriulia.
b) rigis skalaze mocemuli qulebi, romelTa ganawileba daaxloebiT
simetriulia.
g) Sefardebis skalaze mocemuli qulebi, romelTa ganawileba
daaxloebiT simetriulia.
d) rigis skalaze mocemuli qulebi, romelTa ganawileba mkveTrad
dadebiTad asimetriulia.
e) intervalebis skalaze mocemuli qulebi, romelTa ganawileba
daaxloebiT simetriulia.
v) intervalebis skalaze mocemuli qulebi, romelTa ganawileba
mkveTrad dadebiTad asimetriulia.
z) Sefardebis skalaze mocemuli qulebi, romelTa ganawileba
mkveTrad uaryofiTad asimetriulia.
Sejameba
• cvalebadobis sazomebi gviCvenebs, Tu ramdenad gansxvavdebian qulebi
erTmaneTisgan da ganawilebis centraluri tendenciis sazomisgan.
• diapazoni gamoiTvleba Semdegnairad: Xmaqsimaluri _ Xminimaluri.
• kvartilTSorisi gabnevis diapazoni aris intervali ganawilebis 25-e da 75-e
procentilebs Soris.
• SerCevis dispersia S2 gamoiTvleba formuliT ( )∑ − populaciaNXX /2
.
• populaciis dispersiis Sefaseba s2 gamoiTvleba formuliT
( ) ( )∑ −− 1/2
NXX .
• kvadratuli fesvi dispersiidan gvaZlevs standartul gadaxras.
kvadratuli fesvi S2-dan aris SerCevis standartuli gadaxra, kvadratuli
fesvi s2-dan ki _ populaciis standartuli gadaxris Sefaseba.
• populaciis dispersiisa da standartuli gadaxris Sefaseba warmoadgens
cvalebadobis yvelaze xSirad gamoyenebad sazoms.
• cvalebadobis sazomis arCeva damokidebulia gazomvis skalaze, romelzec
mocemulia qulebi, qulaTa sixSireebis ganawilebis formasa da imaze, Tu
ra mizans isaxavs monacemebis Semdgomi analizi.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 20
ZiriTadi terminebi da simboloebi
diapazoni
kvadratebis jami
kvartilTSorisi gabnevis diapazoni
populaciis dispersia (σ 2)
populaciis dispersiis Sefaseba (s2)
populaciis standartuli gadaxra (σ)
populaciis standartuli gadaxris
Sefaseba (s)
kvartilTSorisi gabnevis diapazonis
naxevari
SerCevis dispersia
SerCevis standartuli gadaxra
cvalebadoba
interkvartiluri rangi (IQR)
SIQR
S
S2
SS
s
s2
X25
X75
σ2
kiTxvebi
1. damokidebulia Tu ara studentis mier profesoris kompetenturobis
aRqma profesoris sqesze? davuSvaT, kolejis mamrobiTi sqesis 20
moswavle afasebs mamrobiTi sqesis profesoris kompetenturobas,
romelic maT leqcias utarebs. kompetenturobis Sesafaseblad iyenebT
maswavleblis Sefasebis formas, romelic warmoadgens jamuri Sefasebis
skalas (McCarthy & Schmeck, 1982). am skalaze qulebis diapazonia 12-dan
(Zalze arakompetenturidan) 84-is (Zalze kompetenturis) CaTvliT. Tqvens
mier miRebuli 20 qulaa: _ 50, 65, 44, 47, 52, 55, 35, 43, 43, 50, 60, 40, 45, 50, 45,
57, 47, 53, 54 da 57.
a) ipoveT am ganawilebis Mdn, X , IQR da s.
b) wminda aRweriTi TvalsazrisiT, aris Tu ara Mdn an X am qulebis
centraluri tendenciis saukeTeso sazomi? axseniT Tqveni pasuxi.
2. davuSvaT, axla kolejis mamrobiTi sqesis 20 moswavle afasebs
mdedrobiTi sqesis profesoris kompetenturobas, romelic igive leqcias
atarebs, rasac wina kiTxvaSi aRwerili mamrobiTi sqesis profesori. am
SemTxvevaSi miiReT kompetenturobis Semdegi qulebi: 50, 80, 44, 47, 52, 55,
17, 43, 43, 50, 60, 40, 45, 50, 45, 57, 47, 53, 54 da 57.
a) ipoveT am ganawilebis Mdn, X , IQR da s.
b) sufTad aRweriTi TvalsazrisiT, aris Tu ara Mdn an X am
qulebis centraluri tendenciis saukeTeso sazomi? axseniT Tqveni
pasuxi.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 21
g) qulaTa am erTobliobisTvis s-is mniSvneloba, 1 kiTxvaSi miRebuli
s-is mniSvnelobasTan SedarebiT, mniSvnelovnad gaizarda, Tumca,
ar Secvlila IQR . axseniT miRebuli Sedegi.
3. cvalebadobis romeli sazomia aRwerili qvemoT mocemul TiToeul
debulebaSi?
a) is aRwers intervals, romelic moicavs qulaTa 100%-s.
b) misi mniSvneloba gamoiTvleba 75-e procentilis qulidan 25-e
procentilis qulis gamoklebiT.
g) is aRwers intervals, romelic moicavs qulaTa daaxloebiT 25%-s
medianis zemoT da qvemoT.
d) is aRwers ganawilebis saSualodan qulis saSualo kvadratul
gadaxras da warmoadgens populaciis Sefasebas.
e) is aRwers ganawilebis saSualodan qulis saSualo kvadratuli
gadaxridan kvadratul fesvs da warmoadgens populaciis
Sefasebas.
v) is aRwera intervals, romelic moicavs ganawilebis qulaTa Sua
50%-s.
z) misi mniSvneloba damokidebulia ganawilebaSi Semaval mxolod or
qulaze.
T) misi mniSvneloba gamoiTvleba formuliT: ( )∑ − NXX /2
.
i) misi mniSvneloba gamoiTvleba formuliT: ( ) ( )∑ −− 1/2
NXX .
k) misi mniSvneloba gamoiTvleba formuliT: ( )∑ − NXX /2
.
l) misi mniSvneloba gamoiTvleba formuliT: ( ) ( )∑ −− 1/2
NXX .
m) misi mniSvneloba gamoiTvleba formuliT:. ( ) 2/2575 XX − .
n) misi mniSvneloba gamoiTvleba formuliT: ∑ − populaciaNX /)( 2µ
o) misi mniSvneloba gamoiTvleba formuliT: ∑ − populaciaNX /)( 2µ
4. me-4 Tavis bolos mocemul me-6 SekiTxvaSi warmodgenili iyo grosis
(Gross, 1993) mier Catarebuli gamokiTxvis Sedegebis nawili, romelic
Seexeboda kolejis studentebSi alkoholis miRebis sakiTxs; kerZod,
mocemuli iyo, rom 21 welze ufros mamakacebTan dalevis saSualo
raodenoba kviraSi iyo 23.9, qalebTan ki _ 7.5 daleva kviraSi. am
saSualoebis Tanmxlebi standartuli gadaxrebi iyo 15.4 daleva kviraSi
mamakacebisTvis da 8.7 daleva kviraSi qalebisTvis.
თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 22
a) Tu Tqven iwinaswarmetyvelebT, rom kolejis mamrobiTi sqesis, 21
welze ufrosi tipuri studenti kviraSi 23.9-jer iRebs alkohols,
ra iqneba am winaswarmetyvelebis saSualo Secdoma?
b) SegiZliaT, TqvaT, rom es standartuli gadaxra migviTiTebs
gakeTebuli Sefasebis did an mcire cvalebadobaze?
g) ra informacias gaZlevT es standartuli gadaxra saSualos,
rogorc centraluri tendenciis sazomis, adekvaturobis Sesaxeb
am gamokiTxvaSi miRebuli qulebis ganawilebis SemTxvevaSi?
5. me-4 Tavis bolos mocemul me-7 SekiTxvaSi warmodgenili iyo cxridan
rva qirurgiuli pacientis martivi reaqciis dro zogadi anesTeziis
miRebidan ori saaTis Semdeg. mecxre pacienti ar iyo sakmarisad
gamosuli anesTeziidan, rom gaezomaT reaqciis dro, amitom am cdis
piris monacemi ganisazRvra, rogorc reaqciis usasrulo dro. Tu Tqven
gsurT cxrave qulis aRwera, cvalebadobis romel sazoms gamoiyenebdiT
maTTvis? axseniT sazomis arCevis mizezi.
Tavi VI. normaluri ganawileba, albaToba da
standartuli qulebi
normaluri ganawileba
normaluri ganawileba da bihevioristuli mecnierebebi
normaluri ganawilebis Tvisebebi
normaluri ganawilebis qveS moTavsebuli farTobi
standartuli normaluri ganawileba
standartuli normaluri ganawilebis cxrilis gamoyeneba
z qulebis gamoyenebis magaliTi
albaToba
diskretuli albaTobis ganawilebis Tvisebebi
Teoriuli da empiriuli albaToba
standartuli normaluri ganawileba, rogorc albaTuri
ganawileba
standartuli qulebi
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 2
me-3-5 TavebSi SerCevidan miRebuli qulebis mowesrigebisa da aRweris sakiTxebze
visaubreT. axla ganvixiloT populaciebi da maTi maxasiaTeblebi; kerZod,
populaciis erT-erTi ganawileba _ normaluri ganawileba.Y pirvel TavSi
populacia ganvmarteT, rogorc saerTo maxasiaTeblis mqone adamianebis,
cxovelebis, sagnebisa Tu movlenebis sruli erToblioba. SerCeva aris
populaciidan arCeuli qvesimravle anu qvejgufi. me-3, me-4 da me-5 TavebSi
ganvixileT SerCevidan miRebuli qulebis mowesrigeba da aRwera sixSireTa
ganawilebisa da iseTi aRweriTi statistikebis gamoyenebiT rogoricaa mediana,
saSualo, kvartilTSorisi gabnevis diapazonis naxevari da standartuli gadaxra;
Tumca, rogorc wina TavebSi iTqva, populaciis qulebis aRwerac SesaZlebelia
saSualoTi da standartuli gadaxriT. vinaidan axla vambobT, rom vzomavT mTel
populacias, saSualos aRvniSnavT simboloTi μ (miu, berZnuli anbanis meoce aso),
xolo standartul gadaxras _ simboloTi σ (sigma, berZnuli anbanis meTvramete
aso). garda amisa, miRebul sidideebs axla ukve vuwodebT ara statistikebs,
aramed parametrebs.
parametri aris ricxvi, magaliTad, μ an σ, romelic aRwers populacias.
terminologiis aseTi Secvlis mizani isaa, rom gasagebi iyos, raze vsaubrobT: _
SerCevaze Tu populaciaze. mniSvnelovania am gansxvavebis gageba, radgan
normaluri ganawileba, romelzec qvemoT visaubrebT, xasiaTdeba μ-Ti da σ-Ti da
ara _ X da s-iT. aRniSnuli terminologia mocemulia cxrilSi N6-1.
normaluri ganawileba
bihevioristul mecnierebebSi populaciis erT-erT mniSvnelovan ganawilebas
warmoadgens normaluri ganawileba. normaluri ganawileba aris Teoriuli
maTematikuri ganawileba, romelic aRwers populaciis qulebis fardobiT
sixSireebs. normalurad ganawilebul qulaTa erTobliobis sixSireTa ganawileba
SeiZleba srulad aRiweros, Tu cnobilia ganawilebis saSualo (μ) da
standartuli gadaxra (σ). normaluri ganawileba, romlis μ = 70 da σ = 10,
cxrili N6.1. SerCevisa da populaciis Sesatyvisi aRniSvnebi.
SerCeva populacia
aRweriTi
maxasiaTeblebi statistikebi parametrebi
saSualo
standartuli
gadaxra
dispersia
X
s
s2
μ
σ
σ2
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 3
mocemulia suraTze N6-1. mocemuli ganawileba warmoadgens maTematikur
ganawilebas, romelic ar asaxavs romelime realurad gazomili populaciis
qulebs. realuri populaciis qulebi arasodes aris ise zustad ganawilebuli,
rom ganawileba zustad normaluri iyos; magram zogjer gamosadegia imis daSveba,
rom zogierTi populaciis qulebi daaxloebiT normalurad aris ganawilebuli;
Tumca, winamdebare TavSi normalur ganawilebas ganvixilavT, rogorc Teoriul
maTematikur ganawilebas.
normaluri ganawileba da bihevioristuli mecnierebebi
normaluri ganawileba iakob bernulis (Jakob Bernoulli, 1654-1705), abraam de
moiavris (Abraham de Moivre, 1667-1754), pier remond de monmortis (Pierre Remond de
Montmort, 1678-1719), karl fridrix gausisa (Karl Friedrich Gauss, 1777-1855) da sxvebis
SromebSi ganviTarda. maTi interesis sagani iyo sxvadasxva azartul TamaSebSi
mosalodneli Sedegebis albaTobebis gamoTvla da astronomiul da fizikur
cdebSi gazomvebis Secdomebis ganawilebebis ageba. normaluri ganawileba maleve
gadaiqca statistikosebisTvis mniSvnelovan ganawilebad, radgan statistikaSi
arsebuli mravali problemis gadaWra mxolod normaluri ganawilebis daSvebis
Semdeg gaxda SesaZlebeli.
normaluri ganawileba bihevioristuli mecnierebebis warmomadgenlebisTvisac
mniSvnelovania, radgan gamarTlebulia daSveba imisa, rom mravali damoukidebeli
normaluri ganawileba (Normal distribution): _ Teoriuli maTematikuri ganawileba, romelic
aRwers populaciis qulebis fardobiT sixSireebs.
40 -3σ
70 μ
50 -2σ
60 -1σ
100 +3σ
90 +2
80 +1σ
.10
.20
.30
.40
qulebi
fardobiTi sixSire
suraTi N6.1. normaluri ganawileba, romlis μ =70 da σ =10.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 4
cvladiT ganpirobebuli nebismieri qcevis gazomvis Sedegebi popoulaciaSi
daaxloebiT normalurad iqneba ganawilebuli. adamianis mravali unari, iqneba es
maTematikuri Tu verbaluri unarebi, ganpirobebulia damoukidebeli faqtorebis
mTeli rigiT: _ genetikuri TandayolilobiT, mSoblebis mxridan miRebuli
waxalisebiT, ganaTlebiT, konkretuli maswavlebliT, socialuri mdgomareobiT,
kulturuli Zalebis zemoqmedebiT da a.S. Sesabamisad, aseTi unarebis
mocemulobis xarisxi mosalodnelia, rom populaciaSi daaxloebiT normalurad
iyos ganawilebuli. Cven imitom viyenebT sityvas „daaxloebiT“, rom realurad
miRebuli qulebi arasodes araa zustad normalurad ganawilebuli.
normaluri ganawilebis Tvisebebi
normaluri ganawilebis Tvisebebia: _ simetriuloba, asimptoturoba da
uwyvetoba.
simetriuloba
Teoriuli normaluri ganawileba simetriulia modis, medianisa da saSualos
mimarT; Sesabamisad, normalur ganawilebaSi moda, mediana da saSualo erTmaneTis
tolia.
asimptoturoba
yuradReba miaqcieT, rom suraTze N6.1 gamosaxuli normaluri ganawilebis
boloebi (kudebi) rac ufro daSorebulia μ-sgan, miT metad uaxlovdeba X RerZs
anu abscisas, Tumca, ganawileba asimptoturia, anu ramdenadac ar unda daSordes
μ-s, kudebi mainc ar Seexeba X RerZs.
uwyvetoba
normaluri ganawileba uwyvetia plius da minus usasrulobebs Soris
moTavsebuli yvela qulisTvis. es niSnavs, rom nebismieri ori qulisTvis
moiZebneba mesame qula, romelic maT Soris mdebareobs.
normaluri ganawilebis qveS moTavsebuli farTobi
normaluri ganawileba fardobiT sixSireTa Teoriuli ganawilebaa. gavixsenoT
me-3 Tavidan, rom qulis fardobiTi sixSire aris ganawilebaSi qulis Sexvedris
sixSire gayofili ganawilebaSi Semaval qulaTa saerTo raodenobaze. amrigad,
fardobiTi sixSire gamoisaxeba SefardebiT da ganawilebis saerTo dagrovili
fardobiTi sixSire aris 1.0 (ix. cxrili N3.4, (i) sveti). vinaidan normaluri
ganawileba fardobiTi sixSireebis ganawilebaa, ganawilebis qveS moTavsebuli
sruli farTobi 1.0-s udris; amitom normalur ganawilebaSi qulaTa fardobiTi
sixSireebi dagrovilia 1.0-mde.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 5
yvela normaluri ganawilebaSi qulaTa garkveuli nawili moTavsebulia
saSualos garSemo garkveul intervalSi. nebismier normalurad ganawilebul
populaciaSi qulaTa .3413 wili moTavsebulia intervalSi μ-sa da μ-s plus erT
standartul gadaxras Soris, anu qulaTa 34.13% moTavsebulia intervalSi (μ, μ+1σ).
vinaidan ganawileba simetriulia, qulaTa igive procenti moTavsebulia
intervalSi (μ_1σ, μ). saSualodan erT da or standartul gadaxras Soris anu
intervalSi (μ_2σ, μ_1σ) moTavsebulia qulaTa 13.59%. intervalSi (μ_2σ, μ+2σ)
moTavsebulia qulaTa (13.59+34.13+34.13+13.59)%=95.44%. normaluri ganawilebis
intervalebSi Semaval qulaTa fardobiTi sixSirebi naCvenebia suraTze N6-2. es
damokidebuleba ZalaSia nebismieri normaluri ganawilebis SemTxvevaSi, garda im
SemTxvevebisa, rodesac μ da σ zogierT konkretul mniSvnelobas iRebs.
garda amisa, normaluri ganawilebis qveS moTavsebuli farTobebi SeiZleba
Seikribos. ase magaliTad, saSualos minus erTi standartuli gadaxra da
saSualos plius erTi standartuli gadaxra anu intervali (μ_1σ, μ+1σ) moicavs
qulaTa (.3413+.3413)-s anu .6826 wils. analogiurad, saSualos minus ori
standartuli gadaxra da saSualos plius ori standartuli gadaxra anu (μ_2σ,
μ+2σ) intervalSi Sedis qulaTa .1359+.3413+.3413+.1359 anu .9544 wili. qulaTa ufro
didi intervali, romelic Seadgens sam standartul gadaxras saSualos orive
mxares anu intervali (μ_3σ, μ+3σ), moicavs qulaTa (.0215+.1359+.3413+.3413+.1359+.0215)-s
anu .9974 wils. es damokidebulebac naCvenebia surTze N6.2. es fardoba ucvleli
rCeba μ-sa da σ-s mniSvnelobebisgan damoukideblad.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 6
aRniSnuli mimarTebebis saCveneblad, davuSvaT, rom qulaTa populacia
normalurad aris ganawilebuli. ganawilebis μ=100 da σ =15. ganawilebaSi qula 85
aris erTi standartuli gadaxra saSualos marcxniv (anu μ_1σ = 100-15=85), 115 qula
ki _ erTi standartuli gadaxra saSualos marjvniv (anu μ+1σ = 100+15=115).
Sesabamisad, ganawilebaSi qulaTa .6826 wili Sedis intervalSi (85, 115).
analogiurad, qulaTa .9544 wili moTavsebuli iqneba 70-sa da 130 qulas Soris,
intervalSi (μ_2σ, μ+2σ), xolo qulaTa .9974 wili _ 55-sa da 145 qulas Soris,
intervalSi (μ_3σ, μ+3σ).
kidev erTi magaliTi. davuSvaT, gvaqvs qulaTa normalurad ganawilebuli
populacia, romlis μ=50 da σ =5. am ganawilebaSi qulaTa .6826 wili moTavsebulia
intervalSi 45-sa da 55-s Soris anu intervalSi (μ_1σ, μ+1σ), qulaTa .9544 wili _
intervalSi 40-sa da 60-s Soris anu intervali (μ_2σ, μ+2σ) da qulaTa .9974 wili _
intervalSi 35-sa da 65-s Soris anu intervali (μ_3σ, μ+3σ).
amrigad, normalur ganawilebaSi μ-sa da σ-s Semdegi damokidebulebaa: _
saSualos orive mxares mdebare konkretuli intervalebi ganawilebis qulaTa
konkretul wils da konkretul procents moicaven.
intervali
intervalSi
Semaval qulaTa
wili
intervalSi
Semaval qulaTa
procenti
-3σ -2σ -1σ μ +1σ +2σ +3σ
suraTi N6.2. normaluri ganawilebis konkretul
intervalebSi Semaval qulaTa fardobiTi sixSireebi.
.0013 .0013 .0215
.1359
.1334 .1334
.1359 .0215
.6826
.9544
.9974
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 7
(μ_1σ)-dan (μ+1σ)-mde (μ_2σ)-dan (μ+2σ)-mde (μ_3σ)-dan (μ+3σ)-mde
.6826
.9544
.9974
68.26 95.44 99.74
Semdeg qveTavSi viswavliT cxrilis gamoyenebas imis gansasazRvrad, qulaTa ra
nawili xvdeba normaluri ganawilebis saSualos irgvliv mdebare konkretul
intervalebSi. ase rom, ar aris saWiro aq warmodgenili mniSvnelobebis
damaxsovreba; Tumca, sasurveli da gamosadegi iqneba miaxloebiTi mniSvnelobebis
damaxsovreba, radgan amocanebis amoxsnisas SeZloT Tqvens mierve warmoebuli
gamoTvlebis sisworis Semowmeba. amrigad, amocanebis amoxsnisas gamogadgebaT, Tu
daimaxsovrebT, rom:
• intervali (μ_1σ)-dan (μ+1σ)-mde daaxloebiT moicavs qulaTa 68%-s.
• intervali (μ_2σ)-dan (μ+2σ)-mde daaxloebiT moicavs qulaTa 95%-s.
• intervali (μ_3σ)-dan (μ+3σ)-mde daaxloebiT moicavs qulaTa 99%-s.
SeamowmeT Tqveni codna VI.1
1. ganmarteT: _ asimptoturi, uwyveti, μ, normaluri ganawileba, parametri,
populacia, s, SerCeva, SerCeva, σ, statistika, X .
2. daasaxeleT normaluri ganawilebis sami Tviseba.
3. ratom ver iqneba adamianTa realuri populaciisgan miRebuli qulebi
zustad normalurad ganawilebuli? miniSneba: gaixsneT normaluri
ganawilebis Tvisebebi.
4. davuSvaT, gaqvT qulaTa normalurad ganawilebuli populacia, romlis μ
= 200 da σ = 20. miuTiTeT, qulaTa ra wili Seva qvemoT mocemul TiToeul
intervalSi (gamoiyeneT suraTi N6.2).
intervali
a) 220-220;
b) 200-240;
g) 200-260;
e) 180-200;
f) 160-200;
v) 140-200;
z) 180-260;
T) 160-240;
i) 140-260;
k) 220-240;
l) 220-260;
m) 140-160.
standartuli normaluri ganawileba
normaluri ganawilebis saSualosa da standartuli gadaxris codniT viRebT
zust informacias konkretul intervalebSi qulaTa ganawilebis Sesaxeb. vinaidan
normaluri ganawileba maTematikuri gantolebiT ganisazRvreba, Cven SegviZlia
zusti fardobiTi sixSiris gansazRvra ganawilebis qulaTa nebismieri
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 8
intervalisTvis; Tumca, aq Semdegi problemis winaSe vdgebiT: _ Cven unda
gamoviyenoT rTuli maTematikuri gamosaxuleba, sadac Seva qulebis CvenTvis
saintereso ganawilebis μ-sa da σ-is mniSvnelobebi. sabednierod, arsebobs am
problemisTvis gverdis avlis martivi gza: _ normaluri ganawileba, romlis
saSualoa μ da standartuli gadaxra _ σ (magaliTad, μ = 20, σ = 2) unda gardavqmnaT
standartuli formis normalur ganawilebad, romlis μ = 0, σ = 1. amis Semdeg, Tu
mxolod erT standartul normalur ganawilebas gamoviyenebT (romlis μ = 0, σ = 1),
Cven SevZlebT gantolebis amoxsnas qulis mocemuli mniSvnelobisTvis da misi
Sesabamisi fardobiTi sixSiris tabulirebas cxrilSi. SemdgomSi, Cven normaluri
ganawilebis gantolebis gamoyenebis nacvlad SegviZlia, ubralod cxrilSi
movZebnoT mocemuli qulis fardobiTi sixSire, an dagrovili fardobiTi sixSire.
standartuli normaluri ganawilebis
saSualo nulis (μ = 0), standatruli gadaxra
ki _ 1-is (σ = 1) tolia. nebismieri μ-sa da σ-is
mqone normalurad ganawilebuli cvladisTvis konkretuli qula (magaliTad, X)
SeiZleba gardaiqmnas standartuli normaluri ganawilebis qulad, Tu mas
SevitanT gamosaxulebaSi
( ) σµ /−= Xz ,
sadac z _ qulidan z-is realurad miRebuli1 mniSvneloba;
X _ sawyisi anu originaluri qula;
μ _ saSualo;
σ _ standartuli gadaxra im normaluri ganawilebisa, saidanac aRebulia X
qula.
mocemul formulaSi ricxviTi
mniSvnelobebis Setanis Semdeg X-is
mniSvnelobisgan miRebuli z-is
mniSvnelobas z realurs vuwodebT da zrealuri-iT aRvniSnavT. am terminologias
imis naTelsayofad viyenebT, rom z-is mniSvneloba formulaSi (zrealuri) ricxviTi
mniSvnelobebis SetaniTaa gamoTvlili Tu a.1. cxrilSia napovni. aRniSnuli
cxrilis Sesaxeb qvemoT visaubrebT. z-s xSirad standartul normalur gadaxras
uwodeben, radgan misi mniSvneloba standartul normalur ganawilebaSi qulis
mniSvnelobas gamoxatavs. standartuli normaluri gadaxra warmodgenilia
suraTze N6-3. suraTze naCvenebia qulaTa ganawileba mocemul z qulebs Soris.
yuradReba miaqcieT, rom vinaidan z = +1 miuTiTebs saSualodan erTi standartuli
1 SemdgomSi mTel teqstSi „realurad miRebulis“ nacvlad SemoklebiT gamoiyeneba
„realuri“ (mTargmnelis SeniSvna).
standartuli normaluri ganawileba
(Standard normal distribution): _ normaluri
ganawileba, romlis μ = 0, σ = 1.
standartuli normaluri gadaxra (Standard
normal deviate): _ normaluri ganawileba,
romlis μ = 0, σ = 1.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 9
gadaxriT zemoT mdebare qulaze, z = 0-dan z = +1-mde SualedSi xvdeba qulaTa 3413
nawili, anu igive raodenoba, rac μ-sa da (μ+1)-s Soris suraTze 6.2.
აკლია სურათი 6-3 suraTi N6.3. standartuli normaluri ganawileba, romlis μ = 0 da σ = 1.0. naCvenebia z-is
or dadgenil mniSvnelobas Soris moxvedril qulaTa wili.
standartuli normaluri ganawilebis gamoyenebis sailustraciod ganvixiloT
magaliTi. davuSvaT, 115 qula (anu X = 115) aviReT qulaTa normalurad
ganawilebuli erTobliobidan, romlis μ = 100, σ = 15. es qula gadaviyvanoT
zrealuri-Si formuliT
( )
0.11515
15100115
+==
−=
−=
σµXzrealuri
sawyisi 115 qulis gardaqmniT miRebuli standartuli normaluri qulaa +1.0.
zrealuri = +1.0 niSnavs, rom sawyisi qula (anu 115) aris im normaluri ganawilebis
erTi standartuli gadaxriT saSualos (anu 100) zemoT, saidanac is aviReT.
qulis gadayvana standartuli ganawilebis z qulaSi gviCvenebs, Tu ramdeni
standartuli gadaxriT aris sawyisi qula (anu X) daSorebuli saSualosgan.
magaliTad, aRebuli 115 qula +1.0σ erTeuliT aris daSorebuli ganawilebis
saSualodan. ganvixiloT cxrilSi N6.2 mocemuli qulebi da maTi Sesabamisi
zrealuri-is mniSvnelobebi. es qulebi miRebulia normaluri ganawilebidan, romlis
μ = 75 da σ =10. SeecadeT, gamoTvaloT zogierTi mocemuli qula. yuradReba
miaqcieT, rom zrealuri-is mniSvneloba
nebismieri qulisTvis ubralod
warmoadgens imas, Tu standartuli
gadaxris ramdeni erTeuliT aris
daSorebuli qula standartuli
normaluri ganawilebis saSualosgan
(romelic nulis tolia) zemoT an
qvemoT.
105 100 95 90 85 80 75 70 65 60 55 50 45
+3.0 +2.5 +2.0 +1.5 +1.0 +0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0
cxrili N6.2. normaluri ganawilebis (μ =75
da σ =10) qulebis gadayvana standartuli
ganawilebis z qulebSi.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 10
aqve unda gagafrTxiloT, rom X
qulis gadayvana z qulaSi mxolod im
SemTxvevaSi gviCvenebs sawyisi qulis zust mdebareobas standartul normalur
ganawilebaSi, Tu es sawyisi qula aRebulia qulaTa normalurad ganawilebuli
simravlidan. Tu qula aRebulia ara normaluri ganawilebidan, aramed iseTi
ganawilebidan, rogoricaa, magaliTad, marTkuTxa an bimodaluri ganawileba,
qulis gadayvana z qulaSi ver aqcevs sawyis qulebs normalurad ganawilebuls.
standartuli ganawilebis cxrilis gamoyeneba
qulebis standartuli normaluri ganawilebis z qulebSi gadayvanis
upiratesoba isaa, rom z-is mniSvnelobebs Soris moxvedrili qulebis Tanafardoba
cnobilia da mocemulia cxrilSi. es Tanafardobebi mocemulia danarTis
cxrilSi a-1. am cxrilis nawili warmodgenilia N6.3 cxrilis saxiT. (a) svetSi
mocemulia z-is mniSvnelobebi, romlebic miRebulia ricxviTi mniSvnelobebis
SetaniT formulaSi
( )σµ−
=Xzrealuri
yuradReba miaqcieT, rom Cven viyenebT aRniSvnas zrealuri, rodesac vsaubrobT z-
is nedli qulidan (magaliTad, X-dan) miRebul mniSvnelobaze, xolo aRniSvna z,
indeqsis „realuri“-is gareSe, Seesabameba a-1 cxrilis (a) svetSi mocemul z-is
mniSvnelobas. (b) svetSi mocemulia farTobi z = 0-sa da z-is (a) svetSi mocemul
mniSvnelobas Soris. svetis Tavze warmodgenil grafikze es farTobi gamuqebulia.
(g) svetSi mocemulia farTobi z-is (a) svetSi mocemuli mniSvnelobidan
usasrulobamde. aqac svetis Tavze mocemul grafikze gamuqebulia es farTobi.
vinaidan normaluri ganawileba simetriulia, z-is uaryofiTi mniSvnelobebiT
SemosazRvruli farTobebi z-is dadebiTi mniSvnelobebis identuria. amrigad,
cxrilSi ar aris warmodgenili z-is uaryofiTi mniSvnelobebi.
cxrilis gamoyenebis sailustraciod, ganvixiloT qulaTa normalurad
ganawilebuli simravle, romlis μ =100 da σ =15. davuSvaT, am ganawilebidan aviReT
qula 115. ganawilebis qulaTa ra wili iqneba 115-is toli an masze naklebi? am
amocanis amosaxsnelad, Cven 115 qula gadagvyavs z qulaSi, anu
( ) 0.115
100115+=
−=
−=
σµXzrealuri
qula (X) z
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 11
(a) (b) (g)
0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10
.3289 .3315 .3340 .3365 .3389 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621 .3643
.1711 .1685 .1660 .1635 .1611 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379 .1357
nawyveti amoRebulia II.1 cxrilidan, normaluri ganawileba da masTan dakavSirebuli ganawilebebi. CRC Handbook of Tables for Probability and Statistics (meore gamocema). COpyright 1968, CRCPress, Inc., Boca Raton, Florida. gamoiyeneba nebarTviT.
yuradReba miaqcieT, rom vinaidan z-is mniSvneloba nedli 115 qulidan miiReba,
mas zrealuri-s vuwodebT, z-is cxrilSi N6-3 mocemuli mniSvnelobisgan
gansasxvaveblad. saZiebeli qulis TvalsaCinod gamosaxvisTvis sasargebloa
standartuli normaluri ganawilebis daxazva da masze nedli qulebisa da am
qulebis Sesabamisi z qulebis miTiTeba. ganawilebis nedli qulebi da maTi
Sesabamisi z qulebi naCvenebia suraTze N6-4. amocanis amosaxsnelad saWiroa, rom
miviRoT ganawilebis farTobi z = _ ∞-dan z = +1.0-mde. ganawilebis qulaTa erTi
naxevari (anu 0.5000) z = 0-s qvemoT aris moTavsebuli. cxrilis N6.3 (b) sveti
gviCvenebs, rom ganawilebis qulaTa .3413 moTavsebulia z = 0-sa da z = +1.0-s Soris
(ix. suraTi N6.4, (g) RerZi). amrigad, amocanis pasuxia qulaTa z = _ ∞-dan z = 0-mde
moTavsebul wils (.5000) plius qulaTa z = 0-dan da z = +1.0-mde moTavsebuli wili
cxrili N6.3. farTobebi normaluri ganawilebis qveS z = 0.95-dan z = 1.10-mde.
(a) (b) (g)
0
0
0
0
+z +z
-z -z -z
z-is mniSvneloba
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 12
(.3413 a-1 cxrilis (b) svetidan), anu .8413. 115-is toli an masze nakleb qulaTa
wilia .8413 anu 84.13%.
აკლია სურათი 6-4 suraTi N6.4. normalur ganawilebaSi, romlis μ = 100 da σ =15, Semavali qulis (115)
adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs
(a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _∞-dan z =
0-mde da z = 0-dan z = +1-mde.
a-1 cxrilis wakiTxvisas Zalze advilia Secdomis daSveba, amitom yovelTvis
sasargebloa, rom amocanis amoxsniT miRebuli paxusi SeadaroT a-1 cxrilSi
mocemul mniSvnelobebs. Secdomis Tavidan asacileblad, SegiZliaT daaxloebiT
gansazRvroT pasuxi imis mixedviT, rom normalurad ganawilebul qulaTa erT
mesamedze odnav meti (anu 33%-ze odnav meti) moTavsebulia ganawilebis saSualosa
da erT standartul gadaxras Soris. amiT gansazRvravT, rom Tqveni pasuxi
(50+33)%-ze anu 83%-ze odnav meti unda iyos. miRebuli pasuxi 84.13% Seesabameba
aseT Sefasebas.
davuSvaT, axla gvsurs, rom vicodeT, qulaTa ra nawilia 115-is toli an masze
meti. amis gamosaTvleli ori gza arsebobs:
(1) vinaidan Cven ukve viciT, rom qulaTa 84.13% tolia an naklebia 115-ze,
normaluri ganawilebis qveS darCenili farTobi unda migviTiTebdes im qulaTa
raodenobaze, romelic 115-is toli an masze meti iqneba. amitom saZiebel pasuxs
miviRebT, Tu ganawilebis mTlian farTobs 1.000-s gamovaklebT .8413-s, anu qulaTa
1.000_.8413=.1587 nawili tolia an metia 115-ze.
(2) Tu winaswar ar gvaqvs gamoTvlili, rom qulaTa .8413 nawili tolia an
naklebia 115-ze, maSin amocanis amosaxsnelad unda ganvsazRvroT, rom saZiebeli
nawili aris ganawilebis farTobi z = +1.0-is Semdeg. cxrilis N6-3 (a) svetSi
vpoulobT z = 1.00-is mniSvnelobas, Semdeg vnaxulobT mis gaswvriv (g) svetSi
mocemul mniSvnelobas, sadac warmoadgenilia z-is zemoT qulaTa wili. z = +1.0-
isTvis es mniSvnelobaa .1587. amrigad, qulaTa .1587 nawili anu 15.87% tolia an
metia 115-ze.
amave ganawilebis magaliTze ganvixiloT kidev erTi amocana. qulaTa ra
nawilia 82 qulis toli an masze naklebi? am SemTxvevaSic 82 gadagvyavs z qulaSi:
zrealuri = (82_100)/15 = _1.2. es zrealuri gviCvenebs, rom 82 qula mdebareobs
ganawilebis saSualodan 1.2 standartuli gadaxriT qvemoT. ganawilebis nedli
qulebi da zrealuri = _1.2 naCvenebia suraTze N6-5. Semdeg gamovTvliT z = _1.2-is
qvemoT mdebare qulaTa wils. yuradReba miaqcieT, rom naxazis mixedviT mxolod
imis gansazRvra SegviZlia, rom es nawili Zalze mcire iqneba. vinaidan
standartuli normaluri ganawileba simetriulia, imdeni qula iqneba
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 13
moTavsebuli z = _1.2-is qvemoT, ramdenic z = +1.2-is zemoT; amitom viTvliT z = +1.2-
is zemoT; Tumca, es mniSvneloba ar aris mocemuli cxrilSi N6-3. amitom viyenebT
normaluri ganawilebis qveS mdebare farTobis wilebis srul cxrils (ix.
cxrili a-1). (a) svetSi vpoulobT z = 1.2-s da vnaxulobT mis gaswvriv (g) cxrilSi
mocemul mniSvnelobas. amrigad, qulaTa .1151 nawili anu 11.51% tolia an naklebia
82-ze. standartuli normaluri ganawilebis gamoyenebis kidev erTi magaliTi
warmodgenilia qveTavSi „magaliTi N6.1“.
აკლია სურათი 6-5 suraTi N6.5. normalur ganawilebaSi, romlis μ = 100 da σ =15, Semavali qulis (82)
adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs
(a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _∞-dan z =
_1.2-mde.
magaliTi N6.1
standartuli normaluri ganawilebis gamoyeneba
vTqvaT, gvaqvs normalurad ganawilebuli populacia, romlis μ = 80, da σ =5.
amocana
qulaTa ra nawilia 88 qulis toli an masze meti?
saWiro statistika: ( ) σµ /−= Xzrealuri
amoxsna
88 qula unda gadaviyvanoT zrealuri-Si. Semdeg cxrilis Na-1 gamoyenebiT
vipoviT zrealuri-is tol an masze met qulaTa wils. z-is formulaSi CavsvaT
ricxviTi mniSvnelobebi:
6.158
58088
+==−
=realuriz .
a-1 cxrilis (g) sveti gviCvenebs, rom standartuli normaluri ganawilebis z =
+1.6-is toli an masze meti farTobia .0548. aqedan gamomdinare, 88-is tol an masze
met qulaTa wili iqneba .0548 anu 5.48%. amrigad, populaciis yoveli 1000 qulidan
54.8 tolia an metia 88-ze.
amocana
ganxiluli ganawilebis qulaTa ra nawilia moTavsebuli 83-sa da 87-s Soris?
amoxsna
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 14
orive qula _ 83 da 87 unda gadaviyvanoT zrealuri qulebSi. Semdeg viTvliT
standartuli normaluri ganawilebis farTobs zrealuri-is or mniSvnelobas
Soris. ricxviTi mniSvnelobebis Casmis Semdeg miviRebT:
zrealuri 83-isTvis 6.053
58083
+==−
= ;
zrealuri 87-isTvis 4.157
58087
+==−
= .
nedli qulebi da maTi Sesabamisi zrealuri qulebi mocemulia suraTze N6-6.
zrealuri = +0.6-isTvis a-1 cxrilis (b) svetSi vnaxulobT, rom z = 0-sa da z = 0.6-s
Soris mdebare qulaTa wilia .2257. zrealuri= +1.4-isTvis z = 0-sa da z = +1.4-s Soris
mdebare qulaTa wilia .4192. ganawilebis farTobi, romelic z-is am or
mniSvnelobas Soris mdebareobs, gamoxatavs 83 da 87 qulebs Soris moTavsebul
qulaTa wils. am farTobis gamosaTvlelad .4192-s vaklebT .2257-s. miRebuli
pasuxia .1935 (ix. suraTi N6.6, (g) RerZi). amrigad, qulebis wili, romelic Tavsdeba
83-sa da 87-s Soris intervalSi, aris .1935 anu 19.35%. yoveli 1000 qulidan 193.5
moTavsebulia 83 qulasa da 87 qulas Soris.
აკლია სურათი 6-6 suraTi N6.6. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semavali qulebis (83 da 87)
adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs
(a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z =
+0.6-mde da z = 0-dan z = +1.4-mde.
amocana
qulaTa ra nawilia moTavsebuli 65-sa da 74-s Soris?
amoxsna
es amocana wina amocanis msgavsad ixsneba. qulebi 65 da 74 unda gadaviyvanoT
zrealuri qulebSi. Semdeg viTvliT standartuli normaluri ganawilebis farTobs
zrealuri-is or mniSvnelobas Soris. ricxviTi mniSvnelobebis Casmis Semdeg
miviRebT:
zrealuri 65-isTvis 0.3515
58065
−=−
=−
= ;
zrealuri 74-isTvis 2.156
58074
−=−
=−
= .
nedli qulebi da maTi Sesabamisi zrealuri qulebi mocemulia suraTze N6.7. a-1
cxrilis (b) svetSi vnaxulobT, rom qulaTa .4987 moTavsebulia z = 0-sa da z = _3.0-
s Soris. zrealuri = _1.2-isTvis qulaTa .3849 nawili moTavsebulia z = 0-sa da z =
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 15
_1.0-s Soris. z = _3.0-sa da z = _1.2-s Soris moTavsebul qulaTa wilis
gamosaTvlelad .4987-s gamovaklebT .3849-s da miviRebT .1138-s (ix. suraTi N6.7, (g)
RerZi). qulaTa wili 65-dan 74-mde intervalSi aris .1138 anu 11.38%. populaciaSi
yoveli 1000 qulidan 113.8 moTavsebulia 65-sa da 74-s Soris.
აკლია სურათი 6-7 suraTi N6.7. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semavali qulebis (65 da 74)
adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs
(a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z =
-3.0-mde da z = 0-dan z = -1.2-mde.
amocana
qulaTa ra wili Seadgens ganawilebis Sua 80%-s?
amoxsna
amocanis amosaxsnelad a-1 cxrilSi unda vnaxoT (b) sveti, sadac
warmodgenilia saSualos orive mxares mdebare qulaTa .40 nawili anu 40%; am
qulebisTvis ganvsazRvravT z-is mniSvnelobas da Semdeg z-is formulis amoxsniT
viRebT X-is mniSvnelobebs. suraTze N6-8 naCvenebia qulaTa saZiebeli wilebi. z-is
sapovnelad, romelic moicavs z = 0-sa da mis mniSvnelobas Soris moTavsebul
qulaTa .40 wils, (b) svetSi veZebT .40-Tan yvelaze miaxloebul wils. es aris .3997,
romelic Seesabameba z = 1.28-s. amrigad, z = 0-sa da z = + 1.28-s Soris moTavsebuli
qulebis Sexvedris fardobiTi sixSirea .3997 da z = 0-sa da z = _ 1.28-s Soris
moTavsebuli qulebic igive fardobiTi sixSiriT (.3997) gvxvdeba. z = _ 1.28-sa da z =
+ 1.28-s Soris moTavsebuli qulebis saerTo raodenobaa .3997+.3997=.7994 anu
daaxloebiT .80. bolo nabijia X-is mniSvnelobebis gamoTvla, romlebic z = _ 1.28-
sa da z = + 1.28-s Seesabameba. am mniSvnelobebis gamosaTvlelad z-is, μ-sa da σ-s
ricxviTi mniSvnelobebi CavsvaT formulaSi zrealuri=(X_μ)/σ da gamovTvaloT X-is
mniSvnelobebi.
აკლია სურათი 6-8 suraTi N6.8. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semaval qulaTa Sua 80%-is
povna standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs (a) nedl
qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z = -1.28-mde
da z = 0-dan z = +1.28-mde.
zrealuri= +(1.28)-isTvis
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 16
58028.1 −
=+X
gantoleba amovxsnaT X-is mimarT
( )( )
4.86804.6
8028.1580528.1
=+=
+=−=+
XX
zrealuri= _1.28-isTvis
58028.1 −
=−X
gantoleba amovxsnaT X-is mimarT
( )( )
6.73804.6
8028.1580528.1
=+−=
+−=−=−
XX
Sesabamisad, qulaTa 80% moTavsebulia 73.6-sa da 86.4-s Soris.
z qulebis gamoyenebis magaliTi
xSirad iTvleba, rom didi raodenobiT adamianebze Catarebul standartizebul
testebSi miRebuli qulebi daaxloebiT normalurad aris ganawilebuli da z-is
mniSvnelobebiT gamoixateba. amitom cnobilia xolme, ganawilebis danarCen
qulebTan mimarTebaSi zustad sad mdebareobs mocemuli qula. magaliTad,
iarnoldma, grimma da miuserma (Yarnold, Grimm & Mueser, 1986) ikvlies, Tu ramdenad
ukavSirdeba A da B tipis qcevis paternebi socialur konformulobas. adamianebis
azriT, A tipis qceva gulisxmobs Sejibrianobas, agresiulobas, mouTmenlobasa da
swrafad gaRizianebadobas. B tipis individebi xasiaTdebian rogorc
araSejibrianebi, araagresiulebi, momTmenebi, romlebic mSvidad xvdebian
imedgacruebebsa da frustraciebs. A da B tipis qcevis paternebis Sefasebis erT-
erTi sazomia jenkinsis aqtiurobis kiTxvari (Jenkins, Zyzanski & Rosenman, 1979),
romelic tipur aqtivobebs Seexeba. masSi Sedis iseTi kiTxvebi, rogoricaa,
magaliTad:
dagisignalebiaT Tu ara frustraciis gamosaxatad avtomanqaniT sacobSi
moxvedrisas?
A tipis individebi pasuxoben „xSirad“ an „TiTqmis yovelTvis“, xolo B tipis
individebi _ „xandaxan“ an „arasodes“.
iarnoldma da sxvebma gamoTqves hipoTeza, rom xazis sigrZis Sefasebisas A
tipis individebi ufro naklebad daeqvemdebarebodnen jgufis gavlenas da
arasodes xandaxan xSirad TiTqmis yovelTvis
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 17
wavidodnen kompromisze, vidre _ B tipis individebi. kvlevis am hipoTezis
Sesamowmeblad maT SearCies individTa ori jgufi: _ jenkinsis aqtiurobis
kiTxvaris qulebis mixedviT, ukiduresad A tipis individebi da ukiduresad B
tipis individebi. maTi monacemebiT, jenkinsis aqtiurobis kiTxvaris saSualo z
qula A tipis individebiTvis iyo +1.22 (dadebiTi z qula gviCvenebs A tipis qcevas),
xolo B tipis individebisTvis _ _1.38 (uaryofiTi z qula gviCvenebs B tipis qcevas).
TiToeuli jgufis saSualo z qulebis Sesaxeb informacia saSualebas gvaZlevs,
zustad ganvsazRvroT TiToeuli jgufis mdebareoba jenkinsis aqtiurobis
kiTxvaris A _ B tipis wrfeze. a-1 cxrilis (g) svetSi vnaxavT, rom standartul
normalur ganawilebaSi z-is mniSvnelobaTa mxolod .11 nawili tolia z = +1.22-is,
an aRemateba mas. amrigad, z = +1.22 A tipis qcevis jgufis saSualos qulaTa 89-e
procentilze ayenebs. analogiurad, z = _1.38-is SemTxvevaSi vxedavT, rom B tipis
wrfeze qulaTa mxolod .08 nawilia moTavsebuli am mniSvnelobis qvemoT.
ukiduresad A tipis qcevis da B tipis qcevis mqone individebi sxvadasxva jgufSi
Tavsdebodenen. miRebuli Sedegebi adasturebs hipoTezas: _ B tipis individebi 2-
jer metad eqvemdebarebian jgufis gavlenas da midian konsensusze, vidre A tipis
cdis pirebi.
SeamowmeT Tqveni codna VI.2
1. ganmarteT: _ standartuli normaluri gadaxra, standartuli normaluri
ganawileba, z, zrealuri.
2. X-is, μ-sa da σ-s TiToeuli mniSvnelobisTvis gamoTvaleT zrealuri. Semdeg
cxrilis Na-1 gamoyenebiT ipoveT farTobi: _ (1) z = 0-sa da z-s Soris; (2)
z-is zemoT.
3. amoxseniT Semdegi amocanebi „magaliTi N6-1“-
Si warmodgenili midgomis gamoyenebiT.
davuSvaT, Tqven CaatareT farTomasStabiani
kvleva studentebSi da ikvlieT verbaluri
analogiebiT msjelobis unari. vTqvaT,
testis qulebi normalurad aris
ganawilebuli, ganawilebis μ = 72 da σ = 8.
a) ganawilebis qulaTa ra wilia 86-is toli
an masze meti?
b) ganawilebis qulaTa ra wilia moTavsebuli 80-sa da 90-s Soris?
g) ganawilebis qulaTa ra wilia moTavsebuli 60-sa da 70-s Soris?
μ σ X
a. 73 4.2 81
b. 127 11.6 121
g. 17 1.3 18
d. 50 3.0 39
e. 5 0.4 5.7
v. 256 35.0 212
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 18
d) qulaTa ra diapazons moicavs testSi miRebul qulaTa ganawilebis
Sua 90%?
albaToba
Cven ganvixileT normaluri ganawileba, rogorc fardobiT sixSireTa
Teoriuli ganawileba, Tumca, is, amavdroulad, uwyveti cvladis albaTur
ganawilebasac warmoadgens. es cnebebi statistikis mniSvnelovan komponents _
albaTobas _ ukavSirdeba; Tumca, saxelmZRvanelos Semdgom TavebSi albaTobis
mxolod yvelaze mniSvnelovani aspeqtebia warmodgenili. am qveTavSi gavecnobiT
albaTobis cnebas, romelic saWiroa albaTuri ganawilebis gagebisa da
albaTobis statistikaSi gamoyenebisaTvis.
yoveldRiur cxovrebaSi ganxorcielebuli qmedebebi albaTobis intuitur
wvdomas uzrunvelyofs. magaliTad, migiRiaT rodesme qvemoT ganxilul magaliTSi
aRwerilis msgavsi werili?
saswrafo Setyobineba _ upasuxeT dauyovnebliv. Tqven moigeT qvemoT CamoTvlili
prizebidan erT-erTi:
principulad axali modelis manqana, fasi $36.000.
orkviriani ufaso mogzauroba havaize, fasi $6.000.
farTo ekraniani stereo televizori, fasi $1995.
mikrotalRovani Rumeli, fasi $375.
umaRlesi xarisxis fotoaparati, fasi $75.
kompiuteri, fasi $749.
mogebuli prizis wasaRebad Tqven ubralod unda mobrZandeT CvenTan da waiRoT
kuTvnili prizebi.
mdidruli manqanisa Tu orkviriani dasvenebis perspeqtivam SeiZleba
dagafiqroT, Tumca, lamazi ocneba gagiqrebaT, Tu waikiTxavT meore gverdze
wvrilad dabeWdil teqsts, sadac naxavT TiToeuli saCuqris mogebis Sanss,
romelic ase gamoiyureba:
davuSvaT, TiToeul monawiles
erTi prizis mogebis toli Sansi
aqvs. rogor fiqrobT, romeli
prizis mogeba gendomebodaT
yvelaze metad? romeli prizis
mogeba gendomebodaT yvelaze
naklebad? albaTobis Teoria
saerTod rom ar gqondeT naswavli,
prizi
saCuqrebis
raodenoba
(1,000,000 monawile)
1. manqana
2. dasveneba
3. televizori
4. mikrotalRovani Rumeli
5. fotoaparati
6. kompiuteri
1
20
34
66
999,845
34
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 19
Tqven, albaT, mainc ityodiT, rom manqanaze fiqrs azri ar aqvs da isev
fotoaparatis arCeva sjobia. milioni monawilidan mxolod erTi moigebs manqanas
maSin, roca milionidan 999,845-s Sexvdeba fotoaparati. cxadia, rom, manqanasTan
SedarebiT, ufro metia fotoaparatis mogebis Sansi.
davuSvaT, gekiTxebian, Tu ras udris albaToba imisa, rom televizori moigoT.
mosalodnelia, rom daaxloebiT ase upasuxebT: _ „34 Sansi milionidan“.
faqtobrivad, Tqven am pasuxiT miuTiTeT, rom konkretuli prizis mogebis
albaToba gamoiTvleba am prizis raodenobis gayofiT mosageb prizTa saerTo
raodenobaze, anu
raodenoba saerTo prizTa gasacem
raodenoba prizis konkretulialbaToba mogebis prizis konkretuli = .
Tu am formulas rigiT mesame prizis mimarT gamoviyenebT, miviRebT
televizoris mogebis albaTobas anu 34/1,000,000=.000034. Tu am formulas TiToeuli
prizis mimarT gamoviyenebT, maTTvis albaTobebis Semdeg ganawilebas miviRebT:
Tqven, albaT, SeamCnevT, rom prizis
mogebis albaTobis formula Zalian hgavs
fardobiTi sixSiris formulas, romelic me-3
TavSi ganvixileT. SegviZlia vTqvaT, rom es
formula fardobiTi sixSiris formulis
gamoyenebis specifikuri SemTxvevaa.
Tu prizebis ganawilebaSi TiToeul
SesaZlo xdomilobas Sedegs vuwodebT, maSin
am ganawilebaSi 1,000,000 SesaZlo Sedegi
gveqneba. TiToeuli Sedegi Tanabrad albaTuria da TiToeulis ganxorcielebis
albaToba 1/1,000,000 anu .000,001 iqneba. CvenTvis saintereso Sedegebis erTobliobis
ganxorcielebas xdomilobas uwodeben. magaliTad, Tu ganvixilavT me-5 prizis
mogebis xdomilobas, mosalodnelia, rom es xdomiloba 999,845 SemTxvevaSi
ganxorcieldes. Tu me-3 an me-4 prizis mogebis xdomiloba gvainteresebs, maSin
vnaxavT, rom me-3 prizisTvis 34 SemTxvevaa, me-4 prizisTvis ki _ 66, rac jamSi am
xdomilobis ganxorcielebis 100 SemTxvevas, SesaZlo Sedegs Seadgens. yuradReba
miaqcieT, rom me-5 prizis mogebis xdomiloba 1,000,000 SesaZlo SemTxvevidan 999,845
Sedegia, magram imave 1,000,000 mxolod 20 SemTxvevaSia mosalodneli me-2 prizis
mogeba. ufro zogadad, Tu populaciaSi xdomilobis ganxorcieleba ar aris
damokidebuli sxva xdomilobebis ganxircielebaze, maSin xdomilobis
ganxorcielebis albaToba (p) ganisazRvreba, rogorc
( )raodenoba saerTo axdomilobaT SesaZlo ipopulaciaS
raodenoba aSemTxvevaT xelSemwyob xdomilobis ipopulaciaSxdomiloba =p
prizi
saCuqrebis
raodenoba
(1,000,000 monawile)
mogebis
albaToba
1
2
3
4
5
6
1
20
34
66
999,845
34
.000001
.000020
.000034
.000066
.999845
.000034
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 20
simbolo p gamoiyeneba xdomilobis
ganxorcielebis albaTobis aRsaniSnad.
am formulis gamoyenebiT warmoebuli
gamoTvlis Sedegi, rogorc wesi,
aTwiladiT gamoixateba. magaliTad,
p(prizi 4) = .000066
miuxedavad amisa, albaTobebi
procentebiT an wiliTac SeiZleba
gamoisaxos. es formula gansazRvravs
diskretuli xdomilebebis Sedegebs,
anu iseTi xdomilebebis albaTobas,
romelTac aqvT mniSvnelobaTa
sasruli raodenoba, rogoricaa,
magaliTad, milion monawileze prizis
saxiT gacemul manqanaTa raodenoba. amrigad, TiToeuli prizis miRebis
albaTobebi qmnian diskretul albaTur ganawilebas.
diskretuli albaTuri ganawilebis kidev erTi magaliTis saxiT cxrilSi N3.3
mocemuli gamocdasTan dakavSirebuli SfoTvis 100 qula ganvixiloT. davuSvaT,
gvainteresebs, ramdenia albaToba imisa, rom am qulebidan SemTxveviT arCeuli
erTi qula moTavsebuli iqneba 45-sa da 49-s Soris SualedSi. es sakiTxi 45-sa da
49-s Soris qulis moxvedris xdomilobis albaTobis gansazRvras gulisxmobs.
amrigad, am xdomilobaSi moiazreba is Sedegebi, rodesac aRebuli qula iqneba 45,
46, 47, 48 da 49. cxomilobis am albaTobis gamosaTvlelad Cven unda vipovoT am
qulebTan dakavSirebuli Sedegebis saerTo raodenoba, risi gakeTebac am qulebis
cxrilSi N3.4 mocemuli ganawilebis mixedviT SegviZlia. cxrili gviCvenebs, rom
45-49 intervalSi 15 qulaa. vinaidan cnobilia, rom mocemul ganawilebaSi sul 100
qulaa, albaToba imisa, rom SemTxveviT aRebuli qula moTavsebuli iqneba 45-sa da
49-s Soris, aris
.15.100/15)4945( ==−− Soris s da sa qulap
aseve, SeiZleba vTqvaT, rom es albaToba:
.15.)4945( =≤≤ Xp
sadac ≤ _ niSnavs „naklebia an tolia“. ≤ simbolos win mdgomi ricxvi
naklebia an tolia ricxvis, romelic mosdevs am simbolos.
X _ SemTxveviT arCeuli qula.
mocemuli gamosaxuleba ase ikiTxeba: „albaToba imisa, rom X tolia an metia
45-ze (anu 45≤X) da tolia an naklebia 49-ze (anu X≤49) aris .15.“ es albaToba
pirdapir cxrilSi N3.4 SeiZleba moviZioT am qulebisTvis v svetSi mocemuli
Sedegi (outcome) _ albaTobis ganawilebaSi
TiToeuli SesaZlo varianti.
xdomiloba (event) _ albaTobis ganawilebaSi
Sedegebis konkretuli simravlis
ganxorcieleba.
xdomilobis ganxorcielebis albaToba
(probability of occurrence of en event – p) _
raodenoba saereTo
Sedegebis SesaZlo
raodeoba Sedegebis
iSemadgenel xdomilobis
xdomiloba =)(p
diskretuli Sedegebi (discrete outcomes) _
mniSvnelobebaTa sasruli raodenobis mqone
Sedegebi ganawilebaSi.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 21
fardobiTi sixSireebis gamoyenebiT, romelic gviCvenebs, rom 45-sa da 49-s Soris
moTavsebuli qulebis fardobiTi sixSire .15-ia. yuradReba miaqcieT, rom es
albaToba gamocdasTan dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebis
fardobiTi sixSiris histogramidan da fardobiTi sixSiris poligonidanac
(Sesabamisad, suraTi 3.5 da 3.6) SeiZleba vipovoT.
diskretuli albaTuri ganawilebis Tvisebebi
albaTobebis ganawilebas aqvs Tvisebebi, romelTac Tqven, albaT, ukve icnobT.
pirveli Tviseba isaa, rom xdomilebis albaToba meryeobs 0.00-dan +1.00-mde.
xdomileba, romlis albaToba udris nuls, mosalodnelia, rom namdvilad ar
ganxorcieldeba da xdomileba, romlis albaToba udris erTs, mosalodnelia,
rom namdvilad ganxorcieldeba. suraTze N6.9 warmodgenilia albaTobis skala da
individis cxovrebaSi zogierTi xdomilebis ganxorcielebis albaTobebi.
zogierTi xdomileba sruliad mosalodnelia cxovrebaSi da misi albaToba 1.00-is
tolia, zogierTi ki _ sruliad SeuZlebeli da maTi albaToba 0.00-is tolia.
albaTobis diapazoni 0.00-dan 1.00-mde Seesabameba fardobiTi sixSireebis
ganawilebis Tvisebas _ dagrovili fardobiTi sixSireebi (crƒ) icvleba 0-dan 1.0-
mde diapazonSi (ix. cxrili N3.4, sveti (i)).
meore Tviseba isaa, rom albaTur ganawilebaSi Sedegebis saerTo
raodenobasTan dakavSirebuli calkeuli albaTobebis jami 1.00-s Seadgens.
albaTuri ganawilebis es maxasiaTebeli iolad SeiZleba davinaxoT cxrilSi N3.4:
_ gamocdasTan dakavSirebuli SfoTvis kiTxvaris qulebis dagrovili fardobiTi
sixSireebi 1.00-s Seadgenen (ix. i sveti). Tu Tqven Lazy Acres Vacation Resort-s ewveviT,
prizis mogebis labaToba 1.00 iqneba.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 22
suraTi 6.9. albaTobebis cxrili da zogierTi xdomilebis ganxorcielebis albaToba.
albaToba
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
albaToba imisa, rom Tqven mokvdebiT.
.86 albaToba imisa, rom qali cocxali iqneba 60 wlis asakSi (yoveli 100 dabadebuli qalidan 86 cocxalia 60 wlis asakSi, p=86/100=.86).1
.81 albaToba imisa, rom mamakaci cocxali iqneba 60 wlis asakSi (yoveli 100 dabadebuli mamakacidan 81 cocxalia 60 wlis asakSi, p=81/100=.81).1
.64 albaToba imisa, rom aSS-is SemTxveviT SearCeul zrdasrul moqalaqes eqneba, sul mcire, erTi sakredito baraTi.2
albaToba imisa, rom agdebisas 50-TeTriani borjRalze dajdeba (p=1/2=.5)
.077 banqos dastidan valetis amoRebis albaToba (p=4/52=.077).
.000000005 totalizatorSi mogebis miaxloebiTi albaToba.3
1 1980 Commissioners Standard Ordinary (CSO) Mortality Table (The Wizards, 1989). 2 The Way, 1985. 3 Artis, 1993
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 23
mesame Tviseba isaa, rom
urTierTgamomricxavi Sedegebis momcveli
xdomilobis albaToba SeiZleba am
xdomilobis Semadgeneli Sedegebis albaTobebis SekrebiT vipovoT.
urTierTgamomricxavi Sedegebi is Sedegebia, romlebic ver ganxorcieldeba
erTdroulad. zemoT ganxilul magaliTSi individis mier me-2 da me-5 prizebis
mogeba urTierTgamomricxavi xdomilobebia. mas SeuZlia moigos erTi an meore
prizi, magram ara _ orive erTdroulad. analogiurad, gamocdasTan dakavSirebuli
SfoTvis kiTxvarSi individi erTdroulad ver miiRebs 53 da 67 qulas. es Sedegebi
urTierTgamomricxavia. magaliTad, davuSvaT, rom gvainteresebs, ramdenia
albaToba imisa, rom gamocdasTan dakavSirebuli SfoTvis kiTxvaris qulebis
ganawilebidan SemTxveviT aRebuli erTi qula 45 an masze meti iqneba? am kiTxvaze
pasuxis gasacemad cxrilis N3.4 e sveti unda vnaxoT, sadac martiv sixSireTa
ganawilebaa mocemuli. yuradReba miaqcieT, rom am svetSi 45-is toli an masze
meti 28 qulaa. es ricxvi e svetSi mocemuli, 45-is toli an masze meti qulebis
sixSireebis dajamebiTaa miRebuli (anu 15+5+3+2+1+2=28). amrigad, 45-is toli an
masze meti qulis SemTxveviT arCevis albaTobaa 28/100 anu .28. albaTobis tolobis
gamoyenebiT:
.28.)45( =≥Xp
es albaToba N3.4 cxrilidanac SeiZleba miviRoT: _ 45-is toli anda masze meti
qulebisTvis v svetSi mocemuli fardobiTi sixSireebis SekrebiT (anu
.15+.05+.03+.02+.01+.02=.28).
es magaliTi gviCvenebs, rom Tu ori an meti Sedegi urTierTgamomricxavia,
maSin erT-erTi SedegiT xdomilobis ganxorcielebis albaToba TiToeuli Sedegis
albaTobebis dajamebiT gamoiTvleba. albaTobis terminebiT Tu vityviT, Tu A da
B albaTuri ganawilebidan aRebuli ori urTierTgamomricxavi Sedegia, maSin
erTis an meoris ganxorcielebis albaToba A-s ganxorcielebis albaTobisa da B-s
ganxorcielebis albaTobis jamis tolia, anu:
).()()( BpApBAp +=an
es damokidebuleba cnobilia, rogorc aditiurobis wesi (anu damatebis,
Sevsebis) urTierTgamomricxavi Sedegebis albaTobisTvis.
am formulis gamoyenebis kidev erTi magaliTi: _ davuSvaT, isev SemTxveviTad
virCevT gamocdasTan dakavSirebuli SfoTvis kiTxvarSi miRebul erT qulas da
gvainteresebs, ramdenia albaToba imisa, rom moTavsebuli iqneba 20-sa da 24-s, an
70-sa da 74-s Soris. cxrilSi N3.4 mocemuli sixSireTa ganawilebis gamoyenebiT
vnaxavT, rom 20-sa da 24-s Soris, an 70-sa da 74-s Soris SemTxveviT qulis arCevis
albaToba 20-24 intervalSi moTavsebuli qulebis fardobiTi sixSirisa (anu .01-is)
urTierTgamomricxavi Sedegebi (Mutually
exclusive outcomes) _ Sedegebi, romelic ver
ganxorcieldeba erTdroulad.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 24
da 70-74 intervalSi moTavsebuli qulebis fardobiTi sixSiris (anu .02-is) jamia
anu sruli albaToba .03-s Seadgens. albaTobis gamosaxulebiT:
.03.02.01.)70()2420()702420(
=+=−−+−−=
=−−−−
mde74 danmde dan
qula arCeuli Soris s74 da sa an s, da sa
ppp
aditiurobis es wesi Zalze mniSvnelovani gaxdeba SemdgomSi _ am Tavis
bolosa da Semdeg TavebSi, sadac hipoTezebis Semowmebaze iqneba saubari.
Teoriuli (klasikuri) da empiriuli albaToba
xdomilebis albaTobis gansazRvris ori gza arsebobs: _ Teoriuli da
empiriuli. zogierTi xdomilobisTvis Cven SegviZlia davaskvnaT misi xelSemwyob
SemTxvevaTa albaTobis Sesaxeb. magaliTad, 50-TeTriani monetis agdebisas
borjRalis gamosvlis Sesaxeb SeiZleba msjelobiT davaskvnaT. 50-TeTriani
monetis agdebisas ori SesaZlo Sedegi gvaqvs: _ borjRalis an safasuris gamoCena.
Tu moneta sworia, TiToeuli maTgani erTnairad mosalodnelia. Sesabamisad,
borjRalis mosvliT dasrulebuli erTi agdebis albaToba, ori SesaZlo
Sedegidan, erT-erTia, anu p(borjRalis mosvla) = 1/2 = .5. Teoriuli albaTobis
modelebi xSirad am magaliTze gacilebiT rTulia. Semdeg qveTavSi ganvixilavT
standartul normalur ganawilebas, rogorc albaTur Teoriul ganawilebas.
albaTuri Teoriuli ganawileba (Theoretical probability distribution) _ Teoriuli albaTobis
modelis gamoyenebiT miRebuli albaTuri ganawileba.
albaTuri klasikuri ganawileba (Empirical probability distribution) _ xdomilobis realurad
ganxorcielebis sixSiris daTvliT miRebuli albaTuri ganawileba.
zogierTi xdomilebis SemTxvevaSi Teoriulad ver vimsjelebT misi
ganxorcielebis albaTobis Sesaxeb; magaliTad, Teoriuli msjelobiT ver
davaskvniT, Tu ras udris albaToba imisa, rom Tu qalbatoni xarT, Tqven
cocxali iqnebiT 60 wlis asakSi. am albaTobis gamosaTvlelad gamoviyenebT
xdomilobis ganxorcielebis xangrZlivi SeswavliT miRebul empiriul
albaTobebs. magaliTad, sicocxlis dazRvevis industriaSi aSS-is moqalaqeTa
sicocxlis xangrZliobis sixSiris gansasazRvrad specialuri komisiis mier
SemuSavebul bunebrivi sikvdilianobis standartul cxrilebs iyeneben (The Wizards,
1989). es cxrili empiriulad aris Sedgenili xangrZlivi drois manZilze
sxvadasxva asakis mamakacebisa da qalebis sikvdilianobis sixSiris Canawerebis
warmoebis Sedegad. cxrili gviCvenebs, rom yoveli 10,000,000 cocxali qalidan
8,603,801 jer kidev cocxalia 60 wlis asakSi. amrigad, dabadebisas albaToba imisa,
rom aSS-is moqalaqe qali jer kidev cocxali iqneba 60 wlis asakSi, iqneba
p(60 wlis asakis cocxali qali) 86.000,000,10801,603,8
== .
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 25
cxadia, p-is es mniSvneloba Seicvleba, Tu Seicvleba sikvdilianobis sixSire
janmrTelobis dacvisa da kvebis gaumjobesebis Sedegad.
bihevioristul mecnierebaSi aris orive (Teoriuli da empiriuli) albaTobis
gamoyenebis SemTxvevebi, Tumca, winamdebare saxlmZRvaneloSi aqcenti keTdeba iseT
Teoriul ganawilebaze, rogoricaa standartuli normaluri ganawileba.
Semdgom TavebSi statistikuri daskvnis iseT problemebs SevexebiT rogoricaa
populaciidan aRebuli SerCevidan miRebul sazomebze dayrdnobiT am populaciis
Sesaxeb daskvnebis gakeTeba; Tumca, aseTi daskvnebis gamotanisas verasodes
viqnebiT absoluturad darwmunebulni maT sisworeSi; amrigad, Cveni daskvnebi
albaTobis terminebiT iqneba formulirebuli.
standartuli normaluri ganawileba, rogorc albaTuri ganawileba
normaluri ganawileba fardobiTi sixSireebis Teoriuli ganawilebaa. vinaidan
normalur ganawilebaSi warmodgenili fardobiTi sixSireebi populacias
efuZneba, normaluri ganawileba albaTobebis ganawilebac aris; Tumca, is odnav
gansxvavebuli albaTuri ganawilebaa, vidre axlaxans ganxiluli diskretuli
albaTuri ganawileba. normalur ganawilebaSi Semavali X qula uwyveti cvladia,
romelmac SeiZleba miiRos mniSvnelobaTa usasrulo raodenoba Tavisi
cvalebadobis diapazonSi. magaliTad, davuSvaT, rom gvaqvs sxeulis wonis
amsaxveli qulebis Teoriuli normaluri ganawileba. ras udris albaToba imisa,
rom vipoviT individs, romelic zustad 123 funts (58 kg-s) iwonis? wona uwyveti
cvladia da zogierTi adamiani SeiZleba iwonides 123.3 funts, 123.36 funts, an
123.364 funtsac ki. Teoriulad, Cven SeiZleba iseTi individic vipovoT, romelic
123.3642193 funts iwonis, Tu SevZlebT ase zustad gazomvas. SeiZleba, rom 123.3
funti ganvixiloT, rogorc 123 funti? am TvalsazrisiT, ra SeiZleba iTqvas
123.00001 funtze? principSi, Cven verasodes miviRebT zustad 123 funts, Tu
yovelTvis ufro zustad gazomvas vcdiT da yovel jerze vnaxavT, rom gazomvis
Sedegi ar aris zustad 123 funts.
Tu am mimarTulebiT vimsjelebT, uwyveti cvladis zusti mniSvnelobis miRebis
albaToba nulis toli iqneba, magram garkveul intervalSi monacemis moxvedris
albaToba ar aris nulis toli. miuxedavad imisa, rom ar SegviZlia zustad 123
funtis miReba, SegviZlia miviRoT wona 122.5 funtsa da 123.5 funts Soris
intervalSi. magaliTad, ganurCevlad imisa, aTwiladSi mZimis Semdeg ramdeni
Tanrigis sizustiT SegviZlia gazomva, 123.36 funti (anda 123.3642193 funtic ki)
yovelTvis moxvdeba 122.5 funtsa da 123.5 funts Soris intervalSi. amrigad,
normaluri ganawilebis saxiT mocemuli albaTobis funqcia uzrunvelyofs ara X-
is diskretuli mniSvnelobebis albaTobebs, aramed imis albaTobas, rom X-is
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 26
mocemuli mniSvneloba moxvdeba konkretul intervalSi. albaToba gamoisaxeba
ganawilebis qveS mdebare farTobiT, romelic SemosazRvrulia intervaliT.
normaluri ganawilebis albaTuri ganawilebis saxiT gamoyenebis
sailustraciod ganvixiloT Semdegi amocana. davuSvaT, iciT, rom qulaTa
erToblioba normalurad aris ganawilebuli da ganawilebis μ = 150 da σ = 8. Tu
Tqven ganawilebidan SemTxveviT airCevT qulas, ras udris albaToba imisa, rom
aRebuli qula moTavsebuli iqneba 150-sa da 158-s Soris intervalSi? am kiTxvaze
pasuxis gasacemad, pirvel rigSi, mocemuli ganawileba unda gardavqmnaT
standartul normalur ganawilebad, romlis μ = 0 da σ = 1; amisaTvis viyenebT
formulas z = (X_ μ)/ σ. amrigad, 150 qula gaxdeba z = 0, xolo 158 qula _ z = +1.0.
Sesabamisad, dasmuli kiTxva gadaiqceva kiTxvad imis Sesaxeb, Tu ramdenia z = 0-sa
da z = +1.0-s Soris intervalSi moTavsebuli qulis miRebis albaToba. am kiTxvis
pasuxia standartuli normaluri ganawilebis qveS mdebare da z = 0-sa da z = +1.0-s
Soris intervaliT SemosazRvruli farTobi anu .3413 (ix. a-1 cxrilis (b) sveti).
Sesabamisad, 150-sa da 158-s Soris intervalSi moTavsebuli qulis miRebis
albaTobaa .3413.
magaliTi N6.2
standartuli normaluri ganawilebis rogorc albaTuri ganawilebis
gamoyeneba
davuSvaT, Tqven gaqvT magaliTSi N6.1 aRwerili normalurad ganawilebuli
populacia, romlis μ = 80 da σ = 5. qvemoT mocemuli amocana N6.1 magaliTSi
mocemuli amocanis msgavsia, Tumca, aq mogeTxovebaT ara qulaTa wilis, aramed
maTi albaTobebis gamoTvla. amocanis amoxsna gagiadvildebaT, Tu magaliTis N6.1
Tanmxleb suraTs mimarTavT.
amocana
ras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula 88-is
toli an masze meti iqneba?
saWiro statistika: ( ) σµ /−= Xzrealuri
amoxsna
88 qula unda gadaviyvanoT zrealuri-Si. Semdeg a-1 cxrilis gamoyenebiT vipoviT
qulaTa wils, romelic tolia an metia zrealuri-ze. es wili warmoadgens saZiebel
albaTobas. zrealuri-is formulaSi ricxviTi mniSvnelobebis Setanis Semdeg
miviRebT:
6.158
58088
+==−
=realuriz .
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 27
a-1 cxrilis (g) sveti gviCvenebs, rom standartuli normaluri ganawilebis
qveS mdebare farTobi, romelic tolia an metia z = +1.6-ze, aris .0548. amrigad,
albaToba imisa, rom qula toli an meti iyos 88-ze, aris .0548 anu, Tu albaTobis
saxiT CavwerT, p(X≥88) = .0548.
amocana
ras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula
moTavsebuli iqneba 83-sa da 87-s Soris?
amoxsna
orive qula _ 83 da 87 unda gadaviyvanoT zrealuri qulebSi. Semdeg vpoulobT
standartuli normaluri ganawilebis qveS farTobs zrealuri-is or mniSvnelobas
Soris (ix. suraTi N6.6). ricxviTi mniSvnelobebis Setanis Semdeg miviRebT:
zrealuri 83-isTvis 6.053
58083
+==−
= ;
zrealuri 87-isTvis 4.157
58087
+==−
= .
cxrilis Na-1 (b) svetidan vxedavT, rom zrealuri = +0.6-isTvis z = 0-sa da z =
+0.6-s Soris intervalSi moTavsebul qulaTa wilia .2257. zrealuri = +1.4-isTvis z =
0-sa da z = +0.6-s Soris intervalSi moTavsebul qulaTa wilia .4192. ganawilebis
farTobi z-is am or mniSvnelobas Soris gvaZlevs 83-sa 87-s Soris moTavsebul
qulaTa wils. am farTobis misaRebad .4192-s vaklebT .2257-s da viRebT .1935-s.
miRebuli wili warmoadgens saZiebel albaTobas. amrigad, 83-sa da 87-s Soris
intervalSi qulis moxvedris albaTobaa .1935; albaTobis terminebiT,
p(83≤X≤87)=.1935.
amocana
ras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula
moTavsebuli iqneba 65-sa da 74-s Soris?
amoxsna
orive qula _ 65 da 74 unda gadaviyvanoT zrealuri qulebSi. Semdeg vpoulobT
standartuli normaluri ganawilebis farTobs zrealuri-is or mniSvnelobas Soris
(ix. suraTi N6.7). ricxviTi mniSvnelobebis Setanis Semdeg miviRebT:
zrealuri 65-isTvis 0.3515
58065
−=−
=−
= ;
zrealuri 74-isTvis 2.156
58074
−=−
=−
= .
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 28
Tu vnaxavT a-1 cxrilis (b) svets da gavixsenebT, rom standartuli normaluri
ganawilebis simetriulobidan gamomdinare, erTnairi mimarTebaa z-is rogorc
dadebiTi, ise _ uaryofiTi qulebis SemTxvevaSi, vnaxavT, rom z = 0-sa da z = _3.0-s
Soris SualedSi moTavsebulia qulaTa .4987 nawili. zrealuri = _1.2-isTvis ki
qulaTa .3849 nawili moTavsebulia z = 0-sa da z = _1.2-s Soris SualedSi.
Sesabamisad, Cven miviRebT z = _3.0-sa da z = _1.2-s Soris moTavsebul qulaTa wils,
Tu .4987-s gamovaklebT .3849-s, rac .1138-is tolia. es wili warmoadgens saZiebel
albaTobas; aqedan gamomdinare, p(65≤X≤74)=.1138.
SeamowmeT Tqveni codna VI.3
1. ganmarteT: _ diskretuli xdomileba, Sedegi, empiriuli albaToba,
xdomiloba, urTierTgamomricxavi Sedegebi, xdomilobis albaToba,
Teoriuli albaToba.
2. romeli formuliT gamoiTvleba diskretuli xdomilobis ganxorcielebis
albaToba?
3. velosipedis mosagebad SeiZineT latareis bileTi. bileTidan SeityveT,
rom yovel 700 gayidul bileTze gaTamaSdeba mxolod erTi velosipedi.
ras udris albaToba imisa, rom Tqven pirvelive gaTamaSebisas moigoT
velosipedi?
4. X-is, μ-sa da σ-s qvemoT mocemuli TiToeuli mniSvnelobisTvis
gamoTvaleT z-is mniSvneloba. Semdeg a-1 cxrilis gamoyenebiT moZebneT
albaToba imisa, rom qula toli an meti
iqneba mocemul qulaze.
5. Tqven CaatareT farTomasStabiani kvleva
studentebSi da ikvlevdiT verbaluri
analogiebiT msjelobis unars. vTqvaT,
testis qulebi normalurad aris
ganawilebuli, ganawilebis μ = 72 da σ = 8.
a) ras udris albaToba imisa, rom
ganawilebaSi Semavali qula 84-is toli an masze meti iqneba?
b) ras udris albaToba imisa, rom ganawilebaSi Semavali qula
moTavsebuli iqneba 78-sa da 86-s Soris?
g) ras udris albaToba imisa, rom ganawilebaSi Semavali qula
moTavsebuli iqneba 64-sa da 870-s Soris?
standartuli qulebi
μ σ X
a. 73 4.2 81
b. 127 11.6 121
g. 17 1.3 18
d. 50 3.0 39
e. 5 0.4 5.7
v. 256 35.0 212
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 29
Cven vnaxeT, rom SegviZlia normalurad ganawilebuli nedli qulis aReba da
misi gadayvana standartuli normaluri ganawilebis qulad Semdegi formulis
gamoyenebiT:
σµ−
=Xz .
z aris standartuli qula, romlis gamoyeneba SesaZlebelia ori sxvadasxva
normaluri ganawilebidan aRebuli qulebis Sesadareblad. magaliTad, vTqvaT,
individma erT testSi miiRo 75 qula, romlis μ = 70 da σ = 5, xolo meore testSi,
romlis μ = 100 da σ = 10, miiRo 110 qula. davuSvaT, rom orive testSi miRebuli
qulebi normalurad aris ganawilebuli. TiToeuli qulisTvis (75 da 110) zrealuri
aris +1. orive testSi individma miiRo saSualoze erTi standartuli gadaxriT
meti qula. Sesabamisad, misi mdebareoba sxvebis mier miRebul qulebTan
mimarTebaSi erTnairi iqneba orive testis SemTxvevaSi: _ man miiRo saSualoze
erTi standartuli gadaxriT meti qula.
SerCevidan miRebuli qulebis msgavsi gardaqmnis ganxorcieleba maSinac
gamosadegia, rodesac qulebi ar aris normalurad ganawilebuli, rac Semdegi
formulis gamoyenebiT aris SesaZlebeli:
SXXz −
= ,
sadac X = saZiebeli qula.
X = qulaTa SerCevis saSualo.
S = SerCevis standartuli gadaxra, romelic, Tavis mxriv,
( )N
XXS ∑ −=
2
.
am formuliT gamoTvlil z-is
mniSvnelobas xSirad standartul qulas
uwodeben. es formula qulas gardaqmnis
ricxvad, romelic gviCvenebs, Tu ramdeni standartuli gadaxriT aris
daSorebuli mocemuli qula SerCevis saSualodan. amis sailustraciod, davuSvaT,
rom xuT students miveciT ori _ statistikis da istoriis _ testi da miviReT
N6.4 cxrilSi mocemuli nedli da z qulebi. standartuli qulebi miRebul iqna
zemoT mocemuli formulis gamoyenebiT. magaliTad, anas mier statistikaSi
miRebuli nedli qulaa 91. testis saSualo iyo 72.0, xolo S _ 14.2. amitom anas z
qula iqneba
standartuli qula (Standard score): _ qula,
romelic miiReba formuliT ( ) SXXz /−= .
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 30
34.12.140.19
2.140.7291
+==−
=−
=S
XXzrealuri
rodesac viciT standartuli
qula, warmodgena gvaqvs klasis
saSualosTan anas qulis
mimarTebaze. es aris 1.34
standartuli gadaxra saSualos
zemoT.
standartuli qulebis
gamoyenebiT Cven martivad
SevadarebT erTmaneTs or
sxvadasxva testSi miRebul
qulebs, risi gakeTebac
SeuZlebelia nedli qulebis
SemTxvevaSi. magaliTad, joni statistikis testis mixedviT, moTavsebulia
saSualodan 1.06 standartuli gadaxriT zemoT, xolo istoriis testis mixedviT _
0.50 standartuli gadaxriT qvemoT SerCevis saSualodan. dadebiTi standartuli
qulebi gviCvenebs SerCevis saSualos zemoT mdebare qulebs, nulis toli
standartuli qulis (rogoricaa daviTis mier istoriaSi miRebuli qula)
SemTxvevaSi qula SerCevis saSualos tolia, xolo uaryofiTi standartuli
qulebi migviTiTebs SerCevis saSualos qvemoT mdebare qulebze; Tumca, yuradReba
unda miaqcioT, rom vinaidan es z SerCevis qulebidan aris miRebuli, mas ar eqneba
igive Tvisebebi normaluri ganawilebis TvalsazrisiT, rac z = (X_μ)/σ formuliT
miRebul z-s axasiaTebs. miuxedavad amisa, standartuli qulebi xSirad gamoiyeneba
testis Sedegebis warmodgenisas, rodesac aRiwereba individis fardobiTi
mdgomareoba im sxva individebTan mimarTebaSi, vinc igive testi gaiara. unda
aRiniSnos, rom standartul qulebSi gadayvana ar cvlis qulaTa ganawilebis
formas. Tu nedli qulebis ganawileba asimetriulia, maSin standartuli qulebis
ganawilebac asimetriuli iqneba.
Sejameba
• normaluri ganawileba fardobiTi sixSireebis Teoriuli ganawilebaa.
gazomvis Sedegad miRebul qulaTa arc erTi erToblioba ar aris zustad
normalurad ganawilebuli, Tumca, xSirad gamosadegia imis daSveba, rom
qcevis gazomvis zogierTi Sedegi daaxloebiT normalurad aris
ganawilebuli.
cxrili 6.4. xuTi studentis mier statistikisa
da istoriis testebSi miRebuli nedli qulebi
da standartuli qulebi.
testi
statistika istoria
pirovneba nedli
qula z qula
nedli
qula z qula
amalia
denisi
jefi
hansi
maTTi
91
62
87
56
64
+1.34
-0.70
+1.06
-1.13
-0.56
93
79
73
90
60
+1.18
0.00
+0.50
+0.92
-1.60
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 31
• normaluri ganawilebidan aRebuli qula SeiZleba standartuli normaluri
ganawilebis, romlis µ=0 da σ=1, qulad gardaiqmnas. amisaTvis gamoiyeneba
formula
z = (X_μ)/σ.
• diskretuli xdomilebis albaToba gamoiTvleba formuliT
raodenoba saerTo axdomilobaT ipopulaciaS
raodenoba aSemTxvevaT xelSemwyob xdomilobis ipopulaciaSxdomiloba =)(p
• nebismier albaTuri ganawilebisTvis Sedegis albaToba .00-sa da 1.00-s Soris
unda iyos moTavsebuli.
• calkeuli albaTobebis jami, romlebic albaTur ganawilebaSi Sedegebis
saerTo raodenobas ukavSirdeba, 1.00-s Seadgens.
• Tu A da B urTierTgamomricxavi Sedegebia, maSin ).()()( BpApBAp +=an
• standartuli normaluri ganawileba albaTuri ganawilebaa, romelic
gviCvenebs ganawilebis garkveul intervalSi qulebis moxvedris
albaTobebs. misi gamoyeneba SesaZlebelia normalurad ganawilebuli
qulebis SemTxvevaSi rogorc konkretul intervalSi moxvedrili qulebis
wilis, ise _ intervalebSi maTi moxvedris albaTobebis gansasazRvrad.
• SerCevidan miRebuli nedli qulebis standartul qulebSi gadayvana
SesaZlebelia formuliT
SXXz −
= .
es formula qulas gardaqmnis ricxvad, romelic gviCvenebs, Tu ramdeni
standartuli gadaxriT aris is daSorebuli saSualosgan. am dros ar
icvleba nedli qulebis ganawilebis forma.
ZiriTadi terminebi da simboloebi
asimptoturi ganawileba
albaToba
diskretuli xdomiloba
diskretuli xdomilobis
empiriuli albaToba
empiriuli albaTuri
ganawileba
Teoriuli albaToba
Teoriuli albaTuri
ganawileba
normaluri ganawileba
parametri
populacia
simetriuli ganawileba
standartuli normaluri
gadaxra
standartuli normaluri
ganawileba
standartuli qula
statistika
urTierTgamomr
icxavi
Sedegebi
uwyveti
ganawileba
Sedegebi
SerCeva
xdomiloba
xdomilobis
ganxorcie
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 32
lebis albaToba
μ
σ
s
X
z
zrealuri
kiTxvebi
1. inteleqtis testebSi xSirad Sedis ricxvebis Tanmimdevrobis
dalagebis, instruqciebis Sesrulebis, leqsikonis, analogiebis
danaxvis, suraTis dasrulebis, logikuri msjelobisa da
maTematikuri gamoTvlebis warmoebis unarebis Sefaseba. axseniT,
ratom aris mosalodneli, rom inteleqtis testis qulebi
populaciaSi daaxloebiT normalurad iqneba ganawilebuli.
2. reaqciis (anu gamRizianeblis miwodebasa da masze reaqciis
ganxorcielebas Soris) drois gazomvis Sedegebi, rogorc wesi, ar
aris normalurad ganawilebuli. metic, miRebul qulaTa ganawilebas
xSirad Sebrunebuli J-is forma aqvs, sadac qulaTa umravlesoba
SedarebiT mcirea, umciresoba ki _ didi. axseniT, ratom aris
mosalodneli, rom qulebi swored ase iqneba ganawilebuli.
3. wlebis manZilze dagrovda zrdasrulTa veqsleris inteleqtis
skalis monacemebi. davuSvaT, miRebuli qulebi normalurad aris
ganawilebuli da ganawilebis μ = 100 da σ = 15.
a) inteleqtis testis qulaTa ra nawili iqneba 115-is toli an masze
meti?
b) inteleqtis testis qulaTa ra wili iqneba 70-ze naklebi?
g) inteleqtis testis qulaTa ra wili iqneba 60-ze naklebi?
d) inteleqtis testis qulaTa ra wili iqneba moTavsebuli 100-sa da
120-s Soris?
e) inteleqtis testis qulaTa ra wili iqneba moTavsebuli 75-sa da
100-s Soris?
v) romel intervalSi iqneba moTavsebuli testis qulaTa Sua 80%?
z) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,
rom misi inteleqtis qula moTavsebuli iyos 90-sa da 120-s
Soris?
T) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,
rom misi inteleqtis qula 80-is toli an masze naklebi iyos?
i) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,
rom misi inteleqtis qula 125-is toli an masze meti iyos?
k) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,
rom misi inteleqtis qula 140-is toli an masze meti iyos?
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 33
4. Tqven sxva 64 adamianTan erTad gamogiZaxes sasamarTloSi, rogorc
nafici msajulobis erT-erTi kandidati. nafic msajulTa SesarCevad
furclebze dawerili saxelebi yuTSi Cayares da saTiTaod amoiRes.
a) ras udris albaToba imisa, rom pirvelive jerze Tqveni saxeli
amoiRon?
b) Tu Tqveni saxeli ar aRmoCnda pirvel rva amoRebul furcelze,
ras udris albaToba imisa, rom mecxre amoRebul furcelze
Tqveni saxeli eweros, Tu gaviTvaliswinebT, rom pirveli rva
amoRebuli furceli aRar daabrunes yuTSi?
5. nacionalurma gazeTma USA Today-m gamoaqveyna statia saTauriT
„Sansebi“ (31 dekemberi, 1985), sadac mocemuli iyo Semdegi wlis
manZilze individis cxovrebaSi garkveuli xdomilebebis
ganxorcielebis Sansebi. qvemoT CamoTvlili xdomilebebisTvis
gamoTvaleT TiToeuli maTganis ganxorcielebis albaToba.
a) doRis dros mravali firma iRebs fsonebs. erT-erTi garbenze
yovel 3000 ganacxadze erTi prizi gaicemoda. ras udrida
albaToba imisa, rom am garbenze nebismieri erTi monawile
moigebda prizs?
b) bolo wlebi mZime gamodga fermerebisTvis. weliwadSi yoveli
5314 fermeridan erTi tovebda Tavis bizness. ras udris
albaToba imisa, rom mimdinare wels fermerma miatovos biznesi?
g) mraval Cvengans uocnebia popularuli romanis daweraze, Tumca,
gamomcemlobaSi erTi wlis manZilze Sesuli yoveli 4637 wignis
moculobis mqone xelnaweridan mxolod erTi ibeWdeba. Tqven
dawereT romani. ras udris albaToba imisa, rom is daibeWdeba
mimdinare wels?
6. 6-1 magaliTSi Cven ganvixileT iarnoldisa da sxvebis (Yarnold et al.,
1986) kvleva, A da B tipis individebis Sedarebis Sesaxeb. A da B
tipis individebis misaRebad iarnoldma da misma kolegebma cdis
pirebi jenkinsis aqtiurobis kiTxvaris qulebis mixedviT SearCies.
davuSvaT, rom jenkinsis aqtiurobis kiTxvarSi populaciis saSualoa
0 da populaciis standartuli gadaxra _ 10. vTqvaT, Tqven
dainteresdiT da ganagrZeT iarnoldis kvleva. A tipis cdis
pirebidan Tqven gsurT iseTi cdis pirebis SerCeva, romelTac
jenkinsis aqtiurobis kiTxvarSi miRebuli aqvT +13-ze meti qula,
xolo B tipis cdis pirebidan _ visac miRebuli aqvT -13-ze naklebi
qula. davuSvaT, jenkinsis aqtiurobis kiTxvaris qulebi normalurad
aris ganawilebuli.
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 34
a) ras udris albaToba imisa, rom SemTxveviT arCeul individs
jenkinsis aqtiurobis kiTxvarSi eqneba +13-ze meti qula?
b) ras udris albaToba imisa, rom SemTxveviT arCeul individs
jenkinsis aqtiurobis kiTxvarSi eqneba -13-ze naklebi qula?
g) ras udris albaToba imisa, rom SemTxveviT arCeul individs
jenkinsis aqtiurobis kiTxvarSi eqneba -13-sa da+13-s Soris
intervalSi moTavsebuli qula?
7. davuSvaT, amerikeli 21-30 wlis mamakacebis saSualo wonaa 156 funti
(70.36 kg), standartuli gadaxra ki _ 8 funti (4.63 kg).
a) 21-30 wlis mamakacTa ra nawili iwonis 144 funts (65 kg) an ufro
naklebs?
b) 21-30 wlis mamakacTa ra nawili iwonis 170 funts (77 kg) an ufro
mets?
g) 21-30 wlis mamakacTa ra nawilis wona iqneba 140-dan (63.5 kg) da
172 funtamde (78 kg)?
d) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba
imisa, rom misi wona iyos 144-dan (65 kg) 168 funtamde (76 kg)?
e) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba
imisa, rom misi wona 150 funti (68 kg) an ufro naklebi iyos?
v) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba
imisa, rom misi wona 160 funti (72.6 kg) an ufro meti iyos?
8. davuSvaT, ikvlieT eqvsi studenti; gainteresebdaT kviraSi video
TamaSebze saSualod daxarjuli dro (saaTebSi) da semestruli
niSnebis saSualo. vTqvaT, miiReT Semdegi monacemebi:
a) gamoTvaleT saSualo da SerCevis
standartuli gadaxra qulaTa
TiToeuli jgufisTvis da Semdeg
TiToeuli miRebuli qula
gadaiyvaneT standartul qulaSi.
b) lelas video TamaSebisTvis
daxarjuli drois standartuli
qulaa nuli. ras geubnebaT es qula
yoveli kviris manZilze video TamaSebisTvis daxarjuli drois Sesaxeb?
g) ra informacias gawvdiT ias semestruli niSnebis saSualos
standartuli qula misi semestruli niSnebis saSualos Sesaxeb?
d) ras gviCvenebs semestruli niSnebis saSualos uaryofiTi
standartuli qula?
studenti
video
TamaSebze
daxarjuli
dro
semestruli
niSnebis
saSualo
aleqsi loreni ivona lavali boni iasoni
6 8 0 15 3 16
2.5 2.0 3.9 3.4 3.1 1.3
თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 35
e) ras gviCvenebs video TamaSebze daxarjuli drois dadebiTi
standartuli qula?
9. cota xnis win literaturasa da istoriaSi gamocda CaabareT. Tqveni
niSnebi da klasis saSualo da standartuli gadaxra (S) aseTia:
literatura istoria
Tqveni qula
klasis saSualo
klasis S
88
83
6
70
65
5
TanaklaselebTan SedarebiT romel gamocdaSi gaqvT ukeTesi Sedegi?
ratom?
10. cxrilSi warmodgenilia aSS-is Census Bureau-is (marti, 1998) monacemebi,
romlis mixedviTac, ojaxSi wevrebis raodenobis fardobiTi
sixSireebi Semdegnairad nawildeba:
a) Tu SemTxveviT avarCevT erT
ojaxs, ramdenia albaToba imisa,
rom masSi zustad oTxi wevri
iqneba?
b) Tu SemTxveviT avarCevT erT
ojaxs, ramdenia albaToba imisa,
rom masSi eqvsi an meti wevri
iqneba?
11. qvemoT mocemulia aSS-Si 1998 wel amerikeli qmrebis asakis sixSireTa
ganawileba (U.S. Census Bureau, 1998, marti).
a) Tu SemTxveviT avarCevT erT ojaxs,
ramdenia albaToba imisa, rom qmari
44 wlis an ufro axalgazrda iqneba?
b) Tu SemTxveviT avarCevT erT ojaxs,
ramdenia albaToba imisa, rom qmari
34 wlis an ufri axalgazrda, anda 75
wlis da ufro xnovani iqneba?
ojaxis
wevrTa
raodenoba
fardobiTi
sixSire
2 3 4 5 6
7 da meti
.427
.230
.206
.092
.029
.016
asaki fardobiTi
sixSire
24 an naklebi 25-34 35-44 45-54 55-59 60-61 62-64 65-69 70-74 75-79 80-84
85 an meti
.023
.178
.257
.217
.079
.029
.040
.062
.049
.039
.019
.008
Tavi VII. statistikis gamoyeneba daskvnisa da
SefasebisTvis
X rogorc μ-s wertilovani Sefaseba
Caunacvlebeli Sefaseba
Zalmosili Sefaseba
Sefasebis sizuste
populaciis dispersiisa da standartuli gadaxris
Sefaseba
dispersia
standartuli gadaxra
saSualos SerCeviTi ganawileba: _ Sefasebis sizustis
gansazRvra
saSualos SerCeviTi ganawilebis Tvisebebi
saSualos standartuli Secdomis Sefaseba
faqtorebi, romlebic gavlenas axdenen Xs -is
mniSvnelobaze
Xs -is gamoyeneba
ganawilebis tipebis mimoxilva
populaciis saSualos intervaluri Sefaseba
ndobis intervalis ageba
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 2
winamdebare TavSi ganvixilavT populaciis parametrebis SefasebisTvis
aRweriTi statistikis gamoyenebas. rogorc pirvel TavSi aRvniSneT,
qcevaTmecnierebebis warmomadgenlebs xSirad surT SerCevaSi miRebuli Sedegebis
ufro farTo populaciaze ganzogadeba. magaliTad, saarCevno kampaniaSi CarTul
organizaciebs surT arCevnebis Sedegebis prognozireba, magram isini ver
gamokiTxaven amomrCevelTa did populaciaSi Semaval yvela amomrCevels. isini
zomaven amomrCevelTa populaciidan aRebul SerCevas da Semdeg axdenen miRebuli
Sedegebis populaciaze ganzogadebas. analogiurad, bevr SemTxvevaSi SeuZlebelia
did populaciaSi alkoholis moxmarebis, Sobadobis kontrolisadmi
damokidebulebis, avtomanqanaSi usafrTxoebis Rvedebis gamoyenebis an nebismieri
sxva qcevis gazomva. Sefaseba unda moxdes sakvlevi populaciidan aRebuli
SerCevidan miRebuli statistikebis safuZvelze. aRweriTi statistikebis
safuZvelze populaciis parametrebis SefasebisTvis saWiroa SerCevis monacemebze
dayrdnobiT daskvnebis gakeTeba. daskvna aris raRac cnobilis safuZvelze raRac
ucnobze msjelobis procesi. statistikuri daskvna aris populaciis ucnobi
parametrebis Sefaseba SerCevis cnobil statistikebze dayrdnobiT. winamdebare
TavSi Tavdapirvelad ganvixilavT SerCevis saSualos ( )X , rogorc populaciis
saSualos (μ) Sefasebis, Tvisebebs; Semdeg gavecnobiT populaciis dispersiasa (s2)
da standartul gadaxras (s), rogorc, Sesabamisad, populaciis dispersiisa (σ2) da
standartuli gadaxris (σ) Sefasebebs; bolos ki SevexebiT X -is mier μ-s
Sefasebis sizustis sakiTxs.
statistikuri daskvna (Statistical inference) _ SerCevisgan miRebuli statistikis mixedviT
populaciis Sefaseba.
X , rogorc μ-s wertilovani Sefaseba
SerCevis saSualo ( )X xSirad
gamoiyeneba, rogorc populaciis saSualos
(μ) wertilovani Sefaseba. wertilovani
Sefaseba aris parametris mniSvnelobis
Sefaseba statistikis mniSvnelobiT (anu erTi ricxviT). davubrundeT me-3 TavSi
ganxilul SfoTvis gazomvas. konsultantma Tavisi universitetis studentTa
populaciidan SearCia 100 studenti da gazoma maTi gamocdebTan dakavSirebuli
SfoTva. SerCevis qulaTa saSualo warmodgenilia cxrilSi N3.3 da Seadgens 41.64-
s, xolo standartuli gadaxra _ 8.66-s. SerCevis saSualo 41.64 SeiZleba
gamoyenebul iqnas, rogorc μ-s (gamocdebTan dakavSirebuli SfoTvis im
wertilovani Sefaseba (Point estimation): _
parametris mniSvnelobis Sefaseba
statistikis mniSvnelobiT (anu erTi
ricxviT).
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 3
populaciis saSualo, saidanac SerCeva iyo aRebuli) Caunacvlebeli da
Zalmosili Sefaseba.
Caunacvlebeli Sefaseba
Caunacvlebeli Sefaseba aris statistika,
romelic monacemTa usasrulod didi
raodenobisTvis Sesafasebeli parametris
mniSvnelobis tolia. amrigad, Caunacvlebel
SefasebasTan saqme gvaqvs maSin, rodesac
garkveuli moculobis SemTxveviT SerCevaTa
usasrulo raodenobis aRebisas statistikis mniSvnelobaTa saSualo Sesafasebeli
parametris tolia. SerCevis saSualo ( )X imitom aris μ-is Caunacvlebeli
Sefaseba, rom Tu Cven populaciidan aviRebT N moculobis yvela SesaZlo
SemTxveviT SerCevas, maSin SerCevaTa saSualoebis saSualo μ-s toli iqneba.
SerCevis dispersia S2 imitom ar aris σ2-is Caunacvlebeli Sefaseba, rom is
sistematurad gadaixreba misgan, Tumca, populaciis dispersiis Sefaseba (s2) σ2-is
Caunacvlebel Sefasebas warmoadgens.
Zalmosili Sefaseba
Zalmosili Sefaseba aris statistika,
romlisTvisac albaToba imisa, rom es
statistika ufro axlos iqneba
Sesafasebel parametrTan, izrdeba
SerCevis moculobis zrdasTan erTad.
vinaidan is Zalmosili Sefasebaa, SerCevis 25 quliT gamoTvlil saSualos ufro
meti albaToba aqvs ufro axlos iyos μ-sTan, vidre SerCevis 5 qulaze
dayrdnobiT gamoTvlil saSualos. am maxasiaTeblis gacnobiT intuiciurad
gveqneba warmodgena SerCevis saSualoze. rac ufro didia populaciidan aRebuli
SerCevis moculoba, miT meti qulaa CarTuli X -is gamoTvlaSi da, Sesabamisad,
mosalodnelia, rom X ukeTesad Seafasebs μ-s. Tu Cven SevZlebT iseTi SerCevis
aRebas, romelic moicavs mTel sakvlev populacias, maSin X igive iqneba, rac μ
da, aqedan gamomdinare, Sefasebisas ar gveqneba Secdoma.
Sefasebis sizuste
SerCevis saSualo μ-s Caunacvlebeli da Zalmosili Sefasebaa, magram Cven
uyuradRebod ar unda davtovoT is faqti, rom Sefaseba mxolod miaxloebiTi
gamoTvlaa. nebismieri Sefasebisas SeuZlebelia, rom X zustad μ-s toli iyos. is,
Caunacvlebeli Sefaseba (Unbiased
estimation): _ statistika, romlis
mniSvnelobaTa saSualo,
gamoTvlili usasrulod bevri
SerCevidan, Sesafasebeli
parametris mniSvnelobis tolia.
Zalmosili Sefaseba (Consistent estimation): _
statistika, romlisTvisac albaToba imisa,
rom es statistika ufro axlos iqneba
Sesafasebel parametrTan, izrdeba SerCevis
moculobis zrdasTan erTad.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 4
Tu ramdenad kargad afasebs X μ-s, damokidebulia SerCevis meTodze, SerCevis
moculobasa da populaciaSi qulaTa cvalebadobaze.
SerCevis meTodi
populaciidan aRebuli nebismieri SerCeva garkveul azrs gviqmnis am
populaciis maxasiaTeblebis Sesaxeb. faqtobrivad, informaciis uqonlobis
pirobebSi X -is mixedviT SegviZlia „saukeTeso varaudi gamovTqvaT“ Sesabamisi μ-s
mniSvnelobis Sesaxeb, magram raTa darwmunebuli viyoT, rom X warmoadgens μ-s
karg Sefasebas, SerCeva unda xasiaTdebodes igive maxasiaTeblebiT, riTac
sakvlevi populacia. arsebobs reprezentaciuli SerCevis meTodebis mTeli rigi,
romlebic mowodebulia imis uzrunvelsayofad, rom SerCeva iyos im populaciis
reprezentaciuli, saidanac is iqna aRebuli. Cven ganvixilavT or meTods.
martivi SemTxveviTi SerCeva. martivi
SemTxveviTi SerCeva (romelsac xSirad
SemTxveviT SerCevasac uwodeben)
ganisazRvreba, rogorc populaciidan
wevrebis SerCevis iseTi procesi,
rodesac:
1. populaciis yvela wevrs
SerCevaSi moxvedris toli Sansi aqvs.
2. erTi wevris arCeva populaciis nebismieri sxva wevris arCevisgan
damoukideblad xdeba.
SemTxveviTi SerCeva miukerZoebeli SerCevaa. Sesabamisad, SerCevis
maxasiaTeblebi ar unda gansxvavdebodes im populaciis maxasiaTeblebisgan raime
sistematuri da mudmivi gziT, saidanac es SerCeva iqna aRebuli; Tumca,
SemTxveviTi SerCeva ar iZleva imis garantias, rom konkretuli SerCeva zustad
populaciis reprezentaciuli iqneba. vinaidan SerCevebi, Cveulebriv, populaciaze
gacilebiT mcire moculobisaa, Cven ar unda movelodeT, rom SerCevis
maxasiaTeblebi zustad iseve iqneba ganawilebuli, rogorc populaciaSi.
zogierTi SemTxveviTi SerCeva ufro reprezentaciuli iqneba populaciisa, vidre
sxvebi. Tumca, SemTxveviTi SerCeva uzrunvelyofs, rom, zogadad (anu usasrulod
didi raodenobis SerCevaTa SemTxvevaSi), aseTi SerCevebi populaciis
reprezentaciuli iqneba.
martivi SemTxveviTi SerCeva (Simple random
sampling): _ populaciidan wevrebis SerCevis
iseTi procesi, rodesac populaciis
TiToeul wevrs SerCevaSi moxvedris toli
Sansebi aqvT da erTi wevris arCeva
populaciis nebismieri sxva wevris
arCevisgan damoukideblad xdeba.
stratificirebuli SemTxveviTi SerCeva (Stratified random sampling): _ SerCevis meTodi,
rodesac populaciis wevrebi erTiandeba homogenur qvejgufebSi, romelTac stratebi
ewodeba. populaciis wevrebi SemTxveviT airCevian stratidan im proporciiT, romliTac
es strata populaciaSi gvxvdeba.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 5
stratificirebuli SemTxveviTi SerCeva. stratificirebuli SemTxveviTi
SerCevisas populaciis weverebi erTiandebian homogenur qvejgufebSi, romelTac
stratebi ewodebaT. strata gulisxmobs msgavsi obieqtebis Sres an dones. aqedan
gamomdinare, stratebSi Semavali wevrebi homogenurebi anu msgavsni arian
zigierTi iseTi maxasiaTeblis mixedviT rogoricaa sqesi, asaki, anda ganaTlebis
done. Semdeg populaciis wevrebs SemTxveviTad irCeven stratidan im proporciiT,
romliTac es strata populaciaSi gvxvdeba. amis sailustraciod ganvixiloT
erTi universitetis studentebis populacia. am populaciaSi sxvadasxva SesaZlo
homogenuri qvejgufia, magaliTad, qvejgufebi swavlis xangrZliobis mixedviT: _
swavlis pirveli, meore, mesame da meoTxe wlis studentebi. TiToeul am
qvejgufSi Semavali wevri homogenuri anu erTmaneTis msgavsia iseTi
maxasiaTeblis mixedviT, rogoricaa swavlis xangrZlivoba (wlebiT). cxadia, rom
mravali sxva SesaZlo qvejgufi arsebobs, romelic dajgufebulia iseTi
maxasiaTeblebis mixedviT rogoricaa, magaliTad, sqesi, ZiriTadi sagani an naxevar
ganakveTze muSaoba.
davuSvaT, studentTa stratificireba swavlis xangrZlivobis (wlebiT)
mixedviT movaxdineT da TiToeul strataSi (anu doneze an qvejgufSi) moxvda
studentTa Semdegi raodenoba: _ pirveli weli _ 40%, meore weli _ 25%, mesame
weli _ 20 % da meoTxe weli _ 15%. am populaciidan martivi SemTxveviTi SerCeva
gvaZlevs imas, rom swavlis TiToeuli wlisTvis populaciidan arCeul studentTa
proporcia analogiuri proporciiTve iqneba warmodgenili mraval SerCevaSi,
magram aseTi proporcia ver gveqneba nebismier erT konkretul SerCevaSi.
stratificirebuli SemTxveviTi SerCevisas individebis SerCeva xdeba stratis
populaciaSi Sexvedris sixSiris proporciulad. Tu meore wlis studentebi
populaciis 25%-s Seadgenen, maSin SerCevis 25% SemTxveviT SeiCeva am stratidan.
rodesac SesaZlebelia populaciaSi stratebis gamoyofa da Semdeg rodesac
konkretuli stratis wevroba ukavSirdeba gasazom qcevas, maSin
stratificirebuli SemTxveviTi SerCeva mosalodnelia, rom ufro
reprezentaciul SerCevas gvaZlevdes, vidre SemTxveviTi SerCeva.
SerCevis moculoba
Zalmosili Sefasebis ganxilvisas aRvniSneT, rom rac ufro didia SerCeva, miT
metia albaToba imisa, rom SerCevis saSualo ufro axlos iyos populaciis
saSualos mniSvnelobasTan. amrigad, Zalmosili Sefasebisas SerCevis moculoba
miRebuli Sefasebis sizustes ukavSirdeba.
populaciis sazomvis cvalebadoba
ganvixiloT mcired albaTuri SesaZlebloba, rodesac populaciis yvela
wevris qula erTmaneTis tolia. am SemTxvevaSi TiToeuli qula populaciis
saSualos toli, xolo populaciis σ nulis toli iqneba. am pirobebSi naTelia,
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 6
rom nebismieri moculobis SerCevidan miRebuli X μ-s toli iqneba. SerCevis
SefasebaSi ar gveqneba Secdoma. meore mxriv, ganvixiloT populacia, romelic
gazomvis Sedegad miRebuli qulebis didi cvalebadobiT xasiaTdeba da romlis σ
Zalian didia. miuxedavad imisa, rom es iSviaTad SeiZleba moxdes, erT SemTxveviT
SerCevaSi SeiZleba SemTxveviT mTlianad dabali qulebi miviRoT. am dros
miRebuli X mniSvnelovnad gansxvavebuli iqneba μ-isgan. analogiurad, μ-is
SedarebiT cudi Sefaseba gveqneba, Tu populaciidan SemTxveviT, ZiriTadad,
maRali qulebi avirCieT. amrigad, X -is, rogorc μ-is wertilobani Sefasebis,
sizuste populaciaSi cvalebadobis zomas ukavSirdeba. rac ufro didia
populaciis standartuli gadaxra, miT naklebia albaToba imisa, rom konkretuli
SerCevis saSualo populaciis saSualos kargi Sefaseba iqneba.
araSemTxveviTi SerCeva
rogorc vnaxeT, SerCevis meTodi gadamwyvetia imisaTvis, Tu ramdenad iqneba
SerCeva populaciis reprezentaciuli; Tumca, sazogadoebrivi azris mraval
gamokiTxvaSi araSemTxveviTi SerCevebi gamoiyeneba. xanmokle komunikaciebis
epoqaSi, rogorc Cans, mravali Cvengani dainteresebulia imiT, Tu ras fiqroben
adamianebi ama Tu im sakiTxis Sesaxeb. am interesis dasakmayofileblad xanmokle
gamokiTxvebis sruliad axali industria Seiqmna. gazeTebi da televiziebi
mimdinare dRis aqtualur Temebze mokle satelefono gamokiTxvebs gvTavazoben.
TiToeuli aseTi gamokiTxviT informacia SerCevidan _ anu im adamianebisgan,
romlebic gamokiTxvaSi monawileoben _ grovdeba, magram ramdenad gamosadegia
aseTi gamokiTxvebi im populaciis Sesaxeb daskvnebis gasakeTeblad, romlidanac
es SerCevaa aRebuli? samwuxarod, arc ise gamosadegia ramdenime mosazrebis gamo.
pirveli sirTule isaa, rom populacia zustad unda iqnes gansazRvruli.
gavixsenoT pirveli Tavidan, rom populacia saerTo maxasiaTeblis mqone
adamianTa, cxovelTa, saganTa an movlenaTa sruli erTobliobaa. SerCevis
mixedviT populaciaze daskvnis gasakeTeblad aucileblad unda vicodeT, Tu ra
saerTo maxasiaTebeli aqvs mocemul populacias, raTa davrwmundeT, rom SerCevis
yvela wevrs aqvs es maxasiaTebeli. meore: _ SerCeva populaciis reprezentaciuli
unda iyos, rac SerCevis sistematuri procesis ganxorcielebas gulisxmobs. arc
erTi es moTxovna ar sruldeba, rodesac individebi TviTon iReben gazeTsa Tu
televiziaSi darekvis an internetiT azris gamoTqmis gadawyvetilebas. magaliTad,
erT-erT internet-gamokiTxvaSi amerikis moqalaqeebs ekiTxebodnen, Tu ramdenad
Sors apirebdnen samogzaurod wasvlas Svebulebis dros; azris gamoTqma
paraskevobiT an dasvenebis dReebSi iyo SesaZlebeli. gamokiTxvaSi daaxloebiT
20,000 adamianma miiRo monawileoba, romelTagan 78% miuTiTebda, rom ar hqondaT
dagegmili, Tu rogor gaatarebda Svebulebas. ramdenad SegviZlia, darwmunebulni
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 7
viyoT, rom es ricxvi realurad asaxavs amerikelTa Svebulebis dros samogzauro
gegmebs? samwuxarod, am gamokiTxvis SerCeva TviTSerCeulia da mxolod im
adamianebisgan Sedgeba, romlebic dainteresdnen pasuxis gacemiT. metic, vinaidan
bevri adamiani ukve wasuli iyo samogzaurod manamde, sanam gamokiTxva
Catardeboda, isini verafers upasuxebdnen Svebulebis samomavlo gegmebis Sesaxeb
dasmul SekiTxvas. amrigad, Cven ver ganvsazRvravT im populacias, romlidanac es
SerCevaa aRebuli; Sesabamisad, am SerCevidan populaciis Sesaxeb daskvnis gakeTeba
aramizanSewonilia. aseTi gamokiTxvebi mecnierulad gamousadegaria.
SeamowmeT Tqveni codna VII.1
1. ganmarteT: _ Zalmosili Sefaseba, martivi SemTxveviTi SerCeva,
statistikuri daskvna, stratificirebuli SemTxveviTi SerCeva,
Caunacvlebeli Sefaseba.
2. davuSvaT, SfoTviTi aSlilobis mqone individTa populaciaSi 27%-s aqvs
generalizirebuli SfoTviTi aSliloba, 23%-s _ agorafobia, 19%-s _
martivi fobia, 22%-s _ panikuri aSliloba da 9%-s _ obsesiur-
kompulsuri aSliloba. Tqven am populaciidan SemTxveviTi SerCevis
stratificirebas da stratad aSlilobebis tipis gamoyenebas apirebT.
a) ra proporciiT iqneba warmodgenili TiToeuli strata Tqvens
SerCevaSi?
b) Tu Tqven aiRebT 100 individisgan Semdgar SerCevas, ramdeni adamiani
moxvdeba TiToeul strataSi?
g) Tu Tqven aiRebT erT SeTxveviT SerCevas, ramdeni adamiani moxvdeba
TiToeul strataSi?
populaciis dispersiisa da standartuli gadaxris Sefaseba
dispersia
populaciis dispersia (σ2) mexuTe TavSi ganisazRra, rogorc:
( )populaciaN
X∑ −=
22 µ
σ ,
Tumca, rogorc ukve aRvniSneT, Cveulebriv, SeuZlebelia populaciis yvela
wevris gazomva. aqedan gamomdinare, populaciis dispersiis (σ2) Sefaseba, rogorc
wesi, SerCevaze dayrdnobiT xdeba. rogorc me-5 TavSi vnaxeT, σ2-is Caunacvlebeli
da Zalmosili Sefaseba gamoiTvleba formuliT
( )1
2
2
−
−= ∑
NXX
s .
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 8
yuradReba miaqcieT, rom wiladis mniSvneli aris ara _ N, aramed _ (N_1).
rogorc me-5 TavSi vTqviT, SerCevis dispersia, romelic gamoiTvleba formuliT
( )N
XXS ∑ −
=2
2,
aris σ2-is Canacvlebuli Sefaseba. rogorc wesi, is bolomde ver (naklebad)
afasebs σ2-is mniSvnelobas. aqedan gamomdinare, Tu calke ar aRvniSneT, dispersiis
saxiT yovelTvis s2-s gamoviyenebT.
standartuli gadaxra
σ-is Sesafaseblad populaciis standartuli gadaxris Sefaseba s gamoiyeneba. s2-
isgan, rogorc σ2-is Sefasebisgan, gansxvavebiT, s ar aris σ-is Caunacvlebeli
Sefaseba. metic, s odnav arazustia σ-s Sefasebisas; magram vinaidan s-is
SemTxvevaSi Canacvleba mcirea da kidev ufro mcirdeba SerCevis moculobis
zrdasTan erTad, mkvlevarTa umravlesoba σ-is Sefasebis saWiroebisas s-s iyenebs.
saSualos SerCeviTi ganawileba: _ Sefasebis sizustis gansazRvra
populaciis saSualos wertilovani Sefasebis sizuste damokidebulia rogorc
SerCevis moculobaze, ise _ populaciis standartul gadaxraze. garda im
SemTxvevebisa, rodesac SerCeva moicavs populaciis yvela qulas an rodesac
populaciis qulebi ar xasiaTdeba cvalebadobiT, μ-s Sefasebisas X -s yovelTvis
Tan sdevs Secdoma. miuxedavad imisa, rom Cven darwmunebiT arasodes viciT, Tu
ramdenad zustia populaciis parametris konkretuli Sefaseba, Sefasebisas
mosalodneli Secdomis zomis gansazRvra SerCeviTi ganawilebis cnebis
gamoyenebiT SegviZlia.
SerCeviTi ganawileba aris populaciidan
aRebuli N moculobis mqone yvela SesaZlo
SerCevidan miRebuli statistikis
mniSvnelobebis albaTuri Teoriuli
ganawileba. amrigad, saSualos SerCeviTi
ganawileba aris X -is mniSvnelobebis
ganawileba, rodesac populaciidan
aRebulia N moculobis mqone yvela
SesaZlo SerCeva. SerCeviTi ganawileba
Teoriulad ganisazRvreba maTematikuri
gantolebiT, vinaidan Cven verasodes
SevZlebT populaciidan yvela SesaZlo SerCevis aRebas. Semdeg TavebSi
warmodgenili TiToeuli statistikuri kriteriumi am statistikis SerCeviTi
SerCeviTi ganawileba (Sampling
distribution): _ populaciidan aRebuli N
moculobis mqone yvela SesaZlo
SerCevidan miRebuli statistikis
mniSvnelobebis albaTuri Teoriuli
ganawileba.
saSualos SerCeviTi ganawileba
(Sampling distribution of the mean): _ X -is
mniSvnelobebis ganawileba, rodesac
populaciidan aRebulia N moculobis
mqone yvela SesaZlo SerCeva.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 9
ganawilebis codnas saWiroebs. es SerCeviTi ganawilebebi maTematikuri
gantolebebiT miRebuli Teoriuli SerCeviTi ganawilebebia.
miuxedavad imisa, rom statistikuri kriteriumebi Teoriul SerCeviT
ganawilebas iyeneben, masSi CarTul cnebaTa sailustraciod Cven saSualos
ramdenime empiriul SerCeviT ganawilebas ganvixilavT. sityva „empiriuli“ niSnavs
dakvirvebads an dakvirvebis Sedegad miRebuls. aqedan gamomdinare, saSualos
empiriuli SerCeviTi ganawileba SeiZleba miviRoT populaciidan N moculobis
mqone ramdenime SerCevis aRebiT, TiToeuli SerCevisTvis X -is gamoTvliTa da
Semdeg miRebuli saSualoebis sixSireTa ganawilebis agebiT. amis sailustraciod,
aviRoT 100 qulisgan Semdgari populacia, sadac qulebi meryeobs 8-dan 20-mde
diapazonSi da romlis μ = 14.0 da σ = 2.14. populaciis 100 qulis sixSireTa
poligoni warmodgenilia suraTze N7.1. rogorc suraTze xedavT, qulaTa
ganawileba simetriuli da unimodaluria.
0
2
4
6
8
10
12
14
16
18
20
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
qulaTa mniSvnelobebi
six
Sir
e
saSualos empiriuli SerCeviTi ganawileba sami sxvadasxva moculobis mqone
SerCevidan miviReT: _ N = 2, N = 5 da N = 10. es SerCevebi kompiuteris daxmarebiT
populaciidan martivi SemTxveviTi SerCevis gziT miviReT. SerCevis TiToeuli
moculobisTvis SerCeul iqna 100 SemTxveviTi SerCeva; TiToeuli maTganisTvis
gamovTvaleT saSualo. saSualos empiriuli SerCeviTi ganawilebebi miRebuli iyo
sami sxvadasxva zomis SerCevisgan: _ N = 2, N = 5 da N = 10, rogorc naCvenebia
suraTze N7.2. TiToeuli (a), (b) da (g) ganawileba warmoadgens konkretuli
moculobis mqone SerCevis (N = 2, N = 5 da N = 10, Sesabamisad) saSualos
empiriul SerCeviT ganawilebas.
სურათი 7-2 არასრულია
suraTi N7.1. sixSireTa ganawileba 100 qulisgan Semdgari
populaciisTvis, romlis μ = 14.0 da σ = 2.14.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 10
saSualos SerCeviTi ganawilebis Tvisebebi
suraTze N7.2 mocemuli empiriuli SerCeviTi ganawileba saSualos Teoriuli
SerCeviTi ganawilebis ramdenime Tvisebis ilustraciaa. gavixsenoT, rom sixSireTa
ganawilebis aRwera formis, centraluri tendenciisa da cvalebadobis sazomebis
terminebiT aris SesaZlebeli. vinaidan saSualos SerCeviTi ganawileba sixSireTa
ganawilebas warmoadgens, isic SeiZleba aRiweros formis, saSualosa da
standartuli gadaxris terminebiT.
saSualos SerCeviTi ganawilebis forma
suraTze N7.2 mocemuli sami SerCeviTi ganawileba SedarebiT simetriuli da
unimodaluria. simetriidan gadaxra ubralod saSualo mniSvnelobebis
SemTxveviTi cvalebadobiT aixsneba. yuradReba miaqcieT, rom ganawileba miT ufro
wamaxvilebuli da naklebad gafantulia, rac ufro izrdeba SerCevis moculoba
(SeadareT SerCevebi, romelTa moculobebia N = 2 da N = 10). principi, romelsac
centraluri zRvris Teorema ewodeba, saSualos SerCeviT ganawilebas Seexeba:
SerCevis moculobis zrdasTan erTad saSualos SerCeviTi ganawileba normalur
ganawilebas uaxlovdeba. saSualos SerCeviTi ganawilebis normalur
ganawilebasTan aseT miaxloeba xdeba maSin, rodesac populaciis qulaTa
ganawilebis forma simetriuli an asimetriulia.
centraluri zRvris Teorema (Central limit theorem): _ maTematikuri Teorema, romlis Tanaxmad,
SerCevis moculobis zrdasTan erTad saSualos SerCeviTi ganawileba normalur
ganawilebas uaxlovdeba.
suraTi N7.2. 7.1 suraTze mocemuli populaciis qulebidan aRebuli 100
SerCevis saSualoebis sami empiriuli SerCeviTi ganawileba. SerCevis
moculobaa: _ (a) N = 2, (b) N = 5 da (g) N = 10.
(a) N = 2 (b) N = 5 (g) N = 10
SerCevis saSualos mniSvneloba ( )X
sixSire
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 11
ramdenad didi unda iyos N imisaTvis, rom saSualos SerCeviTma ganawilebam
normalur ganawilebasTan miaxloeba daiwyos? am kiTxvaze pasuxi damokidebulia
populaciaSi qulaTa ganawilebis formaze. populacia, saidanac Cven SerCevebi
aviReT, ar aris normalurad ganawilebuli, magram is unimodaluri da
simetriulia (ix. suraTi N7.1). aseTi ganawilebebisTvis saSualos SerCeviTi
ganawileba, rodesac SerCevis moculoba iseTi mcirea, rogoricaa N = 5 an N = 10,
normaluri ganawilebis formasTan miaxloebas iwyebs (ix. suraTi N7.2, (b) da (g)).
Tu populaciaSi qulaTa ganawileba asimetriulia, maSin normalur
ganawilebasTan misaaxloeblad SerCevis moculoba saSualos SerCeviTi
ganawilebisTvis ufro didi unda iyos. mravali statistikosi Tvlis, rom Tu
populaciis ganawileba ar aris Zalian asimetriuli, SerCevis moculoba N = 30
aris is moculoba, rodesac saSualos SerCeviTi ganawileba axlos iqneba
normalur ganawilebasTan.
centraluri zRvris Teoremaze dafuZnebuli saSualos Teoriuli SerCeviTi
ganawileba warmodgenilia suraTze N7.3. yuradReba miaqcieT, rom es ganawileba
simetriuli, unimodaluri da normaluria. Semdeg TavebSi daskvniTi statistikis
kriteriumebis ganxilvisas swored am da sxva Teoriul SerCeviT ganawilebebs
gamoviyenebT. empiriuli SerCeviTi ganawilebebi, rogoricaa, magaliTad, suraTze
N7.2 warmodgenili ganawilebebi, mxolod SerCeviTi ganawilebis cnebis
asaxsnelad gamoiyeneba.
Xµ _ saSualos Teoriuli SerCeviTi ganawilebis saSualo.
XX _ saSualos empiriuli SerCeviTi ganawilebis saSualo.
saSualos SerCeviTi ganawilebis saSualo
Teoriul SerCeviT ganawilebaSi SerCevis saSualoebis saSualo (aRiniSneba
Xµ -iT) populaciaSi Semaval calkeul qulaTa saSualos (μ) toli iqneba, xolo
saSualos empiriul SerCeviT ganawilebaSi SerCevis saSualoebis saSualo
(aRiniSneba XX -iT) ar iqneba zustad populaciis saSualos toli. es mosazreba
naCvenebia suraTze N7.2 warmodgenil empiriul SerCeviT ganawilebaSi. sam
SerCeviT ganawilebaSi SerCevis saSualoebis saSualo ar aris zustad 14.0-is (anu
sawyisi populaciis saSualos) toli. (a), (b) da (g) ganawilebebisTvis saSualos
mniSvnelobebia 96.13=XX , 14.14=XX da 99.13=XX .
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 12
SerCevis Secdoma da saSualos standartuli Secdoma
SerCevis Secdoma. suraTze N7.2 mocemuli
TiToeuli SerCevis saSualo warmoadgens im
populaciis saSualos Sefasebas, saidanac isini
iqnen aRebuli (am magaliTSi μ = 14). rogorc
xedavT, zogierTi SerCevis saSualo ufro zustad afasebs μ-s, vidre sxvebi.
SerCevis Secdoma aris is sidide, romliTac konkretuli SerCevis saSualo
gansxvavdeba populaciis saSualosgan (anu µ−X ). Tu X zustad imdenia,
ramdenic μ anu Tu µ=X da amitom 0=− µX , maSin SefasebaSi ar gveqneba
SerCevis Secdoma. suraTi N7.2 gviCvenebs, rom SerCevis Secdoma SerCevis
moculobas ukavSirdeba: _ rac ufro didia SerCevis moculoba, miT naklebia
SerCevis Secdoma. suraTze N7.2a mocemuli 100 SerCevis saSualos mniSvnelobebi
(sadac TiToeuli saSualo or qulas efuZneba) ufro cvalebadia, vidre N7.2b da
N7.2g suraTze mocemuli SerCevis saSualoebi, romelic, Sesabamisad, 5 an 10
quliT aris gamoTvlili. amrigad, SerCevis Secdoma ufro TvalsaCinoa suraTze
N7.2a, vidre N7.2b anda suraTze N7.2g.
saSualos SerCeviT
ganawilebaSi SerCevis
saSualoebis cvalebadobis
sidide aRiwereba SerCevis
saSualoebis mniSvnelobebiT
gamoTvlili standartuli
gadaxriT. am standartul
gadaxras saSualos
standartul Secdomas an ubralod standartul Secdomas uwodeben; amiT mas
nedli qulebis populaciiT gamoTvlili standartuli gadaxrisgan ganasxvaveben.
SerCevis Secdoma (Sampling error): _
sidide, romliTac konkretuli
SerCevis saSualo gansxvavdeba
populaciis saSualosgan.
saSualos standartuli Secdoma, Xσ (Standard error of
the mean): _ saSualos SerCeviTi ganawilebis
standartuli gadaxra, romelic miiReba σ-as SerCevis
moculobis kvadratul fesvTan SefardebiT.
saSualos standartuli Secdomis Sefaseba, Xs
(Estimated standard error of the mean) _ s-is gamoyenebiT σ-as,
saSualos standatruli Secdomis Sefaseba.
-3 Xσ -2 Xσ -1 Xσ Xµ +1 Xσ +2 Xσ +3 Xσ
suraTi N7.3. saSualos Teoriuli SerCeviTi ganawileba,
romlis saSualoa Xµ da standartuli Secdoma Xσ
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 13
saSualos standartuli Secdoma, romelic aRiniSneba simboloTi Xσ , populaciis
standartuli gadaxrisa da SerCevis moculobidan kvadratuli fesvis Sefardebis
tolia. formulis saxiT:
NX
σσ ==Secdoma istandartul saSualos .
sadac σ aris populaciaSi Semaval qulaTa standartuli gadaxra, xolo N _
SerCevis moculoba. saSualos standartuli Secdoma Xσ iseTive standartuli
gadaxraa, rogoric σ, im gansxvavebiT, rom Xσ SerCevis saSualoebis ganawilebis
standartuli gadaxraa da ara _ populaciaSi Semaval nedl qulaTa standartuli
gadaxra (rasac σ warmoadgens).
SexedeT suraTze N7.3 mocemul saSualos Teoriul SerCeviT ganawilebas.
yuradReba miaqcieT, rom misi saSualo aRniSnulia simboloTi Xµ , xolo misi
standartuli gadaxra _ simboloTi Xσ . me-6 TavSi ganvixileT normaluri
ganawilebis qveS mdebare farTobi da avxseniT, Tu rogor unda vipovoT qulaTa
wili ganawilebis qveS mdebare farTobebiT. vinaidan saSualos SerCeviTi
ganawileba normaluri ganawilebaa, aqac ZalaSi rCeba es damokidebuleba, Tumca,
am SemTxvevaSi Cven gvaqvs ara intervalebSi moxvedrili nedli qulebi, aramed _
SerCevis saSualoebi. ase magaliTad, SerCeviTi ganawilebis saSualos qvemoT erT
standartul Secdomasa da SerCeviTi ganawilebis saSualos zemoT erT
standartul Secdomas Soris intervali (anu intervali XX σµ 1− -dan XX σµ 1+ -mde)
moicavs SerCevaTa saSualoebis .6826-s, intervali XX σµ 2− -dan XX σµ 2+ -mde _
SerCevaTa saSualoebis .9544-s, xolo intervali XX σµ 3− -dan XX σµ 3+ -mde _
SerCevaTa saSualoebis .9974-s. yuradReba miaqcieT, rom am intervalebis
aRniSvnisas gamoviyeneT simbolo Xµ , radgan saSualos SerCeviT ganawilebasTan
gvaqvs saqme. vinaidan µµ =X , Cven SegviZlia, rom Xµ -is nacvlad µ gamoviyenoT,
rogorc es Semdeg magaliTSia naCvenebi.
am damokidebulebis gamoyenebis sailustraciod, davuSvaT, rom qulaTa
populaciis saSulo μ = 50 da σ = 6, xolo am populaciidan SemTxveviT aRebulia
N = 9 moculobis SerCevebi. am SerCeviTi ganawilebisTvis saSualos standartuli
Secdoma gamoiTvleba formuliT
239
96
====NX
σσ ,
miRebuli mniSvneloba 2=Xσ warmoadgens σ = 6 mqone populaciidan aRebuli
SerCevebis (romelTa N = 9) saSualos SerCeviTi ganawilebis standartul
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 14
gadaxras. amrigad, populaciidan aRebuli N = 9 SerCevebisTvis saSualos
SerCeviT Teoriul ganawilebaSi SerCevaTa saSualoebis .6826 xvdeba XX σµ 1− -dan
(anu 50-2=48) XX σµ 1+ -mde (anu 50+2=52), anu 48-dan 52-mde intervalSi; XX σµ 2− -dan
(anu 50-4=46) XX σµ 2+ -mde (anu 50+4=54), anu 46-dan 54-mde intervali moicavs
SerCevaTa saSualoebis .9544-s, xolo XX σµ 3− -dan (anu 50-6=44) XX σµ 3+ -mde (anu
50+6=56), anu 44-dan 56-mde intervali _ X -is mniSvnelobaTa .9974-s. es informacia
X -is, rogorc μ-s Sefasebis, sizustis Sesafaseblad gamogvadgeba.
standartuli normaluri sxvaobis gamoyeneba saSualos SerCeviT ganawilebaSi
saSualos SerCeviTi ganawileba normaluri ganawilebaa, romlis saSualoa Xµ ,
xolo standartuli Secdoma _ Xσ . Cven saSualos SerCeviTi ganawilebis X -is
mniSvnelobis standartuli normaluri ganawilebis qulad gardaqmna SegviZlia
formuliT
X
XXz
σµ−
=realuri .
am z qulis gamoyeneba SerCevis saSualos mimarT isevea SesaZlebeli, rogorc
calkeuli qulis mimarT (ix. Tavi 6). amis ilustracias warmoadgens magaliTi N7-1.
magaliTi N7.1
saSualoebis SerCeviTi ganawilebis gamoyeneba
amocana
davuSvaT, gvaqvs qulaTa normalurad ganawilebuli populacia, romlis μ = 80
da σ = 12. Cven vgegmavT, rom am populaciidan aviRoT didi raodenobiT
SemTxveviTi SerCevebi, romelTa moculoba N = 16 da gamovTvaloT TiToeuli
SerCevis saSulo. SerCevaTa saSualoebis ra nawili iqneba 82-is toli an masze
meti?
amoxsna
am amocanis amosaxsnelad saWiroa aRvniSnoT, rom saSualos SerCeviTi
ganawileba normaluri ganawilebaa, romlis 80=Xµ da 0.34/1216/12 ===Xσ .
Teoriuli SerCeviTi ganawilebis nebismier intervalSi moxvedrili SerCevaTa
saSualoebis wili Semdegnairad gamoiTvleba: _ SerCevis saSualos mniSvneloba
(anu 82) gadaiyvaneba standartuli normaluri ganawilebis mniSvnelobad da a-1
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 15
cxrilSi moiZebneba misi Sesatyvisi wili. SerCevis saSualo zrealuri-is
mniSvnelobad gardaiqmneba formuliT
67.0.3
20.38082
+==−
=−
=X
XXz
σµ
realuri .
.
SerCevebis saSualoebisa da zrealuri-is mniSvnelobebi, rodesac SerCevis
saSualoa 82, mocemulia suraTze N7.4. a-1 cxrilis (g) sveti gviCvenebs, rom z =
+0.67-is zemoT mdebare farTobia .2514 (ix. suraTi N7.4, (g) RerZi). amrigad,
mosalodnelia, rom SerCevaTa saSualoebis .2514 nawili 82-is toli an masze meti
iyos. es daskvna albaTobis terminebiT ase Caiwereba:
( ) 2514.82 =≥Xp
აკლია სურათი 7.4
saSualos standartuli Secdomis Sefaseba
saSualoebis SerCeviTi ganawileba warmoadgens SerCevaTa saSualoebis
mosalodneli cvalebadobis Sefasebis meTods, magram naTlad Cans mocemuli
codnis gamoyenebis sazRvari. populaciis standartuli gadaxra σ populaciis
parametria, populaciis nebismieri parametri ki, rogorc wesi, ucnobia. maS, rogor
ganisazRvreba saSualos standartuli Secdoma, Tu ar viciT σ?
am kiTxvaze pasuxi SerCevis monacemebze dayrdnobiT σ-s SefasebaSi Zevs
(rogorc μ SevafaseT X -iT). am SemTxvevaSi saSualos standartuli Secdomis
Sefasebis misaRebad populaciis standartuli gadaxris Sefaseba (s) gamoiyeneba.
saSualos standartuli Secdomis Sefaseba, romelic aRiniSneba simboloTi Xs ,
efuZneba populaciis standartul gadaxras da formulis saxiT gamoisaxeba:
NssXX ===σ Sefaseba Secdomis istandartul saSualos ,
sadac s gamoiTvleba SerCevis qulebidan, xolo N SerCevis moculobaa. amrigad,
Xs warmoadgens saSualos SerCeviTi ganawilebis standartuli gadaxris
Sefasebas. samecniero statiaSi standartul Secdomas xSirad SE (da ara Xs )
simboloTi aRniSnaven.
suraTi N7.4. SerCevis saSualos (82) mdebareoba saSualos SerCeviT ganawilebaSi,
romlis μ = 80 da Xσ = 3. X RerZis warwerebi gviCvenebs: _ (a) SerCevebis
saSualos mniSvnelobebs, (b) SerCevebis saSualos mniSvnelobebis Sesabamis z
qulebs, (g) SerCevaTa saSualoebis wils, romelic 82-is tolia an metia masze.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 16
mniSvnelovania Xσ -sa da Xs -is Soris arsebuli gansxvavebis gageba. Xσ
fiqsirebuli mniSvnelobaa, romelic SeiZleba ganisazRvros maSin, rodesac
cnobilia σ; Tumca, radgan CvenTvis iSviaTad aris σ cnobili, amitom, Cveulebriv,
Xσ -is Sefaseba Xs -iT xdeba. vinaidan Xs SerCevis s-is mniSvnelobas eyrdnoba,
sxvadasxva SerCevaSi misi mniSvneloba sxvadasxva iqneba. erTi da imave
populaciidan aRebuli sxvadasxva SemTxveviTi SerCeva sxvadasxva standartul
gadaxras mogvcems, sxvadasxva SerCevaSi qulaTa SemTxveviTi gansxvavebis gamo.
miuxedavad amisa, Xσ -is Sefasebis erTaderT safuZvels SerCevis qulebi
warmoadgenen.
faqtorebi, romlebic gavlenas axdenen Xs -is mniSvnelobaze
Xs -is formula Ns / gviCvenebs, Tu rogor ukavSirdeba SerCeviT ganawilebaSi
SerCevaTa saSualoebis cvalebadoba (1) populaciaSi Semavali qulebis
cvalebadobas da (2) SerCevis moculobas. rac ufro cvalebadia populaciis
qulebi, miT meti iqneba SerCevis qulaTa s; amasTan, s-is gazrdasTan erTad Xs -ic
izrdeba; Tumca, SerCevis moculobis (N) gazrdiT Xs -is formulis mniSvnelic
izrdeba, rac, Tavis mxriv, standartuli Secdomis Semcirebas ganapirobebs.
amrigad, qulaTa mcire cvalebadobis mqone populaciidan aRebuli didi
moculobis SemTxveviTi SerCevebis pirobebSi mcire Xs -is miRebas unda
movelodeT. miuxedavad imisa, rom mkvlevars SeuZlia SerCevis moculobis
kontroli, xSirad praqtikulad SeuZlebelia populaciis qulebis cvalebadobis
Semcireba. amrigad, qcevaTmecnierebis warmomadgenlebi Xs -is mniSvnelobis
Sesamcireblad xSirad didi zomis SerCevebs iyeneben.
Xs -is gamoyeneba
standartuli Secdoma saSualoebis SerCeviT ganawilebaSi SerCevis Secdomis
sididis gasazomad gamoiyeneba. Sesabamisad, Xs SeiZleba gamoviyenoT imis
gansasazRvrad, Tu ramdenad kargad afasebs SerCevis miRebuli saSualo
populaciis saSualos. rac ufro mcirea Xs , miT ufro darwmunebulni SegviZlia
viyoT, rom X mniSvnelovnad ar gansxvavdeba populaciis saSualosgan. me-8 TavSi,
Xs -is gamoyenebiT, detalurad aRvwerT X -iT μ-is Sefasebis sizustes; axla ki,
qvemoT ganxiluli sakiTxebis ganxilvis gasaioleblad, davuSvaT, rom CvenTvis
Xσ cnobilia, Tumca, unda gvaxsovdes, rom praqtikaSi Xσ ar aris cnobili da
mis nacvlad Xs unda gamoviyenoT.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 17
ganawilebis tipebis mimoxilva
cxrilSi N7-1 gaerTianebulia aqamde ganxiluli ganawilebis tipebi da maTi
maxasiaTeblebis aRmniSvneli simboloebi. ganawileba SeiZleba aigos nedli
qulebiT, an populaciidan aRebuli SerCevebis saSualoebiT.
• nedli qulebis ganawileba populaciis yvela wevris gazomvis Sedegad
miiReba. Tu Cven gvaqvs populaciis gazomvis saSualeba, maSin SevZlebT
populaciis saSualos (μ), standartuli gadaxrisa (σ) da dispersiis (σ2)
gamoTvlas.
• populaciidan aRebuli SerCevis nedli qulebis ganawileba SerCevis
saSualos _ X -is, standartuli gadaxris _ s-isa da dispersiis _ s2-is
gamoyenebiT aRiwereba. es statistikuri sidideebi im populaciis
Sesabamisi parametris aRsawerad SeiZleba iqnes gamoyenebuli,
romlidanac es SerCevaa aRebuli.
cxrili N7.1. ganawilebis tipebi da maTi maxasiaTeblebi.
sazomi
ganawilebis
tipi miiReba saSualo gadaxra dispersia
nedli qulebi
populacia populaciis yvela wevris
gazomvis Sedegad.
μ σ σ2
SerCeva populaciidan aRebuli
SerCevis gazomviT. X s s2
saSualos SerCeviTi ganawileba
Teoriuli Teoriulad maTematikuri
gantolebiT. Xµ Xσ X
2σ
empiriuli empiriulad populaciidan
SerCevebis aRebiT. XX Xs Xs2
• saSualoebis SerCeviTi ganawileba aris populaciidan aRebul SerCevaTa
saSualoebis ganawileba. SerCevis mocemuli moculobis SemTxvevaSi
saSualoebis Teoriuli SerCeviTi ganawileba matematikuri gantolebiT
miiReba. saSualoebis Teoriuli SerCeviTi ganawilebis saSualo
gamoixateba Xµ -iT, romelic im populaciis μ-is tolia, romlidanac
SerCeva iqna aRebuli. saSualoebis SerCeviTi ganawilebis standartuli
gadaxra gamoixateba Xσ -iT, xolo dispersia _ X2σ -iT. rogorc wesi, μ-
sa da Xσ -is Sefaseba X -isa da Xs -is safuZvelze xdeba.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 18
• saSualoebis empiriuli SerCeviTi ganawileba populaciidan SerCevebis
realurad aRebiTa da Semdeg am SerCevaTa saSualoebis sixSireTa
ganawilebis agebiT miiReba. empiriuli SerCeviTi ganawileba, ZiriTadad,
mowodebulia SerCeviTi ganawilebis cnebis gagebis gasamartiveblad. maT,
Cveulebriv, ar iyeneben realur kvlevaSi.
SeamowmeT Tqveni codna VII.2
1. ganmarteT: _ centraluri zRvariTi Teorema, saSualos standartuli
Secdomis Sefaseba, Xµ , SerCeviTi ganawileba, saSualos SerCeviTi
ganawileba, SerCevis Secdoma, Xσ , saSualos standartuli Secdoma.
2. aris Tu ara ( )∑ −= NXXS /22 formuliT miRebuli S2-is mniSvneloba σ2-
is Caunacvlebeli Sefaseba? Tu ar aris, ratom?
3. aris Tu ara ( ) ( )∑ −−= 1/22 NXXs formuliT miRebuli s2-is mniSvneloba
σ2-is Caunacvlebeli Sefaseba? Tu ar aris, ratom?
4. aris Tu ara ( ) ( )∑ −−= 1/2
NXXs formuliT miRebuli s-is mniSvneloba
σ-is Caunacvlebeli Sefaseba? Tu ar aris, ratom?
5. ra gansxvavebaa saSualos empiriul da Teoriul SerCeviT ganawilebebs
Soris?
6. CamoayalibeT centraluri zRvariTi Teorema.
7. davuSvaT, inteleqtis testSi miRebuli qulebis populaciidan, romlis μ
= 100 da σ = 15, Tqven 100 SemTxveviTi SerCevis aRebas apirebT, romelTa
moculoba N = 25.
a) SerCevaTa saSualoebis ra nawili iqneba moTavsebuli 97-sa da 103-s
Soris?
b) SerCevaTa saSualoebis ra wili iqneba 103-ze meti?
g) SerCevaTa saSualoebis ra wili iqneba 103-ze naklebi?
d) SerCevaTa saSualoebis ra wili iqneba 105-ze meti?
e) SerCevaTa saSualoebis ra wili iqneba moTavsebuli 100-sa da 105-s
Soris?
8. davuSvaT, gaqvT qulaTa SerCeva, romelic monitorze moZravi
obieqtisTvis kursoris midevnebis Sefasebis Sedegad miiReT; Tqven
afiqsirebdiT cdis piris mier samizne obieqtze kursoris SeCerebis dros.
am qulebis ganawilebis s = 18.0 wami.
a) gamoTvaleT Xs -is mniSvneloba, Tu SerCevis moculobaa 4.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 19
b) gamoTvaleT Xs -is mniSvneloba, Tu SerCevis moculobaa 9.
g) gamoTvaleT Xs -is mniSvneloba, Tu SerCevis moculobaa 36.
d) Xs -is mniSvnelobasa da SerCevis moculobas Soris arsebul ra
damokidebulebasTan gvaqvs saqme?
2. davuSvaT, wina kiTxvaSi aRweril amocanaSi miRebul qulaTa sami
gansxvavebuli SerCeva gaqvT. maTTvis s-is mniSvnelobebia 10, 15 da 20,
Sesabamisad. SerCevis moculobaa 25 (N =25).
a) TiToeuli SerCevisTvis gamoTvaleT Xs -is mniSvneloba.
b) qulaTa cvalebadobasa da Xs -is mniSvnelobas Soris arsebul ra
damokidebulebasTan gvaqvs saqme?
3. qulaTa ori sxvadasxva _ A1 da A2 _ populaciidan erTnairi moculobis
ori SerCevis aRebas apirebT. A1 populaciis σ = 17.0 da A2 populaciis σ
= 30.0. rogor fiqrobT, romeli populaciidan aRebuli SerCevis Xs -is
mniSvneloba iqneba meti? axseniT Tqveni pasuxi.
populaciis saSualos intervaluri Sefaseba.
SerCevis saSualo μ-is Caunacvlebeli da
Zalmosili Sefasebaa. Tu Cven im
populaciis saSualos vafasebT, saidanac
SerCeva iqna aRebuli, Sefasebis saukeTeso
wertilovani Sefaseba X iqneba; magram,
rogorc ukve aRvniSneT, wertilovan
Sefasebas Tan axlavs Secdoma; μ-is
mniSvneloba ar iqneba X -is mniSvnelobis toli. Sefasebis sizusteSi rom ufro
metad viyoT darwmunebulni, Cven SegviZlia, SevafasoT ara marto μ-is
wertilovani mniSvneloba, aramed avagoT ndobis intervalic, romelSic Seva μ-is
mosalodneli mniSvnelobebi. magaliTad, gamocdebTan dakavSirebuli SfoTvis
kiTxvaris qulebis populaciis saSualos zustad 41.64-quliT Sefasebis nacvlad,
mkvlevarma is SeiZleba μ-is Semcveli, 39.94-dan 43.34-mde intervaliT Seafasos.
ndobis done anu albaTobis sidide SeiZleba Tan axldes am Sefasebas; magaliTad,
mkvlevars SeuZlia 95%-iT darwmunebuli iyos, rom es intervali moicavs μ-s.
intervalis qveda (magaliTad, 39.9) da zeda (magaliTad, 43.3) qulebs ndobis
sazRvrebi ewodebaT. axla vnaxoT, rogor aigeba aseTi intervalebi da ras
niSnavs, rodesac vambobT, rom eqsperimentatori 95%-iT darwmunebulia, rom
intervali moicavs μ-is mniSvnelobas.
ndobis intervali (Confidence interval): _
qulis mniSvnelobaTa diapazoni,
romelic, savaraudod, moicavs μ-is
mniSvnelobas ndobis garkveul doneze.
ndobis sazRvrebi (Confidence limits): _
ndobis intervalis SemomsazRvreli
qveda da zeda qulebi.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 20
ndobis intervalis ageba
davubrundeT suraTs N7.3, romelzec mocemulia saSualos SerCeviTi Teoriuli
ganawileba. gavixsenoT, rom am SerCeviTi ganawilebis standartuli gadaxraa Xσ
da rom is normaluri ganawilebaa. am codnaze dayrdnobiT zemoT vnaxeT, rom
XX σµ 1− -dan XX σµ 1+ -mde intervali moicavs am populaciidan aRebul SerCevaTa
saSualoebis .6826 nawils. davuSvaT, axla vsvamT kiTxvas: _ Tu Cven aviRebT erT
SerCevas da misTvis X -s gamovTvliT, ras udris albaToba imisa, rom X moxvdeba
XX σµ 1− -dan XX σµ 1+ -mde intervalSi? pasuxia .6826 anu .6826-isTvis SerCevaTa
saSualoebis mniSvnelobebi moTavsebulia XX σµ 1− -sa da XX σµ 1+ -s Soris.
Sesabamisad, Tu Cven aviRebT SerCevas, albaToba imisa, rom misi saSualo moxvdeba
XX σµ 1− -sa da XX σµ 1+ -s Soris, aris .6826-is toli. yuradReba miaqcieT, rom
kiTxvaze pasuxis gasacemad Cven intervali Xµ -is irgvliv (anu μ-is irgvliv,
radgan µµ =X ) avageT da gamoviyeneT Xσ -is mniSvneloba, Semdeg ki normaluri
ganawilebis TvisebebiT visargebleT; magram SemTxvevaTa umravlesobaSi μ ucnobia
da amitom mas X -iT vafasebT. davuSvaT, μ-s irgvliv intervalis asagebad igive
procedura gamoviyeneT, magram μ-is Sesafaseblad X -s mivmarTeT. amrigad, miviReT
XX σ1− -sa da XX σ1+ -s Soris intervali. axla ukve aRar vsvamT kiTxvas: _ Sedis
Tu ara X -is mniSvneloba am intervalSi? Cven ukve viciT X . amjerad kiTxva
aseTia: _ moicavs Tu ara es intervali μ-s? am kiTxvaze pasuxis gacema albaTobis
terminebiTac SegviZlia. Tu albaToba imisa, rom XX σµ 1− -dan XX σµ 1+ -mde
intervali moicavs X -s, aris .6826, maSin, marTebulia isic, rom albaToba imisa,
rom XX σ1− -dan XX σ1+ -mde intervali moicavdes μ-s, aris .6826.
XX σ1− -dan XX σ1+ -mde intervali aris 68%-iani ndobis intervali (.6826
davamrgvaleT da procentis misaRebad 100-ze gavamravleT). am intervalis
sazRvrebi anu ukiduresi mniSvnelobebi ndobis sazRvrebs warmoadgenen. Cven am
intervalis interpretireba Semdegnairad SegviZlia: _ Tu Cven mravaljer aviRebT
SerCevebs, TiToeuli maTganisTvis gamovTvliT saSualos da Semdeg TiToeuli X -
sTvis avagebT XX σ1− -dan XX σ1+ -mde intervals, maSin am intervalTa 68%-Si Seva
μ-is mniSvneloba. yuradReba miaqcieT, rom vinaidan μ populaciis parametria da
amitom konkretuli mniSvneloba aqvs, is an Sedis intervalSi, an _ ara.
ganvixiloT am msjelobis ricxviTi magaliTi.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 21
davuSvaT, Cven gvaqvs populacia, romlis μ = 40 da σ = 10. am populaciidan
aviReT SerCeva, romlis moculobaa 25 (N = 25) da gamovTvaleT 5.41=X . Semdeg
gamovTvaloT 0.225/10/ === NX σσ . μ-sTvis 68%-iani ndobis intervali miviReT
XX σ1− -dan XX σ1+ -mde intervalis povniT. ricxviTi mniSvnelobebis 5.41=X da
2=Xσ Setanis Semdeg miviRebT ndobis intervals 39.5-dan (anu
5.3925.411 =−=− XX σ ) 43.5-mde (anu 5.4325.411 =+=+ XX σ ). Cven SegviZlia 68%-iT
darwmunebulni viyoT, rom es intervali Seicavs μ-s mniSvnelobas da am
SemTxvevaSi es asea. axla davuSvaT, rom X 41.5-is nacvlad 44.0-is tolia, maSin
68%-iani ndobis intervali iqneba 42.0-dan (anu 0.4220.441 =−=− XX σ ) 46.0-mde (anu
0.4620.441 =+=+ XX σ ). axlac SegviZlia 68%-iT darwmunebulni viyoT, rom es
intervali Seicavs μ-s, magram am SemTxvevaSi intervali ar Seicavs μ = 40-s.
cxadia, rom praqtikaSi zustad ar viciT, Seicavs ndobis intervali μ-s Tu ara,
vinaidan Cven ar viciT μ-is mniSvneloba. CvenTvis μ rom cnobili yofiliyo, aRar
dagvWirdeboda misi mniSvnelobis Sefaseba. Cven mxolod garkveuli doniT
SegviZlia viyoT darwmunebulni, rom intervali Seicavs μ-s mniSvnelobas.
kvlevaSi, rogorc wesi, 95%-iani an 99%-iani (da ara 68%-iani) ndobis
intervalebi gamoiyeneba. am intervalebis sazRvrebis dadgena standartuli
normaluri ganawilebis (romlis μ = 0 da σ = 1) Tvisebebis gamoyenebiT xdeba. a-1
cxrilSi vnaxulobT, rom z = _1.96-dan 0-mde intervali Seicavs z-is mniSvnelobaTa
.4750-s da z = 0-dan z = +1.96-mde intervali, aseve, Seicavs z-is mniSvnelobaTa .4750-s.
amrigad, z = _1.96-dan z = +1.96-mde intervalSi moTavsebuli iqneba z-is
mniSvnelobaTa .95 (anu .4750+.4750=.95). Sesabamisad, populaciisTvis 95%-iani ndobis
intervali iqneba
mde dan −+−− XX XX σσ 96.196.1 .
99%-iani ndobis intervalisTvis vnaxavT, rom z = _2.58-dan 0-mde intervali
Seicavs z-is mniSvnelobaTa .4951-s da z = 0-dan z = +2.58-mde intervali, aseve,
Seicavs z-is mniSvnelobaTa .4951-s. amrigad, z = _2.58-dan z = +2.58-mde intervalSi
moTavsebuli iqneba z-is mniSvnelobaTa .99 (anu .4951+.4951=.9902 anu .99). Sesabamisad,
populaciisTvis 99%-iani ndobis intervali iqneba
mde dan −+−− XX XX σσ 58.258.2
magaliTSi, sadac 5.41=X da 2=Xσ , 95%-iani ndobis intervali iqneba:
( ) ( ) mde21.961.5 dan −+−+ 4296.15.41 ,
rac udris
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 22
mde3.921.5 dan −+−+ 492.35.41 anu mde5.4 dan −− 458.37 .
99%-iani ndobis intervali iqneba:
( ) ( ) mde22.581.5 dan −+−+ 4258.25.41
rac udris
mde51.5 dan −+−+ 16.416.55.41 anu mde.7 dan −− 4634.36 .
am msjelobis erTma aspeqtma SeiZleba dagabnioT. Cven imitom vafasebT
populaciis parametrs μ-s, rom ar SegviZlia am parametris mniSvnelobis misaRebad
mTeli populaciis gazomva; Tumca, ndobis intervalebis agebisas Xσ -is
gamosaTvlelad viyenebT meore parametrs σ-s. saidan unda vicodeT σ-is
mniSvneloba, Tu ar viciT μ? pasuxia: _ ar unda vicodeT. Xσ unda SevafasoT Xs -
iT. Cven intervaluri Sefasebis axsnis gasamartiveblad davuSviT, rom viciT Xσ ,
magram praqtikaSi ndobis sazRvrebis dasadgenad Xs ufro gamoiyeneba, vidre _
Xσ . Xs -is gamoyenebis SemTxvevaSi ndobis intervalebi odnav didia, vidre Xσ -is
gamoyenebisas, Tumca, intervaluri Sefasebis cneba ar icvleba ganurCevlad imisa,
Xσ -s viyenebT,YTu Xs -s. ndobis intervalebis agebisas Xs -is gamoyenebaze me-8
TavSi visaubrebT.
magaliTi N7.2
ndobis intervalis ageba, rodesac Xσ cnobilia
amocana
populaciidan, romlis σ = 13.0 aviReT 30-quliani SerCeva. SerCevis saSualoa
71.0 μ-sTvis ipoveT 95%-iani ndobis intervali.
amoxsna
vpoulobT Xσ -s da a-1 cxrilSi veZebT z-is mniSvnelobas, romelic
gansazRvravs standartuli normaluri ganawilebis farTobis 95%-is Semcvel
intervals.
37.2477.5
0.1330
0.13====
NX
σσ .
a-1 cxrilSi vnaxulobT, rom standartul normalur ganawilebaSi z = _1.96-dan
z = +1.96-mde intervali Seicavs ganawilebis farTobis 95%-s. amrigad, μ-sTvis 95%-
iani ndobis intervalia
mde dan −+−− XX XX σσ 96.196.1 .
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 23
X -isa da Xσ -is ricxviTi mniSvnelobebis Casmis Sedegad miviRebT intervals
( ) ( ) mde dan −+−− 37.296.10.7137.296.10.71
anu
mde dan −+−− 65.40.7165.40.71
rac 66.35-dan 75.65-mde intervalis tolia. μ-sTvis 95%-iani ndobis intervalia
66.35-dan 75.65-mde.
SeamowmeT Tqveni codna VII.3
1. ganmarteT: _ ndobis intervali, ndobis sazRvrebi, intervaluri Sefaseba.
2. populaciidan, romlis σ=10.0 aiReT 25-quliani SerCeva. X -is mniSvnelobaa
73.0.
a) ipoveT μ-s wertilovani Sefaseba.
b) aageT 95%-iani ndobis intervali μ-sTvis.
g) aageT 99%-iani ndobis intervali μ-sTvis.
(d), (e) da (v) kiTxvebisTvis davuSvaT, rom SerCevis moculoba (25-is nacvlad)
iyo 100 (N = 100).
d) ipoveT μ-s wertilovani Sefaseba.
e) aageT 95%-iani ndobis intervali μ-isTvis.
v) aageT 99%-iani ndobis intervali μ-isTvis.
z) N = 100-ze dafuZnebuli ndobis intervali ufro mcirea, vidre _ N =
25-ze dafuZnebuli ndobis intervali. axseniT, ra ganapirobebs am
gansxavebas.
Sejameba
• statistikuri daskvna aris populaciis ucnobi parametrebis Sefaseba
SerCevis cnobil statistikebze dayrdnobiT.
• X aris μ-s Caunacvlebeli da Zalmosili Sefaseba.
• zusti Sefasebis uzrunvelsayofad SerCeva im populaciis reprezentaciuli
unda iyos, saidanac is iqna aRebuli. reprezenaciulobas ganapirobebs
SerCevis aRebis meTodi.
• s2 aris σ2-is Caunacvlebeli da Zalmosili Sefaseba.
• s gamoiyeneba σ-is Sesafaseblad, miuxedavad imisa, rom is odnav bolomde
ver afasebs mas.
• saSualos SerCeviTi ganawileba mocemuli moculobis SerCevebis saSualos
Teoriuli ganawilebaa.
თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 24
• saSualos standartuli Secdoma aris saSualos SerCeviTi ganawilebis
standartuli gadaxra.
• Xσ gamoiTvleba formuliT NX /σ . umravles SemTxvevaSi is fasdeba
formuliT NssX /= .
• SerCevis moculobis zrdasTan erTad saSualos SerCeviTi ganawileba
normalur ganawilebas uaxlovdeba. amrigad, Xσ SeiZleba gamoyenebul iqnas
X -ze dayrdnobiT μ-s Sefasebis ndobis intervalebis dasadgenad.
• ndobis intervali aris monacemTa diapazoni, romelic, savaraudod, moicavs
μ-s mniSvnelobas ndobis garkveul doneze. ndobis intervalis
SemomsazRvreli qveda da zeda qulebs ndobis sazRvrebi ewodebaT.
ZiriTadi terminebi da simboloebi
dispersia
martivi SeTxveviTi SerCeva
ndobis intervali
ndobis sazRvrebi
saSualos standartuli
Secdoma
saSualos SerCeviTi ganawileba
statistikuri daskvna
SerCevis Secdoma
Caunacvlebeli
Sefaseba
centraluri zRvariTi
Teorema
Zalmosili Sefaseba
intervaluri Sefaseba
wertilovani Sefaseba
Xµ
Xσ
2σ
Xs
s
s2
X
XX
Tavis damateba: ZiriTadi simboloebi da formulebi
me-4-dan me-7 Tavis CaTvliT mTeli rigi mniSvnelovani statistikuri
simboloebi da formulebi warmogidgineT. aRniSnuli masalis gagebisa da
daswavlis gasamartiveblad mas cxrilis saxiT gTavazobT.
simbolo formula ganmarteba
N SerCevaSi Semaval qulaTa raodenoba anu SerCevis
moculoba.
Npopulacia populaciaSi Semaval qulaTa raodenoba anu populaciis
moculoba.
X i an X SerCevis an populaciis calkeuli qula.
X
NX∑
SerCevis saSualo. X aRniSnavs SerCevaSi Semaval
calkeul qulas.
თავი 7. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 25
μ
populaciaNX∑
populaciis saSualo. X aRniSnavs populaciaSi Semaval
calkeul qulas.
S2 ( )N
XX∑ − 2
SerCevis dispersia. es dispersia ar aris populaciis
dispersiis Caunacvlebeli Sefaseba.
S ( )N
XX∑ − 2
SerCevis standartuli gadaxra. es standartuli gadaxra
SerCevis aRweriTi statistikaa, magram is ar gamoiyeneba
poulaciis standartuli gadaxris SefasebisTvis.
s2 ( )1
2
−
−∑N
XX
populaciis dispersiis Sefaseba. es dispersia
populaciis dispersiis Caunacvlebeli Sefasebaa.
s ( )1
2
−
−∑N
XX
populaciis standartuli gadaxris Sefaseba.
standartuli gadaxris es formula yvelaze xSirad
gamoiyeneba standartuli gadaxris gamosaTvlelad. Tu
gansakuTrebulad ar aris aRniSnuli, saxelmZRvanelos
teqstSi swored es standartuli gadaxraa gamoyenebuli.
SS ( )∑ −2
XX kvadratebis jami.
σ2 ( )populaciaN
X∑ − 2µ
populaciis dispersia. X aRniSnavs populaciaSi Semaval
calkeul qulas.
σ ( )populaciaN
X∑ − 2µ
populaciis standartuli gadaxra. X aRniSnavs
populaciaSi Semaval calkeul qulas.
Xµ saSualos SerCeviTi Teoriuli ganawilebis saSualo. es
saSualo populaciis saSualos (μ) tolia.
XX saSualos SerCeviTi empiriuli ganawilebis saSualo.
Xσ
Nσ
N moculobis SerCevis saSualos standartuli Secdoma.
saSualos standartuli Secdoma aris saSualos
SerCeviTi ganawilebis standartuli gadaxra.
Xs
Ns
N moculobis SerCevis saSualos standartuli Secdomis
Sefaseba.
z
σµ−X
standartuli normaluri gadaxra. am formuliT xdeba
normalurad ganawilebuli nedli qulebis gadayvana
standartuli normaluri ganawilebis z qulebSi.
z
SXX −
standartuli qula. am formuliT xdeba nedli qulis
gadayvana z qulaSi, romliTac aRiwereba, SerCevis
saSualodan ramdenad zemoT an qvemoT imyofeba nedli
თავი 7. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 26
qula. am formuliT miRebul z qulebs ver gamoviyenT
cxrilSi Na-1 mocemul standartul normalur
ganawilebasTan.
z
X
Xσ
µ−
am formuliT xdeba SerCevis saSualos gadayvana
standartuli normaluri ganawilebis z qulaSi. am
formuliT miRebuli zrealuri-is mniSvneloba SeiZleba
gamoyenebul iqnes cxrilSi Na-1 mocemul standartul
normalur ganawilebasTan.
kiTxvebi
1. romeli statistikaa romeli parametris Sefaseba?
statistika parametri
a) s μ
b) X Xσ
g) s2 σ
d) Xs σ2
2. μ saSualos mqone populaciidan aiReT ori SerCrva. pirveli SerCevis
moculoba N = 10, meore SerCevis moculoba N = 24. rogor fiqrobT,
romeli SerCevis saSualo iqneba μ-sTan ufro axlos? ratom?
3. arCevnebis Sedegebis prognozirebis mizniT mkvlevari qalaqSi
gamokiTxvas atarebs.man icis, rom qalaqis amomrCevelTa populaciis 34%
Seadgens registrirebul demokrats, 29% _ registrirebul
respublikels, xolo 37%-s ara aqvs politikuri mikuTvnebuloba.
SerCevis romeli tipi (martivi Tu stratificirebuli SemTxveviTi
SerCeva) uzrunvelyofs amomrCevelTa ufro metad reprezentaciul
SerCevas? ratom?
4. XXI saukunis mecnierebs globaluri daTbobisa da am movlenaSi
adamianis qcevis SesaZlo wvlilis sakiTxebi ainteresebT; Tumca,
ramdenad sainteresoa es Temebi Cveulebrivi saSualo adamianisTvis?
CvenTvis ukve cnobili internet-gamkiTxvis gverdze ganTavsda SekiTxva
imis Sesaxeb, Tu ramdenad ainteresebT gverdis damTvaliereblebs
globaluri daTbobis Tema. daaxloebiT 25,000 adamianma upasuxa dasmul
SekiTxvas da maTgan 56%-ma aRniSna, rom ainteresebs, danarCenma
respondentebma ki miuTiTes, rom maTTvis es Tema uinteresoa.
yoveldRiurad, sul mcire asobiT aTasi adamiani aTvalierebs am veb
თავი 7. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 27
gverds. ra daskvnis gakeTeba SegiZliaT am gamokiTxvis mixedviT imis
Sesaxeb, Tu ramdenad ainteresebT adamianebs globaluri daTbobis
problema? axseniT Tqveni pasuxi.
5. kolejSi mwevelebis populaciaSi yoveldRiurad moweuli sigaretis
raodenobis Sesafaseblad, Tqven SemTxveviT SearCieT 16 studenti im
studentebidan, vinc sakuTar Tavs mwevelad miiCnevs. Tqven naxeT, rom am
jgufis SerCevis saSualoa 9.0 sigareti dReSi.
a) gansazRvreT populaciis saSualos wertilovani Sefaseba.
b) davuSvaT, rom σ = 3.1 sigareti dReSi. ipoveT saSualos 95%-iani
ndobis intervali.
6. davuSvaT, rom kolejis studentebis populaciaSi saSualo Zilis
xangrZlivobaa 7.1 sT, xolo σ = 1.2 sT. Tqven SemTxveviT airCieT 16
studenti da gansazRvreT am SerCevis Zilis saSualo xangrZlioba
(saaTebSi).
a) ra iqneba N = 16 moculobis SerCevis saSualos standartuli
Secdoma?
b) ras udris albaToba imisa, rom Tqveni SerCevis saSualo 8.0 sT-is
toli an masze meti iyos?
g) ras udris albaToba imisa, rom Tqveni SerCevis saSualo 6.8 sT-is
toli an masze naklebi iyos?
d) ras udris albaToba imisa, rom Tqveni SerCevis saSualo
moTavsebuli iyos 6.5-sa da 7.7 sT-s Soris?
7. ra gansxvavebaa s-sa da Xs -s Soris?
8. cukermanma (Zuckerman, 1979) SegZnebebis Ziebis gasazomi skala Seqmna.
romelTac SegrZnebebis Ziebis maRali maCveneblis mqone adamianebs male
bezrdebaT dawyebuli aqtivobebi da Zalian esaWiroebaT axal da zogjer
riskian saqmianobebSi CarTva. skalaze qulebi 10-dan 40-mde meryeobs.
maRali qula gviCvenebs SegrZnebebis Ziebis maRal xarisxs. davuSvaT,
kolejis aTleti studentebis poulaciis saSualoa 23.0, xolo σ = 4.0.
Tqven am skalaze kolejis aTleti studentebis xuTi (a, b, g, d da e)
SerCeva gazomeT. SerCevebi sxvadasxva moculobis iyo, rogorc qvemoT
aris mocemuli:
SerCeva N
a
b
g
d
4
9
16
25
თავი 7. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 28
e 100
TiToeuli SerCevisTvis
a) gamoTvaleT Xσ .
b) ipoveT albaToba imisa, rom Tqveni SerCevis saSualo 25-is toli
an masze meti iqneba.
g) Xσ -isa da N-is ra kavSiris ilustraciaa es magaliTi.
d) TiToeuli SerCevisTvis ipoveT albaToba imisa, rom Tqveni
SerCevis saSualo moTavsebuli iqneba 23-sa da 25-s Soris.
Tavi VIII. statistikuri hipoTezebis Semowmebis
Sesavali: z kriteriumi da erTi SerCevis t kriteriumi
statistikuri hipoTezis Semowmeba
parametruli da araparametruli kriteriumebi
statistikuri hipoTezebi
kriteriumis statistika: z
mniSvnelovnobis doneebi: statistikuri iSviaToba
gadawyvetilebis miReba statistikuri hipoTezebis Sesaxeb
kriteriumis statistikisa da mniSvnelovnobis donis
gamoyenebiT
statistikuri hipoTezis Semowmebis etapebis Sejameba
erTi SerCevis t kriteriumi
t statistikis SerCeviTi ganawileba
statistikuri hipoTezis Semowmeba erTi SerCevis t
kriteriumiT
erTi SerCevis t kriteriumis Sedegebis prezentacia
mniSvnelovani sakiTxebi statistikuri hipoTezis
SemowmebaSi
ratom ar viRebT nulovan hipoTezas?
I da II gvaris Secdomebi statistikuri kriteriumebis
gamoyenebisas
statistikuri da mecnieruli mniSvnelovnoba
ndobis intervalebis ageba populaciis saSualosaTvis t
statistikis gamoyenebiT
wina TavebSi ganvixileT statistikis gamoyeneba monacemTa organizebis, aRweris
mizniT da populaciis parametrebis Sesafaseblad. am TavSi ki ganvixilavT
statistikis gamoyenebas hipoTezis Semowmebis mizniT.
ganvixiloT Semdegi problema. adamianuri faqtori aris bihevioristuli
mecnierebis sfero, romelic adamianis SesaZleblobebisadmi manqanisa da teqnologiis
Sesatyvisobas swavlobs. adamianuri faqtoris Semswavleli fsiqologebi xSirad
atareben adamianTa fizikuri maxasiaTeblebis gazomvebs, raTa Seqmnan skamebis,
kontrolis panelisa da monitoris iseTi dizaini, romelic adamians moergeba.
magaliTad, warmoidgineT, rom xarT satvirTo manqanebis mwarmoebeli kompaniis
TanamSromeli, romelic pasuxismgebelia Sida dizainze. wlebis manZilze skamebi
Tqvens satvirTo manqanebSi iseTnairad iyo mowyobili, rom Seesatyviseboda mZRolebis
populacias, romelTa saSualo muxlqveSa simaRle iyo 42 santimetri (sm) da σ _ 3 sm.
fexis muxlqveSa simaRle aris vertikaluri manZili mjdomare mZRolis muxlidan
iatakamde.
axlaxan Tqvenma kompaniam daiwyo satvirTo manqanebis gayidva mTel msoflioSi
da miiRo saCivrebi skamebis arakomfortulobis Taobaze. Tqven ivaraudeT, rom
SesaZloa zogierT qveyanaSi mZRolebis muxlqveSa simaRle gansxvavdeba im
populaciisagan, romlisTvisac Seqmnili iyo es skamebi. am hipoTezis Sesamowmeblad,
Tqven aiyvaneT satvirTo manqanis 25 mZRolisagan Semdgari SerCeva, romelic im qveynis
mZRolebis populaciisagan damoukideblad da SemTxveviTad iyo SerCeuli,
romelTaganac iyo miRebuli saCivrebi. am SerCevis mZRolebis saSualo muxlqveSa
simaRle aris 44 sm. aSkaraa, rom am SerCevis saSualo _ 44 sm ar aris igive, rac im
populaciis saSualo, romelic 42 sm-is tolia. magram ras gveubneba es gansxvaveba
populaciis Sesaxeb? arsebobs ori SesaZlebloba. pirveli aris is, rom populacias,
romlidanac iyo aRebuli realurad SerCeva, saSualo muxlqveSa simaRle 42 sm-is
toli aqvs, xolo Tqveni SerCevis saSualo _ 44 sm gansxvavdeba am µ-gan mxolod
SerCevis Secdomis gamo. SemTxveviT Tqven SearCieT iseTi SerCeva, romlis saSualo
muxlqveSa simaRle iyo 42 sm-ze meti.
meore SesaZlebloba isaa, rom SerCevis saSualo asaxavs populacias iseTi µ-iT,
romelic gansxvavdeba 42 sm-isagan. mZRolebis populacias am qveyanaSi aqvs saSualo
muxlqveSa simaRle, romelic gansxvavdeba 42 sm-sagan. am ori SesaZleblobidan romeli
unda airCioT rogorc im situaciis amsaxveli, romelic arsebobs im qveyanaSi,
saidanac iyo aRebuli SerCeva?
Cven gvaqvs problema, romelic moiTxovs statistikuri hipoTezis Semowmebas,
raTa gavakeToT arCevani, Tu romelia ufro metad mosalodneli im ori SesaZlo
situaciidan. statistikuri terminebiT, es problema moiTxovs statistikuri hipoTezis
Semowmebas imis gadasawyvetad, mniSvnelovnad gansxvavdeba Tu ara SerCevis saSualo _
44 sm populaciis saSualosagan, romelic 42 sm-is tolia. suraTi #8.1 am problemis
ilustracias warmoadgens.
suraTi #8.1. ori SesaZlo situacia, romelic SeiZleba ganapirobebdes SerCevis realurad miRebul saSualos.
SerCevis realuri saSualo = 44 sm qvemoT mocemuli ori SesaZleblobidan romeli ganapirobebs SerCevis am saSualos?
.
statistikuri hipoTezis Semowmeba
parametruli da araparametruli kriteriumebi
parametruli kriteriumebi
parametrulia kriteriumebi Tu hipoTezaSi aris
gamoTqmuli varaudi populaciis iseTi parametrebis
Sesaxeb rogoricaa, magaliTad, µ da σ2. parametruli
kriteriumebi xSirad garkveul pirobebs
warmoadgenen, rogorc WeSmarit daSvebas im
populaciis monacemebisaTvis, romlidanac aris
aRebuli SerCeva. roca gavecnobiT konkretul
parametrul kriteriumebs, agreTve, unda ganvsazRvroT am kriteriumebisaTvis
gakeTebuli daSvebebi. am TavSi gavecnobiT statistikuri hipoTezis Semowmebas,
romelic iyenebs or gansxvavebul parametrul kriteriums, erTi SerCevis z kriteriums
parametruli kriteriumi
(parametric test) _ statistikur
kriteriums ewodeba parametruli,
Tu hipoTezaSi aris gamoTqmuli
varaudi populaciis parametris
Sesaxeb.
populaciis saSualo, romlidanac SerCeva iyo aRebuli, aris 42 sm. SerCevis saSualo gansxvavdeba populaciis saSualosagan SerCevis Secdomis gamo.
populaciis saSualo, romlidanac SerCeva iyo aRebuli, ar aris 42sm. SerCevis saSualo gansxvavdeba populaciis saSualosagan ara SerCevis Secdomis gamo, aramed sxva mizeziT.
rogor vupasuxoT SekiTxvas: gamoviyenoT statistikuri kriteriumi, raTa ganvsazRvroT,
gansxvavdeba Tu ara X =44 sm mniSvnelovnad µ=42 sm-sagan.
da erTi SerCevis t kriteriums. sxva parametrul kriteriumebs gavecnobiT IX-XII
TavebSi.
araparametruli kriteriumebi
araparametrulia kriteriumebi, Tu
hipoTezaSi ar aris gamoTqmuli
varaudi populaciis parametris
Sesaxeb. Cveulebriv, araparametruli
kriteriumebi ar akeTeben daSvebebs im populaciis monacemebis ganawilebis Sesaxeb,
romlidanac aris aRebuli SerCeva. am mizeziT, isini xSirad miekuTvnebian e.w.
ganawilebisagan Tavisufal kriteriumebs. araparametruli kriteriumebi xSirad
gamoiyeneba, roca parametruli kriteriumebisaTvis aucilebeli daSvebebi ar
Seesabameba monacemebs. parametruli kriteriumebis didi nawilisaTvis, magram ara
yvela kriteriumisaTvis, arsebobs alternatiuli araparametruli kriteriumebi igive
monacemebis gasaanalizeblad. ramdenime araparametruli kriteriumi warmodgenilia me-
13 TavSi. statistikuri hipoTezis Semowmebisas gamoyenebuli ZiriTadi cnebebis
umravlesoba erTnairia parametruli da araparametruli kriteriumisaTvis.
statistikuri hipoTezebi
statistikuri hipoTezis
Semowmeba, iqneba is parametruli
Tu araparametruli, iwyeba
statistikuri hipoTezis
CamoyalibebiT. parametruli
kriteriumisaTvis statistikuri
hipoTeza aris debuleba
populaciis parametris Sesaxeb. es
debuleba SeiZleba WeSmariti iyos
da SeiZleba _ ara. is ubralod
imisaTvis keTdeba, raTa safuZveli
daudos Semowmebas. magaliTad, Cven SegviZlia CamovayaliboT hipoTeza imis Sesaxeb,
rom muxlqveSa simaRlis saSualo im populaciisaTvis, romlidanac ayvanili gvyavda
25 kaciani SerCeva, aris 42 sm. realurad populaciis saSualo SesaZloa iyos 42 sm an
ar iyos. Cven ubralod vvaraudobT, rom is 42 sm-ia da am hipoTezas SemdgomSi
SevamowmebT. statistikuri kriteriumebi moiTxovs, rom formulirebuli iyos ori
hipoTeza, nulovani da alternatiuli.
araparametruli kriteriumi (nonparametric test) _
statistikur kriteriums ewodeba araparametruli,
Tu hipoTezaSi ar aris gamoTqmuli varaudi
populaciis parametris Sesaxeb.
statistikuri hipoTeza (statistical hypothesis) _
debuleba populaciis parametris
Sesaxeb(parametruli kriteriumis Sesaxeb).
nulovani hipoTeza (null hypothesis) - aris im pirobis
Camoyalibeba, romelic mecniers populaciis
Sesaxeb apriorulad WeSmaritad miaCnia. es aris
hipoTeza, romelic mowmdeba statistikuri
kriteriumiT.
alternatiuli hipoTeza (statistical alternative hypothesis)
_ aris debuleba imis Sesaxeb, Tu ra unda iyos
WeSmariti, Tu nulovani hipoTeza mcdaria.
nulovani hipoTeza
nulovani hipoTeza aris im pirobis Camoyalibeba, romelic mecniers populaciis
Sesaxeb apriorulad WeSmaritad miaCnia. es aris hipoTeza, romelic mowmdeba
statistikuri kriteriumiT. nulovani hipoTeza, Cveulebriv, aRiniSneba simboluri an
pirobiTi formiT H0-is gamoyenebiT imis aRsaniSnavad, rom SemoTavazebulia iseTi
hipoTeza, rogoricaa µ=42 smis. nulovani hipoTezis _ µ=42 sm gamoxatva statistikur
CanawerSi Semdegnairad SegviZlia H0:µ=42sm, da es gamoiTqmis aseTnairad: `aS nuli: miu
tolia 42 sm-is”.
alternatiuli hipoTeza
alternatiuli hipoTeza aris debuleba imis Sesaxeb, ra unda iyos WeSmariti, Tu
nulovani hipoTeza mcdaria. magaliTad, Tu H0 : µ=42 sm ar aris WeSmariti, maSin
alternatiuli unda iyos, rom µ ar aris 42 sm-is toli. es alternatiuli hipoTeza
aRiniSneba H1-iT da iwereba Semdegnairad: H1: µ ≠ 42 sm. es gamoiTqmis aseTnairad: `aS
erTi: miu ar aris 42 sm-is toli“.
statistikuri hipoTezebis Tvisebebi
TiToeuli statistikuri kriteriumi moiTxovs nulovani da alternatiuli
hipoTezebis formulirebas. es hipoTezebi iseTnairad unda iyos warmodgenili, rom
akmayofilebdnen Semdeg pirobebs:
• isini arian urTierTgamomricxavni: urTierTgamomricxavi niSnavs, rom orive
hipoTeza ar SeiZleba iyos WeSmariti erTdroulad. magaliTad, SeuZlebelia H0 :
µ= 42 sm da H1 : µ ≠ 42 sm iyos erTdroulad WeSmariti erTi da imave
populaciisaTvis.
• isini moicaven parametris yvela SesaZlo mniSvnelobas, romelic CarTulia
hipoTezaSi. H0 : µ=42 sm da H1 : µ ≠42 sm akmayofilebs am moTxovnas; maT Soris
moTavsebulia µ-s yvela SesaZlo mniSvneloba CvenTvis saintereso
populaciisaTvis.
am Tvisebebis gamo mocemuli ori hipoTezidan mxolod erTi unda warmoadgendes
populaciisaTvis WeSmarit pirobas. magaliTad, populaciis saSualo an aris 42 sm-
is toli da H0 : µ= 42 sm aris WeSmariti, an ar aris toli da maSin alternatiuli
hipoTeza H1 : µ≠ 42 sm-s aris WeSmariti.
amrigad, ganxiluli problemisaTvis statistikuri hipoTezebia:
H0 : µ= 42 sm (nulovani hipoTeza)
H1 : µ ≠ 42 sm (alternatiuli hipoTeza)
statistikuri hipoTezebis funqcia
nulovani hipoTezis funqciaa statistikur kriteriumSi daadginos is pirobebi,
romlis drosac statistikis SerCeviTi ganawileba iyo miRebuli. me-7 TavSi Cven
ganvsazRvreT saSualos SerCeviTi ganawileba, rogorc X -is mniSvnelobebis
ganawileba, roca N moculobis yvela SesaZlo SerCeva populaciidan iyo aRebuli.
magaliTad, Tu vivaraudebT, rom H0:µ=42 sm aris WeSmariti da σ=3 sm, Cven SegviZlia
miviRoT X -is SerCeviTi ganawileba am populaciis iseTi SerCevebisaTvis, romelTa
moculoba aris 25. suraTze #8.2 mocemuli es SerCeviTi ganawileba aris normaluri
ganawileba populaciis saSualoTi µ X =42.0 sm, standartuli SecdomiT _ Xσ -iT, sadac
6.053
253
====NXσσ
am ganawilebaSi SerCeviTi saSualo _ 44 sm µ X -isagan daSorebulia saSualos sami
standartuli SecdomiT. Sesabamisad, im pirobebSi, roca H0:µ=42 sm-s, iseTi SerCevis
albaToba, romlis saSualoa 44 sm, aris Zalian mcirea, Tuki populaciidan aRebuli
SerCevis moculoba iqneba 25. amitomac Cven ufro mivemxrobodiT nulovani hipoTezis _
H0:µ=42 sm-is uaryofas, Tuki aRebuli SerCevis saSualo iqneboda 44 sm. X -is aseTi
mniSvneloba aris statistikurad iSviaTi xdomiloba, roca H0 aris WeSmariti. magram
sanam gadavwyvetdeT sabolood ganvixiloT Tu ara H0 rogorc WeSmariti hipoTeza,
Cven unda:
• davadginoT albaToba imisa, rom SeiZleba miviRoT X -is iseTi mniSvneloba,
romelic iqneba 44 sm-is toli an masze meti, Tu H0:µ=42 sm iqneba WeSmariti. es
nabiji moiTxovs kriteriumis statistikis gamoyenebas;
• ganvsazRvroT ras vgulisxmobT statistikurad iSviaT xdomilobaSi. es nabiji
moiTxovs mniSvnelovnobis donis gansazRvras.
suraTi #8.2. saSualos
Teoriuli SerCeviTi
ganawileba agebuli N=25
moculobis toli
SerCevebisaTvis µ X =42.0 sm-
is da _ Xσ =0.6-s mqone
populaciidan.
-3σ X -2σ X -1σ X µ X +1σ X +2σ X +3σ X 40.2 40.8 41.4 42.0 42.6 43.2 43.8
muxlqveSa simaRle (sm)
kriteriumis statistika: z
kriteriumis statistika aris ricxvi, romelic nulovani hipoTezis Semowmebis
saSualebas gvaZlevs da gamoiTvleba SerCevis monacemebze dayrdnobiT. magaliTad,
Cven gvinda kriteriumi, raTa davadginoT albaToba imisa, rom SerCeviTi saSualo
iqneba 44 sm-is toli an masze meti, Tu H0 : µ=42 sm aris WeSmariti. radgan suraTze #8.2
ilustrirebuli SerCeviTi saSualos ganawileba aris normaluri ganawileba µ=42 sm-
iTa da Xσ =0.6, Cven SegviZlia X -is nebismieri mniSvnelobis standartizacia:
z =X
Xσ
µ−
sadac:
X =SerCeviTi saSualo
µ= populaciis hipoTeturi saSualo
Xσ =saSualos standartuli Secdoma
z-is mniSvneloba gansazRvravs SerCeviTi saSualos mdebareobas standartul
normalur ganawilebaSi. aRsaniSnavia, rom es
formula identuria im formulisa, romelic
mocemuli iyo z-isaTvis me-7 TavSi.
Tu X =44 sm µ-s hipoTeturi mniSvneloba _ 42 sm,
da Xσ =0.6, maSin zrealuris mniSvnelobaa
zrealuri= 33.36.0
26.04244
+==−
=−
X
Xσ
µ
z statistikuri sidide gansazRvrulia rogorc zrealuri, radgan is miRebulia SerCevis
X mniSvnelobebze dakvirvebiT. z statistikis SerCeviTi ganawilebidan, romelic
mocemulia cxrilSi #a-1, Cven vpoulobT, rom albaToba imisa, rom z iqneba toli an
meti +3.33-ze aris 0.0004 (ix. g sveti cxrilSi #a-1 z=3.33-saTvis). Sesabamisad X =44 sm an
masze meti mniSvnelobis miRebis albaToba 25-is toli moculobis SerCevisaTvis aris
0.0004, Tu H0:µ=42sm-is aris WeSmariti. sxva sityvebiT rom vTqvaT, Tu Cven SemTxeviT
SevarCevT 25-is toli moculobis 10000 SerCevas populaciidan, romlis µ=42 sm da
σ=3, maSin mosalodnelia, rom 10000 SerCevidan mxolod oTxs eqneboda SerCeviTi
saSualo 44-is toli an masze meti. amrigad, Tuki H0:µ=42sm aris WeSmariti, iseTi
SerCevis arseboba, romlis X tolia 44 sm-is, iqneboda statistikurad iSviaTi
xdomiloba. Tuki Cven miviRebdiT am X , Cven namdvilad uarvyofdiT nulovan hipoTezas,
radgan is miRebulia iseTi populaciidan, romlis saSualo muxlqveSa simaRle aris
42 sm da miviRebdiT alternatiul hipoTezas, rom µ ≠ 42 sm.
kriteriumis statistika (test
statistic) aris ricxvi, romelic
nulovani hipoTezis Semowmebis
saSualebas gvaZlevs da
gamoiTvleba SerCevis monacemebze
dayrdnobiT.
aseTi gadawyvetilebis miRebis azri isaa, rom Tu populaciis saSualo aris 42 sm,
maSin iseTi SerCevis arseboba, romlis saSualo aris 44 sm, SesaZlebeli iqneboda
mxolod 4-jer yoveli aRebuli 10 000 SerCevisaTvis. marTalia, X =44 sm aris
SesaZlebeli xdomiloba, Tuki H0 : µ=42 sm-a, magram is mainc ar aris maRali albaTobis
mqone xdomiloba. misi miReba ufro metad mosalodneli iqneboda, Tuki populaciis
saSualo gansxvavebuli iqneboda 42sm-isagan. amrigad, Cven uarvyaviT H0 da miviReT
H1. es gadawyvetileba SeiZleba mcdaric iyos. Cven SegveZlo migveRo X =44 sm, maSinac
ki, roca µ=42 sm-ia, magram es iqneboda naklebad mosalodneli xdomiloba.
davuSvaT, miviReT SerCeviTi saSualo, romelic 42.5 sm-is tolia, nacvlad 44 sm-
isa. aq z toli iqneba:
zrealuri= 83.06.05.0
6.0425.42
+==−
cxrili #a-1-is g svetSi vipoviT, rom albaToba imisa, rom z iqneba toli an meti
+0.83-ze aris 0.2033. amrigad, Tu H0 aris WeSmariti da Cven aviRebT 10000 SerCevas,
romlis moculoba aris 25-is toli, amaTgan 2033-s eqneba X -is mniSvneloba 42.5-is
toli an masze meti. aris Tu ara 42.5 sm-is toli SerCeviTi saSualo statistikurad
iSviaTi xdomiloba, Tu H0 : µ=42sm aris WeSmariti? ra gadawyvetilebas miiRebT am
SemTxvevaSi nulovani hipoTezis Sesaxeb? uaryofT H0 : µ=42sm hipoTezas da miiRebT H1 :
µ ≠ 42sm hipoTezas? Tu ver uaryofT H0-s da amitom ar miiRebT H1-s? imisaTvis rom
vupasuxoT am SekiTxvas, Cven unda ganvsazRvroT zrealuris romeli mniSvneloba unda
miviCnioT statistikurad iSviaT Sedegad, UTuki H0 aris WeSmariti . statistikuri
iSviaToba ganisazRvreba im mniSvnelovnobis donis dadgeniT, romelic iqneba
miRebuli.
mniSvnelovnobis doneebi: statistikuri iSviaToba
mniSvnelovnobis done – esaa albaTobis is mniSvneloba,
romelic iZleva nulovani hipoTezis uaryofis pirobas
statistikur kriteriumSi. statistikur kriteriumSi
mniSvnelovnobis doneebi mocemulia rogorc alfas anu
α-s mniSvneloba (berZnuli anbanis pirveli aso).
Cveulebriv, bihevioristebi iyeneben alfas doneebs,
romlebic tolia 0.05 an 0.01. amrigad, Tu α=0.05, maSin
zrealuris mniSvnelobebi, romlebic 100-dan 5-jer an
naklebad xorcieldeba, iTvleba iSviaT xdomilobebad, Tu H0 aris WeSmariti, amitom
Cven varCevT H0-is uaryofasa da H1-is miRebas.
mniSvnelovnobis done
(significance level) _ esaa
albaTobis is mniSvneloba,
romelic iZleva nulovani
hipoTezis uaryofis pirobas
statistikur kriteriumSi.
alfa (α): mniSvnelovnobis
donis sidide (albaToba).
statistikuri hipoTezebis Sesaxeb gadawyvetilebis miReba kriteriumis
statistikisa da mniSvnelovnobis donis gamoyenebiT
mas Semdeg, rac avirCevT kriteriumis statistikas da mniSvnelovnobis dones, Cven
ukve mzad varT miviRoT gadawyvetileba statistikuri hipoTezis Sesaxeb. es moicavs
sam safexurs:
1. kriteriumis statistikis SerCeviT ganawilebaSi uaryofis aris an areebis
mdebareobis gansazRvra.
2. kriteriumis statistikis mniSvnelobis gamoTvla SerCevidan miRebuli
monacemebis safuZvelze.
3. statistikuri hipoTezebis Sesaxeb gadawyvetilebis miReba imis safuZvelze,
xvdeba Tu ara kriteriumis statistika uaryofis areSi.
uaryofis areebis mdebareobis
gansazRvra
uaryofis are warmoadgens
kriteriumis statistikis (SerCeviTi
ganawilebis) im mniSvnelobebs,
romelTa albaToba tolia an naklebia
α-ze nulovani hipoTezis WeSmaritebis
pirobebSi. magaliTad, uaryofis are
gansazRvravs z-is mniSvnelobebs,
romlebic uzrunvelyofen H0-is
uaryofas.
z statistikis SerCeviTi
ganawileba Tavisi uaryofis areebiT, romlebic ganlagebulni arian ganawilebis
TiToeul kudTan , naCvenebia suraTze #8.3. z-is statistikurad iSviaTi mniSvnelobebi
SeiZleba Segvxvdes ganawilebis orive kudTan. magaliTad, z-is mniSvnelobebi +3 an -3
Tanabrad SeuZlebelia, roca H0 WeSmaritia. z-is rogorc didi dadebiTi, aseve didi
uaryofiTi mniSvneloba ganxiluli problemisaTvis SeiZleba saeWvod xdides H0 : µ=42
sm-is WeSmaritebas. amrigad, uaryofis areebi z statistikisaTvis mdebareobs SerCeviTi
ganawilebis orive mimarTulebiT. kriteriumis statistika uaryofis areebiT
kriteriumis statistikis SerCevis ganawilebis orive mimarTulebiT ganisazRvreba
rogorc ormxrivi an aramimarTulebiTi kriteriumi. zrealuris mniSvnelobis
ganTavseba romelime uaryofis areSi araalbaTuri, an iSviaTia, Tuki H0 WeSmaritia,
magram amis albaToba ufro didia H1-is WeSmaritebis pirobebSi. amitom, Tuki zrealuri
moxvdeba uaryofis romelime areSi, H0 iqneba uaryofili da H1 miRebuli.
uaryofis are (rejection region) - kriteriumis
statistikis (SerCeviTi ganawilebis) is
mniSvnelobebi, romelTa albaToba tolia an
naklebia α-ze nulovani hipoTezis
WeSmaritebis pirobebSi. Tuki kriteriumis
statistika moxvdeba uaryofis areSi, maSin H0
uaryofili iqneba.
ormxrivi kriteriumi (two-tailed test) - statistikuri kriteriumi, roca uaryofis
aris asagebad gamoyenebulia kriteriumis
statistikis SerCevis ganawilebis orive kudi.
calmxrivi kriteriumi (one-tailed test)-
SeiZleba iseTi SemTxvevac moxdes, rom Cven garkveuli mizeziT uaryofis are z-is
SerCeviTi ganawilebis mxolod erTi mimarTulebiT ganvaTavsoT. am SemTxvevaSi
gamoiyeneba calmxrivi kriteriumi an mimarTulebis mqone kriteriumi.
suraTi #8.3. z statistikis SerCeviTi ganawileba gviCvenebs uaryofis areebs ormxrivi
kriteriumisaTvis, roca α=0.05. Tu zrealuris mniSvneloba xvdeba erT-erT
uaryofil areSi, H0 unda uarvyoT.
calmxrivi kriteriumis upiratesoba isaa, rom zrealuris ufro mcire mniSvneloba
SeiZleba moxvdes uaryofis areSi. calmxriv kriteriums ufro detalurad mogvianebiT
ganvixilavT am TavSi. am SemTxvevaSi ki ganvixilavT ormxrivi kriteriumis magaliTs.
Kkritikuli mniSvnelobebi. Kkritikuli mniSvnelobebi
konkretuli ricxviTi mniSvnelobebia, romlebic adgenen
uaryofis aris sazRvrebs. z-is kritikuli mniSvnelobebi
mocemulia cxrilSi #a-1 g sveti. pirveli nabiji am
kritikuli mniSvnelobis povnaSi aris mniSvnelovnobis
donis SerCeva, rogoricaa α=0.05. ormxrivi kriteriumisaTvis uaryofis areebi
ganlagebulia z-is SerCeviTi ganawilebis orive kudTan. amitomac Cven gvWirdeba are
ganawilebis TiToeul kudTan, romelSic moxvedra tolia α-s naxevrisa an 0.5/2, rac
tolia 0.025. z-is ra mniSvnelobebi akmayofileben moTxovnas, rom z-is SerCeviTi
ganawilebis 0.025 wili moxvdes am mniSvnelobebis miRma? cxrilSi #a-1 Cven gavyvebiT
g svets, sanam ar vipoviT ares, romelic mdebareobs z-is im mniSvnelobis gaswvriv,
romelic tolia 0.025. roca vipoviT am ares, wavikiTxavT z-is Sesatyvis mniSvnelobas
a svetSi. z-is es mniSvneloba iqneba 1.96. amrigad, α=0.05-saTvis ormxrivi kritikuli
mniSvnelobebia z = -1.96 da z=+1.96. z-is es kritikuli mniSvnelobebi (simbolurad
aRvniSnoT zkritikuli) gansazRvraven uaryofis areebis mdebareobas z-is SerCeviT
ganawilebaSi, rogorc es naCvenebia suraTze #8.3.
kritikuli mniSvnelobebi
(critical value) _ konkretuli
ricxviTi mniSvnelobebi,
romlebic adgenen
uaryofis aris sazRvrebs.
-3 -2 -1 0 +1 +2 +3 z-is mniSvnelobebi
H0 uaryofa uaryofis are 0.025
H0 uaryofa uaryofis are 0.025
H0 miRebis zona
z=-1.96 z=+1.96 z-is kritikuli z-is kritikuli mniSvneloba mniSvneloba
gadawyvetilebebi statistikuri hipoTezebis Sesaxeb
H0 hipoTezis uaryofisa Tu miRebis Sesaxeb gadawyvetilebis miReba xdeba
zrealursa da zkritikuli mniSvnelobaTa SedarebiT. zrealuris is mniSvneloba, romelic
tolia an naklebia zkritikuli =-1.96 an tolia an metia zkritikuli=+1.96 xvdeba uaryofis
areSi. zrealuris mniSvnelobebi, romlebic xvdebian uaryofis areSi, arian
statistikurad iSviaTni, Tuki H0 aris WeSmariti , magram arian ufro mosalodneli im
SemTxvevebSi, roca H1 aris WeSmariti. amrigad, roca zrealuri xvdeba uaryofis areSi,
amas mivyavarT H0-is uaryofamde. Tuki uarvyofT H0-s, maSin unda miviRoT H1. Tu
zrealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da Tuki H0-s ver
uarvyofT, maSin ver miviRebT H1-s.
aRsaniSnavia, rom Cven viyenebT zrealuris mniSvnelobas, raTa miviRoT H0-is Sesaxeb
gadawyvetileba; uarvyoT H0, TuU CavTvaloT, rom H0-s ver uarvyofT. gadawyvetileba,
romelic miiReba H0-is Sesaxeb, SemdgomSi gansazRvravs gadawyvetilebas, romelic
miiReba H1-is Sesaxeb. Tu Cven uarvyofT H0-s, maSin unda miviRoT H1. piriqiT, Tu H0-s
ver uarvyofT, ver miviRebT H1-s.
muxlqveSa simaRlis magaliTs Tu davubrundebiT, zrealuri= + 3.33 aris ufro meti,
vidre zkritikuli=+1.96, amitom is xvdeba maRal uaryofis areSi. Sesabamisad, Cven unda
uarvyoT H0 : µ=42sm da miviRoT H1 : µ≠42sm. Cven uarvyofT nulovan hipoTezas imis
Sesaxeb, rom SerCeva iyo aRebuli im populaciidan, romlis saSualo 42.0 sm-is tolia.
radgan SerCevis saSualo, X =44.0 sm, aris ufro didi, vidre populaciis hipoTeturi
saSualo _ 42 sm, Cven gamogvaqvs daskvna, rom im qveyanaSi, romlidanac iyo aRebuli
SerCeva, mZRolebis saSualo muxlqveSa simaRle aris ufro meti, vidre X =42 sm.
saukeTeso wertilovani Sefaseba am populacis saSualosaTvis aris SerCevis saSualo,
X =44.0. erT-erTi mizezi imisa, rom satvirTo manqanebis skamebi SeiZleba yofiliyo
arakomfortuli, isaa, rom zogierT qveyanaSi mZRolebis populacias ufro grZeli
muxlqveSa simaRle aqvs, vidre im skamebis simaRle, romelic maTTvis iyo Seqmnili.
SevajamoT, rogor unda miviRoT gadawyvetileba statistikuri hipoTezebis
Sesaxeb:
• Tu zrealuri xvdeba uaryofis areSi, maSin uarvyofT H0-s da viRebT H1-s.
• Tu zrealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da ar
viRebT H1-s.
statistikurad mniSvnelovani gansxvaveba
statistikurad mniSvnelovania gansxvaveba, roca
kriteriumis statistikis realuri mniSvneloba xvdeba
uaryofis areSi da uarvyofT H0-s. z kriteriumisaTvis
stastikurad mniSvnelovani gansxvaveba gviCvenebs, rom
Cven miviReT gadawyvetileba imis Sesaxeb, rom SerCevis
saSualo aRebulia im populaciidan, romlis saSualo
gansxvavdeba populaciis hipoTeturi saSualosagan.
statistikurad mniSvnelovani gansxvaveba xSirad warmodgenilia Semdegi formiT:
zrealuri =+3.33, p≤0.05. simbolo ≤ niSnavs naklebia an tolia, amrigad, p≤0.05 ikiTxeba
rogorc “p naklebia an toli 0.05”. xSirad naklebobis niSani < gamoiyeneba ≤ niSnis
nacvlad, ase rom z aris warmodgenli, rogorc zrealuri=+3.33, p<0.05. orive SemTxvevaSi
naCvenebia, rom H0 uaryofilia 0.05 mniSvnelovnobis doneze, radganac zrealuris miRebis
albaToba (p), roca H0 WeSmaritia, tolia an naklebia 0.05-s. es gadawyvetilebebi da
daskvnebi, romelic miRebulia statistikurad mniSvnelovani gansxvavebidan z
kriteriumis saSualebiT Sejamebulia cxrili #8.1-is marcxena svetSi.
cxrili #8.1. SerCevis saSualosa da populaciis hipoTeturi saSualos Sefasebisas
gadawyvetilebebisa da daskvnebis Sejameba statistikuri hipoTezebis Semowmebis Sesaxeb z
kriteriumis gamoyenebiT. gamoyenebulia 0.05 mniSvnelovnobis done.
Tu zrealuri xvdeba
uaryofis areSi α=0.05-saTvis, maSin:
Tu zrealuri ar xvdeba
uaryofis areSi α=0.05-saTvis, maSin:
zrealuris miRebis albaToba naklebia an toli
0.05, an p<0.05.
H0 uaryofilia.
H1 miRebulia. SerCevis saSualo aRebulia
populaciidan, romlis saSualo gansxvavebulia
populaciis hipoTeturi saSualosagan.
gansxvaveba X -sa da hipoTetur µ-s Soris
statistikurad mniSvnelovania 0.05 doneze.
miiReba gadawyvetileba imis Sesaxeb, rom X -sa
da hipoTetur µ-s Soris gansxvavebis mizezi ara
SerCevis Secdomaa, aramed sxva ram.
zrealuris miRebis albaToba metia an toli 0.05,
an p>0.05
H0 ar aris uaryofili
H1 ar aris miRebuli
gansxvaveba X -sa da hipoTetur µ-s Soris
aris statistikurad aramniSvnelovani 0.05
doneze
miiReba gadawyvetileba imis Sesaxeb, rom
SerCevis Secdoma yvelaze damajerebelad
xsnis gansxvavebas X -sa da hipoTetur µ-s
Soris.
statistikurad
mniSvnelovani gansxvaveba
(statistically significant difference)
_ kriteriumis statis-tikis
realuri mniSvneloba
xvdeba uaryofis areSi da
H0 uaryofilia.
aramniSvnelovani gansxvaveba
gansxvaveba SerCevis saSualosa da populaciis
saSualos Soris aris aramniSvnelovani, Tuki
kriteriumis statistikis realuri mniSvneloba ar
xvdeba uaryofis areSi. nulovani hipoTeza ar aris
uaryofili da realurad miRebuli gansxvaveba X -sa
da hipoTetur µ-s Soris miewereba mxolod SerCevis
Secdomas. aramniSvnelovani gansxvaveba aRiniSneba p>0.05, rac mosdevs z realuris
ricxviT mniSvnelobas. gadawyvetilebebi da daskvnebi, romelic miRebulia z
kriteriumis saSualebiT statistikurad aramniSvnelovani gansxvavebidan, Sejamebulia
cxrilis #8.1 marcxena svetSi.
statistikuri hipoTezis Semowmebis etapebis Sejameba
yvela statistikuri kriteriumi moiTxovs Semdegs:
• kriteriumis statistikis arCevas, rogoricaa magaliTad, z;
• ori statistikuri hipoTezis formulirebas, H0 da H1;
• kriteriumis statistikis SerCeviTi ganawilebis miRebas H0-is WeSmaritebis
pirobebSi;
• mniSvnelovnobis donis SerCevas;
• kriteriumis statistikis kritikuli mniSvnelobis an mniSvnelobebis povnas;
• kriteriumis statistikis SerCeviT ganawilebaSi uaryofis aris an areebis
mdebareobis gansazRvras;
• statistikuri hipoTezebis Sesaxeb gadawyvetilebis miRebis wesebis
formulirebas;
• kriteriumis statistikis mniSvnelobis gamoTvlas SerCevis monacemebiT;
• gadawyvetilebis miRebas H0-is uaryofisa Tu misi ver uaryofis Sesaxeb imis
safuZvelze, rom kriteriumis statistikis realuri mniSvneloba xvdeba Tu
ara uaryofis areSi;
• gadawyvetilebis gamotanas H1-is miRebisa Tu ar miRebis Sesaxeb H0-isaTvis
miRebuli gadawyvetilebis safuZvelze.
statistikuri hipoTezebis Semowmebis Sesavali SeiZleba rTuli mogeCvenoT, magram
is male gaxdeba martivi, radgan TiToeuli statistikiuri kriteriumisaTvis Semdeg
erTi da igive etapebia Sesasrulebeli.
aramniSvnelovani gansxvaveba (nonsignificant difference) _
kriteriumis statistikis
realuri mniSvneloba ar xvdeba
uaryofis areSi da nulovani
hipoTeza ar aris uaryofili
SeamowmeT Tqveni codna VIII-1
1. gansazRvreT: alfa done, α, alternatiuli hipoTeza, kritikuli mniSvneloba, H0,
H1, aramniSvnelovani gansxvaveba, nulovani hipoTeza, erTmxrivi kriteriumi,
uaryofis are, mniSvnelovnobis done, statistikuri hipoTeza, statistikurad
mniSvnelovani gansxvaveba, kriteriumis statistika, ormxrivi kriteriumi,
zkritikuli, zrealuri.
2. gansazRvreT statistikuri hipoTezebis ori Tviseba.
3. CawereT H0 da H1 pirobiTi niSnebiT Semdegi debulebebisaTvis:
a) veqsleris inteleqtis testis qulebisaTvis µ aris 100.
b) 25-dan 54 wlamde qalebis populaciisaTvis televizoris yurebaze
daxarjuli saSualo dro dReSi 4.4 saaTis tolia.
g) martiv stimulze 19 wlis mamakacis reaqciis saSualo dro aris 423.0
miliwami.
4. qvemoT moyvanili TiToeuli monacemisaTvis µ warmoadgens populaciis saSualos,
Xσ saSualos standartul Secdomas am populaciisaTvis da X _ SerCevis
saSualos, romelic aRebulia am populaciidan. qvemoT mocemul mniSvnelobaTa
TiToeuli rigisaTvis gansazRvreT H0 da H1 da ipoveT zrealuris mniSvneloba.
Semdeg cxrili #a-1-is gamoyenebiT gansazRvreT, xvdeba Tu ara zrealuri ormxriv
uaryofis areSi 0.05 mniSvnelovnobis doneze da miuTiTeT Tqveni gadawyvetileba
H0-isa da H1-saTvis.
5. gadawyviteT Semdegi amocana 0.05
mniSvnelovnobis donisa da ormxrivi
kriteriumis gamoyenebiT. populaciis
saSualo inteleqtis skalisaTvis aris
100 da σ=15. Tqven xarT dawyebiTi skolis
xelmZRvaneli soflis skolaSi. Tqveni
skolis 121 moswavlis X am skalaze tolia 103.4-is. gansxvavdeba Tu ara Tqveni
skolis moswavleTa saSualo mniSvnelovnad populaciis saSualosagan? Tqvens
pasuxSi miuTiTeT:
a) romel statistikas gamoiyenebT;
b) statistikuri hipoTezebi;
g) mniSvnelovnobis done;
d) zkritikuli;
e) uaryofis areebis mdebareoba;
v) zrealuri;
z) Tqveni gadawyvetilebani H0-is da H1-is Sesaxeb;
µ Xσ X
a) 50 4.7 40
b) 100 25 70
g) 143 0.4 143.9
d) 87 2.9 92
T) Tqveni gadawyvetilebani Tqveni moswavleebis saSualos Sesaxeb populaciis
saSualosTan _ 100 mimarTebaSi.
erTi SerCevis t kriteriumi
praqtikaSi σ xSirad fasdeba s-is _ SerCevaSi qulebis standartuli gadaxris
saSualebiT. amitom Xσ -c fasdeba Xs -is saSualebiT. roca Cven Xσ -s Xs -is
saSualebiT vafasebT, viRebT axal statistikas _ t-s, romelic ganisazRvreba, rogorc
t =Xs
X µ−
Tu Xs -is naclad CavsvamT s/√N ,
t =Ns
X/
µ−
sadac X = SerCevis saSualoa
µ= populaciis hipoTeturi saSualoa
Xs = saSualos standartuli Secdomis Sefaseba, romelic miRebuli iyo
populaciis standartuli gadaxris Sefasebis _ s-is saSualebiT
N= SerCevis moculoba.
t kriteriumi SeiZleba iyos gamoyenebuli im
gansxvavebis statistikuri mniSvnelovnobis
Sesamowmeblad, romelic arsebobs SerCevis saSualosa
da populaciis hipoTetur saSualos Soris, roca Xσ
Sefasebulia Xs -is saSualebiT. radgan mxolod erTi
SerCevis saSualos gulisxmobs, statistikas uwodeben
erTi SerCevis t-s, iseTi t kriteriumisagan gansxvavebis
mizniT, romelic ori SerCevis saSualos moicavs, rac
me-9 TavSi iqneba ganxiluli.
t kriteriumi SemoRebuli iyo britaneli maTematikosis uiliam sili gosetis mier
me-20 saukunis dasawyisSi. gosetma gamosca Tavisi naSromi fsevdonim “studentis”
gamoyenebiT. amitomac t kriteriums xSirad uwodeben `studentis t kriteriums”.
t-s SerCeviTi ganawileba
t statistikis gamoyenebisaTvis unda vicodeT misi SerCeviTi ganawileba. t-s
SerCeviTi ganawileba gansxvavdeba z-is SerCeviTi ganawilebisagan. radgan z statistika
iyenebs populaciis parametrs Xσ -s, mas aqvs mxolod erTi SerCeviTi ganawileba,
erTi SerCevis t kriteriumi
(one-sample t test) _
t kriteriumi, romelic
gamoiyeneba SerCevis
saSualosa da populaciis
hipoTetur saSualos Soris
gansxvavebis statistikuri
mniSvnelovnobis
Sesamowmeblad, roca σ X
Sefasebulia s X -is saSualebiT.
romelic warmodgenilia cxrilSi #a-1. t kriteriumi ki, radgan iyenebs populaciis s
standartul gadaxris Sefasebas, ganisazRvreba, rogorc
s=1
)( 2
−
−∑N
XX
s-is gamoTvla gulisxmobs SerCevis N moculobas. amitom t-s SerCeviTi ganawileba
icvleba SerCevis moculobasTan erTad, an ufro zustad rom vTqvaT, SerCevis
Tavisuflebis xarisxTan erTad.
Tavisuflebis xarisxi
Tavisuflebis xarisxi (romlis abreviaturaa df)
ganisazRvreba rogorc qulebis raodenoba, romelic
Tavisufalia cvalebadobis dros statistikis
gamoTvlisas. moviyvanT ramdenime magaliTs imis ukeT
gasagebad, Tu ras niSnavs `qulebis raodenoba, romelic
Tavisufalia cvalebadobis dros” statistikis gamoTvlisas. davuSvaT, rom
Tqven gTxoves amoirCioT sami qula testSi. Tqven SeiZleba airCioT
nebismieri sami qula, vTqvaT, 75, 62 da 97. am davalebisas ar arsebobda raime
SezRudva qulebis arCevanSi; TiToeuli qula SeiZleba Tavisufali iyos
Tavisi cvalebadobis dros. Sesabamisad qulebis am simravlisaTvis
Tavisuflebis xarisxi df aris N=3. ganvixiloT meore davaleba, roca Tqven
gTxoven airCioT sami qula, magram iseTi, romelTa jami unda iyos 254.
davuSvaT Tqven airCieT 81 da 78, rogorc pirveli ori qula; maT SeiZleba
hqondeT nebismieri mniSvneloba, romelsac Tqven isurvebT. es ori qula
SeiZleba Tavisufali iyos Tavisi cvalebadobis dros. SeiZleba mesame qula
aseve Tavisuflad Seicvalos? ara. radgan sami qulis jami unda iyos 254,
mesame qula unda iyos 95; arc erTi sxva qulis SemTxvevaSi es jami ar
miiReba. amitom, Tu Tqven giTxres, rom qulaTa jami aris 254, ori qula
SeiZleba Tavisuflad Seicvalos, xolo mesame aris gansazRvruli im ori
qulis mniSvnelobiT, romlebic Tavisuflad SeiZleba Seicvalos da am sami
qulis jamiT. Sesabamisad, Tu Tqven iciT sami qulis jami, maSin arsebobs
N_1=3_1=2 Tavisuflebis xarisxi am sami qulisaTvis.
Tu gamoviyenebT am argumentacias SerCeviTi saSualos gamoTvlisas, X -s
aqvs N df. yvela qula qulaTa erTobliobaSi aris Tavisufali
cvalebadobisas, roca viTvliT X -s. magaliTad, Tqven gTxoves gamoiTvaloT
SerCevis saSualo 5 qulis erTobliobisaTvis, aqedan 4 aris mocemuli: 7, 10,
Tavisuflebis xarisxi
(degrees of freedom): qulaTa raodenoba, romelic Tavisufalia cvalebadobis dros statistikis gamoTvlisas.
3 da 6. mxolod am informaciis mocemulobisas Tqven ver SesZlebTY
saSualos povnas. mexuTe qulad SeiZleba CaTvaloT nebismieri mniSvneloba
da Tqven ar gecodinebaT mexuTe qulis mniSvneloba mxolod am oTxi
mniSvnelobis codnis safuZvelze. amrigad, yvela N qula (sadac N=5
rogorc am magaliTSi) aris Tavisufali cvalebadobisas, roca xdeba
SerCevis saSualos gamoTvla.
s-is Tavisuflebis xarisxi aris N_1. amis sailustraciod davuSvaT, rom
Tqven gTxoves gamoTvaloT s 5 qulis mqone SerCevis safuZvelze.
populaciis standartuli gadaxris Sefaseba ganisazRvreba, rogorc
1
)( 2
−
−= ∑
NXX
s
davuSvaT, rom xuTidan oTx qulas aqvs Semdegi mniSvnelobebi: 13, 15, 13 da 12
da am xuTi qulis saSualo aris 13.4. Cven viciT, agreTve, rom 0)( =−∑ XX
Tuki Cven gamovaklebT X am oTxi qulidan TiToeuls, maSin Cven miviRebT
)( XX −
13 _ 13.4= - 0.4
15 _ 13.4= +1.6
13 _ 13.4= - 0.4
12 _ 13.4= -1.4
jami = -0.6
am oTxi )( XX − sxvaobis jami tolia -0.6. roca mexuTe qula iqneba CarTuli
am erTobliobaSi, es jami toli iqneba nulis. amrigad, mexuTe qulam unda
mogvces )( XX − =+0.6, +0.6 rom davumatoT -0.6, jami toli iqneba nulis. Tuki
X5 _ 13.4= +0.6, maSin X5 toli unda iyos 14. Sesabamisad, Tu X cnobilia am
xuTi qulis erTobliobisaTvis, maSin am xuTi qulidan mxolod oTxi qulis
mniSvnelobaa Tavisufali cvalebadobisas. mexuTe qula ver Seicvleba
Tavisuflad. misi mniSvneloba aris mkacrad gansazRvruli im SezRudviT,
romelic saSualos mniSvnelobiT iqna Semotanili. radgan X unda iyos
cnobili, raTa gamoviTvaloT s, yvela qula, garda erTisa, aris Tavisufali
cvalebadobis dros s-is gamoTvlisas. amrigad, roca viTvliT s an s2 NN
qulaTa erTobliobas aqvs N-1 Tavisuflebis xarisxi df.
Cven SegviZlia Tavisuflebis xarisxis Sesaxeb es codna s-is gamosaTvlel
formulaSi gamoviyenoT, raTa is ufro mokled CavweroT. radgan ∑ − 2)( XX
, agreTve, kvadratebis jams (SS) uwodeben, da N-1 asaxavs Tavisuflebis
xarisxs SS-is gamoTvlisas, populaciis standartuli gadaxris Sefaseba
SeiZleba gamovsaxoT rogorc kvadratuli fesvi kvadratebis jamidan
gayofili Tavisuflebis xarisxze:
dfSSs = .
momdevno TavebSi Cven vnaxavT, rom s-is es gamosaxuleba sakmaod xSirad
gamoiyeneba.
erTi SerCevis t kriteriumis Tavisuflebis xarisxi
axla gamoviyenoT df-is Sesaxeb Cveni codna t kriteriumisaTvis. saSualos
standartuli Secdomis Sefaseba, Xs gamoiTvleba rogorc s/√N. radgan s-s aqvs
N-1 df, Xs -s, agreTve, aqvs N-1 df. amrigad, erTi SerCevis t statistikis SerCeviTi
ganawilebisaTvis Tavisuflebis xarisxi aris, agreTve N-1.
t SerCeviTi ganawileba
TiToeuli df-isaTvis arsebobs t-s sxvadasxva Teoriuli ganawileba. radgan
potenciurad qulaTa nebismieri raodenoba SeiZleba iyos miRebuli SerCevaSi,
df SeiZleba icvlebodes 1-dan usasrulobamde. rogorc potenciuri SerCeviTi
ganawilebis magaliTi, suraTi #8.4 gviCvenebs t ganawilebebs, roca df aris 2, 4,
da 60.
Tumca t ganawilebis forma icvleba Tavisuflebis xarisxis
cvalebadobasTan erTad, is normaluri ganawilebis msgavsia ramdenime aspeqtSi.
TiToeuli t ganawileba simetriuli da unimodaluria da aqvs saSualo,
romelic nulis tolia. ganvixiloT, ratom Cndeba es Tavisebureba. roca
nulovani hipoTeza, iseTi rogoricaa: H0:µ=populaciis saSualos raime
hipoTeturi mniSvnelobisa, WeSmaritia, t-s mosalodneli mniSvneloba iqneba
nuli, radgan ( X _µ)-s saSualo im SerCeviT ganawilebaSi, romelic iyo aRebuli
aseTi populaciidan iqneba nuli. magram, maSinac roca H0 WeSmaritia, µ−_X -s
mniSvnelobebi Seicvleba nulis toli saSualos garSemo SerCevis Secdomis
gamo. ufro metic, mosalodnelia, rom _X iqneba µ-ze meti, TiTqmis iseve xSirad,
rogorc, is iqneba µ-ze naklebi. mosalodnelia, rom, roca nulovani hipoTeza
WeSmaritia, µ−_X mniSvnelobebi iseve xSirad iqneba dadebiTi, rogorc
uaryofiTi. amrigad, t-s mosalodneli mniSvnelobebi simetriulad iqneba
ganlagebuli nulis toli saSualos garSemo.
Tavisuflebis xarisxis usasrulo raodenobisaTvis (Teoriuli da ara realuri
SesaZlebloba) t SerCeviTi ganawileba aris normaluri ganawilebis msgavsi.
yvela sxva t ganawilebebi arian ufro farTod gaSlilebi, vidre normaluri
ganawileba.
ratom aris ufro swori Tavisuflebis xarisxis da ara N-is gamoyeneba? Tu
df erTi SerCevis t kriteriumisaTvis aris mocemuli NN-1 saSualebiT, ratom ar
SeiZleboda t SerCeviTi ganawilebis warmodgena N-is terminebiT df-is magivrad?
erTi SerCevis t statistikisaTvis Cven ver miviRebT verc erT upiratesobas df-
is gamoyenebiT N-is magivrad, im df-saTvis, romelic yovelTvis erTiT naklebia
N-ze. df-is gamoyenebis upiratesoba gamoCndeba me-9 TavSi, sadac Cven gagacnobT
sxva t statistikas, romelic gamoiyeneba ori SerCevis saSualoTa Sesadareblad.
am t-saTvis df N-2 da ara N-1-ia. toli df-saTvis orive t kriteriums aqvs erTi
da igive SerCeviTi ganawileba. igive magaliTebs sxva statistikebis
ganxilvisas momdevno TavebSi SevxvdebiT.
SeamowmeT Tqveni codna VIII-2
1. dawereT erTi SerCevis t statistikis formula.
2. ratom uwodeben t statistikas zogjer studentis t-s?
3. Tqven mocemuli gaqvT 7 qula: 73, 42, 51, 68, 49, 62 da 57. ramdeni
Tavisuflebis xarisxi aqvs am qulebis simravles, raTa gamoiTvaloT
a) X b) s da g) s2
4. Tqven gegmavT gamoTvaloT erTi SerCevis t statistika Semdegi
moculobebis SerCevebze. TiToeuli SerCevisaTvis, ipoveT df: a) N=10, b) N=16,
g) N=7, d) N=54, e) N= 120, v) N=30
5. Tqven gaqvT t-s ori Teoriuli ganawileba, erTi 13 df da meore 27 df.
romeli ganawileba iqneba ufro dabali da gaSlili?
6. ra pirobebSi iqneba t statistikis SerCeviTi ganawileba z SerCeviTi
ganawilebis identuri?
statistikuri hipoTezebis Semowmeba t-s saSualebiT erTi SerCevis
SemTxvevaSi
erTi SerCevis t saSualebas gvaZlevs ganvsazRvroT, gansxvavdeba Tu ara X
hipoTeturi µ-sagan ufro metad, vidre es iqneboda mosalodneli mxolod
SerCevis Secdomis SemTxvevaSi, roca σ aris gamoTvlili s-is saSualebiT. t-s
saSualebiT statistikuri hipoTezis Semowmebis procedura identuria imisa,
rac iyo z statistikisaTvis. aucilebeli nabijebi moicavs Semdegs:
• H0-is da H1-is formulireba;
• t statistikis SerCeviTi ganawilebis miReba H0-is WeSmaritebis
pirobebSi;
• mniSvnelovnobis donis SerCeva;
• t-s kritikuli mniSvnelobis an mniSvnelobebis povna;
• uaryofis aris an areebis mdebareobis dadgena t-s SerCeviT ganawilebaSi;
• gadawyvetilebis miReba statistikuri hipoTezebis Sesaxeb;
• trealuris mniSvnelobis gamoTvla SerCevis monacemebis safuZvelze;
• gadawyvetilebis miReba H0-is da H1-is Sesaxeb trealuris mniSvnelobaze
dayrdnobiT.
am nabijebs asaxavs Semdegi amocanis magaliTi. davuSvaT, bavSvTa fsiqologma,
romelic ikvlevda sakiTxs, rogor atarebdnen dros mesame klaseli bavSvebi,
romlebic CarTulni iyvnen gakveTilebis Semdgom bavSvebis movlis programaSi,
aRmoaCina, rom kviris dReebSi bavSvebi uyureben televizors saSualod 83
wuTis ganmavlobaSi, romlis s 26 wuTis tolia. davuSvaT, agreTve, cnobilia,
rom dReSi televizoris yurebaze daxarjuli drois saSualo mesame klaselTa
populaciisaTvis Seadgens 95 wuTs. aris Tu ara SerCevis saSualo, romelic
tolia 83 wuTis, mniSvnelovnad gansxvavebuli populaciis saSualosagan _ 95
wuTisagan? radgan σ aris gamoTvlili s-is saSualebiT, am kiTxvaze pasuxis
gasacemad unda iyos gamoyenebuli erTi SerCevis t kriteriumi.
statistikuri hipoTezebis formulireba
statistikuri hipoTezebi erTi SerCevis t statistikisaTvis identuria imis, rac
iyo gamoyenebuli z statistikis SemTxvevaSi. nulovani hipoTeza H0
gansazRvrulia rogorc populaciis saSualos hipoTeturi mniSvneloba.
alternatiuli hipoTeza, H1 ganisazRvreba rogorc debuleba, rom populaciis
saSualo ar aris is, rac navaraudevia H0-is SemTxvevaSi. magaliTad,
statistikuri hipoTezebis pirobiTi aRniSvna aris
H0:µ=95 wuTi
H1:µ≠95 wuTi
t statistikis SerCeviTi ganawilebis miReba
t-s SerCeviTi ganawileba damokidebulia t-s Tavisuflebis xarisxebze, romelic
gamoiTvleba SerCevis qulebis safuZvelze. df erTi SerCevis t-saTvis mocemulia
N-1 saxiT. im saSualosaTvis, romelic efuZneba 30 monacems, df t-saTvis aris
30-1=29. mniSvnelovnobis donisa da uaryofis areebis arCevis Semdeg,
Tavisuflebis xarisxis am maCvenebels gamoviyenebT t-s SerCeviTi ganawilebis
cxrilSi tkritikulis sapovnelad.
mniSvnelovnobis donis arCeva
mniSvnelovnobis done aris albaTobis mniSvneloba, romelic uzrunvelyofs
H0-is uaryofas. statistikuri hipoTezis testirebisas Cveulebriv gamoiyeneba
0.05 an 0.01 mniSvnelovnobis done. magaliTad, Cven viyenebT α=0.05.
uaryofis areebis gansazRvra t kriteriumisaTvis: ormxrivi kriteriumi
roca H0 WeSmaritia uaryofis are moicavs t ganawilebis iseT mniSvnelobebs,
romlis albaToba aris α-s toli an masze naklebi, amrigad, uaryofis are
gansazRvravs trealuris iseT mniSvnelobebs, romlebic uaryofis areSi moxvdebian.
ormxriv anu aramimarTulebiT kriteriumSi t-s uaryofis areebi
mdebareoben Teoriuli SerCeviTi ganawilebis orive kudze, rogorc es
naCvenebia suraTze #8.4. roca H0 WeSmaritia, X -is mniSvnelobaTa umetesoba
mxolod mcired iqneba gansxvavebuli µ-sagan. amrigad, t-s mniSvnelobaTa
umravlesoba dajgufebuli iqneba ganawilebis SuaSi nulis garSemo. SerCevis
Secdomis gamo, t-s zogierTi mniSvneloba iqneba ganawilebis romelime boloSi,
maSinac ki roca H0 WeSmaritia. es mniSvnelobebi miiReba, roca t-s mricxveli
(e.i. µ−_X sxvaoba) metia mniSvnelze (anu Xs ). magram t-s aseTi didi
mniSvnelobebi ufro iSviaTi xdomilobebia nulovani hipoTezis WeSmaritebis
pirobebSi.
meore mxriv, Tuki H1 aris WeSmariti da X asaxavs SerCevas im
populaciidan, romlis saSualo gansxvavdeba µ-sagan, maSin ufro xSirad
mosalodnelia µ−_X -is da, maSasadame t-s didi mniSvnelobebi. amrigad,
uaryofis are mdebareobs t-s Teoriuli SerCeviTi ganawilebis TiToeul kudze
da mTeli α iyofa or nawilad kudebs Soris. Tuki α =0.05, maSin TiToeuli
uaryofis are moicavs albaTobas α/2, an 0.025. Tuki SerCevis qulebis mixedviT
gamoTvlili t-s mniSvneloba, (gansazRvruli, rogorc trealuri ) xvdeba romelime
uaryofis areSi, maSin nulovani hipoTeza iqneba uaryofili da miiReba
alternatiuli hipoTeza. uaryofis aris mdebareobis gansazRvriT t ganawilebis
orive kudze, eqsperimentatori iTvaliswinebs orive SesaZleblobas, romelic
SeiZleba iyos nagulisxmevi alternatiul hipoTezebSi: roca X metia µ-ze da
roca X naklebia µ-ze. uaryofis am or areSi moxvedris sruli albaToba
udris 0.05, roca H0 WeSmaritia.
suraTi #8.4
t ganawilebis ormxrivi uaryofis are, roca df aris 29 da α=0.05. trealuri
mniSvneloba xvdeba uaryofis erT-erT areSi, rasac mivyevarT H0 –is uaryofamde
da viRebT H1-s.
t-s SerCeviTi ganawilebis uaryofis ori are, roca df aris 29, naCvenebia
suraTze #8.4. Tu df tolia 29-is da H0 WeSmaritia, maSin albaToba imisa, rom t
iqneba toli an naklebi, vidre -2.045, aris 0.025 da albaToba imisa, rom t iqneba
t=-2.045 t=+2.045 t-s kritikuli t-s kritikuli mniSvneloba mniSvneloba
-3 -2 -1 0 +1 +2 +3 t-s mniSvneloba
H0 uaryofa uaryofis are 0.025
H0 uaryofa uaryofis are 0.025
H0 miRebis zona
toli an meti +2.045-ze isev 0.025-ia. trealuris nebismieri mniSvneloba, romelic
tolia an naklebia -2.045-ze, an tolia, an metia +2.045-ze xvdeba uaryofis areSi.
es Sedegi miiReba xdomilobaTa mxolod 5 procentis SemTxvevaSi, roca
nulovani hipoTeza WeSmaritia. mniSvnelobebi -2.045 da +2.045 miRebuli iyo t-s
SerCeviTi ganawilebis kritikul mniSvnelobaTa cxrilidan. am cxrilis
gamoyeneba SemdgomSi iqneba ganmartebuli.
t kritikul mniSvnelobaTa cxrilis gamoyeneba
FmniSvnelovnobis calkeuli donisaTvis saWiroa t-s kritikul mniSvnelobebis
gansazRvra (aRiniSneba, rogorc tkritikuli.), romlebic damokidebulni arian
trealuris df-ze. t ganawilebis grafikebi naTlad gviCvenebs, rom t ganawilebis
forma icvleba Tavisuflebis xarisxebis cvalebadobasTan erTad, magram am
grafikebidan Znelia ipovo tkritikulis zusti mniSvneloba. amitom tkritikuli
mniSvnelobebi 0.05 da 0.01 mniSvnelovnobis doneebisaTvis gamoTvlilia da
Setanilia cxrilebSi.
cxrili #8.2 asaxavs tkritikulis mniSvnelobebs α=0.05 da 0.01. ormxrivi
kriteriumisaTvis. cxrili #8.2 aris danarTSi mocemuli cxrili #a-2-is
nawili. pirvel svetSi mocemulia Tavisuflebis xarisxebi t
ganawilebebisaTvis. α=0.05 da 0.01 svetebis nebismier striqonSi mocemulia
tkritikuli mniSvnelobebi am mniSvnelovnobis donis Sesatyvisi Tavisuflebis
xarisxisaTvis. magaliTad, roca α=0.05 tkritikuli ormxrivi mniSvneloba im
trealurisaTvis, romelsac aqvs 29-is toli Tavisuflebis xarisxi, aris 2.045.
miaqcieT yuradReba, rom am mniSvnelobas ara aqvs win + an _ niSani. is
warmodgenilia rogorc t-s absoluturi mniSvneloba. Tumca uaryofis areebis
mdebareobis gansazRvrisaTvis, es mniSvneloba unda iyos warmodgenili rogorc
-2.045 da +2.045. analogiurad α=0.01-saTvis ormxrivi tkritikuli mniSvneloba,
romlis Tavisuflebis xarisxi 29-is tolia udris 2.756. aRsaniSnavia, rom 0.01
mniSvnelovnobis done ufro mkacri kriteriumia H0-is uaryofisaTvis, vidre
α=0.05. nebismieri realuri µ−_X sxvaoba unda iyos ufro didi, raTa uarvyoT
H0, roca 0.05-s nacvlad miRebuli iqneba 0.01 mniSvnelovnobis done.
cxrili #8.2. t ganawilebis kritikuli mniSvnelobebi α=0.05 da α=0.01-saTvis.
mniSvnelobebi mocemulia ormxrivi kriteriumisaTvis.
df α=0.05 α=0.01
1 12.607 63.657
2 4.3030 9.925
3 3.182 5.841
4 2.776 4.604
5 2.571 4.032
6 2.447 3.707
7 2.365 3.499
8 2.306 3.355
9 2.262 3.25
10 2.228 3.169
11 2.201 3.106
12 2.179 3.055
13 2.160 3.012
14 2.145 2.977
15 2.131 2.947
16 2.120 2.921
17 2.110 2.898
18 2.101 2.878
19 2.093 2.861
20 2.086 2.845
21 2.080 2.831
22 2.074 2.819
23 2.069 2.807
24 2.064 2.797
25 2.060 2.787
26 2.056 2.779
27 2.052 2.771
28 2.048 2.763
29 2.045 2.756
30 2.042 2.750
40 2.021 2.704
60 2.000 2.660
120 1.980 2.617
∞ 1.960 2.576
30-ze ufro meti Tavisuflebis xarisxisaTvis mxolod calkeuli
kritikuli mniSvnelobebi aris mocemuli cxrilSi. roca df trealurisaTvis metia
30-ze, girCevT, rom tkritikulis mniSvneloba gamoTvlili iyos im df-s safuZvelze,
romelic cxrilSi yvelaze axlos imyofeba, magram amasTanave naklebia am
trealuris df-ze. magaliTad, TuU df=35, cxrili #8.2 gviCvenebs trealuris
mniSvnelobebs 30 da 40 df-isaTvis, magram ara maT Soris mdebare
mniSvnelobebisaTvis. amitomac tkritikulis gansazRvrisaTvis, roca df=35, Cven
viyenebT 30-is tol df (is tolia 2.042 α=0.05) nacvlad imisa, rom gamogveyenebina
40 df. am proceduris Sedegad, mniSvnelovnobis done iqneba odnav dabali, vidre
0.05. df=40 rom gamogveyenebina, maSin tkritikulis mniSvnelovnobis done iqneboda
odnav maRali, vidre 0.05. Tu trealuri xvdeba uaryofis areSi df=30-saTvis, maSin
is, agreTve, xvdeba uaryofis areSi df=40-saTvisac. Tuki trealuri (magaliTad,
trealuri =2.033) moxvdeba cxrilSi im mniSvnelobaTa Soris, romlebic
Seesabamebian df-is momdevno dabal (mag., tkritikuli=2.045, roca df aris 30) da
momdevno maRal mniSvnelobebs (mag. tkritikuli=2.021, roca df aris 40), maSin
tkritikulis ufro zusti mniSvneloba SeiZleba gamoviTvaloT cxrilis
mniSvnelobebidan.
t uaryofis areebis mdebareobis gansazRvra: calmxrivi kriteriumi
ormxriv SemowmebaSi H0-s uaryofen roca trealuri xvdeba t-s ganawilebis
romelime boloSi. magram, roca sakvlevi hipoTeziT nawinaswarmetyvelebia
saSualoTa Soris mimarTulebiTi kavSiri, (mag., Tu Cvens magaliTSi mkvlevari
varaudobs, rom X iqneba ufro naklebi, vidre µ) da ar arsebobs interesi
navaraudevis sapirisipiro SedegSi (mag., Tu X aris ufro meti, vidre µ), maSin
zogierTi mkvlevari iyenebs calmxriv kriteriums. calmxriv anu mimarTulebis
mqone kriteriumSi uaryofis are, romelic SemosazRvrulia α-s sruli
mniSvnelobiT, ganlagebulia t ganawilebis im kudis mxareze, romelic
Seesatyviseba kvlevis hipoTeziT navaraudevi Sedegis mimarTulebas. tkritikuli
mniSvnelobebi calmxrivi kriteriumisaTvis 0.01 da 0.05 mniSvnelobaTa
doneebiT warmodgenilia, agreTve, cxrilSi #a-2. df=29-saTvis, calmxrivi
tkritikuli 0.05 mniSvnelovnobis doneze aris 1.699. calmxrivi uaryofis are, roca
tkritikuli=-1.699 naCvenebia suraTze #8.5, t ganawilebis marcxena kudze. es
uaryofis are am magaliTisaTvis Sesatyvisi iqneboda, Tu davuSvebdiT Semdeg
statistikur hipoTezebs:
H0 : µ ≥42
H1 : µ<42
suraTi aklia (Reject _ H0 uaryofa, Do not reject _ H0 miRebis zona, Rejection region - uaryofis are, value of t_ t-s mniSvneloba, critical value of t _ t-s kritikuli mniSvneloba) suraTi # 8.5
t ganawilebis calmxrivi uaryofis are, roca df aris 29 da α=0.05. trealuris
mniSvneloba xvdeba uaryofis areSi, rasac mivyavarT H0 –is uaryofamde da
viRebT H1.
rodis unda iyos gamoyenebuli calmxrivi kriteriumi?
calmxrivi uaryofis aris upiratesoba isaa, rom tkritikuli iqneba ufro mcire,
radgan mTeli uaryofis are mdebareobs ganawilebis mxolod erT kudSi.
amitomac ufro advilia H0-is uaryofa, radgan X -sa da µ Soris mcire
gansxvavebisas trealuri xvdeba uaryofis areSi. TumcaRa am upiratesobas Tan
axlavs SezRudvac. gadawyvetileba imis Sesaxeb, Tu romeli uaryofis are
iqneba gamoyenebuli, calmxrivi Tu ormxrivi, miiReba im dros, roca xdeba
kvlevis dagegmva da sanam daiwyeba monacemTa Segroveba da analizi. Tuki
miRebuli iqneba calmxrivi uaryofis are, maSin eqsperimentatori ver SeZlebs
uaryos H0 saSualoTa Soris miRebuli im sxvaobis safuZvelze, romelic am
hipoTezis sapirispiro mimarTulebisaa, radgan aseTi SedegisaTvis uaryofis
are ar gvaqvs. calmxriv da ormxriv kriteriumebs ufro dawvrilebiT me-9
TavSi ganvixilavT. praqtikaSi ormxrivi kriteriumebi yvelaze farTod
gamoyenebadia; amitomac, isini gamoyenebulni arian am Tavis yvela magaliTSi.
trealuris gamoTvla
trealuris gamoTvla moiTxovs X -is, µ da Xs mniSvnelobaTa Casmas formulaSi.
magaliTad, 30 mesameklaselisaTvis televizoris yurebis X =83 wuTs, s=26wuTs
da N=30. µ-s hipoTeturi mniSvneloba tolia 95 wuTis. Xs -is mniSvneloba
miRebulia s/√N-dan; amrigad,
Xs =477.526
3026
= ,
rac tolia 4.75. amis Sedegad,
trealuri =X
Xσ
µ−
= 526.275.412
75.49583
−=−
=−
roca df aris N-1=29 .
gadawyvetilebani statistikuri hipoTezebis Sesaxeb
statistikur hipoTezaTa Sesaxeb gadawyvetilebis miRebis wesebi erTi da igivea
yvela statistikuri kriteriumisaTvis. Sesabamisad, erTi SerCevis t
kriteriumisaTvis:
• Tuki trealuri xvdeba uaryofis areSi, maSin H0 iqneba uaryofili da
H1 miRebuli;
• Tuki trealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da
H1 ar iqneba miRebuli.
tkritikulis ormxrivi mniSvneloba, roca df=29 da α=0.05, aris 2.045. Sesabamisad,
rogorc naCvenebia suraTze #8.5, trealuri, romelic tolia an naklebia -2.045-ze
an tolia, an metia +2.045-ze, xvdeba uaryofis areSi. trealuri _2.526 aris ufro
Sors mdebare (eqstremaluri), vidre tkritikuli =-2.045 da amitomac xvdeba
ganawilebis marcxena kudze mdebare uaryofis areSi. Cven uarvyofT H0:µ ≠ 95
wuTs da viReb H1:µ ≠ 95. gansxvaveba SerCeviT saSualosa da populaciis
saSualos Soris aris statistikurad mniSvnelovani 0.05 doneze. Cven vaskvniT,
rom SerCevis saSualo, romelic tolia 83 wuTis aris im populaciidan,
romlis µ naklebia 95 wuTze. im mesameklaselebis mier televizoris yurebaze
daxarjuli dro, romlebic CarTulni iyvnen gakveTilebis Semdgom bavSvebis
movlis programaSi, aris mniSvnelovnad naklebi, vidre populaciis saSualo,
romelic tolia 95 wuTisa dReSi.
Tuki trealuri xvdeba or kritikul mniSvnelobas _ (-2.045) da +2.045 Soris
(magaliTad, X aris 90 wuTi da amrigad, trealuri =-1.056), maSin H0 -s ver uarvyofT
da ver miviRebT H1-s. realuri gansxvaveba SerCeviT saSualosa da populaciis
hipoTetur saSualos Soris iqneboda aramniSvnelovani. am SemTxvevaSi ar
gveqneboda sakmarisi mtkicebuleba imisaTvis, raTa CagveTvala, rom H0 ar aris
WeSmariti.
gadawyvetilebani statistikuri hipoTezebisa da im daskvnebis Sesaxeb,
romlebic miRebuli iqneboda romelime statistikurad mniSvnelovani an
aramniSvnelovani erTi SerCevis t kriteriumisaTvis aris Sejamebuli cxrilSi
#8.3.
daSvebebi erTi SerCevis t kriteriumisaTvis
erTi SerCevis t kriteriumi ayalibebs Semdeg daSvebebs SerCevis
monacemebisaTvis:
• monacemebi SerCevaSi SemTxveviTad arian miRebuli populaciis
monacemebidan da TiToeuli monacemi damoukidebelia nebismieri sxva
monacemisagan;
• monacemebi populaciis SerCevaSi normaluradaa ganawilebuli.
praqtikaSi, SerCevidan aRebulma monacemebma SeiZleba srulyofilad ar
daakmayofilos es daSvebebi, Tumca mkvlevrebi ganagrZoben am
monacemebisaTvis t kriteriumis gamoyenebas. am sakiTxs ufro
dawvrilebiTY me-9 TavSi ganvixilavT.
cxrili #8.3
erTi SerCevis t kriteriumis gamoyenebiT statistikuri hipoTezebis Semowmebis
Sesaxeb gadawyvetilebebisa da daskvnebis Sejameba. gamoyenebuli iyo 0.05
mniSvnelovnobis done.
magaliTi #8-1
erTi SerCevis t kriteriumis gamoyeneba
amocana
fsiqologebma wamoayenes hipoTeza, rom zogierTi individi Sejibrebis dros
gaurbis did warmatebas, raTa Tavi aaridos misi Tanatolebis
kritikulobasa da jgufidan gariyvas (gandevnas). TviTSefasebis kiTxvari,
romelic zomavs warmatebisagan garidebas, SemuSavebuli iyo Ggudisa da
gudis mier (Good and Good, 1973). davuSvaT cnobilia, rom kolejis
damamTavrebeli klasis studenti gogonebisaTvis populaciis saSualo am
kiTxvaris mixedviT iyo 8.6 qula. qulebis diapazoni, romelic SeiZleboda
yofiliyo miRebuli am kiTxvaris mixedviT icvleboda 0 qulidan, rac
niSnavda warmatebisaTvis Tavis aridebis dabal motivacias, 28 qulamde, rac
niSnavda warmatebisaTvis Tavis aridebis maRal motivacias. konsultantma
moawyo konkurentuli situacia. man misca warmatebisaTvis Tavis aridebis
motivaciis sazomi skala kolejis damamTavrebeli klasis student 33
gogonas, vinc gaira es seminari da miiRo saSualo 8.1 da s=1.31. gansxvavdeba
Tu trealuri xvdeba
uaryofis areSi α=0.05-saTvis, maSin:
Tu trealuri ar xvdeba
uaryofis areSi α=0.05-saTvis, maSin:
trealuris miRebis albaToba naklebia an
toli 0.05, an p≤0.05
H0 aris uaryofili
H1 aris miRebuli. SerCevis saSualo
aRebulia populaciidan, romlis
saSualo gansxvavebulia populaciis
hipoTeturi saSualosagan.
gansxvaveba X -sa da hipoTetur µ-s Soris
aris statistikurad mniSvnelovani 0.05
doneze.
miiReba gadawyvetileba imis Sesaxeb, rom
X -sa da hipoTetur µ-s Soris
gansxvavebis mizezi aris ara SerCevis
Secdoma, aramed sxva mizezi.
trealuris miRebis albaToba metia an
toli 0.05, an p>0.05
H0 ar aris uaryofili
H1 ar aris miRebuli
gansxvaveba X -sa da hipoTetur µ-s
Soris aris statistikurad
aramniSvnelovani 0.05 doneze
miiReba gadawyvetileba imis Sesaxeb,
rom SerCevis Secdoma yvelaze
damajerebelad xsnis gansxvavebas
X -sa da hipoTetur µ-s Soris.
Tu ara es X mniSvnelovnad µ=8.6-sagan 0.05 mniSvnelovnobis doneze?
gamoiyeneT ormxrivi kriteriumi.
saWiro statistika: radganac σ gamoiTvleba s-is saSualebiT, am SemTxvevaSi
unda gamoviyenoT erTi SerCevis t kriteriumi.
daSvebebi:
1. individebi, romlebmac gamokiTxvaSi miiRes monawileoba, SemTxveviTad
iyvnen SerCeulebi kolejis damamTavrebeli klasis studenti gogonebis
populaciisagan;
2. warmatebis Tavidan aridebis motivaciis qulebi arian normalurad
ganawilebulni populaciis SerCevaSi.
statistikuri hipoTezebi: H0: µ =8.6; H1: µ ≠8.6.
mniSvnelovnobis done: α=0.05
df t-saTvis: N-1=33-1=32
t-s kritikuli mniSvneloba: tkritikuli =2.042. t-s kritikuli mniSvneloba, roca
df=32 ar aris warmodgenili cxrilSi #a-2. Cven gamoviyeneT tkritikuli df-is
ufro dabali yvelaze axlo mdgomi mniSvneloba, df=30.
uaryofis areebi: trealuris mniSvnelobebi, romlebic arian toli an naklebi -
2.042, an arian toli, an meti +2.042.
gamoTvla:
228.0745.5
31.13331.1
=
=
=Xs
amitom, 193.2228.0
5.0228.0
6.81.8−=
−=
−=
−=
xsXt µ
realuri
gadawyvetileba: trealuri ufro eqstremaluria, vidre t kritikuli = -2.042 da
xvdeba uaryofis areSi. amrigad, Cven uarvyofT H0-s da viRebT H1-s.
daskvna: warmatebisagan garidebis motivaciis saSualo kolejis damamTavrebeli
klasis studenti gogonebis im SerCevisaTvis, vinc daeswrnen seminarebs
(kerZod, X =8.1) mniSvnelovnad naklebia, vidre kolejis damamTavrebeli klasis
studenti gogonebis populaciis saSualo _ 8.6.
erTi SerCevis t kriteriumis Sedegebis prezentacia
amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvanelos (1994)
stili moiTxovs daskvniTi statistikis kriteriumebis Sedegebis
warmodgenas, raTa specialuri forma mieces statistikuri kriteriumis,
kriteriumis df-is, statistikis realuri mniSvnelobis, gamoyenebuli alfa
donis da statistikis realuri mniSvnelobis albaTobisa da eqstremaluri
mniSvnelobis Caweras nulovani hipoTezis WeSmaritebis pirobebSi. aseve
saWiroa, Sesatyvisi aRweriTi statistikis _ SerCeviTi saSualosa da
standartuli gadaxris warmodgena. Cven giCvenebT Sedegebis prezentaciis am
stils erT-erTi sanimuSo magaliTis saxiT, romelic exeba kvlevas
mesameklaseli moswavleebis mier televizoris yurebaze daxarjuli drois
Sesaxeb.
saSualo dro, romelsac 30 mesameklaseli xarjavda yoveldRiurad
televizoris yurebaze iyo 83 wuTi (SD=26 wuTis). alfa tolia 0.05-is da
ormxrivi kriteriumisaTvis erTi SerCevis t kriteriumi gviCvenebs, rom
SerCeviTi saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi
saSualosagan, romelic tolia 95 wuTis; t(29)= -2.53, p<0.05.
am Sedegebis prezentaciisas:
α=0.05 gviCvenebs am kriteriumis mniSvnelovnobis dones;
t(29) gansazRvravs testis statistikas rogorc t-s. es t
gansazRvravs trealurs, Tumca indeqsi `realuri” SeiZleba ar
axldes. radgan t ganisazRvreba, rogorc erTi SerCevis t, Cven
viciT, rom kriteriumi gulisxmobs SerCevis saSualos
Sedarebas populaciis hipoTetur saSualosTan. Tavisuflebis
xarisxi trealurisaTvis naCvenebia frCxilebSi. amrigad, Cven viciT,
rom t-s gamoTvlisas gamoyenebuli iyo 30 monacemi (radgan
df=N-1, amitom 29=30-1);
=-2.53 trealuris mniSvneloba (ara tkritikulisa, romelsac vpoulobT
cxrilSi #a-2). t-s mniSvneloba unda iyos gamosaxuli ricxviT,
romelsac eqneba mZimis Semdeg sul cota ori cifri;
p<0.05 gviCvenebs Semdegs:
a) p da naklebobis niSani (p<) gviCvenebs, rom trealuris miRebis
albaToba H0-is WeSmaritebis pirobebSi aris naklebi an toli
0.05-s. es mniSvneloba aris trealuris albaToba an ufro Sors
mdebare trealuris albaToba, roca H0 aris WeSmariti;
b) H0: µ =95.0 wuTi aris uaryofili,
H0: µ ≠95.0 wuTi aris miRebuli;
g) gansxvaveba SerCevis saSualosa da hipoTetur µ-s Soris
aris statistikurad mniSvnelovani;
d)SerCevis saSualo ganixileba rogorc aRebuli im
populaciidan, romlis saSualo naklebia hipoTetur µ=95
wuTze;
e) Tu p>0.05 iqneba SedegebSi warmodgenili, maSin metobis
niSani aRniSnavs imas, rom H0 ar aris uaryofili da X ar
gansxvavdeba mniSvnelovnad hipoTeturi µ-sagan
mniSvnelovnobis 0.05 doneze.
SeamowmeT Tqveni codna VIII-3
1. dawereT H0 da H1 erTi SerCevis t kriteriumisaTvis qvemoT mocemuli
TiToeuli debulebisaTvis.
a) Ramis Zilis saSualo xangrZlivoba populacisaTvis aris 493 wuTi.
b) depresiisaken midrekilebis Sefasebis kiTxvarSi populaciis saSualo aris
36.1.
g) populaciis saSualo standartizebuli inteleqtis testisaTvis aris 100.
2. cxrilis #a-2 gamoyenebiT, ipoveT tkritikulis mniSvnelobebi miTiTebuli
df-sa da α gaTvaliswinebiT. gamoiyeneT ormxrivi kriteriumi.
df α
a. 6 0.05
b. 17 0.01
g. 23 0.05
d. 32 0.05
e. 39 0.05
v. 75 0.01
z. 103 0.05
T. 256 0.05
3. miuTiTeT, xvdeba Tu ara trealuris Semdegi mniSvnelobebi uaryofis
areSi. Semdeg miuTiTeT Tqveni gadawyvetileba H0 da H1-saTvis. gaxsovdeT, rom
Tqven unda airCioT Semdegi: an ver uaryofT H0-s, an uaryofT H0-s, da ar
miiRebT H1-s, an miiRebT H1-s. gamoiyeneT α=0.05 da ormxrivi kriteriumi. df
TiToeuli trealurisaTvis mocemulia frCxilebSi.
a. trealuri(19)= +3.014
b. trealuri(19)=-1.989
g. trealuri(16)=-2.179
d. trealuri(16)=+2.04
e. trealuri(30)=-3.961
b. trealuri(30)=+2.743
z. trealuri(6)=-2.364
T. trealuri(6)=+2.447
i. trealuri(9)=-2..262
k. trealuri(37)=+2.938
l. trealuri(42)=-2.41
4. gansazRvreT daSvebebi, romlebic safuZvlad udevs erTi SerCevis t
kriteriums.
5. amoxseniT Semdegi amocana da upasuxeT qvemoT mocemul kiTxvebs. gamoiyeneT
ormxrivi kriteriumi da 0.05 mniSvnelovnobis done. Tqven gainteresebT Tu
ramdenad kargad SeuZlia adamians Seafasos dro saaTis gareSe. Tqven sTxoveT
11 cdis pirs SeefasebinaT drois 7 wuTiani intervali da eniSnebinaT TqvenTvis
misi gasvlis dro saaTis gareSe. am 11 Sefasebis saSualo iyo 5.8 wuTi, s=0.8
wuTi. aris Tqveni SerCevis saSualo mniSvnelovnad gansxvavebuli populaciis
saSualosagan, romelic tolia 7 wuTis? Tqvens pasuxSi miuTiTeT Semdegi:
a. kriteriumi, romelsac gamoiyenebT
b misi gamoyenebisas gakeTebuli daSvebebi
g. statistikuri hipoTezebi
d. mniSvnelovnobis done
e. kriteriumis statistikis kritikuli mniSvneloba
v. uaryofis areebi
z. trealuris mniSvneloba
T. Tqveni gadawyvetilebani statistikuri hipoTezebis Sesaxeb
i. Tqveni daskvnebi SerCevis saSualos Sesaxeb
6. es savarjiSo warmoadgens t kriteriumis gamoyenebis Sedegebs, romelic
adaptirebulia gamoqveynebuli kvlevis Sedegebidan. TiToeuli warmodgenili
SedegisaTvis upasuxeT Semdeg SekiTxvebs:
a. ras udris SerCevis saSualos mniSvneloba?
b. ras udris populaciis standartuli gadaxris Sefasebis mniSvneloba?
g. romeli t kriteriumi gamoiyeneba am kvlevaSi?
d. ra aris H0 da H1 am t kriteriumisaTvis?
e. ramdeni monacemi iyo miRebuli am kvlevaSi?
v. risi tolia trealuris mniSvneloba am kvlevaSi?
z. risi tolia df am trealurisaTvis?
T. risi tolia tkritikuli α=0.05-sa da ormxrivi uaryofis aresaTvis?
i. uaryofT H0-s, Tu ver uaryofT H0-s?
k. miiRebT H1-s, UTu ar miiRebT H1-s?
l. daaskvniT Tu ara, rom SerCevis saSualo mniSvnelovnad gansxvavdeba
populaciis hipoTeturi saSualosagan?
angariSi I: morbenali mamakacis saSualo wona iyo 74 kg (SD=7.6 kg), α=0.05 da
ormxrivi t kriteriumisaTvis erTi SerCevis t kriteriumma gviCvena,
rom SerCevis saSualo ar gansxvavdeba mniSvnelovnad populaciis
hipoTeturi saSualosagan, romelic tolia 75 kg-is 25 wlis
mamakacebisaTvis; t(5)= -1.48, p<0.05.
angariSi II: araletaluri daavadebebis mqone 65 wlis da zemoT pacienti
mamakacebis saavadmyofoSi gatarebuli dReebis saSualo Seadgenda 10.4 dRes
(SD=2.3). α=0.05 da ormxrivi kriteriumisaTvis erTi SerCevis t kriteriumma
gviCvena, rom SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis
hipoTeturi saSualosagan, romelic tolia 8.7 dRisa; t(24) = +3.7, p<0.01.
mniSvnelovani sakiTxebi statistikuri hipoTezis SemowmebaSi
ratom ar viRebT nulovan hipoTezas?
Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin Cven ar
uarvyofT nulovan hipoTezas. am gadawyvetilebas xSirad ase gamovTqvamT:
`Cven ver uarvyofT nulovani hipoTezas”. Tqven SeiZleba gainteresebT, Tu
ratom vxmarobT am rTul (Znelad gasageb) frazas; rogorc wesi, Tu Cven ar
uarvyofT raimes, maSin Cven viRebT mas. ratom ar SegviZlia ubralod miviRoT
nulovani hipoTeza, Tuki kriteriumis statistika ar xvdeba uaryofis areSi? am
kiTxvaze pasuxis gasacemad Cven unda davubrundeT magaliTs, romelic
satvirTo manqanebSi skamis simaRles exeboda. skami morgebuli iyo im
mZRolebis populaciisaTvis, visi muxlqveSa simaRlec iyo 42 sm da σ=3.0sm.
davuSvaT, rom Tqven gyavT satvirTo manqanis mZRolebis 25 kaciani SerCeva da
aRmoCnda, rom am mZRolebis saSualo muxlqveSa simaRle aris 42,5 sm. zrealuris
mniSvneloba am SerCevisaTvis iqneba +0.83.
zrealuri = 83.06.05.0
25/3425.42
+=+=−
es zrealuri ar xvdeba uaryofis areSi, romelic gansazRvrulia zkritikuls plus an
minus 1.96 α=0.05-is mniSvnelovnobis doneze, amitom is aris aramniSvnelovani.
Cven ver uarvyofT H0:µ=42sm-is. magram niSnavs Tu ara es gadawyvetileba imas,
rom im populaciis saSualo, romlidanac aris aRebuli es SerCeva unda iyos
zustad 42sm? ra Tqma unda, ara; SerCevis saSualo 42.5 sm ar niSnavs imas, rom
populaciis saSualo, romlidanac SerCeva aris aRebuli, unda iyos 42 sm-is
toli. meore mxriv, X =42.5 sm, agreTve, ar gvaZlevs imis mtkicebis saSualebas,
rom µ ar SeiZleba iyos 42 sm toli, anu X =42.5 sm ar gvaZlevs imis mtkicebis
saSualebas, rom Cven unda uarvyoT H0: µ =42 sm, magram arc imis mtkicebis
saSualebas gvaZlevs, rom µ zustad 42 sm-is tolia. is gvaCvenebs mxolod imas,
rom Cven ar SegviZlia uarvyoT H0: µ =42 sm.
davuSvaT, rom X toli iyo 44 sm, ase rom zrealuri =+3.33. es zrealuri aris
statistikurad mniSvnelovani 0.05 doneze da Cven uarvyofT H0: µ =42 sm-s da
miviRebT H1: µ ≠42sm-s. am SemTxvevaSi sworia miviRoT H1, radgan H1 amtkicebs,
rom µ aris garkveuli mniSvneloba, romelic gansxvadeba 42sm-sagan da X =44sm
gviCvenebs, rom µ marTlac aris garkveuli mniSvneloba, romelic gansxvavdeba
42sm-sagan. am SemTxvevaSi, SerCevis saSualo 44sm uzrunvelyofs µ-s saukeTeso
wertilovan Sefasebas mZRolebis im populaciisaTvis, saidanac iyo aRebuli
SerCeva.
swori daskvnebi statistikuri hipoTezebis Sesaxeb SeiZleba Semdegnairad
CamovayaliboT:
• TuU kriteriumis statistika xvdeba uaryofis areSi, maSin H0 uaryofili
iqneba da miRebuli iqneba H1;
• Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin H0-s ver
uarvyofT da ver miviRebT H1-s.
pirveli da meore gvaris Secdomebi statistikuri kriteriumebis
gamoyenebisas
statistikuri kriteriumi mkvlevars aZlevs saSualebas
uaryos H0 (1) an ver uaryos H0. (2) gadawyvetilebis
siswore damokidebulia imaze, aris Tu ara nulovani
hipoTeza faqtiurad WeSmariti populaciisaTvis. pirveli
gvaris Secdoma - Secdoma statistikuri gadawyvetilebis
miRebis dros, romelic gamoixateba nulovani hipoTezis
uaryofaSi, maSin roca is realurad WeSmaritia. meore
gvaris Secdoma - Secdoma statistikuri gadawyvetilebis
miRebis dros, romelic gamoixateba imiT, rom ar xdeba
nulovani hipoTezis uaryofa, maSin roca is realurad
mcdaria da alternatiuli H1 hipoTeza ki WeSmariti.
cxrili #8.4 gviCvenebs am Secdomebs da SesaZlo swor
gadawyvetilebebs, romelic SeiZleba miviRoT
statistikuri kriteriumis gamoyenebisas.
pirveli da meore gvaris Secdomebi xdeba imis gamo, rom statistikuri
gadawyvetilebis miReba albaTuri xasiaTiaa. TuU H0 aris WeSmariti, pirveli
gvaris Secdomis albaToba udris α-s. amitom, swori daskvnis albaToba da
imis albaToba, rom H0 ar iqneba uaryofili, maSin roca is WeSmaritia, aris
mocemuli 1-α-s saSualebiT. meore gvaris Secdomis albaToba udris β-s. aqedan
pirveli gvaris Secdoma
(Type I error) Secdoma statistikuri gadawyvetilebis miRebis dros, romelic gamoixateba nulovani hipoTezis uaryofaSi, maSin roca is realurad WeSmaritia. meore gvaris Secdoma (Type II error) - Secdoma statistikuri gadawyvetilebis miRebis dros, romelic gamoixateba imiT, rom ar xdeba nulovani hipoTezis uaryofa, maSin roca is realurad mcdaria da alternatiuli H1 hipoTeza ki WeSmariti.
beta(β): meore gvaris Secdomis albaToba.
gamomdinare, H1 -is sworad miRebis albaToba, roca H1 WeSmaritia aris
mocemuli 1-β-s saSualebiT. mniSvneloba 1-β aris statistikuri kriteriumis
simZlavre. amitomac statistikuri kriteriumis simZlavre an ubralod
simZlavre
aris H0-is uaryofis albaToba, roca H0 ar aris
samarTliani da H1 aris WeSmariti. β mniSvneloba
damokidebulia mTel rig faqtorebze da Cven am
faqtorebs me-9 TavSi ganvixilavT. TiToeuli es
albaToba gansazRvrulia cxrilSi #8.4.
I gvaris Secdoma xdeba mxolod maSin, roca H0 aris WeSmariti. Aamitom, I
gvaris Secdomis daSvebis albaToba pirdapir damokidebulia arCeuli
mniSvnelovnobis doneze. gaixseneT, rom mniSvnelovnobis done gviCvenebs
kriteriumis statistikis mniSvnelobis miRebis albaTobas (mag., an z an t),
romelsac mivyavarT H0-is uaryofamde, roca H0 aris realurad WeSmariti.
Sesabamisad, roca statistikuri kriteriumisaTvis aris arCeuli α, iqmneba I
gvaris Secdomis daSvebis riski. roca α=0.05 WeSmariti H0-is SecdomiT
uaryofis albaToba tolia 0.05. analogiurad, roca α=0.01, maSin arsebobs
mxolod 1 Sansi 100-dan, rom daSvebul iqnes I gvaris Secdoma H0-is uaryofiT,
roca is realurad WeSmaritia. mecnierebi, Cveulebriv, irCeven ufro frTxil
gzas, raTa garantirebuli iyos I gvaris Secdomis daSvebis dabali albaToba
maT kvlevebSi. Sesabamisad, α-s mniSvnelobad irCeven xolme mis dabal
mniSvnelobebs, rogoricaa, α=0.05 an α=0.01.
cxrili #8.4
Secdomebi da swori gadawyvetilebani statistikur gadawyvetilebebSi da
TiToeul SedegTan dakavSirebuli albaToba
Eeqsperimentatoris gadawyvetileba
H0-s ver uarvyofT H0 uaryofilia, H1
miRebulia
H0 WeSmaritia swori gadawyvetileba:
p=1-α
I gvaris Secdoma:
p=α
WeSmariti situacia
populaciaSi
simZlavre (Power) – H0-is uaryofis albaToba, roca H0 ar aris samarTliani da H1 ki aris WeSmariti. statistikuri kriteriumis
simZlavre mocemulia 1-β-s saSualebiT.
H1 WeSmaritia II gvaris Secdoma:
p=β
swori gadawyvetileba:
p=1-β
statistikuri da mecnieruli mniSvnelovnoba
statistikur kriteriumSi mniSvnelovnoba gviCvenebs mxolod imas, rom H0 aris
uaryofili. Tu H0 uaryofilia, maSin statistikuri kriteriumi arc miRebuli
Sedegis mizezs xsnis, arc imas gviCvenebs, Tu ramdenad mniSvnelovania,
mecnieruli TvalsazrisiT, es gansxvaveba. mTeli pasuxismgebloba rCeba
mkvlevars, romelmac yuradRebiT unda Seamowmos kvlevis pirobebi da
miRebuli monacemebis xasiaTi.
kriteriumis statistikis realuri mniSvnelobebi, romlebic ar xvdeba
uaryofis areSi iTvleba aramniSvnelovnad da ara umniSvnelod. umniSvnelo
gulisxmobs rom raimes ara aqvs mniSvneloba. maSin, roca aramniSvnelovani
gansxvaveba SeiZleba mecnierulad iyos Zalian mniSvnelovani. magaliTad,
davuSvaT, rom gakveTilebis Semdeg bavSvebis movlis centris mier
SemoTavazebuli iyo programa, romelic miznad isaxavda Seemcirebina is dro,
romelsac bavSvebi atarebdnen televizoris yurebisas. Tuki X da µ
Sedarebisas aRmoCnda aramniSvnelovani gansxvaveba, es niSnavs, rom programa
araefeqturia da ver cvlis im dros, romelsac isini atareben televizoris
yurebisas da es SeiZleba mkvlevrisaTvis iyos mniSvnelovani Sedegi.
populaciis saSualosaTvis ndobis intervalebis ageba t statistikis gamoyenebiT
ndobis intervalebi populaciis saSualosaTvis gansazRvruli iyo me-7
TavSi rogorc intervalebi, romlebSic SeiZleba iyos moTavsebuli populaciis
saSualo Cveni darwmunebulobis garkveuli xarisxiT. im TavSi Cven avageT es
intervalebi X -is, z-is da Xσ -is gamoyenebiT. magaliTad, Tu Xσ cnobilia, 95-
procentiani ndobis intervali populaciis saSualosaTvis mocemulia
intervaliT
X -1.96 Xσ -dan X +1.96 Xσ -mde.
xSirad Xσ ucnobia da misi mniSvneloba gamoiTvleba Xs -iT. roca Xs
gamoiyeneba Xσ -is gamosaTvlelad, ndobis intervali agebulia t statistikis
da ara z-is gamoyenebiT. t-sa da Xs -is gamoyenebiT 95-procentiani ndobis
intervali populaciis saSualosaTvis mocemulia intervaliT
X _(t0.05)( Xs )-dan X + (t 0.05)( Xs )-mde,
sadac X =realuri SerCevis saSualos
Xs = im saSualos gamoTvlili standartuli Secdoma, romelic
miRebulia s/√N-dan
t 0.05= tkritikulis ormxrivi mniSvnelobisa 0.05 mniSvnelovnobis doneze N-1
Tavisuflebis xarisxiT.
analogiurad, 99%-iani ndobis intervali populaciis saSualosaTvis
mocemulia
X - (t0.01)( Xs )-dan X + (t0.01)( Xs )-mde.
amis sailustraciod, Cven davadgineT 95-procentiani ndobis intervali 30
mesameklaselis mier televizoris yurebisas gatarebuli drois
monacemebisaTvis X =83 wuTisa da s=26 wuTisaTvis. radgan N=30, am SerCevisaTvis
Tavisuflebis xarisxi iqneba 29 da ormxrivi tkritikulis(29) mniSvneloba 0.05
mniSvnelovnobis doneze aris 2.045. saSualos standartuli Secdoma, Xs , tolia
26/√30, anu 4.75 wuTis. amrigad, 95-procentiani ndobis intervali µ-saTvis iqneba
83-(2.045)(4.75)-dan 83+(2.045)(4.75),
an Tu CavatarebT maTematikur gamoTvlebs, miviRebT ndobis intervals
83-9.7-dan 83+9.7 wuTamde,
rac 73.3-dan 92.7 wuTamde intervalis tolia. albaToba imisa, rom µ
moTavsebulia am intervalSi aris 0.95.
Tumca ndobis intervalebi ar aris farTod ganxiluli kvlevebSi, amJamad
mkvlevrebs urCeven xolme CarTon isini TavianT kvlevebSi. (Streiner, 1996; Wilkinson
da sxva, 1999). magaliTad, gvartnei-gibma (Gwartney-Gibb, 1986) Seiswavla wyvilebis
qorwinebamde Tanacxovreba saarqivo masalebis gamoyenebiT, romelic aRebuli
iyo qorwinebaze Tanxmobis ganacxadebidan, raTa mieRo demografiuli
informacia im wyvilebze, romlebic qorwinebamde erTad cxovrebdnen da
romlebic ar cxovrobdnen. misi kvlevis erT nawilSi, man warmoadgina
qorwinebis asakis saSualo maCvenebeli im mamakacebisaTvis da qalebisaTvis,
romlebic qorwinebamde Tanacxovrobdnen da imaTTvisac, vinc qorwinebamde ar
Tanacxovrobdnen. asakis saSualos Tan axlda 95-procentiani ndobis intervali
populaciis saSualosaTvis. rogorc magaliTi, gvartnei-gibma warmoadgina
Semdegi Sedegebi im qalebisaTvis, romlebic qorwinebamde Tanacxovrobdnen da
ar Tanacxovrobdnen da amasTanave ar iyvnen adre namyofi qorwinebaSi.
Qqorwinebamde
Tanacxovrobdnen
Qqorwinebamde ar
Tanacxovrobdnen
Qqorwinebis saSualo asaki 806 877
22.8 21.7
s 4.6 4.4
95-procentiani ndobis
intervali µ-saTvis
22.48-dan 23.12-mde 21.41-dan 21.29-mde
Cven SegviZlia viyoT 95 procentiT darwmunebulebi, rom qorwinebis
asakis µ im qalebisaTvis, romlebic qorwinebamde Tanacxovrobdnen, xvdeba 22.48-
dan 23.12 wlebis SualedSi im populaciisaTvis, romelic SerCeuli iyo
gvartnei-gibis mier. xolo im qalebisaTvis, romlebic qorwinebamde ar
Tanacxovrobdnen, µ-s 95- procentiani ndobis intervali aris 21.41-dan 21.29
wlamde. ndobis es ori intervali ar faravs erTmaneTs, amitom gvartnei-gibsma
daaskvna, rom is qalebi, romlebic qorwinebamde Tanacxovroben, Cveulebriv
qorwindebian erTi wliT gvian, vidre is qalebi, romlebic qorwinebamde ar
Tanacxovroben im populaciaSi, romelic mis mier iyo Seswavlili.
SeamowmeT Tqveni codna VIII-4
1. axseniT, Tu ratom ar iRebs mkvlevari H0-s, Tuki trealuri ar xvdeba
uaryofis areSi.
2. upasuxeT Semdeg kiTxvebs:
a. ra aris I gvaris Secdoma?
b. risi tolia I gvaris Secdomis daSvebis albaToba, roca H0 aris
WeSmariti?
g. risi tolia I gvaris Secdomis daSvebis albaToba, roca H0 aris
mcdari?
d. ra aris II gvaris Secdoma?
e. SesaZlebelia Tu ara, rom eqsperimentatorma dauSvas orive I da II
gvaris Secdomebi erTi da imave statistikur kriteriumSi? axseniT
Tqveni pasuxi.
3. ra gansxvavebaa statistikur mniSvnelovnobasa da mecnierul
mniSvnelovnobas Soris?
4. eqsperimentatori swavlobda, Tu ramden dros xarjavdnen mozardebi
video TamaSebze video salonebSi. aRmoCnda, rom TamaSis saSualo dro dReSi
Seadgenda 97 wuTs 23 mozardi biWisaTvis da standartuli gadaxra iyo 39
wuTi. aageT 95- procentiani ndobis intervali mozardi biWebis mier TamaSze
daxarjuli drois populaciis saSualosaTvis.
Sejameba
• statistikuri kriteriumebi SeiZleba iyo parametruli da
araparametruli.
• statistikuri hipoTezis Semowmeba moicavs Semdeg nabijebs:
kriteriumis statistikis arCeva (mag. z an t)
• H0-is da H1-is formulirebas. H0 gamoiyeneba kriteriumis statistikis
SerCeviTi ganawilebis dasadgenad.
• mniSvnelovnobis donis dadgena. mniSvnelovnobis done,
gansazRvruli α mniSvnelobiT, aris albaTobis mniSvneloba,
romelic adgens nulovani hipoTezis uaryofis kriteriums.
• kriteriumis statistikis kritikuli mniSvnelobis an mniSvnelobebis
gansazRvra cxrilis gamoyenebiT. kriteriumis statistikis
kritikuli mniSvnelobebi gansazRvraven uaryofis aris sazRvars an
sazRvrebs.
• uaryofis aris an areebis mdebareobis gansazRvra. es uaryofis are
warmoadgens SerCevis ganawilebaSi kriteriumis statistikis
mniSvnelobebs, romelTa miRebis albaToba α-s tolia an masze
naklebia, Tu H0 WeSmaritia.
• statistikuri hipoTezebis Sesaxeb gadawyvetilebis miRebis wesebis
formulireba. es wesebia:
_ Tu kriteriumis statistika xvdeba uaryofis areSi, maSin unda
uarvyoT H0 da miviRoT H1. gansxvaveba SerCevis saSualosa da
populaciis hipoTetur saSualos Soris aris statistikurad
mniSvnelovani.
_ Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin ver
uarvyofT H0-s da ver miviRebT H1-s. gansxvaveba SerCevis saSualosa da
populaciis hipoTetur saSualos Soris aramniSvnelovania.
• kriteriumis statistikis realuri mniSvnelobis gamoTvla SerCevis
monacemebidan.
• H0-isa da H1-is Sesaxeb gadawyvetilebis miReba damokidebulia imaze,
kriteriumis statistikis realuri mniSvneloba moxvdeba Tu ara
uaryofis areSi.
• z statistika, mocemuli z=( X -µ)/ Xσ formuliT, gamoiyeneba saSualosa
da populaciis hipoTetur saSualos Sesadareblad, roca σ
cnobilia.
• t statistika, mocemuli t=( X -µ)/ Xs formuliT, gamoiyeneba saSualosa
da populaciis hipoTeturi saSualos Sesadareblad, roca Xσ
gamoiTvleba Xs saSualebiT.
• t statistikis SerCeviTi ganawileba dmokidebula t-s df-ze, sadac
df=N-1.
• statistikuri daskvnebis gakeTeba imTaviTve albaTuria. I gvaris
Secdoma xdeba WeSmariti H0-is uaryofisas. II gvaris Secdoma xdeba
maSin, roca ar aris uaryofili mcdari H0 .
• I gvaris Secdomis albaToba aris α, xolo II gvaris Secdomis
albaToba aris β.
• statistikuri kriteriumis simZlavre aris H0-is .uaryofis albaToba,
roca H0 .aris mcdari da H1 aris WeSmariti.
• t ganawileba da Xs SeiZleba gamoyenebuli iyos populaciis
saSualosaTvis ndobis intervalebis dasadgenad.
mniSvnelovani terminebi da simboloebi
alfa (α)
alternatiuli hipoTeza (H1)
beta (β)
kritikuli mniSvneloba
Tavisuflebis xarisxi
araparametruli kriteriumebi
aramniSvnelovani gansxvaveba
nulovani hipoTeza
erTi SerCevis t kriteriumi
calmxrivi kriteriumi
paramatruli kriteriumebi
simZlavre
uaryofis are
t-s SerCeviTi ganawileba
mniSvnelovnobis done
statistikuri hipoTeza
statistikurad mniSvnelovani gansxvaveba
tkritikuli
trealuri
kriteriumis statistika
ormxrivi kriteriumi
I gvaris Secdoma
II gvaris Secdoma
z
zrealuri
zkritikuli
kiTxvebi
1. ra moTxovnebi unda daakmayofilos nebismierma statistikurma
hipoTezam?
2. ra daniSnulebas emsaxureba nulovani hipoTeza statistikur
kriteriumSi?
3. ra aris mniSvnelovnobis done?
4. ra aris kriteriumis statistikis kritikuli mniSvneloba?
44 5. ra gadawyvetileba miiReba H0-sa da H1-is mimarTebaSi, Tuki:
a. kriteriumis statistikis realuri mniSvneloba xvdeba uaryofis areSi?
b. kriteriumis statistikis realuri mniSvneloba ar xvdeba uaryofis
areSi?
6. Tqven gTxoves SeamowmoT, gansxvavdeba Tu ara SerCevis saSualo _ 73
mniSvnelovnad populaciis hipoTeturi saSualosagan, romelic tolia
75-s. ra gansazRvravs Tqvens arCevans _ gamoiyenebT z kriteriums, Tu
erTi SerCevis t kriteriums?
7. ra gadawyvetilebas miiRebT SerCevis saSualosa da populaciis
hipoTeturi saSualos mimarTebis Sesaxeb, Tuki:
a. zrealuri xvdeba uaryofis areSi?
b. zrealuri ar xvdeba uaryofis areSi?
8. zrealuris TiToeuli mniSvnelobisaTvis gansazRvreT, moxvdeba Tu ara
zrealuri ormxriv uaryofis areSi mniSvnelovnobis 0.01 doneze. Semdeg
miuTiTeT Tqveni gadawyvetilebis Sesaxeb H0-isa da H1-saTvis. gaxsovdeT,
rom Tqven an (1) H0-s ver uaryofT an unda uaryoT H0 da, an (2) ar miiRoT,
an miiRoT H1. (a) zrealuri = +3.71,
(b) zrealuri = -2.6, (g) zrealuri = +1.74, (d) zrealuri=-3.05, (e) zrealuri = -1.96,
(v) zrealuri= +2.58.
9. populaciis saSualo inteleqtis skalaze jgufisaTvis aris 100 da σ=16.
skolis konsultantma am skalaze Seafasa 17 aTi wlis bavSvi da miiRo
X =94.1. SeasruleT statistikuri Semowmeba am monacemebis gamoyenebiT,
raTa gansazRvroT, aris Tu ara gansxvaveba populaciis saSualosa da
SerCevis saSualos Soris statistikurad mniSvnelovani. gamoiyeneT 0.05
mniSvnelovnobis done da ormxrivi kriteriumi.
10. z kriteriumis Sedegebze dayrdnobiT daadgines, rom arsebobs
mniSvnelovani gansxvaveba SerCevis saSualosa da populaciis saSualos
Soris. axseniT, ras niSnavs statistikuri mniSvnelovnoba am SemTxvevaSi.
11. ra gadawyvetilebas miiRebT SerCevis saSualosa da populaciis
hipoTeturi saSualos mimarTebis Sesaxeb, Tuki:
a. trealuri xvdeba uaryofis areSi?
b. trealuri ar xvdeba uaryofis areSi?
45 12. trealuris TiToeuli mniSvnelobisaTvis gansazRvreT, moxvdeba Tu ara is
uaryofis areSi. CaTvaleT, rom α=0.01 da gamoiyeneT ormxrivi kriteriumi.
df trealuris TiToeuli mniSvnelobisaTvis mocemulia frCxilebSi.
a. trealuri(18) = +3.713
b. trealuri(7) = -3.015
g. trealuri(35) = +2.794
d. trealuri(44) = -3.0.58
e. trealuri(74) = -1.96
v. trealuri(15) = -2.368
13. dedebi, romelTac patara bavSvebi hyavT xSirad Civian, rom maT akliaT
Zili. davuSvaT, rom Tqven gazomeT im cxra dedis Zilis xangrZlivobis
maCveneblebi, romelTac hyavT wlamde bavSvebi da miiReT Semdegi monacemebi
(saaTebSi): 6.4, 7.5, 6.9, 7.3, 7.6, 7.1, 6.5, 7.7, 7.8.
a. gansxvavdeba Tu ara mniSvnelovnad am dedebis Zilis xangrZlivobis
saSualo populaciis hipoTeturi saSualosagan, romelic tolia 7.7 saaTisa
RameSi? gamoiyeneT ormxrivi kriteriumi da 0.05 mniSvnelovnobis done.
b. aRwereT Tqveni Sedegebi im nimuSis Sesabamisad, romelic naCvenebi iyo
am Tavis erT-erT qveTavSi _ erTi SerCevis t kriteriumis Sedegebis
prezentacia.
g. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier miRebul
gadawyvetilebaSi iyos I gvaris Secdoma?
d. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier miRebul
gadawyvetilebaSi iyos II gvaris Secdoma?
e. gamoiTvaleT 95 procentiani ndobis intervali populaciis
saSualosaTvis am monacemebis mixedviT. xvdeba Tu ara populaciis
hipoTeturi saSualo _7.7 saaTi RameSi am intervalSi?
14. mkvlevari samedicino sferoSi swavlobda sakiTxs, axlavs Tu ara
avadmyofobas sxeulis temperaturis mateba. man gazoma im 12 adamianis
sxeulis temperatura, romelsac dausves daavadebis diagnozi da miiRo
Semdegi monacemebi (farengeitebSi): 99.3, 99.1, 100.4, 98.4, 98.2, 98.9, 99.7, 100.1,
100.7, 99, 98.8, 99.2.
a. gansxvavdeba Tu ara am individebis sxeulis temperatura populaciis
hipoTeturi saSualo temperaturisagan _ 98.6F? gamoiyeneT ormxrivi
kriteriumi da 0.05 mniSvnelovnobis done.
46 b. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier
miRebul gadawyvetilebaSi iyos II gvaris Secdoma? ganmarteT Tqveni
pasuxi.
15. TiToeuli qvemoT mocemuli debulebisaTvis miuTiTeT, aris
gadawyvetileba swori, Tu Secdomas Seicavs? Tuki is Seicavs Secdomas,
miuTiTeT, is I gvaris, Tu II gvaris Secdomaa?.
a. mocemuli populaciisaTvis H0 aris WeSmariti, magram
eqsperimentatorma zrealuris safuZvelze uaryo H0.
b. mocemuli populaciisaTvis H1 aris WeSmariti, magram
eqsperimentatorma zrealuris safuZvelze CaTvala, rom H0-s ver uaryofs.
g. mocemuli populaciisaTvis H0 aris WeSmariti da
eqsperimentatorma zrealuris safuZvelze CaTvala, rom H0-s ver uaryofs.
d. mocemuli populaciisaTvis aris WeSmariti da eqsperimentatorma
zrealuris safuZvelze uaryo H0.
1 IXFTavi. eqsperimentis safuZvlebi da saSualoTa Soris gansxvavebis Semowmeba
kvlevis procesis mimoxilva
mecnieruli kiTxvis dasma da kvlevis hipoTezis formulireba
kvlevis sqemis arCeva
cdis pirTa SerCeva
ekvivalenturi jgufebis Seqmna
damoukidebeli cvladis manipulacia
damatebiTi cvladebis kontroli
damokidebuli cvladis gazomva
aqvs Tu ara gavlena eqsperimentul Carevas statistikuri
Semowmebis saWiroebaze?
SerCevis Secdomis Sedegebi eqsperimentisaTvis
or saSualos Soris gansxvavebis statistikuri hipoTezis
Semowmebis mimoxilva
t kriteriumi ori damoukidebeli jgufisaTvis
saSualoTa Soris sxvaobebis SerCeviTi ganawileba
statistikuri Semowmeba tdamoukidebelis gamoyenebiT
simZlavre da tdamoukidebeli
eqsperimentuli Carevis zemoqmedebis siZlieris gazomva
ndobis intervalebis ageba ori damoukidebeli populaciis
saSualoTa Soris gansxvavebisaTvis t statistikis gamoyenebiT
ras amowmebs t kriteriumi realurad
statistikuri mniSvnelovnobisa da mecnieruli mniSvnelovnebis
mimoxilva
diskusia statistikuri hipoTezebis Semowmebis Sesaxeb
t kriteriumis Sedegebis prezentacia
cdis pirTa Sida sqema da t kriteriumi damokidebuli
qulebisaTvis
2 t kriteriumi damokidebuli qulebisaTvis
ndobis intervalebis ageba ori damokidebuli populaciis
saSualoTa Soris gansxvavebisaTvis
me-8 TavSi gavecaniT statistikuri hipoTezis Semowmebas, romlis
saSualebiTac ganvsazRvreT, gansxvavdeba Tu ara mniSvnelovnad SerCevis
saSualo populaciis hipoTeturi saSualosagan. am TavSi gamoviyenebT
statistikuri hipoTezis Semowmebas, raTa erTmaneTs ori SerCevis saSualo
SevadaroT da ganvsazRvroT, gansxvavdebian Tu ara isini erTmaneTisagan
ufro metad, vidre mxolod SerCevis Secdomis Sedegad miRebuli
gansxvavebisas. eqsperimentuli kvlevis Sedegebis analizi statistikuri
hipoTezis Semowmebis am tipis erT-erT gamoyenebas warmoadgens. pirvel da
meore TavebSi eqsperimentis ZiriTadi cnebebi mokled iyo warmodgenili, am
TavSi ki es Tema ufro farTod aris gaSlili. dasawyisSi ganixileba
mxolod cdis pirTa Soris sqemebi, Semdgom ki _ cdis pirTa Sida sqemebi.
kvlevis procesis ganxilva
yvela mecnieruli kvleva iwyeba kiTxviT, romelzec SeiZleba empiriulad
gavceT pasuxi. Cveulebriv, mecnieri am kiTxvaze savaraudo pasuxs kvlevis
hipoTezis saxiT gvTavazobs. kvlevis hipoTeza aris debuleba or an met
cvlads Soris mosalodneli, Tu prognozirebuli kavSiris Sesaxeb. kvlevis
hipoTeza eqsperimentSi aris prognozirebuli mimarTeba damoukidebelsa da
damokidebul cvladebs Soris. gaixseneT, rom damoukidebuli cvladi is
cvladia, romlis manipulirebas gegmavs eqsperimentatori, raTa daadginos
misi gavlena damokidebul cvladze. damokidebuli cvladi Cveulebriv
subieqtis gazomili qcevaa. kvlevis hipoTezis forma uCvenebs mecniers,
kvlevis romeli sqema unda iyos gamoyenebuli. Tuki kvlevis hipoTezaSi
nawinaswarmetyvelebia, rom damoukidebeli cvladis manipulireba gavlenas
axdens damokidebul cvladze, maSin eqsperimenti kvlevis sqemis swori
formaa da mas SeuZlia empiriulad Seamowmos kvlevis es hipoTeza.
martivi eqsperimenti iwyeba cdis pirTa ekvivalenturi jgufebis SeqmniT.
cdis pirebis TiToeul jgufs uqmnian zustad erTnair pirobebs, garda
imisa, rom maT damoukidebeli cvladis sxvadasxva done miewodebaT.
3 damoukidebeli cvladis manipulaciis Sedegad miiReba damokidebuli
cvladis qulebi. Semdeg xdeba maTi analizi, raTa ganisazRvros, axdens Tu
ara damoukidebeli cvladi gavlenas damokidebul cvladze. ganvixiloT am
procesis magaliTi.
yovel wels milionobiT recepti iwereba damamSvidebeli preparatebis
dasaniSnad, romlebic sedatiur saSualebas _ diazepams Seicaven. rogorc
uamrav sxva medikaments, diazepamsac sxvadasxva gverdiTi efeqti aqvs,
rogorc fiziologiuri, ise fsiqologiuri. erT-erTi fsiqologiuri efeqti
isaa, rom diazepami cvlis pirovnebis mier drois xangrZlivi Sualedis
aRqmas (Kleinknecht&Donaldson, 1975). unrig-neervurtma, kaizerma da koenenma
(Unrig-Neervoort, Kaiser, Coenen, 1992) wamoayenes hipoTeza, rom SesaZloa diazepams
igive efeqti aqvs drois xanmokle Sualedebzec. kvlevis am hipoTezaSi
nawinaswarmetyvelebia, rom damoukidebeli cvladi _ medikamentis piroba,
gavlenas moaxdens damokidebul cvladze _ pirovnebis mier drois
xanmokle Sualedis xangrZlivobis Sefaseba. am hipoTezis Sesamowmeblad
unrig-neervurtma da sxvebma cdis pirTa Soris eqsperimenti Caatares.
eqsperimentSi Seqmnili iyo cdis pirTa ori ekvivalenturi jgufi. erT
jgufs, placebo sakontrolo jgufs aZlevdnen abs, romelic ar Seicavda
aqtiur ingredientebs. meore jgufs, eqsperimentul jgufs, aZlevdnen 15
miligram (mg) diazepams. TiToeuli jgufi DSedgeboda 12 cdis pirisagan.
damokidebuli cvladi iyo pirovnebis mier TiTis kakunis amocanis
xangrZlivobis Sefaseba. cdis pirebs sTxovdnen TiTi daekakunebinaT
specialur mricxvelze, sanam eqsperimentatori ar miscemda gaCerebis
brZanebas. kakunis damTavrebis Semdeg cdis pirebs sTxovdnen SeefasebinaT,
Tu ra xnis ganmavlobaSi asrulebdnen isini am davalebas. Tumca orive
jgufi davalebas erTi da imave drois ganmavlobaSi asrulebda, placebo
sakontrolo jgufma es dro Seafasa rogorc 19.8 wm (s=7.6wm), maSin roca
eqsperimentuli jgufis mier Sefasebuli dro iyo 13.6 wm (s=4.8 wm). unrag-
neervurtma da sxvebma statistikurad gaaanalizes es saSualoebi
damoukidebeli jgufebis t kriteriumis gamoyenebiT da aRmoaCines, rom
eqsperimentuli jgufis mier Sefasebuli xangrZlivoba mniSvnelovnad
naklebia placebo sakontrolo jgufis mier Sefasebul xangrZlivobaze.
diazepami gavlenas axdens drois xanmokle intervalebis aRqmaze.
eqsperimentis Catareba monacemTa statistikur gaanalizebamde garkveuli
safexurebis gavlas moiTxovs. es safexurebi erTnairia yvela
4 eqsperimentisaTvis da Sedgeba mecnieruli kiTxvis dasmis, kvlevis hipoTezis
formulirebis, kvlevis sqemis arCevis, cdis pirebis SerCevis, cdis pirTa
ekvivalenturi jgufebis formirebis, gverdiTi cvladebis kontrolis,
damoukidebeli cvladis manipulirebis da damokidebuli cvladis
gazomvisagan.
mecnieruli kiTxvis dasma da kvlevis hipoTezis formulireba
mecnierulia kiTxva, romelic saSualebas iZleva empiriuli monacemebis
Segrovebis safuZvelze miviRoT pasuxi. Cveni magaliTi eyrdnoba Semdeg
mecnierul kiTxvas: `ra gavlenas axdens diazepami drois xanmokle
intervalebis aRqmaze?” aSkaraa, rom es kiTxva saSualebas iZleva masze
pasuxis gasacemad sistematuri dakvirveba Catardes. kvlevis hipoTeza
gvTavazobs savaraudo pasuxs am kiTxvaze. is acxadebs, rom diazepami
gavlenas axdens pirovnebis mier drois xanmokle intervalebis Sefasebaze.
kvlevis sqemis arCeva
kvlevis sqema aris gegma, romelic gamoyenebulia monacemTa Segrovebis
mimarTulebis gansazRvrisaTvis. am magaliTSi kvlevis hipoTezis Semowmeba
moiTxovs cdis pirTa im jgufis Sedarebas, romelic iRebda diazepams, meore
jgufTan, romelic ar iRebda diazepams. pirvel TavSi Cven ganvsazRvreT es
sqema, rogorc cdis pirTa Soris sqema. cdis pirTa Soris sqema umartivesia,
roca arsebobs cdis pirTa mxolod ori jgufi, ise, rogorc ganxilul
eqsperimentSi. es sqema moiTxovs cdis pirebis SerCevas eqsperimentSi
monawileobisaTvis da Semdeg maT ganawilebas ekvivalentur jgufebSi.
cdis pirebis SerCeva
SemTxveviTi SerCeva
cdis pirTa Soris sqema(Between–subject design) – eqsperimenti, romelSic ori an meti jgufi
SemTxveviTi SerCeva (Random sampling)- SerCevis meTodi, romelSic individebi iseTnairad arian arCeulni, rom populaciis TiToeul wevrs aqvs am SerCevaSi moxvedris erTnairi Sansi da erTi wevris SerCeva ar aris damokidebuli populaciis nebismieri sxva wevris SerCevaze.
5 eqsperimenti tardeba cdis pirebze, romlebic populaciidan aris
miRebuli. imisaTvis, rom SegveZlos eqsperimentis Sedegebis ganzogadeba
Cvens mier gamoyenebuli SerCevis farglebs gareT, rekomendebulia cdis
pirTa arCevisaTvis SemTxveviTi SerCevis gamoyeneba. magram, radgan
WeSmaritad SemTxveviTi SerCevebis miReba rTulia, SemTxveviTi SerCeva
iSviaTad miiRweva realur eqsperimentirebisas. ufro xSirad gamoiyeneba
xelmisawvdomi SerCeva.
xelmisawvdomi (SesaZleblobis) SerCeva
xelmisawvdom SerCevaSi xdeba cdis pirebis iseTi SerCeva, romlebic
mkvlevrisaTvis ufro xelmisawvdomi an xelsayrelia. SesaZleblobis SerCeva
aSkarad ar aris adekvaturi, Tuki Cven vcdilobT zustad SevafasoT
populaciis saSualo, rogoricaa, magaliTad, arCevnebis Sedegebis Sefaseba
gamokiTxviT. magram eqsperimentis mizani ar aris populaciis saSualos
konkretuli mniSvnelobis Sefaseba; misi mizania gansazRvros, gasxvavdeba Tu
ara ori an meti jgufi erTmaneTisagan damoukidebeli cvladis gavlenis
Sedegad. Tu damoukidebeli cvladi axdens gavlenas, es gavlena
gavrceldeba nebismier cdis pirze, iqnebian isini SemTxveviTad SerCeulni
Tu ara. amrigad, Tu Seqmnilia ekvivalenturi jgufebi, xelsayreli SerCeva
Cveulebriv ar zRudavs eqsperimentis interpretacias da mniSvnelobas.
ekvivalenturi jgufebis Seqmna
ekvivalenturi jgufebi aris cdis pirTa jgufebi, romlebic ar
gansxvavdebian erTmaneTisgan raime mudmivi an sistematuri maxasiaTebliT
eqsperimentis damoukidebeli cvladis miwodebamde. ekvivalenturi jgufebi
ar gulisxmoben imas, rom TiToeuli jgufis cdis pirebi zustad erTnairebi
arian damoukidebeli cvladis gamoyenebamde. is gulisxmobs imas, rom
individebis damoukidebeli cvladis doneebze gadanawilebisaTvis
xelmisawvdomi (SesaZleblobis) SerCeva (Convenience sampling): cdis pirebis iseTi SerCeva, romelic mkvlevrisaTvis ufro xelmisawvdomi an xelsayrelia.
ekvivalenturi jgufebi (Equivalent groups): cdis pirTa jgufebi, romlebic, savaraudod, ar gansxvavdebian erTmaneTisgan raime mudmivi an sistematuri maxasiaTebliT eqsperimentis damoukidebeli cvladis miwodebamde.
6 gamoyenebuli unda iyos miukerZoebeli procedura. jgufebs Soris arsebuli
nebismieri gansxvaveba damoukidebeli cvladis manipulaciamde mxolod
SemTxveviTi gansxvaveba iqneba. rogorc mogvianebiT avxsniT, t kriteriumi
saSualebas iZleva SevafasoT aseTi SemTxveviTi xdomilobis albaToba.
SemTxveviTi ganawileba
ekvivalenturi jgufebis Seqmnis tipuri meTodi aris cdis pirebis SemTxveviTi
ganawileba jgufebSi. SemTxveviTi ganawileba niSnavs, rom eqsperimentisaTvis
SerCeul nebismier individs aqvs erTnairi Sansi mikuTvnebuli iyos nebismieri
jgufisadmi. kerZod, SemTxveviTi ganawileba akmayofilebs or kriteriums:
1. yoveli individisaTvis nebismier jgufSi moxvedris albaToba
Tanabari unda iyos;
2. erTi cdis piris ganawileba jgufSi ar moqmedebs nebismieri sxva
individis ganawilebaze imave jgufSi.
SemTxveviTi ganawileba uzrunvelyofs imas, rom iseTi cvladebis gavlena,
rogoricaa motivacia, sqesi, SfoTva, inteleqti da asaki Tanabrad iqneba
gadanawilebuli jgufebSi. SemTxveviTi ganawilebis Semdeg jgufebi ar
gansxvavdebian raime sistematuri maxasiaTebliT eqsperimentuli pirobis
Semotanamde.
praqtikaSi, mkvlevrebi sxva moTxovnebsac uyeneben SemTxveviT ganawilebas.
maT SeiZleba surdeT, rom hyavdeT cdis pirTa toli raodenoba TiToeul
jgufSi, iseve, rogorc zemoT ganxilul magaliTSi, an surdeT, rom hyavdeT
qalebisa da mamakacebis toli raodenobisagan Semdgari jgufebi. am
moTxovnebis miuxedavad, es jgufebi mainc erTmaneTis ekvivalenturi rCeba.
damoukidebeli cvladis manipulacia
SemTxveviTi ganawileba (Random assignment): cdis pirTa jgufebSi ganawilebis meTodi, romlis mixedviT, eqsperimentisaTvis SerCeul nebismier individs aqvs nebismier jgufSi ganawilebis toli albaToba da erTi cdis piris ganawileba jgufSi ar
moqmedebs nebismieri sxva individis ganawilebaze imave jgufSi.
damoukidebeli cvladi ( Independent variable) _ cvladi, romliTac manipulireben eqsperimentSi, damokidebul cvladze misi gavlenis gansazRvris mizniT.
7
damoukidebeli cvladi _ cvladi, romliTac manipulireben da romelsac
iyeneben sxvadasxva jgufis cdis pirebis mimarT, raTa gansazRvron misi
gavlena qcevaze. damoukidebel cvlads unda hqondes sul cota ori
gansxvavebuli done. damoukidebeli cvladis done aris damoukidebeli
cvladis erTi mniSvneloba. damoukidebeli cvladi xSirad gansazRvrulia
rogorc faqtori, magaliTad, A faqtori da misi doneebi _ A1 da A2. cdis
pirTa raodenoba TiToeul jgufSi aRvniSnoT nA1 da nA2-iT aqamde cdis
pirTa raodenobas jgufSi N-iT aRvniSnavdiT xolme. radgan amjerad ori
jgufi gvyavs, TiToeul jgufSi cdis pirTa raodenobis aRsaniSnavad
gamoviyenoT nA, xolo cdis pirTa mTeli raodenobis aRsaniSnavad
eqsperimentSi gamoviyenebT N-s.
damoukidebeli cvladis TiToeuli done eZleva TiTo jgufs. magaliTis
saxiT moyvanil eqsperimentSi damoukidebeli cvladi (e.i. faqtori A) aris
medikamentis miRebis piroba; misi ori donea: A1 done _ sakontrolo,
placebos miReba; da A A2 done _ eqsperimentuli _ diazepamis miReba.
amrigad, cdis pirTa erT jgufs (nA1=12) eZleoda placebo _ sakontrolo
medikamenti, xolo meore jgufs (nA2=12) eZleoda diazepami. es cdis pirTa
Soris erTfaqtoriani sqemaa, romelSic xdeba erTi damoukidebeli cvladis
manipulacia da cdis pirTa ori gansxvavebuli jgufis Sedarebaa saWiro
imis dasadgenad, axdens Tu ara damoukidebeli cvladi gavlenas.
damatebiTi cvladebis kontroli
damoukidebeli cvladis done (Level of an independent variable) _ damoukidebeli cvladis erTi mniSvneloba. imisaTvis, rom cvladi gaxdes, damoukidebeli cvlads unda hqondes sul cota ori gansxvavebuli done.
faqtori (Factor) _ damoukidebeli cvladis alternatiuli saxelwodeba.
cdis pirTa Soris erTfaqtoriani sqema (One-factor between subject design): kvlevis sqema, romelSic erTi damoukidebeli cvladis manipulacia xdeba da iqmneba ori an meti jgufi.
damatebiTi cvladebi (Extraneous variables) - damoukidebeli cvladisagan gansxvavebuli nebismieri cvladi, romelsac SeuZlia imoqmedos damokidebul cvladze eqsperimentSi.
8
imisaTvis, raTa davrwmundeT, rom or jgufs Soris arsebuli nebismieri
gansxvaveba mxolod damoukidebeli cvladis moqmedebis Sedegia,
aucilebelia damatebiTi cvladebis kontroli. damoukidebeli cvladisagan
gansxvavebuli nebismieri cvladi, romelsac SeuZlia eqsperimentSi
damokidebul cvladze imoqmedos, aris damatebiTi cvladi. is SeiZleba
warmoiqmnas cdis pirebis iseTi maxasiaTeblebisagan, rogoricaa inteleqtis
done, SfoTvis done, medikamentebisadmi sensitiuroba an motivacia; an im
fizikuri garemos cvlilebebis Sedegad, romelSic tardeba eqsperimenti; an
eqsperimentuli proceduris variaciis Sedegad. imisaTvis, raTa davrwmundeT,
rom sakontrolo da eqsperimentul jgufebs Soris napovni nebismieri
gansxvaveba mxolod damoukidebeli cvladis manipulaciis Sedegia,
damatebiTi cvladebi ar unda icvlebodes sistematurad an mudmivad
damoukidebel cvladTan erTad. Sereul eqsperimentSi xdeba damatebiTi
cvladis sistematuri cvlileba damoukidebel cvladTan erTad. aseT
eqsperimentSi SeuZlebelia damoukidebeli cvladis gavlenis gamijvna
Sereuli damatebiTi cvladis SesaZlo gavlenisagan.
zemoT ganxilul eqsperimentSi jgufebi erTmaneTis identuria da
erTmaneTisagan mxolod damoukidebeli cvladis moqmedebiT gansxvavdebian.
unrig-neervortma da sxvebma placebo sakontrolo jgufi gamoiyenes, raTa
cdis pirebi identur pirobebSi moeTavsebinaT. kvlevebSi, sadac xdeba
medikamentebis gamoyeneba, placebos kontroli warmoadgens samkurnalo
pirobebis simulacias, roca cdis pirebs miewodebaT abi an raime nivTiereba,
romelic ar Seicavs aqtiur ingredientebs. amrigad, orive jgufs
_placebosa da sakontrolo jgufebs miewodebodaT abi. Tu abs miscemdnen
mxolod eqsperimentuli jgufis cdis pirebs, maSin eqsperimenti Sereuli
gaxdeboda. adamianebi SeiZleba gansxvavebulad moiqcnen mxolod imitom, rom
medikamentis zemoqmedebas elodebian. amrigad, roca medikaments vawodebT,
qceva SeiZleba Seicvalos ori mizeziT, (1) Tavad medikamentis moqmedebis
Sereuli eqsperimenti (Confounded experiment) aris eqsperimenti, romelSic damatebiTi cvladi sistematurad icvleba damoukidebel cvladTan erTad.
placebos kontroli (Placebo control) - simulirebuli eqsperimentuli (Carevis) piroba.
9 Sedegad da (2) im molodinis Sedegad, rom medikaments efeqti eqneba. roca
unrig-neervortma da sxvebma placebo sakontrolo jgufs abi misces,
romelic ar Seicavda aqtiur ingredientebs, maT gaakontroles es molodini,
amitom orive jgufs erTnairi molodini hqonda medikamentis efeqtis mimarT.
analogiurad, orive jgufs misces identuri instruqciebi da maT mier
Sefasebuli dro erTnairi meTodiT izomeboda. Serevis Tavidan acileba
uaRresad mniSvnelovania eqsperimentis Catarebis dros.
damokidebuli cvladis gazomva
damokidebuli cvladi aris cvladi, romelic imyofeba damoukidebeli cvladis
gavlenis qveS. damoukidebeli cvladis manipulaciis Semdeg TiToeuli cdis
pirisaTvis unda gaizomos damokidebuli cvladi. am gazomvis Sedegebi
TiToeuli cdis pirisaTvis qulis saxiT gamoixateba. rogorc vnaxeT me-2-5
TavebSi, qulebi Sejamebulia centraluri tendenciisa da cvalebadobis
sazomebiT.
unrig-neervortisa da sxvaTa kvlevis hipoTezis mixedviT, diazepamis
miReba cvlis pirovnebis mier drois aRqmas. damokidebuli cvladi iyo drois
aRqma da izomeboda cdis pirebis mier im drois SefasebiT, romelsac isini
xarjavdnen garkveuli davalebis Sesasruleblad. es Sefasebebi warmodgenili
iyo TiToeuli jgufis saSualosa da standartuli gadaxris saSualebiT.
radgan ori saSualo da ori standartuli gadaxra gvaqvs, maTi erTmaneTisagan
gansasxvaveblad unda gamoviyenoT qveviT miwerili indeqsebi. amrigad, A1
jgufis saSualo iqneba 1AX , xolo s 1A iqneba populaciis standartuli gadaxra
A1 jgufisaTvis, 2AX iqneba A2 jgufis saSualo, xolo 2As ki populaciis
standartuli gadaxra A2 jgufisaTvis. am magaliTSi, 1AX =19.8wm da 1As =7.6wm,
placebo sakontrolo jgufisaTvis, xolo 2AX =13.6wm da 2As =4.8wm eqsperimentuli
jgufisaTvis. saSualoTa Soris gansxvavebis statistikuri mniSvnelovnoba
gaanalizebulia damoukidebeli jgufebis t kriteriumis saSualebiT. am
kriteriums Cven mogvianebiT ganvixilavT amave TavSi.
damokidebuli cvladi (Dependent variable): aris cvladi eqsperimentSi, romelic
damokidebulia damoukidebel cvladze.
10 suraTi #9.1 gviCvenebs eqsperimentis dagegmvisa da Catarebis sxvadasxva
etapebs ganxiluli magaliTis safuZvelze. ganixileT es etapebi am suraTis
gamoyenebiT.
SeamowmeT Tqveni codna IX-1
1. gansazRvreT: Sereva, SesaZleblobis SerCeva, damokidebuli cvladi,
ekvivalenturi jgufebi, damatebiTi cvladi, SemTxveviTi ganawileba,
SemTxveviTi SerCeva.
2. ra aris ekvivalenturi jgufebis Seqmnis mizani?
3. ra gansxvavebaa cdis pirebis SemTxveviT SerCevasa da jgufebSi
SemTxveviT ganawilebas Soris?
4. rodis xdeba Sereva eqsperimentSi?
5. ra sirTuleebs qmnis Sereuli eqsperimenti damoukidebeli cvladis
gavlenis Sesaxeb daskvnis gamotanis dros?
6. eqsperimentatori ikvlevs xmauris gavlenas Tvalis midevnebis amocanis
Sesrulebaze. amocana isaa, rom cdis pirs unda gaeyolebina saCvenebeli joxi
mbrunavi diskis garkveuli wertilisaTvis (warmoidgineT, rom gaayolebT
fanqars mbrunav musikalur firfitas). damokidebuli cvladi aris drois is
odenoba, romlis ganmavlobaSi cdis pirs SeuZlia SeinarCunos saCvenebeli
joxi erT wertilze. eqsperimentatori varaudobs, rom im cdis pirebs,
romlebic am davalebas uxmauro garemoSi asruleben, ufro didi xnis
ganmavlobaSi SeuZliaT SeinarCunon saCvenebeli joxi mbrunav wertilze, vidre
im cdis pirebs, romlebic davalebas xmauris maRali donis pirobebSi
asruleben. qvemoT aRwerili eqsperimentatoris TiToeuli moqmedeba xels
uSlis am eqsperiments damatebiTi cvladis SemotaniT, romelic sistematurad
icvleba damoukidebel cvladTan _ xmauris pirobasTan _ erTad.
eqsperimentatoris TiToeuli moqmedebisaTvis gansazRvreT damatebiTi cvladi
da Semdeg axseniT, ratom aris es eqsperimenti Sereuli.
a. eqsperimentatorma gaanawila aTi 20 wlis mamakaci uxmauro
garemoSi da aTi 60 wlis mamakaci xmaurian pirobebSi.
11 b. eqsperimentatorma waaxalisa uxmauro pirobebSi myofi cdis pirebi,
rom davaleba Zalian kargad SeesrulebinaT da ar waaxalisa
xmaurian pirobebSi momuSave cdis pirebi.
g. eqsperimentatorma mamakacebi uxmauro garemoSi gaanawila, xolo
qalebi _ xmaurian garemo pirobebSi.
d. eqsperimentatorma fizikuri kulturis kolejis studentebi
uxmauro garemoSi gaanawila da socialur mecnierebaTa fakultetis
studentebi _ xmaurian garemo pirobebSi.
e. cdis pirebi, romlebic davalebas uxmauro garemoSi asruleben,
eqsperimentSi 9.00 saaTidan 10.00 saaTamde monawileobdnen da cdis
pirebi, romlebic davalebas xmaurian pirobebSi asruleben,
eqsperimentSi 4.00 saaTidan 5.00 saaTamde monawileobdnen.
suraTi #9.1
eqsperimentis dagegmvisa da Catarebis aspeqtebi cdis pirTa Soris sqemis
gamoyenebiT. es suraTi gviCvenebs, Tu rogor gansazRvravs magaliTis saxiT
ganxiluli eqsperimenti diazepamis gavlenas drois aRqmaze.
24 cdis pirisagan Semdgari SerCevis ayvana, romelic miiRebs monawileobas am eqsperimentSi
cdis pirebi SemTxveviTad arian ganawilebulni, raTa Seiqmnas ekvivalenturi jgufebi.
jgufi
A1 placebo sakontrolo jgufi: 12 cdis piri
A2 eqsperimentuli jgufi: 12 cdis piri
ekvivalentruri jgufebi
12
aqvs Tu ara gavlena eqsperimentul Carevas statistikuri
Semowmebis saWiroebaze?
SerCevis Secdomis Sedegebi eqsperimentisaTvis
Tumca eqsperimentis Catarebis safexurebi Tanmimdevruli da naTelia,
sirTuleebi mainc Cndeba. damoukidebeli cvladis gavlenis Sesaxeb nebismieri
gadawyvetileba unda iyos miRebuli SerCevis saSualoebis erTmaneTTan
SedarebiT. principSi, Tuki SerCevis saSualoebi erTmaneTis ekvivalenturia,
Carevas ar hqonia gavlena. meore mxriv, Tuki SerCevis saSualoebi ar arian
erTmaneTis ekvivalenturni, arsebobs imis mtkicebis safuZveli, rom Carevas
mouxdenia gavlena. magram SerCevis Secdomis gamo am daskvnas ase martivad ver
gamovitanT. me-7 TavSi SerCevis Secdoma ganvsazRvreT rogorc raodenoba,
damatebiTi cvladebi kontrolirdeba
xdeba damoukidebeli cvladis manipulacia
jgufi: A1 miewodeba placebo abi
jgufi: A2 miewodeba diazepamis abi
izomeba damokidebuli cvladi: yvela cdis piri afasebs, Tu ra dro dasWirda amocanis Sesasruleblad
gamoiTvleba centraluri tendenciisa da cvalebadobis sazomi:
1AX =19.8 wm da s 1A =7.6 wm,
gamoiTvleba centraluri tendenciisa da cvalebadobis sazomi:
2AX =13.6 wm da s 2A =4.8 wm
cdis pirTa Soris Sedareba
damoukidebeli cvladis gavlena dadgenilia centraluri tendenciis sazomebis SedarebiT statistikuri kriteriumis saSualebiT.
13 romliTac SerCevis saSualo gansxvavdeba populaciis saSualosagan (anu, X _
µ). Tu aviRebT cdis pirTa or jgufs erTi da imave populaciidan, gavzomavT
qcevas da gamoviTvliT X TiToeuli SerCevisaTvis, SerCevis Secdomis gamo
naklebia imis albaToba, rom am ori jgufis saSualoebi erTmaneTis toli an
populaciis saSualos toli iqneba. suraTze #7.2 warmodgenili saSualos
empiriuli SerCeviTi ganawilebebi kargad aCvenebs am faqts.
ras migvaniSnebs SerCevis Secdoma eqsperimentSi damoukidebeli cvladis
gavlenis Sesaxeb gadawyvetilebis miRebaSi? ubralod erTi da imave
populaciidan aRebuli ori gansxvavebuli jgufis saSualoTa Soris
SemTxveviTi gansxvaveba mosalodnelia nebismier eqsperimentSi, sul erTia, aqvs
Tu ara damoukidebel cvlads gavlena damokidebul cvladze. maSinac ki, roca
damoukidebel cvlads araviTari gavlena ara aqvs, or jgufs Soris
saSualoebi erTmaneTisagan mxolod SerCevis Secdomis gamo gansxvavdebian.
maSin, rogorRa axerxeben bihevioristi mecnierebi Semdeg kiTxvaze pasuxs:
SerCevis saSualoTa Soris gansxvaveba ramdenad didi unda iyos, raTa
sakmarisi safuZveli gvqondes davaskvnaT, rom eqsperimentSi SerCevis Secdomis
garda sxva raRac muSaobs?
am kiTxvaze pasuxi gacema SeiZleba or saSualos Soris gansxvavebis
statistikuri hipoTezis SemowmebiT.
or saSualos Soris gansxvavebis statistikuri hipoTezis
Semowmebis mimoxilva
statistikuri hipoTezis Semowmeba gulisxmobs, rom gamoyenebul meTodSi 1AX
aris SerCeva, romelic aRebulia µ1-is mqone populaciidan, xolo 2AX aris
SerCeva, romelic aRebulia µ2-is mqone populaciidan. Tu davuSvebT, rom ori
populaciis saSualo tolia (anu µ1=µ2), roca damoukidebeli cvladi ar axdens
gavlenas, maSin SegviZlia ganvsazRvroT saSualoTa Soris mosalodneli
gansxvavebis sidide, romelic SecdomiTaa gamowveuli. Semdeg unda SevadaroT
saSualoTa Soris gansxvaveba, romelic miRebuli iyo realur eqsperimentSi, im
gansxvavebas, romelic mosalodnelia mxolod SerCevis Secdomis gamo. Tu
realurad miRebuli gansxvaveba imdenad didia, rom SeuZlebelia is mxolod
SerCevis Secdomis gamo mimeRo, maSin gadavwyvet, rom es gansxvaveba
14 ganpirobebulia ara SerCevis SecdomiT, aramed damoukidebeli cvladis
moqmedebiT.
statistikur SemowmebaSi uaRresad mniSvnelovani sakiTxi gagacaniT. imis
gadasawyvetad, imdenad gansxvavdeba Tu ara erTmaneTisagan ori jgufis
saSualoebi (romelTac damoukidebeli cvladis gansxvavebuli doneebi
miewodebaT), rom es gansxvaveba mivaweroT damoukidebel cvlads, pirvel
rigSi, unda davadginoT saSualoTa Soris SemTxveviTi gansxvavebis sidide, im
gansxvavebisa, romelic mosalodnelia mxolod SerCevis Secdomis gamo. Tuki
saSualoTa Soris realuri sxvaoba Tavsdeba Cveulebrivi SemTxveviTi
gansxvavebis farglebSi, maSin Cven ganvixilavT realur sxvaobas, rogorc
SemTxveviT sxvaobas. damoukidebuli cvladi ar axdens gavlenas damokidebul
cvladze. meore mxriv, Tu saSualoTa Soris realuri sxvaoba sakmarisad didia
imisaTvis, rom misi SemTxveviT miRebis albaToba iSviaTad iyos mosalodneli,
maSin gadavwyvetT, rom SerCevis saSualoebi ar gansxvavdebian erTmaneTisagan
mxolod SerCevis Secdomis gamo; gansxvaveba unda mieweros damoukidebel
cvlads.
gansxvavebuli tipis monacemebisa da kvlevis sqemebisaTvis sxvadasxva
statistikuri kriteriumi gamoiyeneba.UTu ar miviRebT mxedvelobaSi
konkretuli kriteriumis gamoyenebas, statistikur hipoTezaTa Semowmebis
ZiriTadi principebi yvela statistikuri kriteriumis mimarT aris gamosadegi.
SemdgomSi gagacnobT t kriteriums ori damoukidebeli jgufisaTvis. es t
kriteriumi saSualebas gvaZlevs statistikurad SevamowmoT gansxvaveba ori
SerCevis saSualos Soris, roca TiToeuli saSualo miRebulia cdis pirTa
gansxvavebuli jgufidan.
t kriteriumi ori damoukidebeli jgufisaTvis
t kriteriumi ori damoukidebeli jgufisaTvis ganisazRvreba Semdegi
formuliT:
21
2121 )()(AxAx
AAAA
sXXt
−
−−−=
µµelidamoukideb
elidamoukidebt mricxveli gviCvenebs, ramdenad gansxvavdeba ori realuri SerCevis
saSualoebs Soris (anu 21 AA XX − ) sxvaoba ori populaciis saSualoebs Soris
sxvaobisagan (anu )21 AA µµ − , im populaciebisa, romelTa warmomadgenlebic es
15 SerCevebi arian. ufro xSirad varaudoben, rom populaciis saSualoebi tolia.
amrigad, maT Soris gansxvaveba (anu )21 AA µµ − aris nulis toli da
elidamoukidebt mricxveli martivdeba 21 AA XX − -mde. mniSvneli ( 21 AxAxs − ) aris Secdomis
cvalebadobis sazomi, romelsac ewodeba saSualoTa Soris sxvaobis
standartuli Secdoma. saSualoTa Soris sxvaobis standartuli Secdoma aris
saSualoTa Soris sxvaobebis SerCeviTi ganawilebis standartuli gadaxra.
imisaTvis, rom gaviazroT Secdomis cvalebadobis es sazomi, unda SevimuSavoT
or saSualos Soris sxvaobis SerCeviTi ganawilebis koncefcia.
saSualoTa Soris sxvaobis SerCeviTi ganawileba
ganvixiloT eqsperimenti, romelSic damoukidebeli cvladis ori donis (A1 da
A2) manipulacia xdeba. davuSvaT, rom damoukidebel cvlads ara aqvs gavlena
damokidebul cvladze. radgan damoukidebel cvlads ara aqvs gavlena
damokidebul cvladze, SegviZlia SemovitanoT xelovnuri magaliTi ori
populaciidan (populaciebi A1 da A2), romelTac aqvT toli saSualoebi (µ1=µ2).
qulebis SerCevis saSualebiT. warmoidgineT, rom vatarebT am eqsperiments da
TiToeuli populaciidan (Sesabamisad A1 da A2-dan) aRebulia ori SerCeva,
TiToeulis moculoba xuTis tolia (anu nA1=5 da nA2=5). SerCevis es moculoba
aris nebismieri; SerCevis nebismieri moculoba SegveZlo agveRo am magaliTis
ganxilvisaTvis. jer unda gamovTvaloT TiToeuli jgufis saSualo da Semdeg
1AX -s unda gamovakloT 2AX . davuSvaT, es hipoTeturi eqsperimenti 100-jer
gavimeoreT da miviReT 100 21 AA XX − sxvaoba. Tu avagebT 21 AA XX − mniSvnelobebis
sixSireTa ganawilebas, miviRebT saSualoTa Soris sxvaobebis SerCeviT
empiriul ganawilebas. saSualoTa Soris 100 aseTi gansxvavebis SerCeviTi
ganawileba mocemulia suraTze #9.2. es ganawileba aris saSualoTa Soris
sxvaobebis empiriuli SerCeviTi ganawileba, radgan Cven miviReT da
gamoviTvaleT 1AX da 2AX Soris sxvaobebis 100 mniSvneloba. praqtikaSi
mkvlevari ar iTvlis da ar agebs aseTi tipis empiriul SerCeviT ganawilebas.
is iyenebs saSualoTa Soris sxvaobis SerCevis Teoriul ganawilebas, romelic
naCvenebia suraTze #9.3. saSualoTa Soris sxvaobis SerCevis Teoriuli
saSualoTa Soris sxvaobis SerCeviTi ganawileba (Sampling distribution of the difference between means): SerCevis saSualoebs Soris sxvaobis ganawileba, roca populaciidan amoRebulia n moculobis SerCevis yvela SesaZlo wyvili.
16 ganawileba aris SerCevis saSualoTa Soris sxvaobebis ganawileba, roca n
moculobis SerCevebis yvela wyvili populaciidanaa aRebuli. gaixseneT me-7
Tavidan, rom saSualos SerCevis Teoriul ganawilebas aqvs mniSvnelovani
maxasiaTeblebi, rogoricaa misi forma, saSualo da standartuli Secdoma. es
maxasiaTeblebi saSualoTa Soris sxvaobis SerCevis Teoriul ganawilebazec
vrceldeba.
suraTi aklia (Frequency -sixSire, value of _. . .-is mniSvneloba)
suraTi #9.2 ( 21 AA XX − ) saSualoTa Soris sxvaobis SerCevis empiriuli
ganawileba 100 SerCevisaTvis, romelic aRebulia toli saSualoebis mqone
populaciebidan (µ1=µ2)
suraTi aklia ( value of _. . .-is mniSvneloba)
suraTi #9.3 saSualoTa Soris sxvaobis Teoriuli ganawileba, roca µ1=µ2.
saSualoTa Soris sxvaobebis SerCeviTi ganawilebis forma
daakvirdiT, rom SerCevis empiriuli ganawileba suraTze #9.2 SedarebiT
simetriuli da unimodaluria. centraluri zRvris Teoremis Tanaxmad, SerCevis
moculobis zrdasTan erTad saSualoTa Soris sxvaobis SerCeviTi ganawileba
uaxlovdeba normalur ganawilebas, imis miuxedavad, Tu rogoria mis
safuZvlad mdebare populaciis ganawilebis forma. es damaxasiaTebeli
Tavisebureba warmodgenilia Teoriul ganawilebaSi, romelic naCvenebia
suraTze #9.3.
saSualoTa Soris sxvaobebis SerCeviTi ganawilebis saSualo
21 AA XX − sxvaobebis usasrulo simravlis ganawilebis saSualo, X X A1- X A2,
populaciis saSualoebs Soris sxvaobis toli iqneba _ µA1-µA2. amitomac, roca
µA1=µA2 da, aqedan gamomdinare, µA1-µA2=0, realuri 21 AA XX − mniSvnelobebis
saSualo iqneba saboloo jamSi nulis toli da SerCevis saSualoebis realuri
sxvaobebi dajgufdeba populaciis saSualoebs Soris nulovani gansxvavebis
garSemo. es Tviseba kargad Cans ganawilebaSi, romelic naCvenebia suraTze #9.2.
21 AA XX − sxvaobis saSualo aris +0.07. mxolod 100 SerCevis pirobebSic 21 AA XX −
sxvaoba Zalian axlos iqneba populaciis nulovan gansxvavebasTan. Teoriuli
ganawilebisaTvis, romelic naCvenebia suraTze #9.3, 21 AA XX − sxvaoba nulis
tolia.
17 saSualoTa Soris sxvaobebis SerCeviTi ganawilebis standartuli Secdoma
21 AA XX − mniSnvnelobaTa SerCeviTi Teoriuli ganawilebis standartul
gadaxras ewodeba saSualoTa Soris sxvaobis standartuli Secdoma an ufro
martivad, sxvaobis standartuli Secdoma. is aRiniSneba rogorc σ 21 AXAX − da
ganisazRvreba Semdegnairad:
σ 21 AXAX − =2
22
1
12
A
A
A
A
nnσσ
+
aRsaniSnavia, rom standartuli Secdomis gamoTvla xdeba TiToeuli
populaciis saSualos SerCeviTi ganawilebis standartuli Secdomis SejamebiT.
amitom misi mniSvneloba TiToeuli ganawilebis saSualos standartul
SecdomasTan erTad icvleba. praqtikaSi σ2 ucnobia da σ 21 AXAX − mniSvnelobis
gamoTvla xdeba SerCevis dispersiebidan, s2A1 da s2
A2-dan. saSualoTa Soris
sxvaobis standartuli Secdomis Sefasebis misaRebad gamoiTvleba
kombinirebuli an jamuri dispersia, romelic aRiniSneba s2kombinirebuli
simboloTi da miiReba ori SerCevis dispersiidan Semdegi formulis (tolobis)
gamoyenebiT:
s2kombinirebuli =
2nn)1()1(
A2A1
A22
2A12
1
−+−+− snsn AA
es toloba gviCvenebs, rom s2kombinirebuli –is misaRebad Cven vamravlebT TiToeuli
SerCevis dispersias Tavisuflebis xarisxze anu (nA-1)-ze, vumatebT erTmaneTs
miRebul mniSvnelobebs da Semdeg vyofT am jams ori jgufis Tavisuflebis
xarisxebis jamze. gavixsenoT me-7 Tavidan, rom saSualos standartuli
Secdoma, xs , mocemulia s/√N. es formula miiRebs Semdeg saxes: xs =. )/1(( 2 Ns
saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba, gansazRvruli
rogorc
s 21 AXAX − Semdegnairad gamoisaxeba:
saSualoTa Soris sxvaobis standartuli Secdoma an sxvaobis standartuli Secdoma:
21 AA XX − mniSvnelobaTa SerCeviTi Teoriuli ganawilebis standartuli gadaxra.
18
s 21 AXAX −
+=
A2A1
2
n1
n1ulikombinirebs
s2kombinirebuli CavsvaT formulaSi da miviRebT
saSualoTa Soris sxvaobis standartuli Secdomis Sefasebas:
s 21 AXAX − =
+
−+
−+−
A2A1A2A1
22
212
1
n1
n1
2nn)1(()1( AAAA snsn
sxvaobis standartuli Secdomis Sefaseba t statistikis mniSvnelia. amitom t-s
mniSvneloba gviCvenebs, Tu ramdenad daSorebulia standartuli Secdomebis
Sefaseba 1AX -sa da 2AX -s Soris sxvaobis SerCevis ganawilebis saSualosagan _
realurad miRebuli 21 AA XX − sxvaobisagan. magaliTad, davuSvaT, rom realurad
miRebuli 21 AA XX − =3 da s 21 AXAX − =3. trealuris mniSvneloba maSin iqneba 3/3=1-is
toli, rac imas niSnavs, rom realuri 21 AA XX − sxvaoba aris erTi standartuli
SecdomiT daSorebuli 21 AA XX − SerCeviTi ganawilebis saSualosagan. Tu
realuri 21 AA XX − =6 da maSin 21 AA XX − =3 t tolia 6/3 an 2-is, rac imas niSnavs, rom
realuri 21 AA XX − sxvaoba aris ori standartuli SecdomiT daSorebuli 1AX da
2AX Soris sxvaobis SerCeviTi ganawilebis saSualosagan. aRsaniSnavia, rom Tu
damoukidebeli cvladi ar axdens gavlenas da eqsperimentSi moqmedebs mxolod
SerCevis Secdoma, mosalodnelia, rom trealuris mniSvneloba iqneba mcire, maSin
roca damoukidebel cvlads aqvs gavlena da iwvevs 1AX da 2AX Soris arsebul
gansxvavebas, trealuris mniSvneloba ufro didi iqneba.
SeamowmeT Tqveni codna IX-2
1. gansazRvreT:YsaSualoTa Soris sxvaobis SerCevis ganawileba, saSualoTa
Soris sxvaobis standartuli Secdoma, saSualoTa Soris sxvaobis
SerCevis Teoriuli ganawileba.
2. axseniT, ratom ar SeiZleba ubralod SexedoT gansxvavebul
eqsperimentul pirobebSi myofi ori jgufis saSualoTa mniSvnelobebs
saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba (Estimated standard error of the differnce between means): saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba miiReba
s2-is gamoyenebiT σ2-is SefasebisaTvis.
19 Soris gansxvavebas da gadawyvitoT, aris Tu ara es gansxvaveba
ganpirobebuli damoukidebeli cvladis moqmedebiT.
3. gamoTvaleT s 21 AXAX − qulaTa Semdegi erTobliobebisaTvis:
a. b.
d.
statistikuri Semowmeba tdamoukidebelis gamoyenebiT
saSualoTa Soris mniSvnelovani gansxvavebis Sesamowmeblad tdamoukidebelis
gamoyenebisas unda gaviaroT statistikuri hipoTezis Semowmebis qvemoT
mocemuli etapebi, romelic aRwerili iyo me-8 TavSi:
• nulovani hipoTezis, H0-is da alternatiuli hipoTezis H1-is
formulireba.
• H0–is WeSmaritebis daSvebis pirobiT SerCevis ganawilebis miReba. es
ganawileba mocemulia cxilSi #a-2.
jgufi
A1 A2
104 99
111 107
120 105
113 110
108 115
jgufi
A1 A2
77 82
84 86
68 78
71 74
82 69
79
80
jgufi
A1 A2
32 38
37 42
28 29
41 36
35 30
33 37
36
39
34
jgufi
A1 A2
71 85
82 80
63 73
74 76
68 70
76 67
20 • mniSvnelovnobis donis arCeva.
• t kritikuli mniSvnelobis (tkritikulis) dadgena t-s SerCeviTi
ganawilebidan, romelic mocemulia cxrilSi #a-2.
• uaryofis areebi ganlagebulni arian t statistikis SerCeviT Teoriul
ganawilebaze.
• t statistika, romelic ganisazRvreba rogorc trealuri, gamoTvlilia
SerCevis monacemebidan.
• H0–is uaryofis Tu misi ver uaryofis gadawyvetileba keTdeba imis
safuZvelze xvdeba Tu ara trealuri uaryofis areSi.
statistikuri hipoTezebi
parametruli kriteriumisaTvis statistikuri hipoTeza aris debuleba
populaciis parametrebis Sesaxeb. parametrebi, romlebic Seesatyvisebian 1AX da
2AX statistikur kriteriumebs, arian populaciis saSualoebi, µ1 da µ2
Sesabamisad. tdamoukidebelisaTvis statistikuri hipoTezebia:
H0: µ A1=µA2. (nulovani hipoTeza)
H1: µ A1≠µA2 (alternatiuli hipoTeza)
monacemTa ori populaciisaTvis am ori hipoTezidan erT-erTi unda iyos
WeSmariti.
nulovani hipoTeza. nulovani hipoTeza ar gansazRvravs populaciis
parametrebis ricxobriv mniSvnelobebs, romlebic, ra Tqma unda, ucnobia.
is adgens mxolod mas, rom populaciis saSualoebi erTmaneTis tolia.
arsebiTad H0: µ A1=µA2 Seesatyviseba im situacias, roca damoukidebeli cvladi
ar moqmedebs damokidebul cvladze. Tu damoukidebeli cvladi ar axdens
gavlenas damokidebul cvladze, eqsperimentSi monawile yvela cdis piri
yofila erTi da imave populaciis warmomadgeneli da nebismieri realuri
gansxvaveba ori SerCevis saSualoebs Soris ganpirobebulia mxolod SerCevis
SecdomiT. amrigad, H0 ayalibebs situacias, roca SeiZleba iyos miRebuli t
statistikis SerCevis Teoriuli ganawileba. t–s SerCeviTi ganawilebis codniT,
SegviZlia ganvsazRvroT rogoria t-s nebismieri mniSvnelobis albaToba, Tu
gaviTvaliswinebT imas, rom SerCevis saSualoTa Soris gansxvaveba miewereba
mxolod SerCevis Secdomas.
21 alternatiuli hipoTeza. alternatiuli hipoTeza asaxavs iseT situacias, roca
1AX da 2AX gansxvavdeba erTmaneTisagan, radgan TiToeuli jgufis cdis pirebi
warmoadgenen sxvadasxva saSualoebis mqone populaciebs. es situacia xdeba
maSin, roca damoukidebel cvlads aqvs gavlena. Tu damoukidebeli cvlads aqvs
gavlena, da H1 aris WeSmariti, maSin mosalodnelia, rom 1AX da 2AX gacilebiT
ufro metad gansxvavdeba erTmaneTisagan, vidre mosalodneli iqneboda mxolod
SerCevis Secdomis gamo. amitom, Tu H1 aris WeSmariti, trealuris mniSvneloba
meti iqneba trealuris mniSvnelobaze H0-is WeSmaritebis pirobebSi.
eqsperimentis kvlevis hipoTeza dasturdeba H0-is uaryofiTa da H1-is
miRebiT. H0-is uaryofa da H1-is miReba aris gadawyvetileba imis Sesaxeb, rom
SerCevis saSualoebs Soris arsebuli realuri gansxvaveba ganpirobebulia ara
SerCevis SecdomiT, aramed damoukidebeli cvladis gavleniT.
am daskvnebis magaliTad SeiZleba ganvixiloT unrig-neervortisa da
sxvaTa eqsperimenti, romlis kvlevis hipoTezis Tanaxmad, diazepamis miReba
cvlida pirovnebis mier drois aRqmas im jgufTan SedarebiT, romelic iRebda
placebos. es hipoTeza dadasturdeba mxolod im SemTxvevaSi, roca orive
jgufSi Sefasebuli drois saSualoebi asaxavs gansxvavebuli saSualoebis
mqone populaciebs, anu situaciaSi, romelic Camoyalibebulia alternatiul
hipoTezaSi. amrigad, H0-is uaryofa da H1-is miReba kvlevis hipoTezis
mxardaWeraa.
mniSvnelovnobis donis SerCeva
mniSvnelovnobis done aris albaTobis is mniSvneloba, romelic iZleva
nulovani hipoTezis uaryofis pirobas statistikur kriteriumSi. rogorc
viciT, bihevioristi mecnierebi Cveulebriv irCeven alfas 0.05 an 0.01
mniSvnelovnobis dones. isini amis saSualebiT iReben gadawyvetilebas, rom t-s
mniSvnelobebis xdomiloba 100 SemTxvevidan mxolod 5-jer Tu ufro naklebad
an 100 SemTxvevidan mxolod erTxel an ufro naklebad aris statistikurad
iSviaTi SemTxveva H0-is WeSmaritebis pirobebSi. magaliTad, Cven virCevT α=0.05.
uaryofis areebis mdebareobis gansazRvra da t-s kritikuli mniSvnelobebis
povna: ormxrivi kriteriumebi
uaryofis are warmoadgens t-s mniSvnelobebs t-s SerCeviT ganawilebaSi,
romelsac aqvs α-s toli an masze naklebi albaToba H0-is WeSmaritebis
22 pirobebSi. aqedan gamomdinare, uaryofis are gansazRvravs trealuris
mniSnvelobebs, romlebic akmayofileben H0-is uaryofis kriteriums.
rogorc me-8 TavSi iyo ganxiluli, ormxriv an aramimarTulebiT
kriteriumSi, t-s uaryofis areebi ganlagebulia SerCevis Teoriuli
ganawilebis erT-erTi kudis mxareze, rogorc naCvenebi iyo suraTze #9.4. roca
H0 WeSmaritia, t-s mniSvnelobaTa umravlesoba dajgufebulia nulis garSemo,
21 AA XX − mniSvnelobaTa umravlesoba odnav iqneba gansxvavebuli nulisagan.
magram, t-s zogierTi mniSvneloba xvdeba ganawilebis romelime boloSi, maSinac
ki, roca H0 WeSmaritia. t-s es SedarebiT maRali mniSvnelobebi gvxvdeba maSin,
roca t-s mricxveli (anu 21 AA XX − ) mniSvnelze metia (anu sxvaobis standartul
Secdomaze). t-s es maRali mniSvnelobebi iSviaTi xdomilobebia, Tumca mainc
gvxvdebian H0: µ A1=µA2 WeSmaritebis pirobebSic.
Tu H1 WeSmaritia da µ A1≠µA2, maSin 21 AA XX − sxvaobis maRali dadebiTi an
uaryofiTi mniSvnelobebis da amrigad, t-s aseTive mniSvnelobebis xdomiloba
ufro xSirad aris mosalodneli. amitom, t-s uaryofis areebi mdebareoben t-s
SerCeviTi Teoriuli ganawilebis TiToeul kudTan. sruli albaToba, romelic
gansazRvrulia α-s saxiT, gayofilia t ganawilebis orive kuds Soris. Tu
trealuri xvdeba romelime uaryofis areSi, maSin H0 iqneba uaryofili da H1
miRebuli. rogorc erTi SerCevis t kriteriumis SemTxvevaSi, ormxrivi
kritikuli mniSvnelobebi, romelic mocemulia cxrilSi #a-2, gansazRvravs
uaryofis areebis zust mdebareobas. am cxrilis gamosayeneblad Cven unda
vicodeT trealuris Tavisuflebis xarisxi.
tdamoukidebelis Tavisuflebis xarisxi. Tavisuflebis xarisxi asaxavs qulaTa
raodenobas, romelic Tavisufalia cvalebadobis dros statistikis
gamoTvlisas. me-8 TavSi Cven vaCveneT, rom s2 aqvs N-1 df, sadac N aris im
qulaTa raodenoba, romelTa dispersiac gamoiTvleba. tdamoukidebeli
statistikisaTvis, s 21 AXAX − gamoiTvleba TiToeuli jgufis dispersiidan, s2A1
da s2A2-dan. Sesabamisad, tdamoukidebelis df efuZneba s2
A1 da s2A2 -is Tavisuflebis
xarisxebis jams. roca nA1 da nA2 asaxavs A1 da A2 jgufebSi qulaTa
raodenobas , s2A1-s aqvs nA1-1 df da s2
A2 aqvs nA2-1 df. amitom df-is jamuri
mniSvneloba tdamoukidebelisaTvis aris (nA1-1)-s mimatebuli (nA2-1). Tuki N
warmoadgens orive jgufis qulaTa saerToO raodenobas, (e.i N= nA1+ nA2), maSin df
23 tdamoukidebelisaTvis aris N-2. Sesabamisad, roca eqsperimentSi, romelSic 12 qulaa
TiToeul jgufSi (e.i. nA1=12 da nA2=12, amrigad, N=24), df tdamoukidebelisaTvis aris 24-
2=22. cxrilidan #a-2 tkritikuli 22 df-isaTvis da α=0.05 donisaTvis aris 2.074.
tkritikulis es mniSvneloba naCvenebia suraTze #9.4, sadac gansazRvrulia
uaryofis are t ganawilebis TiToeul kudTan.
suraTi #9.4
ormxrivi uaryofis aris ilustracia t ganawilebisaTvis, roca df aris 22 da
α=0.05
uaryofis aris mdebareobis gansazRvra: calmxrivi kriteriumebi
roca kvlevis hipoTeza winaswarmetyvelebs saSualoTa Soris mimarTulebis
mqone kavSirs (mag., rom 1AX aris 2AX -ze meti) da ar arsebobs sapirispiro
Sedegis miRebis interesi (mag., 1AX aris 2AX -ze naklebi), maSin mkvlevrebs
SeuZliaT gamoiyenon calmxrivi kriteriumi an mimarTulebis mqone kriteriumi.
calmxriv kriteriumSi, uaryofis are, romelic moicavs α-s srul mniSvnelobas
ganlagebulia t ganawilebis im kudSi, romelic Seesabameba kvlevis hipoTeziT
nawinaswarmetyvelevi Sedegis mimarTulebas. tkritikulis mniSvnelobebi
calmxrivi kriteriumisaTvis 0.01 da 0.05 mniSvnelovnobis doneebze
warmodgenilia cxrilSi #a-2. roca df aris 22, calmxrivi tkritikuli
mniSvnelovnobis 0.05 doneze aris 1.717. calmxrivi uaryofis are tkritikulTan =
+1.717 naCvenebia t-s ganawilebis marjvena kudTan suraTze #9.5., am uaryofis
arisaTvis gamoyenebuli nulovani da alternatiuli hipoTezebi Semdegia:
H0: µ A1≤µA2
H1: µ A1>µA2.
-3 -2 -1 0 +1 +2 +3 t-s mniSvneloba
H0 uaryofa uaryofis zona 0.025
H0 uaryofa uaryofis zona 0.025
H0 miRebis zona
t=-2.074 t=+2.074 t-is kritikuli t-is kritikuli mniSvneloba mniSvneloba
24 suraTi aklia (Reject _ H0 uaryofa, Do not reject _ H0 miRebis zona, Rejection region - uaryofis are, value of t_ t-s mniSvneloba, critical value of t _ t-s kritikuli mniSvneloba) suraTi #9.5 calmxrivi uaryofis aris ilustracia t ganawilebisaTvis, roca df
22-is tolia da α=0.05.
Tumca arsebobs winaaRmdegobebi calmxrivi kriteriumebis gamoyenebasTan
dakavSirebiT. cxrilSi #a-2 warmodgenili t-s SerCeviTi ganawileba Seqmnilia
im varaudis safuZvelze, rom H0: µ A1=µA2 WeSmaritia. SerCevis es ganawileba
nulis mimarT simetriulia da nulidan dadebiTi da uaryofiTi gansxvavebani
Tanabrad moxdeba. calmxrivi kriteriumisaTvis statistikuri hipoTeza aris an
H0: µ A1≤µA2 da H1: µ A1>µA2.
an
H0: µ A1≥µA2 da H1: µ A1<µA2
magram Cven ar SegviZlia SevqmnaT t-s SerCeviTi ganawileba arcerTi nulovani
hipoTezid, radgan arsebobs usasrulo raodenobis saSualebebi, rom H0: µ A1≤µA2
da H0: µ A1≥µA2 gaxdes WeSmariti. amrigad, SerCeviTi ganawileba H0: µ A1=µA2-
saTvis gamoiyeneba t-s kritikul mniSvnelobaTa gansazRvrisaTvis, rogorc
calmxrivi, aseve ormxrivi kriteriumebisaTvis. am da sxva mizezebis gamo
zogierTi bihevioristi mecnieri calmxrivi t kriteriumis gamoyenebis
winaaRmdegia (Gito, 1977). amitom, am Tavis yvela magaliTSi ormxrivi
kriteriumebi gamoiyeneba.
tdamoukidebelis gamoTvla
tdamoukidebelis ganmartebiTi formulaa:
21
2121 )()(
AXAX
AAAA
ssXXt
−−−−
=µµ
elidamoukideb
nulovani hipoTeza tdamoukidebelisaTvis H0: µ A1=µA2, aris H0: µ A1-µA2=0 ekvivalenturi.
amrigad, µ A1-µA2 sxvaoba aris savaraudod nulis toli da µ A1-µA2 amovardeba
formulidan,
21
21 )(
AXAX
AA
ssXXt
−−
=elidamoukideb
sadac s 21 AXAX − aris mocemuli Semdegi saxiT
25
s 21 AXAX − =
+
−+
−+−
A2A1A2A1
22
212
1
n1
n1
2nn)1(()1( AAAA snsn
amitom, gamoTvlebisaTvis gamoiyeneba Semdegi formula
+
−+
−+−
−=
A2A1A2A1
22
212
1
21
n1
n1
2nn)1(()1( AAAA
AA
snsn
XXt elidamoukideb
am formulis gamoyenebisas Cven gvWirdeba vicodeT mxolod ( AX ) cdis pirTa
raodenoba (nA) da populaciis dispersia (s2A) TiToeuli jgufisaTvis. am
formulis gamoyeneba naCvenebia magaliTSi #9-1.
gadawyvetilebebi statistikuri hipoTezebis Sesaxeb
statistikurad mniSvnelovani gansxvaveba. Tu trealuri xvdeba uaryofis areSi,
maSin
H0: µ A1=µA uaryofilia. H0-is uaryofa niSnavs H1:µ A1≠µA-s miRebas. SerCevis
saSualoebs Soris gansxvaveba statistikurad mniSvnelovania. Cven vaskvniT, rom
ori SerCevis saSualo, romlebsac moicavs trealuri, ar warmoadgens erTi da
imave populaciis saSualos Sefasebebs da maT Soris gansxvaveba
ganpirobebulia raRac sxva mizeziT da ara mxolod SerCevis SecdomiT. Tu
eqsperimenti yuradRebiT iqneba dagegmili da Catarebuli, maSin saSualoTa
Soris gansxvaveba ganpirobebulia damoukidebeli cvladis manipulaciiT. es
gadawyvetilebani da daskvnebi Sejamebulia cxrilis #9.1 marcxena svetSi.
cxrili #9.1
ori damoukidebeli SerCevis t kriteriumis gamoyenebiT statistikuri
hipoTezebis Semowmebis Sesaxeb gadawyvetilebebisa da daskvnebis Sejameba.
gamoyenebuli iyo 0.05 mniSvnelovnobis done.
26
aramniSvnelovani gansxvaveba. Tu trealuri ar xvdeba uaryofis areSi, maSin H0-s
ver uarvyofT. Tu H0-s ver uarvyofT, es gulisxmobs, rom viRebT H1-s.
saSualoTa Soris gansxvaveba ar aris statistikurad mniSvnelovani anu,
aramniSvnelovania. ori SerCevis saSualo ganxiluli iqneba, rogorc aRebuli
im populaciebidan, romelTac aqvT identuri saSualoebi da isini
erTmaneTisagan gansxvavdebian mxolod SerCevis Secdomis gamo. eqsperimentis
Sedegad vaskvniT, rom damoukidebel cvlads ara aqvs gavlena da SerCevis
saSualoebs Soris gansxvaveba SemTxveviTi xasiaTisaa. es gadawyvetilebani da
daskvnebi Sejamebulia cxrilis #9.1 marjvena svetSi.
daSvebebi tdamoukidebelisaTvis
erTi SerCevis t kriteriumi Semdeg moTxovnebs ayenebs SerCevis qulebisaTvis:
1. SerCevis qulebi SemTxveviTadaa miRebuli populaciis qulebidan da
TiToeuli qula damoukidebelia nebismieri sxva qulisagan;
2. qulebi populaciaSi normalurad arian ganawilebulni;
Tu trealuri xvdeba
uaryofis areSi α=0.05-saTvis, maSin:
Tu trealuri ar xvdeba
uaryofis areSi α=0.05-saTvis, maSin:
trealuris albaToba naklebia an toli
0.05, an p<0.05
H0 uaryofilia.
H1 miRebulia. TiToeuli SerCevis
saSualo aRebulia gansxvavebuli
populaciidan.
Ggansxvaveba or saSualos Soris aris
statistikurad mniSvnelovani 0.05
doneze.
miiReba gadawyvetileba imis Sesaxeb, rom
saSualoebs Soris gansxvavebis mizezi
aris ara SerCevis Secdoma, aramed sxva
ram. Kakuratulad Catarebul
eqsperimentSi gansxvaveba miewereba
damoukidebel cvlads.
trealuris albaToba metia an toli
0.05, an p>0.05
H0 ar aris uaryofili
H1 ar aris miRebuli
gansxvaveba or saSualos Soris
aris statistikurad
aramniSvnelovani 0.05 doneze.
miiReba gadawyvetileba imis Sesaxeb,
rom SerCevis Secdoma yvelaze
damajerebelad xsnis gansxvavebas
saSualoebs Soris. Aar arsebobs
damamtkicebeli sabuTi, rom
damoukidebeli cvladi axdens
gavlenas.
27 3. qulaTa dispersiebi populaciebSi aris toli: σ2A1=σ2A2.
me-8 TavSi Cven aRvniSneT, rom t kriteriumis daSvebebi xSirad srulyofilad
ar akmayofileben realur kvlevas. isini xSirad arRveven kanonzomierebas.
miuxedavad amisa, mkvlevrebi mainc agrZeleben t kriteriumis gamoyenebas. ra
aris aseTi darRvevis Sedegebi?
Ppirveli daSvebis moTxovnis Tanaxmad, TiToeuli qula jgufSi
damoukidebelia amave jgufis yoveli sxva qulisagan, anu jgufSi erTi
pirovnebis qulis codna Tqven ar gaZlevT informacias amave jgufis nebismieri
sxva pirovnebis qulis Sesaxeb. es daSveba moiTxovs imas, rom TiToeul
individs mieces damoukidebeli cvladis mxolod erTi done da man
eqsperimentSi mxolod erTi qula mogvces. QqulaTa damoukideblobis es
moTxovna ar SeiZleba iyos darRveuli tdamoukidebelisaTvis. Tu pirovnebas miecema
damoukidebeli cvladis orive done da is mogvcems or qulas, maSin unda iyos
gamoyenebuli t kriteriumi damokidebuli qulebisaTvis, romelic am TavSi
ufro mogvianebiT ganixileba.
Ppirveli daSveba, agreTve, gviCvenebs, rom cdis pirebi unda iyvnen SerCeulni
SemTxveviTad, magram realurad cdis pirebi eqsperimentSi, rogorc wesi,
miRebuli arian xelmisawvdomi SerCevidan da ara SemTxveviTi SerCevidan.
xelmisawvdomi SerCevis gamoyenebis ZiriTadi nakli isaa, rom Cven garkveul
farglebSi SezRuduli varT miRebuli Sedegebis populaciaze
ganzogadebisaTvis.
G
Mmeore (qulaTa normaluri ganawileba) da mesame (dispersiebis toloba,
xSirad iwodeba, rogorc dispersiaTa erTgvarovneba) daSvebebis darRvevebma
SeiZleba Secvalos I gvaris Secdomis daSvebis realuri albaToba alfas
mniSvnelobiT. Mmagram t kriteriums axasiaTebs mdgradoba am daSvebebis
darRvevis sawinaaRdegod. Gmdgradoba niSnavs, rom daSvebebis darRvevas mcire
gavlena aqvs I gvaris Secdomis daSvebis albaTobaze. Bbredlis (Bradley, 1980,
1984) mier Catarebuli kvleva ewinaaRmdegeba mdgradobis aseT aRniSvnas.
Bbredli amtkicebs, rom ar arsebobs pirobebis arc erTi simravle, romlis
drosac es daSvebebi SeiZleba dairRves da am dros t kriteriumis mdgradoba
iyos garantirebuli. magram roca qulebi ar arian normalurad ganawilebulni
da darRveulia dispersiebis tolobis daSvebebi, amas savaraudod eqneba
mdgradoba (Robustness): es termini gamoiyeneba imis aRsaniSnavad, rom statistikuri
kriteriumis daSvebebis darRvevas aqvs mcire gavlena pirveli gvaris Secdomaze.
28 minimaluri gavlena pirveli gvaris Secdomis daSvebis albaTobaze, Tu
daculia Semdegi pirobebi:
• TiToeul jgufSi cdis pirebis raodenoba iqneba toli;
• QqulaTa or ganawilebas daaxloebiT erTi da igive forma aqvs da
ganawilebebs ara aqvs arc Zalian wawvetebuli da arc Zalian gluvi
forma;
• MmniSvnelovnobis done iqneba 0.05 da ara 0.01.
MmagaliTi #9-1
diazepamis gavlenis dadgena drois Sefasebaze tdamoukidebelis gamoyenebiT
amocana
tdamoukidebelis gamoTvlis ilustrirebisaTvis, roca X da s cnobilia,
gamoviyenebT amocanas, romlis winaSec dadgnen unrug-neervorti da sxvebi,
romelic exeboda diazepamis gavlenas drois Sefasebas aRqmaze. U unrig-
neervortsa da sxvebs hyavdaT ori jgufi _ TiToeulSi 12 cdis piri.
Pplacebos sakontrolo jgufis (A1) mier Sefasebuli saSualo dro iyo 19,8 wm,
roca sA1=7.6 wm, eqsperimentuli jgufis mier Sefasebuli saSualo dro iyo 13.6
wm, roca sA2= 4.8 wm. gansxvavdeba Tu ara Sefasebuli saSualo droebi
erTmaneTisagan mniSvnelovnobis 0.05 doneze? gamoiyeneT ormxrivi kriteriumi.
saWiro statistika
+
−+
−+−
−=
A2A1A2A1
22
212
1
21
n1
n1
2nn)1(()1( AAAA
AA
snsn
XXt elidamoukideb
daSvebebi:
1. cdis pirebi SemTxveviTad arian SerCeuli da TiToeuli cdis piri
damoukidebelia nebismieri sxva cdis pirisagan;
2. drois Sefasebis qulebi normalurad arian populaciaSi ganawilebulni;
3. drois Sefasebis qulaTa dispersiebi tolia orive jgufis populaciebSi:
29 statistikuri hipoTezebi: H0: µ A1=µA2
H0: µ A1≠µA2
mniSnvelovnebis done: α=0.05
df t-saTvis: (nA1-1)+( nA2-1)=(12-1)+(12-1)=22, an N-2=24-2=22
t-s kritikuli mniSvneloba: tkritikuli=2.074 cxrilidan #a-2.
uaryofis areebi: trealuris mniSvneloba tolia an naklebia –2.074, an tolia, an
metia +2.074.
gamoTvla:
387.2597.2
2.6122
228.888
2.6
121
121
21212)8.4)(112()6.7)(112(
6.138.1922
+=+
=
+=
+
−+
−+−
−=t
gadawyvetileba:
trealuri xvdeba uaryofis areSi, romelic tolia an metia tkritikuli= +2.074-is
MCven unda uarvyoT H0 da miviRoT H1.
Ddaskvna: ori SerCevis saSualoebi 19.8wm da 13.6wm mniSvnelovnad gansxvavdebian
0,05 doneze. radgan jgufebi ekvivalenturni iyvnen eqsperimentul Carevamde,
saSualoTa Soris gansxvaveba miewereba damoukidebeli cvladis gavlenas. cdis
pirebis mier Sefasebuli saSualo dro diazepamis miRebis pirobebSi ufro
moklea, vidre is dro, romelic cdis pirebma Seafases placebo kontrolis
pirobebSi.
A
magaliTi #9-2
tdamoukidebelis gamoyeneba nedli qulebisaTvis
amocana
gasulma wlebma aCvenes ganaTlebis gaumjobesebis moTxovnileba momavali
profesiisaTvis saWiro unarebis SesaZenad. Aam moTxovnilebis erT-erTi
gamoxmaureba iyo kompiuterebis gamoyeneba laboratoriuli eqsperimentebis
modelirebisaTvis zust mecnierebebSi. Gebanma, askarma da ozkanma (Geban, Askar,
30 Ozkan, 1992) gamoikvlies, ganapirobebs Tu ara aseTi kompiuterulad
modelirebuli eqsperimentebi ufro did miRwevebs qimiaSi, vidre sayovelTaod
miRebuli midgoma, romelSic studentebi asruleben realur laboratoriul
eqsperimentebs. DdavuSvaT, rom Tqven dainteresdiT am kvlevis gameorebiT 28
umaRlesi saswavleblis student-qimikosze. Tqven SemTxeveviTi wesiT
gadaanawileT es studentebi ori toli moculobis jgufSi, Cveulebrivi
swavlebis jgufSi (A1) da kompiuterulad modelirebuli swavlebis jgufSi
(A2). Aam kursis bolos Tqven qimiis codna miRwevis testiT gazomeT, romlis
qulebi meryeobs 0-dan 100-mde. MmiRebuli iyo Semdegi qulebi:
swavlebis meTodi
sayovelTaod
miRebuli (A1)
Kkompiuterulad
(A2)
74 84
76 75
81 94
90 88
67 69
72 90
84 86
87 97
92 81
70 91
78 83
82 92
75 82
85 93
mniSvnelovnad Ggansxvavdeba Tu ara miRwevis testis es qulebi erTmaneTisagan
0.05 doneze? GgamoiyeneT ormxrivi kriteriumi.
saWiro statistika: tdamoukidebeli
daSvebebi:
1. cdis pirebi SemTxveviTad arian SerCeuli da TiToeuli cdis piri
damoukidebelia nebismieri sxva cdis pirisagan;
31 2. qimiaSi miRwevis testis qulebi normalurad arian populaciaSi
ganawilebulni;
3. qimiaSi miRwevis qulaTa dispersiebi tolia orive pirobaSi myofi jgufis
populaciaSi.
statistikuri hipoTezebi: H0: µ A1=µA2
H0: µ A1≠µA2
mniSvnelovnobis done: α=0.05
df t-saTvis: (nA1-1)+( nA2-1)=(14-1)+(14-1)=26, an N-2=28-2=26
t-s kritikuli mniSvneloba: tkritikuli = 2.056 cxrilidan #a-2.
uaryofis areebi: trealuris mniSvneloba tolia an naklebia –2.056 an tolia an
metia +2.056.
gamoTvla: Tavdapirvelad unda vipovoT TiToeuli jgufis saSualo da
standartuli gadaxra. sayovelTaod miRebuli swavlebis meTodis jgufisaTvis,
1AX =79.5 da sA1=7.59 da kompiuteruliad modelirebuli swavlis jgufisaTvis
2AX =86.07 da sA2=7.74. CavsvaT es mniSvnelobebi trealuris gamoTvliT formulaSi
da miviRebT:
266.2899.2
57.6
142
26708.1527
57.6
141
141
21414)74.7)(114()59.7)(114(
07.868.7922
−=−
=
−=
+
−+
−+−
−=t
gadawyvetileba:
trealuri xvdeba uaryofis areSi, romelic tolia an naklebia tkritikuli= - 2.056-
is.M uarvyofT H0 -s da viRebT H1-s .
daskvna: ori SerCevis saSualoebi 79.5 da 86.07 mniSvnelovnad gansxvavdebian 0,05
doneze. saSualoTa Soris gansxvaveba miewereba damoukidebeli cvladis
32 gavlenas. kompiuterulad modelirebuli eqsperimentebi qimiaSi ganapirobeben
ufro maRal miRwevebs, vidre swavlebis Cveulebrivi meTodebi.
SeamowmeT Tqveni codna IX-3
1. gansazRvreT: mimarTulebis mqone kriteriumi, aramimarTulebiTi kriteriumi,
calmxrivi uaryofis are, uaryofis are, tkritikuli, trealuri, tdamoukidebeli,
ormxrivi uaryofis are.
2. gansazRvreT statistikuri Semowmebis etapebi tdamoukidebelisaTvis.
3. dawereT statistikuri hipoTezebi ormxrivi tdamoukidebelisaTvis .
4. eqsperimentis romel situacias Sesabameba H0?
5. eqsperimentis romel situacias Sesabameba H1?
6. davuSvaT Tqven gyavT ori jgufi, romelTa moculobebia nA1 da nA2. SeasruleT
Semdegi savarjiSo SerCevis TiToeuli jgufisaTvis, romelic mocemulia
cxrilSi: (a) ipoveT df t kriteriumisaTvis ori SerCevis safuZvelze, (b) ipoveT
tkritikuli ormxrivi kriteriumisaTvis α=0.05 doneze (Tu cxrilSi ar aris
mocemuli df-is zusti mniSvneloba, maSin isargebleT cxrilSi mocemuli
Semdegi dabali mniSvnelobiT), (g) aRniSneT, xvdeba Tu ara trealuri uaryofis
areSi da (d) aRniSneT Tqveni gadawyvetilebani statistikuri hipoTezebis
Sesaxeb.
SerCevebi nA1 nA2 trealuri,
1 10 10 2.341
2 8 12 1.763
3 15 15 2.12
4 16 14 3.479
5 35 35 2.912
6 19 16 2.007
7 65 64 2.002
8 75 75 2.193
7. gansazRvreT daSvebebi, romelic safuZvlad udevs tdamoukidebelis
gamoyenebas.
8. a. t kriteriums axasiaTebs mdgradoba. ras niSnavs es termini.
b. ra pirobebSi aris t kriteriumi savaraudod mdgradi?
33 9 a. 1 da 2 amocanebisaTvis upasuxeT Semdeg kiTxvebs. gamoiyeneT ormxrivi
kriteriumi da mniSvnelovnobis 0.05 done TiToeuli problemisaTvis.
b. risi tolia trealuris mniSvneloba?
g. risi tolia df t kriteriumisaTvis?
d. risi tolia t kriteriumis mniSvneloba?
e. risi tolia uaryofis areebi trealurisaTvis?
v. xvdeba Tu ara trealuri uaryofis areSi?
z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb?
uaryofT Tu ver uaryofT H0 hipoTezas? miiRebT Tu ara H1 hipoTezas?
T. gamoitanT daskvnas imis Sesaxeb, rom SerCevis saSualoebi erTi da
igivea Tu gansxvavebuli populaciebidanaa?
i. saSualoebs Soris gansxvaveba statistikurad mniSvnelovania?
k. gansxvavdebian jgufebi mniSvnelovnad im drois SefasebaSi, romelsac
isini lodinSi atareben? Tu asea, rogoria am gansxvavebis mimarTuleba?
Aamocana #1. AafasebT Tu ara drois Sualeds rogorc xangrZlivs, roca xarT
raime movlenis molodinSi? DdavuSvaT, am kiTxvaze pasuxis gasacemad Tqven
SemTxveviTi wesiT gaanawileT 36 cdis piri sakontrolo da eqsperimentul
jgufebSi, roca nA1= nA2. orive jgufis cdis pirebs Tqven eubnebiT, rom
CautarebT eqsperiments, romelic gagrZeldeba Zalian cota xani da maT unda
Seavson SfoTvis gasazomi kiTxvari. sakontrolo jgufis cdis pirebs eubnebiT,
rom asistenti mova ramdenime wuTSi, xolo eqsperimentuli jgufis cdis pirebs
atyobinebT, rom asistenti mcire xniT daigvianebs. Oorive jgufis SemTxvevaSi
asistenti brundeba 5 wuTSi. roca asistenti brundeba, cdis pirebs sTxoven
Seafason, ramdeni xani mouwiaT lodini. DdavuSvaT, Tqven miiReT Sefasebuli
droebis Semdegi saSualoebi da standartuli gadaxrebi (wuTebSi) orive
jgufisaTvis:
jgufi
sakontolo eqsperimentuli
X 6.9 8.9
s 1.57 1.71
amocana #2. davuSvaT, rom Tqven dainterebuli xarT igive sakiTxiT, romelic
iyo ganxiluli amocanaSi #1: afasebT Tu ara drois Sualeds, rogorc ufro
34 xangrZlivs, roca raime movlenis molodinSi xarT? Dkvlavac Tqven SemTxveviT
gadaanawileT cdis pirebi or jgufSi. TiToeul jgufSi toli raodenobis
nacvlad Tqven gadaanawileT 13 cdis piri sakontrolo jgufSi da 17
eqsperimentulSi. sxva mxriv, eqsperimenti zemoT aRweril amocanaSi moyvanili
eqsperimentis identuri iyo. davuSvaT, rom Tqven miiReT Semdegi saSualo
Sefasebebi da standartuli gadaxrebi (wuTebSi) Tqveni jgufebisaTvis:
jgufi
sakontolo eqsperimentuli
X 6.6 7.8
s 1.8 1.92
simZlavre da tdamoukidebeli
statistikuri kriteriumis simZlavre gansazRvruli iyo me-8 TavSi rogorc H0-
is uaryofis albaToba roca H0 ar aris samarTliani.
vinaidan H0-is uaryofa niSnavs imas, rom eqsperimentatorma miiRo H1Hda
gadawyvita, rom damoukidebel cvlads gavlena aqvs mis eqsperimentze,
statistikuri kriteriumis simZlavre aris imis albaToba, rom damoukidebeli
cvladis moqmedeba iqneba aRmoCenili eqsperimentSi, roca aseTi moqmedeba
realurad arsebobs. eqsperimentators yovelTvis unda, rom gazardos
statistikuri kriteriumis simZlavre, raTa SeZlos damoukidebeli cvladis
gavlenis SemCneva, roca es gavlena realurad arsebobs.
nulovani hipoTeza aris uaryofili, roca trealuri xvdeba uaryofis areSi
da rac ufro metia trealuri, miT ufro didia albaToba, rom is moxvdeba
uaryofis areSi. Aarsebobs ramdenime saSualeba, raTa gavzardoT trealuris
mniSvneloba da amrigad tdamoukidebelis simZlavre. Cven SegviZlia gaviazroT es
saSualebebi tdamoukidebelis formulis GganxilviT:
21
21
AXAX
AA
sXXt
−
−=elidamoukideb
Aam formulidan Cans, rom arsebobs trealuris gazrdis ori gza: (1) gavzardoT
gansxvaveba 1AX da 2AX Soris da (2) SevamciroT s 21 AXAX − -is mniSvneloba.
35 pirveli midgoma aris damoukidebeli cvladis gavlenis gazrda, anu Cven
gvWirdeba damoukidebeli cvladis imgvari manipulacia, rom 1AX da 2AX Soris
rac SeiZleba didi gansxvaveba miviRoT. am mizeziT, bihevioristi mecnierebi
xSirad iyeneben damoukidebeli cvladis Zalian gansxvavebul doneebs.
s 21 AXAX − mniSvnelobis Sesamcireblad Cven SegviZlia gavakeToT ori ram:
gavzardoT SerCevis moculoba da SevamciroT SerCevis qulebis cvalebadoba.
s 21 AXAX − da SerCevis moculobis kavSiris sailustraciod, davuSvaT, rom orive
s2A1 da s2
A2=36. gaviTvaliswinoT, agreTve, rom s2 es mniSvnelobebi sami
SerCevis moculobidan erT-erTs exeba: nA1=nA2=3, nA1=nA2=6, nA1=nA2=9. amis Semdeg
CavsvaT nA mniSvnelobebi s 21 AXAX − formulaSi:
roca nA1= nA2=3, maSin s 21 AXAX − = 4.9
nA1= nA2=6, maSin s 21 AXAX − = 3.5
nA1= nA2=9, maSin s 21 AXAX − = 2.8
aqedan gamomdinare, vxedavT, rom SerCevis moculobis zrdasTan erTad,
sxvaobis standartuli Secdoma mcirdeba. amrigad, 1AX da 2AX Soris
mocemuli sxvaobisaTvis trealuris mniSvneloba izrdeba SerCevis moculobis
zrdasTan erTad da s 21 AXAX − mcirdeba. xolo, roca trealuri izrdeba, maSin
izrdeba imis albaTobac, rom is moxvdeba uaryofis areSi da HH0 iqneba
uaryofili da H1 _ miRebuli.
zogadad, Carevis nebismieri gavlenis Sesafaseblad ara aqvs mniSvneloba
imas, ramdenad mcirea es gavlena, Tuki SerCevis moculoba sakmaod didia.
amitom, mkvlevars SeuZlia minimumamde daiyvanos meore gvaris Secdoma SerCevis
moculobis zrdiT. magram, Tu gaviTvaliswinebT realobas, SerCevis moculobis
zrda xSirad Zviria da moiTxovs didi drois xarjvas. ufro metic, Tuki
saWiroa TiToeul jgufSi 1000 kaciani SerCeva, raTa davadginoT
damoukidebeli cvladis gavlena, maSin Cven SegviZlia vivaraudoT, rom am
damoukidebel cvlads minimaluri gavlena eqneba qcevaze.
meore nabiji, s 21 AXAX − Sesamcireblad aris TiToeuli jgufis qulebs
Soris cvalebadobis Semcireba. Tu qulebi TiToeul jgufSi naklebad
Seicvleba, maSin s2 Semcirdeba TiToeuli jgufisaTvis da amis Sedegad s 21 AXAX −
, agreTve, Semcirdeba. s 21 AXAX − -is es Semcireba gamoiwvevs trealuris gazrdas da
amrigad imis albaTobis gazrdas, rom trealuri moxvdeba uaryofis areSi. aseTi
kavSirebis gamo eqsperimentatorebi Cveulebriv kargad akontroleben kvlevis
36 Catarebis pirobebs, raTa minimumamde daiyvanon qulaTa cvalebadoba TiToeul
jgufSi.
eqsperimentuli pirobis efeqtis siZlieris gazomva
statistikurad mniSvnelovani gansxvaveba gviCvenebs, rom damoukidebel cvlads
gavlena aqvs eqsperimentis damokidebul cvladze, magram is ar gviCvenebs am
zegavlenis siZlieres an sidides. arsebobs statistikebi, romlebic SeiZleba
gamoviyenoT damoukidebeli cvladis zegavlenis siZlieris sazomad mas Semdeg,
rac eqsperimenti dasrulda. am statistikebs xSirad moixsenieben rogorc
efeqtis siZlieris sazomebs an kavSiris siZlieris sazomebs. statistika,
romelsac SemdegSi ganvixilavT, iwodeba eta kvadratad da is warmodgenilia
η2 saxiT, sadac η aris berZnuli aso eta. eta kvadrati gamoiTvleba trealuris
mniSvnelobidan da Tavisuflebis xarisxidan t-saTvis Semdegi Fformulis
saSualebiT:
dftt
+=
realuri2
realuri2
2η
Tu ganvixilavT magaliTs #9-2, sadac sayovelTaod miRebuli da
kompiuterulad modelirebuli swavlis pirobebis gansxvaveba iyo –2.266, roc
df=29. eta kvadrati trealurisaTvis tolia
16.01348.31
1348.526)266.2(
)266.2(2
2
2
=
==
=+−
−=
=+
=dft
trealuri
2
realuri2
η
eta kvadrati aris imis sazomi, Tu ramdenad amcirebs subieqtis mier
miRebuli qulebis winaswarmetyvelebis Secdomas damoukidebeli cvladis im
donis codna, romelic cdis pirs miewoda. amis sailustraciod, davuSvaT, rom
magaliTSi #9-2 gTxoves iwinaswarmetyveloT cdis piris qula miRwevis testSi,
magram Tqven ar icodiT swavlebis romeli meTodi miewoda mas. am magaliTSi
Tqveni saukeTeso winaswarmetyveleba iqneboda sruli saSualos gamoyeneba 28
cdis pirisaTvis, X G=82.79, rogorc nawinaswarmetyvelebi qulisa. ra Tqma unda,
efeqtis siZliere (Strength of effect): damoukidebeli cvladis siZliere gazomili statistikis gavlenis siZlieriT.
37 es winaswarmetyveleba Seicavs Secdomas da saSualo Secdoma iqneba 28 qulis
standartuli gadaxra sruli saSualos garSemo. aq Cven gamoviyenebT ufro S,
vidre s, radgan η2 aris statistika, romelic aRwers mxolod SerCevas, es ar
aris populaciis mniSvnelobis Sefaseba. S mniSvneloba 28 qulisaTvis sruli
saSualos garSemo aris 8.09. am mniSvnelobis kvadrati [e.i. (8.09)2], tolia 65.448-
is da 28-quliani SerCevis srul dispersiaa.
axla davuSvaT, rom gTxoven iwinaswarmetyveloT cdis piris qula, magram
Tqven amjerad iciT swavlebis is meTodi, romelic cdis pirma gaiara. am
SemTxvevaSi saukeTeso winaswarmetyveleba iqneboda im cdis pirebis qulebis
saSualo, romlebmac es swavlebis meTodi gaiares an 79.5, an 86.07. Tqveni
winaswarmetyveleba kvlav Seicavs Secdomas, magram axla saSualo Secdoma
iqneba im jgufis standartuli gadaxra, romelSic es cdis piri imyofeba. Cven
kvlav viyenebT S; amrigad, SA1=7.32 da SA2=7.46. TiToeuli S-is kvadratSi ayvaniT
miviRebT TiToeuli jgufis SerCevis dispersiebs: S2A1 =53.582 da S2
A2=55.652 am
ori dispersiis saSualo aris 54.617 [mag. (53.582+55.652)/2=109.234/2=54.6170]. es
mniSvneloba aris Secdomis saSualo dispersia, roca swavlebis erTi meTodis
saSualo gamoyenebuli iyo cdis piris qulis winaswarmetyvelebisaTvis.
gansxvaveba im dispersiasa, romelic napovni iyo sruli saSualos gamoyenebiT
(65.448) da im saSualo dispersias Soris, romelic gamoTvlili iyo
eqsperimentuli pirobis saSualoebidan (e.i.54.617) aris 65.448-54.617=10.831.
dispersiis es kleba sruli dispersiidan, roca qulis
winaswarmetyvelebisaTvis gamoiyeneba eqsperimentuli pirobebis saSualoebi
aris 10.831. Tuki am sxvaobas gavyofT srul dispersiaze miviRebT
10.831/65.448=0.165, an Tuki davamrgvalebT aseulebamde, 0.16-s. sxva sityvebiT rom
vTqvaT, roca vici damoukidebeli cvladis done, romelic cdis pirs miewoda
da cdis piris qulis winaswarmetyvelebisaTvis viyeneb sruli saSualos
nacvlad eqsperimentuli jgufis saSualos, maSin winaswarmetyvelebis
dispersia Semcirdeba 0.16 an 16%-iT. es mniSvneloba aris η2. amrigad, η2
gviCvenebs sruli dispersiis Semcirebis proporcias damokidebul cvladSi,
roca vici damoukidebeli cvladis done, romelic miawodes cdis pirs. am
mizeziT, η2 xSirad aris ganmartebuli, rogorc damokidebeli cvladis
dispersiis proporciis maCvenebeli, romelic SeiZleba axsnili iyos
damoukidebeli cvladis mier.
38 η2 mniSvneloba icvleba 0-dan 1-mde. nulis toli η2 gviCvenebs, rom
damoukidebeli cvladis codna ar amcirebs damokidebuli cvladis Secdomis
dispersias. sruli saSualos winaswarmetyveleba iseTive kargi
winaswarmetyvelebaa cdis piris qulisaTvis, rogorc jgufis saSualos
winaswarmetyveleba. ar arsebobs raime damokidebuleba damoukidebeli cvladis
donesa da damoukidebuli cvladis qulas Soris. meore mxriv, roca η2 1-is
tolia, es gviCvenebs, rom damoukidebeli cvladis donis codna gvaZlevs cdis
piris qulis sruli winaswarmetyvelebis saSualebas. jgufis saSualos
saSualebiT cdis piris qulis winaswarmetyvelebisas Secdoma ar iqneba
daSvebuli.
roca praqtikaSi η2 0.1-dan 0.15–mdea, iTvleba, rom eqsperimentuli piroba
Zlier gavlenas axdens. aseT interpretacias mxars uWers bihevioristi
mecnierebis gamoqveynebuli kvlevebic. magaliTad, haasma, uaxterma da
solomonma (Haas, Waechter and Solomon, 1982) gamoiTvales η2 statistikuri
mniSvnelovnobis 11044 SemowmebaSi, romelic moxseniebuli iyo fsiqologiuri
konsultaciis Jurnalis 1970 da 1979 wlebis gamocemebSi. maT aRmoaCines, rom
η2-is mniSvnelobis mediana iyo 0.083. interkvartiluri diapazoni η2
mniSvnelobisaTvis iyo 0.043-dan 0.268-mde. es Sedegi gvaZlevs informacias
bihevioristul mecnierebebSi kavSiris tipur siZliereze da, agreTve, η2
mniSvnelobis Sefasebis safuZvels. amerikis fsiqologiuri asociaciis
publikaciebis saxelmZRvanelos (1994) gvirCevs, rom damoukidebeli cvladis
gavlenis siZlieris sazomebi, iseTi rogoricaa eta kvadrati, unda iyos
CarTuli angariSis warmodgenisas statistikur SemowmebaSi. erT-erTi aseTi
angariSi aRwerilia me-10 TavSi.
magaliTi #9-3
η2 gamoTvla eqsperimentSi
amocana
magaliTSi #9-1, placebo sakontrolo da eqsperimentuli jgufebis mier drois
Sefasebebs Soris gansxvavebis trealuri iyo +2.387 df=22. risi tolia
damoukidebeli cvladis gavlenis siZliere medikamentis miRebis pirobaSi?
amoxsna: gavlenis siZliere mocemulia η2-iT, sadac
39
dftt
+=
realuri
realuri
2
22η
Tu trealuris nacvlad CavsvamT ricxviT mniSvnelobebs da df-s, miviRebT
21.0226978.5
6978.522)387.2(
)387.2(2
22
=
=+
=
=+
=η
daskvna: η2-is mniSvneloba gviCvenebs, rom medikamentis pirobis codna 21%-iT
amcirebs eqsperimentis srul dispersias.
ndobis intervalebis ageba ori damoukidebeli populaciis
saSualoTa sxvaobisaTvis t statistikis gamoyenebiT
populaciis saSualosaTvis ndobis intervali gansazRvruli iyo rogorc
intervali, romelic moicavs populaciis saSualos ndobis garkveuli
xarisxiT. am TavSi Cven davinteresdebiT populaciis saSualoebs Soris
sxvaobiT. am SemTxvevaSi Cven dagvWirdeba ndobis intervali damoukidebeli
populaciis saSualoTa µ A1-µA2 sxvaobisaTvis.. midgoma, romelsac Cven aq
gamoviyenebT msgavsia imisa, romelic gamoyenebuli iyo me-8 TavSi. gansxvaveba
mxolod isaa, rom nacvlad X unda gamoviyenoT 21 AA XX − , xolo nacvlad s X
unda gamoviyenoT s 21 AXAX − . Sesabamisad, 95% ndobis intervali damoukidebeli
populaciis saSualoTa Soris sxvaobisaTvis mocemulia intervaliT:
)()( 215.021 AXAXAA stXX−
−− -dan )()( 215.021 AXAXAA stXX−
+− -mde,
sadac
21 AA XX − = realur SerCevis saSualoTa Soris sxvaobisa
s 21 AXAX − = saSualoTa Soris gansxvavebis Sefasebuli standartuli Secdomisa
t0.05 = tkritikulis ormxrivi mniSvnelobisa mniSvnelobis 0.05 doneze, roca
Tavisuflebis xarisxi tolia N-2.
am formuliT mocemul intervals aqvs 0.95 albaToba.
analogiurad 99%-iani ndobis intervali damoukidebeli populaciebis
saSualoebs Soris sxvaobisaTvis mocemulia Semdegnairad
)()( 211.021 AXAXAA stXX−
−− -dan )()( 211.021 AXAXAA stXX−
−−
40 sadac t0.01 aris tkritikulis ormxrivi mniSvneloba mniSvnelovnobis 0.01 doneze,
roca Tavisuflebis xarisxi tolia N-2-is.
am formulebis gamoyenebis sailustraciod ganvixiloT 95%-iani ndobis
intervalis gamoTvla magaliTisaTvis #9-1, romelSic erTmaneTs udarebdnen
drois Sefasebebs placebo sakontrolo jgufsa da diazepamis miRebis jgufebs
Soris. aq 1AX =19.8wm da 2AX =13.6wm da s 21 AXAX − =2.597wm. am magaliTisaTvis gvaqvs
22 Tavisuflebis xarisxi (ix. magaliTi #9-1), amrigad, t kriteriumis ormxrivi
mniSvneloba 0.05 doneze aris 2.074. 95 procentiani ndobis intervali µ A1-µA2..
aris
(19.8-13.6)-(2.074)(2.597) dan (19.8-13.6)+(2.074)(2.579) wm
an, Tu CavatarebT maTematikur gamoTvlebs,
6.2_5.39 –dan 6.2+5.39 wm-mde
rac tolia
0.8 wm-dan 11.6 wm-mde
albaToba imisa, rom µ1-µA2 mniSvneloba moTavsebulia 0.8-dan 11.6-mde
intervalSi 0.95-a. Cven SegviZlia 95 procentiT viyoT darwmunebuli, rom
placebo sakontrolo da diazepamis jgufebis mier drois Sefasebebis
populaciis
saSualoebs Soris sxvaoba aris 0.8-dan 11.6-mde.
ras amowmebs t kriteriumi sinamdvileSi?
t kriteriumi gvaZlevs informacias, romelic saWiroa im gadawyvetilebis
misaRebad, Tu ramdenadaa ganmtkicebuli kvlevis hipoTeza eqsperimentuli
monacemebiT. Zalian mniSvnelovania imis gacnobiereba, rom kvlevis hipoTeza ar
mowmdeba pirdapiri gziT t kriteriumiT. t kriteriumi aris ubralod procedura
statistikuri hipoTezis H0:µ1=µASesamowmeblad. Tu ar gaviTvaliswinebT
damoukidebel da damokidebul cvladebs, romlebic monawileobs kvlevis
hipoTezaSi, nulovani da alternatiuli hipoTezebi ormxrivi t
kriteriumisaTvis aris yovelTvis H0:µ1=µ da H1:µ1≠µ. meore mxriv, kvlevis
hipoTeza eqsperimentSi aris debuleba, romelic damoukidebel da
damokidebul cvladebs Soris kavSirs winaswarmetyvelebs. rogorc zemoT
moyvanil magaliTSi, debuleba: `mosalodnelia, rom im cdis pirebis mier
Sefasebuli dro, romlebsac miewodaT diazepami ufro naklebi iqneba, vidre
placebo sakontrolo jgufis mier Sefasebuli dro", aris kvlevis hipoTeza.
41 statistikurad mniSvnelovani t realuri aramimarTulebiTi
alternatiuli hipoTezisaTvis (e.i. H1:µ1≠µ) ar warmoadgens aucileblad kvlevis
hipoTezis mxardaWeras. magaliTad, eqsperimentuli hipoTezis mxardaWera
iqneboda im SemTxvevaSi, Tu 1AX , placebo sakontrolo jgufis mier Sefasebuli
drois saSualo iqneboda 2AX -ze, anu eqsperimentuli jgufis mier Sefasebul
saSualoze mniSvnelovnad meti. magram 1AX mniSvnelovnad naklebia 2AX -ze, ar
eTanxmeba kvlvevis am hipoTezas. amitom trealuris statistikuri mniSvnelovnobis
miRebis Semdeg, Cven yovelTvis unda SevamowmoT saSualoTa Soris gansxvavebis
mimarTuleba, raTa davadginoT, eTanxmeba Tu ara is kvlevis hipoTezas.
statistikuri mniSvnelovnobisa da mecnieruli mniSvnelovnebis
mimoxilva
me-8 TavSi Cven mivuTiTeT, rom statistikuri hipoTezis testirebaSi sityva
mniSvnelovani niSnavs mxolod imas, rom nulovani hipoTeza uaryofilia. magram
saSualoTa Soris statistikurad mniSvnelovani gansxvaveba eqsperimentSi ar
aris aucileblad mecnierulad mniSvnelovani. eqsperimentis mecnieruli
mniSvnelovnoba ganisazRvreba manam, sanam monacemebi statistikurad
damuSavdeba. kargad gaazrebul da kargad dagegmil eqsperimentSi, mecnierulad
mniSvnelovani SeiZleba iyos statistikurad mniSvnelovani an statistikurad
aramniSvnelovani gansxvaveba. marTlac, Tu damoukidebelsa da damokidebul
cvlads Soris konkretuli kavSiri aris nawinaswarmetyvelebi, maSin am
kavSiris empiriulad povnis warumateblobas SeiZleba hqondes mecnieruli
mniSvneloba. cudad dagegmil eqsperimentSi ki, romelSic damoukidebeli
cvladis Sereva xdeba damatebiT cvladTan, ori SerCevis saSualoebs Soris
statistikurad mniSvnelovani gansxvavebac ki ar SeiZleba naTlad iyos
axsnili.
diskusia statistikuri hipoTezebis Semowmebis Sesaxeb
rogorc pirvel TavSi aRvniSneT, monacemTa statistikuri analizi ar iqneboda
saWiro, rom ar yofiliyo populaciis wevrebs Soris cvalebadoba. magram aseTi
cvalebadoba aris cxovrebiseuli faqti da iwvevs monacemTa statistikuri
analizis saWiroebas. statistikuri hipoTezebis Semowmeba imisaTvis ganviTarda,
42 raTa daexmaros mecnierebs gadawyvetilebebis miRebaSi cvalebadobis im
situaciaSi, romelic yovelTvis aRiniSneba monacemebSi. Tumca statistikuri
hipoTezebis Semowmeba SeiZleba TqvenTvis konceftualurad rTuli Candes, misi
gamoyeneba monacemebisaTvis male xdeba standartuli da advilad
Sesasrulebeli. SesaZloa mogveCvenos, rom es procesi ar Seicavs araviTar
winaaRmdegobas, magram es ase ar aris. piriqiT, gacxovelebuli diskusia
mimdinareobs misi gamoyenebis damcvelebsa da mowinaaRmdegeebs Soris. (Abelson,
1997; Cohen, 1994; Cortina and Dunlap, 1997; Frick, 1996; Hagen, 1997; Harris, 1997; Hunter, 1997;
Wainer, 1999). statistikur hipoTezebs sxvadasxva kuTxidan akritikeben. zogierTi
zedmetad rTulia imisaTvis, rom Sesaval teqstSi ganvixiloT. arsebiTad,
kritikosTa umravlesoba yuradRebas amaxvilebs Semdeg punqtebze:
• statistikuri hipoTezebis Semowmebisas vamowmebT nulovan hipoTezas,
romlis Tanaxmadac H0: µ A1=µA2 . Tumca naklebad mosalodnelia, rom
nulovani hipoTeza odesme iyos WeSmariti;
• statistikuri hipoTezebis Semowmeba Tavs gvaxvevs diqotomiur
gadawyvetilebas: uarvyoT nulovani hipoTeza, an ver uarvyoT is, rac
efuZneba nebismierad SerCeul α-s mniSvnelobas, Cveulebriv 0.05-s.
• II gvaris Secdomis daSvebis albaToba ar SeiZleba zustad
ganvsazRvroT arc erTi kvlevisaTvis. amrigad, roca
eqsperimentatori H0-s ver uaryofs, man SeiZleba dauSvas II gvaris
Secdoma. Tumca bihevioristi mecnierebi fiqroben, rom II gvaris
Secdomis daSvebis albaToba zogadad ufro naklebad mniSvnelovania,
vidre pirveli gvaris Secdomis daSvebis albaToba, es ar exeba yvela
kvleviT problemas.
• statistikuri hipoTezebis Semowmeba ubiZgebs mkvlevrebs guldasmiT
aRar moekidon maT mier miRebul monacemebs da gamoiyenon
kompiuterizebuli statistikuri kriteriumebi, romelic maT
srulyofilad arc esmiT.
ar arsebobs martivi an arawinaaRmdegobrivi pasuxi am kritikaze.
abelsonma (Abelson, 1997) miuTiTa, rom monacemTa analizi ar aris zusti
procesi, radgan is moicavs adamianis mier gakeTebul Sefasebas.
gadawyvetilebis miRebis yvela procesi, bihevioristul mecnierebebSi
kvlevis Sedegebs exeba, mimdinareobs gazomvisa da SerCevis Secdomis
43 cvalebadobis fonze da imis Sefasebisas gansxvavdeba, Tu ara ori
saSualo erTmaneTisagan sakmarisad, raTa es iyos Carevis efeqtis
maniSnebeli, adamianebi ar arian Secdomis cvalebadobis kargi
Semfaseblebi statistikuri kriteriumis gamoyenebis gareSe. Semdegi
Tavis ori amocana, #10.2 da #10.3 amis naTeli dadasturebaa. cxrilSi
#10.13 mocemuli saSualoTa Soris gansxvavebebi amocanaSi #10.2 erTi
SexedviT didia, ramac SeiZleba migviyvanos mcdar gadawyvetilebamde,
rom damoukidebel cvlads aqvs gavlena. magram am eqsperimentSi
Secdomis cvalebadobis didi odenoba gviCvenebs, rom saSualoTa Soris
gansxvavebis yvelaze Sesatyvis interpretacias warmoadgens misi
ganpirobebuloba SerCevis SecdomiT da ara damoukidebeli cvladis
gavleniT. amis sapirispirod, amocanis #10.3 cxrilSi #10.14
warmodgenil saSualoTa mxolod garegnuli dakvirvebiT Cans, rom isini
ar gansxvavdebian didad erTmaneTisagan, ramac SesaZloa dagvarwmunos
damoukidebeli cvladis araefeqturobaSi. magram aseTi martivi
gadaxedviT SesaZloa gamogvrCes monacemebSi Secdomis cvalebadobis
mcire odenoba. statistikuri kriteriumi ki iTvaliswinebs am
cvalebadobas da gviCvenebs, rom ori jgufis saSualoTa Soris
gasxvaveba saukeTesod SeiZleba aixsnas damoukidebeli cvladis
gavleniT. es ar aris SemTxveviT gansxvaveba. yuradRebiT ganixileT es
amocana, roca mas Semdeg TavSi SexvdebiT.
statistikur hipoTezaTa Semowmebis kritikosebis sapasuxod
amerikis fsiqologTa asociaciam daavala statistikur daskvnaTa
Semswavlel specialur komisias, gaekeTebinaT rekomendaciebi kvlevaSi
statistikuri analizis gamoyenebis Sesaxeb. am komisiam gaakeTa
ramdenime rekomendacia, romelic SesaZlebelia ganvixiloT
statistikuri analizis SesavalSi (Wilkonson&the Task Force on Statistical
Inference, 1999):
• mkvlevrebma dawvrilebiT unda aRweron TavianTi monacemebi
SerCevis moculobebis, centraluri tendenciisa da
cvalebadobis sazomebis CaTvliT;
• ndobis intervalebs unda axldes efeqtis siZlieris sazomi,
rogoricaa η2;
44 • mkvlevrebi kargad unda acnobierebdnen monacemTa analizis
romel programas gamoiyeneben. Sedegebis gadamowmebis mizniT
rekomendebulia monacemTa analizis ori gansxvavebuli
programis gamoyeneba.
am rekomendaciebis gaTvaliswinebiT davubrundeT am TavSi mocemul diskusias
simZlavris Sesaxeb. im diskusiaSi warmodgenili iyo rekomendaciebi kvlevis
simZlavris gasazrdelad da amrigad, II gvaris Secdomis albaTobis
Sesamcireblad.
ueWvelia, rom statistikuri hipoTezebis Semowmebisadmi kiTxvebi da
kritika momavalSic iqneba. savaraudoa, agreTve, rom statistikur hipoTezaTa
Semowmebis gamoyeneba, SesaZloa Secvlili formiT, kvlavac gagrZeldes
bihevioristuli mecnierebebis kvlevebSi, radgan misi umkacresi kritikosebic ki
miuTiTeben, rom amJamad statistikuri hipoTezebis Semowmebis Canacvleba raime
sxva xerxiT SeuZlebelia (Cohen, 1994).
t kriteriumis Sedegebis prezentacia amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvanelos (1994)
stilisa da statistikur daskvnaTa Semswavleli specialuri komisiis
(Wilkonson&the Task Force on Statistical Inference, 1999) rekomendaciebis mixedviT
magaliTis #9-1 Sedegebi gamosacem samecniero SromaSi Semdegi saxiT unda
warmovadginoT:
0.05 mniSvnelovnobis alfa donisa da ormxrivi kriteriumisaTvis placebo
sakontrolo jgufis mier Sefasebuli drois saSualo (M=19.8wm, SD=7.6wm)
mniSvnelovnad aRemateboda eqsperimentuli jgufis mier Sefasebuli drois
saSualos (M=13.6 wm, SD =4.8 wm), t(22)= +2.39, p< 0.05, η2=0.21. ori populaciis
saSualoTa Soris gansxvavebisaTvis 95-procentiani ndobis intervali 0.8-dan 11.6
wm-mde Sualedis tolia.
am Sedegebis prezentaciisas:
α=0.05 gviCvenebs am kriteriumisaTvis arCeul mniSvnelovnobis dones.
t(22) gansazRvravs kriteriumis statistikas, rogorc t-s. es t aris
trealuri, Tumca indeqsi `realuri” SeiZleba ar axldes. radgan
am magaliTSi aRwerili iyo ori jgufi, Cven viciT, rom t
ganisazRvreba, rogorc tdamoukidebeli. Tavisuflebis xarisxi
trealurisaTvis naCvenebia frCxilebSi. amrigad, Sedegebis
45 analizisaTvis gamoyenebuli iyo 24 cdis piris monacemi
(radgan df=N-2)
=+2.39 trealuris mniSvneloba (ara tkritikulisa, romelsac #a-2 cxrilSi
vpoulobT).
p<0.05 gviCvenebs Semdegs:
a) trealuris albaToba H0-is WeSmaritebis pirobebSi aris
naklebi an toli 0.05-is. es mniSvneloba aris trealuris
albaToba, an ufro Sors mdebare trealuris mniSvnelobis
albaToba, roca H0 aris WeSmariti. ar aris aucilebeli igive
procedurebis warmoeba, rac α-s arCevis dros;
b) H0: µ A1=µA2 uaryofilia;
H0: µ A1≠µA2 miRebulia;
g) gansxvaveba SerCevis saSualoebs Soris statistikurad
mniSvnelovania;
d) saSualoTa Soris realurad miRebuli gansxvavebis mizezi
aris ara SerCevis Secdoma, aramed sxva ram. radgan es
eqsperimenti Catarda kontrolirebad pirobebSi, Cven vaskvniT,
rom gansxvaveba ganpirobebuli iyo cdis pirebis mier
medikamentis miRebis pirobiT;
e) Tu SedegebSi warmodgenili iqneba p>0.05, maSin metobis
niSani aRniSnavs imas, rom H0 ar aris uaryofili da ori
SerCevis saSualo ar gansxvavdeba mniSvnelovnad
erTamaneTisagan mniSvnelovnobis 0.05 doneze. Aam SemTxvevaSi
Cven davaskvniT, rom medikamentis miRebis piroba ar axdens
gavlenas drois Sefasebaze;
η2=0.21 dispersiis wili, romelic ganpirobebulia damoukidebeli
cvladiT;
95%-iani ndobis es intervali aris ori populaciis saSualoTa Soris
ndobis inter- sxvaobisaTvis. amis nacvlad, SesaZloa warmodgenili iyos
95
vali=0.8-dan procentiani ndobis intervali TiToeuli saSualosaTvis.
11.6 wm-mde ufro iSviaTad ki, avtorebi ndobis intervalis nacvlad
warmoadgenen saSualos standartul Secdomas.
46
SeamowmeT Tqveni codna IX-4
1. gansazRvreT: efeqtis zoma, simZlavre, kavSiris siZliere, efeqtis siZliere.
2. ori SerCevisaTvis miRebuli iyo qulaTa ori simravle. pirveli
simravlisaTvis sA1 = 6.3 da sA2 =7.6. meore simravlisaTvis sA1 = 11.7 da sA2 = 10.3.
a) qulaTa romel simravles aqvs Secdomis meti cvalebadoba?
b) Tuki H1 WeSmaritia, qulaTa romelma simravlem SeiZleba gamoiwvios
statistikur SemowmebaSi II gvaris Secdoma?
g) qulaTa romeli simravle gamoiwvevs statistikuri kriteriumis ufro
met simZlavres?
3. ori SerCevidan ra moculobis SerCeva gamoiwvevs t kriteriumSi s 21 AXAX − -is
ufro mcire sidides: nA1= nA2=12 Tu nA1= nA2=20? ratom? ra moculobis SerCevas
mivyavarT ufro meti simZlavris statistikuri kriteriumisaken?
4. ipoveT η2 trealuris Semdegi mniSvnelobebisaTvis. TiToeuli t-s Tavisuflebis
xarisxi frCxilebSia mocemuli. TiToeuli trealuri statistikurad
mniSvnelovania 0.05 doneze ormxrivi kriteriumisaTvis.
a) t(8)=2.413
b) t(14)=2.2.84
g) t(20)=2.174
d) t(29)=2.939
e) t(60)=2.306
5. erTmaneTisagan ganasxvaveT kvlevisa da statistikuri hipoTeza.
6. niSnavs TuU ara statistikurad mniSvnelovani ganxvaveba imas, rom
aucileblad mecnieruladac mniSvnelovani Sedegebi iqneba miRebuli? axseniT
Tqveni pasuxi.
7. tdamoukidebelis gamoyenebis Semdegi angariSebi aRebulia gamoqveynebuli
naSromebidan. UupasuxeT Semdeg kiTxvebs TiToeuli angariSisaTvis.
a. damoukidebeli cvladis ramdeni done iyo gamoyenebuli am eqsperimentSi?
b. ramdeni cdis piri monawileobda am eqsperimentSi?
g. risi tolia SerCevis saSualo TiToeuli eqsperimentuli pirobisaTvis?
d. risi tolia populaciis standartuli gadaxris Sefasebis mniSvneloba
Carevis TiToeuli pirobisaTvis?
e. risi tolia trealuris mniSvneloba?
47 v. risi tolia tkritikulis mniSvnelovnebis 0.05 doneze ormxrivi uaryofis
arisaTvis?
z. uaryofT Tu ver uaryofT H0-s ?
T. miiRebT Tu ara H1-s?
i. saSualoTa Soris arsebuli gansxvaveba statistikurad mniSvnelovania, Tu
aramniSvnelovania?
k. ras miawerT saSualoTa Soris realurad miRebul gansxvavebas?
AangariSi #1. 0.05 alfas da ormxrivi kriteriumisaTvis, t kriteriumma gviCvena,
rom cdis pirebi sworad amCneven davalebaTa ufro did procents, roca usmenen
nacnob leqss (M=79.7, SD=8.57), vidre maSin, roca usmenen ucnob leqss (M=73.,
SD=5.85), t(60)=3.6, p<0.01.
AangariSi #2. 0.05 alfasa da ormxrivi kriteriumisaTvis, t kriteriumma gviCvena,
rom 5-wuTiani intervalis Semdeg aRdgenili sityvebis raodenoba (M=22.78,
SD=5.23) ar gansxvavdeba 0.5-wuTiani intervalis Semdeg aRdgenili sityvebis
raodenobisagan (M=24.33, SD=5.14), t(28)= -0.82, p>0.05
cdis pirTa Sida sqema da t kriteriumi damokidebuli
qulebisaTvis
Cveni interesi am sakiTxisadmi maSin gaCnda, roca vixilavdiT cdis pirTa
Soris sqemas, romelSic damoukidebeli cvladis TiToeuli done miewodeboda
gansxvavebuli jgufis cdis pirebs. meore tipis kvlevis sqema aris cdis pirTa
Sida sqema, romelSic cdis pirTa erTi jgufi aris warmodgenili da xdeba
misi gazomva damoukidebeli cvladis TiToeuli donis mixedviT. amrigad, cdis
pirTa Sida sqemaSi TiToeul cdis pirs miewodeba eqsperimentis orive piroba.
ganvixiloT erTfaqtoriani cdis pirTa Sida sqemis Semdegi magaliTi.
adamianebi, romlebic mowevas Tavs anebeben, xSirad aRniSnaven, rom imateben
wonaSi. am hipoTezis Sesawmeblad, bekonam da vazqezma (Becona &Vazquez, 1998)
erTmaneTs mwevelTa wonebi Seadares im momentisaTvis, roca maT daiwyes
sigaretisaTvis Tavis danebebis programaSi monawileoba da 36 Tvis Semdeg maT
aRmoaCines, rom programis im monawileebma, romlebmac SesZles Tavi
cdis pirTa Sida sqema (Within-subject design): kvlevis sqema, romelSic cdis pirTa erTi jgufia da izomeba damoukidebeli cvladis TiTeuli donis mixedviT. cdis pirTa Sida sqemaSi TiToeul cdis pirs miewodeba eqsperimentis orive piroba.
48 daenebebinaT sigaretisaTvis, 2 kg-iT moimates wonaSi da es mateba
statistikurad mniSvnelovani aRmoCnda. davuSvaT, Tqven gsurT gaimeoroT es
kvleva 8 mweveli qalisagan Semdgar jgufze, romelTa asaki meryeobs 20-dan 24
wlamde da romlebic 1 wlis ganmavlobaSi aRar eweodnen sigarets. TiToeuli
pirovneba awones am programis dawyebisas (A1) da mowevisaTvis Tavis danebebidan
12 Tvis Semdeg (A2); amrigad, gamoyenebuli iyo cdis pirTa Sida sqema. davuSvaT,
Tqven daadgineT, individTa wonebi programis dawyebis momentisaTvis da 1
wlis Semdeg, rac naCvenebia cxrilSi #9.2. statistikurad mniSvnelovnad
gansxvavdeba Tu ara am or periodSi gazomili wonebi erTmaneTisagan? am
kiTxvaze pasuxis gasacemad da saSualoTa Sesadareblad gamoviyeneT t
kriteriumi damokidebuli qulebisaTvis.
cxrili #9.2
20-24 wlis qalebis wonebi (funtebSi) mowevisaTvis Tavis danebebis programaSi
dasawyisisaTvis (A1) da mowevisaTvis Tavis danebebidan 1 wlis Semdeg (A2):
cdis piri awonvis momenti (A)
programaSi Sesvlisas (A1) mowevis danebebidan 1 wlis Semdeg
(A2)
1 122.4 127.9
2 117.1 119.4
3 127.8 127.1
4 123.2 126.5
5 112.5 114.6
6 118.7 125.3
7 121.4 119.8
8 131.7 134.2
X A 121.85 124.35
As 6.03 6.13
t kriteriumi damokidebuli qulebisaTvis
cdis pirTa Sida sqemaSi xdeba erTi da igive cdis pirebis maxasiaTeblebis
gazomva orive eqsperimentul pirobaSi; amrigad TiToeul pirobaSi miRebuli
qulebi erTmaneTTan dakavSirebuli iqneba. faqtori, romelic moqmedebs cdis
49 piris qulaze erT pirobaSi, imoqmedebs am cdis pirTa qulaze sxva pirobaSi.
tdamoukidebeli ar aris Sesatyvisi sazomi am sqemisaTvis; aq viyenebT t kriteriums
damokidebuli qulebisaTvis, romelic ganisazRvreba, rogorc
D
AA
sXXt 21 −
=lidamokidebu
sadac sD aris saSualos standartuli Secdoma TiToeul 1AX da 2AX qulaTa
Soris gansxvavebisaTvis. sD-s gamoTvliTi formulaa
)1(/)( 22
−Σ−Σ
=wyvilebiNN
NDDsi
Dwyvileb
wyvilebi
sadac D asaxavs gansxvavebas cdis piris or qulas Soris da Nwyvilebi aris
qulaTa wyvilebis raodenoba, an zustad cdis pirTa raodenoba.
CavsvaT formulaD sD-saTvis, tdamokidebulis gamoTvliT formulaSi, miviRebT
)1(/)( 22
21
−
Σ−Σ
−=
ii
i
AA
NNNDD
XXt
wyvilebwyvileb
wyvileb
lidamokidebu
df aris toli Nwyvilebi_1-is, romelic asaxavs qulaTa wyvilebis im raodenobas,
romelic Tavusufalia misi cvalebadobisas.
am formulis gamoyeneba cxrilSi #9.2 mocemuli qulebisaTvis naCvenebia
cxrilSi #9.3. gamoTvlebisaTvis saWiro safuxurebi Semdegia:
1. vipovoT 1AX da 2AX . magaliTad, es mniSvnelobebi tolia 121.85 da 124.35
mili/sT, Sesabamisad.
2. ipoveT D-s mniSvneloba, amisaTvis TiToeuli cdis piris mier A1
pirobaSi miRebul qulebs gamoakeliT cdis pirTa mier A2 pirobaSi miRebuli
qulebi. gaiTvaliswineT sxvaobis algebruli niSnebi. es nabiji naCvenebia im
svetSi, romelic cxrilSi dasaTaurebulia D asoTi.
3. aiyvaneT D-s TiToeuli mniSvneloba kvadratSi, rogorc naCvenebia D2
svetSi.
4. ipoveT ΣD (D svetis jami; magaliTad, _20) da ΣD2 (D2 svetis jami;
magaliTad, 103.7)
5. gansazRvreT Nwyvilebi magaliTad, Nwyvilebi=8
cxrili #9.3
50 cxrilis #9.2 qulebis mixedviT tdamokidebulis gamoTvla
awonvis momenti (A)
cdis
piri
programaSi
Sesvlisas (A1)
mowevis danebebidan
1 wlis Semdeg (A2)
D D2
1 122.4 127.9 -5.5 30.25
2 117.1 119.4 -2.3 5.29
3 127.8 127.1 +0.7 0.49
4 123.2 126.5 -3.3 10.89
5 112.5 114.6 -2.1 4.41
6 118.7 125.3 -6.6 43.56
7 121.4 119.8 +1.6 2.56
8 131.7 134.2 -2.5 6.25
AX 121.85 124.35 ΣD= -20 ΣD2=103.7
sA 6.03 6.13
)1(/)( 22
21
−
Σ−Σ
−=
ii
i
AA
NNNDD
XXt
wyvilebwyvileb
wyvileb
lidamokidebu =
554.2979.0
5.2)7)(8(
8/)20(7.10335.12485.1212
−=
−=
−−
−
roca df aris 7
6. CasviT ricxobrivi mniSvnelobebi tdamokidebulis gamoTvliT formulaSi.
7. ipoveT df, sadac df= Nwyvilebi -1
trealuri cxrilSi #9.2 mocemuli qulebisaTvis aris -2.2554. trealuris
gamosaTvleli danarCeni procedura tdamokidebulis identuria.. unda avirCioT α
mniSvneloba, vipovoT tkritikulis mniSvneloba cxrilidan #a-2, da uaryofis
areebi, romlebic mdebareoben t SerCeviT ganawilebaze. Tu trealuris
mniSvneloba xvdeba uaryofis areSi, maSin
H0: µ A1=µA2 uaryofilia
da
H1: µ A1≠µA2 miRebulia
51 roca df=7 da mniSvnelovnobis done ki 0.05, tkritikuli ormxrivi kriteriumisaTvis
aris 2.365. uaryofis are aris Seqmnili trealuris mniSvnelobidan, romelic
naklebia an toli –2.365-is, an toli an meti, +2.365-is. trealuri –2.554 naklebia –
2.365-ze; amitom is xvdeba uaryofis areSi; H0 unda uarvyoT da H1 miviRoT.
SerCevis saSualoTa Soris sxvaoba statistikurad mniSvnelovania
mniSvnelovnobis 0.05 doneze. programaSi monawile pirebi ufro mets iwonidnen
mowevis danebebidan erTi wlis Semdeg, vidre programis dawyebamde.
am tdamokidebulisaTvis eta kvadrati gamoiTvleba Semdegnairad:
dftt
+=
realuri
realuri2η
CavsvaT trealuris da df-is ricxobrivi mniSvnelobebi:
48.07)554.2(
)554.2(2
22
=+−
−=η
es eta kvadrati gviCvenebs, rom damokidebuli cvladis cvalebadobis 48
procenti aris ganpirobebuli damoukidebeli cvladiT _ siCqaris sazomi
iaraRis pirobiT am eqsperimentSi.
erT-erTi mniSvnelovani sakiTxi am sqemis ganxilvisas isaa, rom Tumca wonaSi
mateba 1 wlis Semdeg statistikurad mniSvnelovani aRmoCnda, Cven mainc ar
SegviZlia vamtkicoT, rom es iyo gamowveuli sigaretis mowevis danebebis
programiT. adamianebi xSirad imateben wonaSi wlebis matebasTan erTad da
SesaZlebelia cdis pirebs isedac moematebinaT wonaSi, Tuki isini ar
miatovebdnen mowevas. imisaTvis, raTa davrwmundeT, rom wonis mateba marTlac
ganpirobebuli iyo sigaretis danebebiT, 1 wlis Semdeg unda SevamowmoT msgavsi
cdis pirebisagan Semdgari meore jgufis wonebi, romlebmac ar daanebes Tavis
mowevas. aseTi jgufis CarTva am magaliTSi gaarTulebda ganxilvas, magram
bekonsa da vazqezs hyavdaT aseTi jgufi da aRmoCnda, rom maTi wona samwlian
periodSi mniSvnelovnad ar gazrdila.
ndobis intervalebis ageba ori damokidebuli populaciis
saSualoTa Soris gansxvavebisaTvis
95-procentiani ndobis intervali ori damokidebuli populaciis saSualoTa
Soris gansxvavebisaTvis mocemulia Semdegi intervaliT:
52 )()( 05.021 DAA stXX −− dan )()( 05.021 DAA stXX +− -mde,
sadac 21 AA XX − = ori realuri SerCevis saSualoTa Soris gansxvavebas
sD = TiToeuli 1AX da 2AX qulaTa Soris sxvaobis saSualos
standartul Secdomas
t0.05 = tkritikulis ormxriv mniSvnelobas mniSvnelovnobis 0.05
doneze
N-1 Tavisuflebis xarisxisiT
albaToba imisa, rom µ A1-µA2 mniSvneloba am intervalSiaA moTavsebuli, aris
0.95.
am magaliTisaTvis, 1AX =121.85 da 2AX =124.35, Nwyvilebi=8, sD=0.979 (cxrilidan
#9.3) da t(0.05) 7-is toli Tavisuflebis xarisxisaTvis tolia 2.365-is. SerCevis
saSualoTa Soris uaryofiTi sxvaobis miRebis Tavidan asacileblad, Cven 1AX -s
vaklebT 2AX -s. Tu CavsvamT ricxobriv mniSvnelobebs, 95-procentiani ndobis
intervali µ A1 da µA2 Soris sxvaobisaTvis iqneba:
(124.35-121.85)_(2.365)(0.979)-dan (124.35-121.85)+(2.365)(0.979)-mde
romelic tolia 0.18-dan 4.82 funtamde Sualedis. 95-procentiani ndobis
intervali populaciis saSualoebs Soris gansxvavebisaTvis programis
dawyebidan gazomil wonasa da mowevis danebebidan 1 wlis Semdeg gazomil
wonas Soris aris moTavsebuli 0.18-sa da 4.82 funts Soris.
\
SeamowmeT Tqveni codna IX-5
1. axseniT gansxvaveba cdis pirTa Soris kvlevis sqemasa da cdis pirTa Sida
kvlevis sqemas Soris.
2. sul ufro meti adamiani Tvlis sakuTar avtomanqanas Tavisi ofisis nawilad
da transportis moZraobis usafrTxoebis mravali eqsperti miiCnevs, rom
yuradRebis gamfantveli stimulebis zrda iwvevs avariebis ricxvis zrdas.
davuSvaT, gsurT empiriulad gadaamowmoT es problema da 11 adamians sTxoveT
etarebinaT manqana or gansxvavebul xelovnurad Seqmnil situaciaSi:
sakontrolo pirobaSi, roca ar iqneboda yuradRebis gamfantveli stimulebi
53 da eqsperimentul pirobaSi, roca mZRoli isaubrebda mobiluri telefoniT.
izomeboda cdis pirebis reaqciis dro xelovnurad Seqmnil avariul
situaciaze, romelic moiTxovda damuxruWebas. vTqvaT, miiReT Semdegi reaqciis
droebi (miliwamebSi):
gansazRvreT saSualo da standartuli gadaxra TiToeuli pirobisaTvis;
Semdeg gamoiTvaleT tdamokidebeli am qulebisaTvis da upasuxeT Semdeg kiTxvebze:
a. CamoayalibeTY statistikuri hipoTeza tdamokidebelisaTvis am monacemebis
safuZvelze.
b. risi tolia trealuris mniSvneloba?
g. risi tolia df t kriteriumisaTvis?
d. risi tolia trealuris mniSvneloba?
e. risi tolia uaryofis areebi trealurisaTvis?
v. xvdeba Tu ara trealuri uaryofis areSi?
z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb? uaryofT Tu
ver uaryofT H0 hipoTezas? miiRebT Tu ara H1 hipoTezas?
T. gamoitanT daskvnas imis Sesaxeb, rom SerCevis saSualoebi erTi da imave Tu
gansxvavebuli populaciebidanaa?
i. saSualoebs Soris gansxvaveba statistikurad mniSvnelovania?
k. gansxvavdebian pirobebi mniSvnelovnad reaqciis drois mixedviT? Tu asea,
rogoria am gansxvavebis mimarTuleba?
l. risi tolia eta kvadrati am kvlevaSi?
cdis piri piroba
sakontrolo eqsperimentuli
1 426 448
2 501 552
3 525 506
4 478 507
5 537 574
6 452 491
7 394 438
8 443 477
9 491 523
10 382 413
11 572 541
54 m. risi tolia 95-procentiani ndobis intervali populaciis saSualoTa Soris
sxvaobisaTvis?
Sejameba
• eqsperimenti moiTxovs:
o mecnieruli kiTxvis dasmas da kvlevis hipoTezis formulirebas;
o cdis pirTa SerCevas;
o ekvivalenturi jgufebis SerCevas SemTxveviTi ganawilebiT;
o damatebiTi cvladebis kontrolirebas;
o damoukidebeli cvladis manipulirebas;
o damokidebuli cvladis gazomvas.
• SeiZleba tdamoukidebuli gamoyenebuli iyos imis gansazRvrisaTvis, ori
SerCevis saSualoebi gansxvavdeba Tu ara erTmaneTisagan
eqsperimentSi mniSvnelovnad?
• tdamoukidebelis formula aris:
21
21
AA XX
AA
sXXt
−
−=elidamoukideb
sadac, s 21 AXAX − aris saSualoTa Soris sxvaobis standartuli Secdoma;
• statistikuri hipoTezis Semowmeba tdamoukidebelis gamoyenebiT yvela
statistikuri SemowmebisaTvis saerTo safexurebs imeorebs;
• statistikuri kriteriumis simZlavre aris H0-is uaryofis albaToba,
roca H0 ar aris samarTliani;
• t kriteriumis simZlavreze moqmed faqtorebs miekuTvneba
damoukidebeli cvladis gavlenis sidide, cvalebadobis sidide
damokidebuli cvladis gazomvisas da SerCevis moculoba;
• eta kvadrati aris kavSiris siZlieris sazomi SerCevaSi, romelic
trealuridan gaizomeba da gamoiTvleba;
• eta kvadrati gviCvenebs damokidebuli cvladis cvalebadobis im
wils sruli cvalebadobidan, romelic damoukidebeli cvladiT
aixsneba;
• ori damoukidebeli populaciis saSualoTa Soris sxvaobis 95-
procentiani ndobis intervali mocemulia intervaliT:
55 )()( 2105.021 AXAXAA stXX −−− dan )()( 2105.021 AXAXAA stXX −+− -mde
• cdis pirTa Sida sqemaSi cdis pirTa erT jgufs miewodeba TiToeuli
damoukidebuli cvladis yvela done;
• tdamokidebuli SeiZleba gamoyenebuli iyos, roca gvaqvs damoukidebuli
cvladis ori done erTfaqtorian cdis pirTa Sida sqemaSi;
• tdamokidebulis formula aris:
D
AA
sXXt 21 −
=lidamokidebu
sadac sD aris TiToeul XA1 da XA2 qulas Soris sxvaobis saSualos
standartuli Secdoma;
• ori damokidebuli populaciis saSualoTa Soris sxvaobis 99-
procentiani ndobis intervali mocemulia intervaliT:
)()( 2101.021 AXAXAA stXX −−− -dan )()( 2101.021 AXAXAA stXX −+− -mde.
ZiriTadi terminebi da simboloebi
cdis pirTa Soris sqema
SemTxveviTi gansxvaveba
ndobis intervali ori populaciis saSualoTa Soris gansxvavebisaTvis
Sereuli eqsperimenti
xelmisawvdomi (SesaZleblobis) SerCeva
damokidebuli cvladi
mimarTulebis mqone kriteriumi
ekvivalenturi jgufebi
saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba, (s 21 AXAX − )
eta kvadrati (η2)
eqsperimenti
damatebiTi cvladi
faqtori
damoukidebeli cvladi
damoukidebeli cvladis done
95-procentiani ndobis intervali damoukidebeli populaciebis saSualoebs
Soris sxvaobisaTvis
56 99-procentiani ndobis intervali ori damoukidebeli populaciebis saSualoebs
Soris sxvaobisaTvis
aramimarTulebiTi kriteriumi
cdis pirTa Soris erTfaqtoriani sqema
erTmxrivi kriteriumi
placebo kontroli
simZlavre
SemTxveviTi ganawileba
SemTxveviTi SerCeva
mdgradoba
sD
s2kombinirebuli
saSualoTa Soris gansxvavebis SerCeviTi ganawileba
saSualoTa Soris sxvaobis standartuli Secdoma
efeqtis siZliere
tkritikuli t kriteriumi damokidebuli qulebisaTvis (tdamoukidebeli)
t kriteriumi ori damoukidebuli jgufisaTvis (tdamoukidebuli)
trealuri
ormxrivi kriteriumi
cdis pirTa Sida sqema
kiTxvebi
1. davuSvaT, gainteresebT daadginoT, aqvs Tu ara xmaurs raime gavlena
wnevaze. Tqven SemTxveviTad gaanawileT 22 individi sakontrolo da
eqsperimentul jgufebSi (nA1=nA2). uxmauro jgufis cdis pirebi (A1) isvenebdnen
komfortul savarZlebSi wynar oTaxSi 30 wuTis ganmavlobaSi. 30 wuTis Semdeg
maT uzomavdnen sistoluri sisxlis wnevas (anu sisxlis wnevas gulis kunTis
SekumSvebs Soris). xmauriani jgufis (A2) cdis pirebi agreTve isxdnen
savarZlebSi 30 wuTi. drois am Sualedis ganmavlobaSi isini usmendnen didi
qalaqis transportis xmis Canawers pikis saaTebSi. amis Semdeg maT uzomavdnen
57 sisxlis sistolur wnevas. davuSvaT, Tqven miiReT sisxlis wnevis Semdegi
monacemebi vecxliswylis svetis milimetrebSi.
jgufi
uxmauro xmauriani
106 141
117 136
124 124
129 139
115 121
131 119
121 147
115 128
128 115
136 134
127 140
a. ipoveT TiToeuli jgufisaTvis saSualo da standartuli gadaxra.
Semdeg gamoiyeneT t kriteriumi Semdeg kiTxvebze pasuxis gasacemad:
moqmedebs Tu ara xmauri sisxlis sistoluri wnevaze? gamoiyeneT
ormxrivi kriteriumi da mniSvnelovnobis 0.05 done.
b. risi tolia η2-is mniSvneloba am eqsperimentSi? damokidebuli cvladis
cvalebadobis ra wili SeiZleba iyos axsnili damoukidebeli cvladis
donis codniT?
g. ipoveT 95-procentiani ndobis intervali ori populaciis saSualoTa
Soris sxvaobisaTvis.
d. aRwereT Tqveni Sedegebi am Tavis erT-erTi nawilSi mocemuli magaliTis Sesabamisad, romelic exeboda t kriteriumis Sedegebis prezentacias.
2. davuSvaT, rom Tqven miiReT sisxlis wnevis monacemebi, romlebic
mocemulia pirveli kiTxvis gaswvriv. moqmedebs Tu ara sistolur sisxlis
wnevaze xmauri? gamoiyeneT 0.05 mniSvnelovnobis done da ormxrivi
kriteriumi.
jgufi
uxmauro xmauriani
110 116
114 111
58 106 120
117 119
108 123
116 108
120 122
111 115
115 117
109 101
118 116
3. pedersenma (Pedersen, 1987) ivarauda, rom arsebobs sqesTa Soris
gansxvavebebi kolejis studentebSi pirad cxovrebaSi Caurevlobis
moTxovnilebis mixedviT. hipoTeza Semowmebuli iyo pirad
cxovrebaSi Caurevlobis moTxovnilebis sakvlevi kiTxvaris
saSualebiT (Pedersen, 1979) kolejis mdedrobiTi da mamrobiTi sqesis
studentebze. am kiTxvaris erTi aspeqti zomavs ganmartoebis
midrekilebas. am skalis qulebi SeiZleba icvlebodes 6-dan
(arasodes aqvs ganmartoebis moTxovnileba) 30-mde (Cveulebriv
sWirdebaT ganmartoeba). davuSvaT, rom miRebuli iyo Semdegi qulebi
13 mamrobiTi da 17 mdedrobiTi sqesis cdis pirisaTvis.
sqesi
mamrobiTi(A1) mdedrobiTi(A2)
23 16
16 29
13 11
25 14
17 17
21 19
10 18
19 21
24 9
59 16 15
11 17
22 20
14 14
24
23
25
24
a. ipoveT TiToeuli jgufis saSualo da standartuli gadaxra. gamoiyeneT t
kriteriumi Semdeg kiTxvaze pasuxis gasacemad: gansxvavdebian Tu ara
qalebi da mamakacebi ganmartoebis moTxovnilebiT? gamoiyeneT ormxrivi
kriteriumi da mniSvnelovnobis 0.05 done.
b. aRwereT Tqveni Sedegebi am Tavis erT-erT qveTavSi mocemuli magaliTis
Sesabamisad, romelic exeboda t kriteriumis Sedegebis prezentacias.
4. fsiqologebi swavloben iluziebs, raTa ukeTesad axsnan, Tu rogor
aRviqvamT samyaros. erT-erTi magaliTia morinagas gadaxrilobis
iluzia, romelic naCvenebia suraTze #10.3. suraTze gamosaxuli
figuris iluzoruloba isaa, rom mocemuli kuTxeebis wveroebs erT
vertikalur xazze mdebared ar aRviqvamT, Tumca realurad isini
erT vertikalur xazze mdebareoben. ise Cans, TiTqos Sua kuTxis
wertili mdebareobs sxva vertikalur xazze, xolo zeda da qveda
kuTxeebis wertilebi _ sxvaze. deim da kasperCikma (Day and Kasperczyk,
1984) Caatares kvleva, imis dasadgenad, Tu ramdenad SeiZleba am
iluziis ganzogadeba, amisaTvis isini cdis pirebs sTxovdnen
gaesworebinaT Sua kuTxe iseTnairad, rom samive kuTxis wvero erT
vertikalur xazze aRmoCeniliyo. am gasworebis dros daSvebuli
Secdoma izomeboda milimetrebSi (mm). Tu cdis piri samive kuTxis
wverobs sworad ganalagebda, maSin misi Secdoma iqneboda 0 mm da
iluziac ar iqneboda. sxva SemTxvevaSi, Tu cdis piri Siga kuTxes
ise ganalagebda, rom is gamoCndeboda erT vertikalur xazze
mdebared, magram realurad es ase ar iqneboda, eqsperimentatori
iniSnavda Secdomis sidides. davuSvaT, eqsperimentatori
dainteresda imiT, Tu ra gavlenas moaxdens cdis pirebis aRqmaze
iluziis xasiaTis Sesaxeb informaciis miwodeba. eqsperimentators
60 pirvelad hyavda 12 cdis piri, romlebmac gaaswores kuTxis
wertilebi da gazomes TiToeuli maTganis mier gasworebisas
daSvebuli Secdoma. mas Semdeg, rac cdis pirebma gaaswores
wertilebi, maT acnobdnen iluziis xasiaTs da sTxovdnen Tavidan
gaesworebinaT es wertilebi. davuSvaT, rom Secdomebis am ori
gazomvis (milimetrebSi) Sedegad TiToeuli cdis pirisaTvis miiRes
Semdegi Sedegebi.
cdis pirebi
instruqciis piroba (A)
informaciis miwodebamde (A1) informaciis miwodebis
Semdeg (A2)
1 1.3 1.4
2 2.1 2
3 2.5 2.7
4 1.9 2
5 3 3.4
6 2.5 2.8
7 1.8 1.6
8 2.4 2.5
9 3.1 3.3
10 3.4 3.4
11 2.5 2.3
12 2.9 2.8
a. ipoveT saSualo da standartuli gadaxra TiToeuli jgufisaTvis.
Semdeg gamoiyeneT t kriteriumi, raTa upasuxoT Semdeg kiTxvebs:
axdens Tu ara gavlenas iluziis xasiaTis Sesaxeb informaciis
miwodeba aRqmuli iluziis sidideze? gamoiyeneT ormxrivi
kriteriumi da mniSvnelovnobis 0.05 done.
b. aRwereT Tqveni Sedegebi am Tavis erT-erT qveTavSi _ “t kriteriumis
Sedegebis prezentacia” _ aRwerili stilis mixedviT.
61 5. kontrolis lokusi exeba pirovnebis aRqmas, Tu ramdenad akontrolebs is im
SemTxvevebs, romlebic moqmedeben mis cxovrebaze. is pirovneba, visac aqvs
kontrolis eqsternaluri lokusi, gareSe movlenebs miiCnevs gadamwyvetad.
xolo pirovneba internaluri lokusiT aRiqvams movlenebs, rogorc misi
Zlieri kontolis qveS myofs. viem (Wiehe, 1986) ivarauda, rom bavSvebi, romlebic
TavianT mSoblebTan ar izrdebodnen, xasiaTdebodnen ufro meti eqsternaluri
kontroliT, vidre bavSvebi, romlebic izrdebodnen TavianT biologiur
mSoblebTan. am hipoTezis dasamtkiceblad, viem gamoyo mozardebis ori jgufi
adgilobriv samSobiaro saxlSi: 45 orsuli mozardi, vinc ar izrdeboda Tavis
mSoblebTan da dafexmZimda, roca mas zrdidnen mamobili da dedobili da 45
orsuli mozardi, romelic dafexmZimda Tavis mSoblebTan cxovrebis periodSi.
TiToeul mozards misces kontrolis lokusis gasazomi sareitingo skala. vies
angariSSi kontrolis lokusis skalis saSualo dedobilTan da mamobilTan
aRzrdili mozardebis jgufSi iyo 15.76(s=4.65), xolo biologiuri mSoblebis
jgufSi iyo 12.27(s=5.74). am sazomiT maRali qulebi gviCvenebs kontrolis ufro
maRal eqsternalur lokuss.
a. gamoiyeneT t kriteriumi am kiTxvaze pasuxis gasacemad: aris lokusis
kontroli dakavSirebuli mozardebis mSoblebis tipTan? gamoiyeneT
ormxrivi kriteriumi da mniSvnelovnobis 0.05 done.
b. risi tolia η2-is mniSvneloba am eqsperimentSi? damokidebuli
cvladis cvalebadobis ra wili SeiZleba iyos axsnili damoukidebeli
cvladis donis codniT?
g. ipoveT 95-procentiani ndobis intervali ori populaciis
saSualoTa Soris sxvaobisaTvis.
d. aRwereT Tqveni Sedegebi am Tavis erT-erT qveTavSi mocemuli
magaliTis Sesabamisad, romelic exeboda t kriteriumis Sedegebis
prezentacias.
6. hiugi (1985) adarebs wesrigis damrRvevad cnobili kolejis studentebis
akademiuri moswrebis qulebs, romlebic im studentebs, romlebsac ar hqondaT
aseTi SeniSvnebi miRebuli. akademiuri moswrebis qulis saSualo wesrigis
damrRvevi 1794 studentisaTvis iyo 1.68 da s=0.34. 3606 studentisaTvis ki,
romlebic ar arRvevdnen wesrigs, akademiuri moswrebis saSualo iyo 2.09 da
s=0.83.
62 a. gamoiyeneT t kriteriumi, raTa upasuxoT am SekiTxvebs: mniSvnelovnad
gansxvavdeba Tu ara akademiuri moswrebis qulebi? gamoiyeneT ormxrivi
kriteriumi da mniSvnelovnobis 0.05 done.
b. risi tolia η2-is mniSvneloba am eqsperimentSi? damokidebuli cvladis
cvalebadobis ra wili SeiZleba iyos axsnili damoukidebeli cvladis
donis codniT?
g. ipoveT 95-procentiani ndobis intervali ori populaciis saSualoTa
Soris sxvaobisaTvis.
d. aRwereT Tqveni Sedegebi am Tavis erT-erT qveTavSi mocemuli
magaliTis Sesabamisad, romelic exeboda t kriteriumis Sedegebis
warmodgenas.
7. mravali studenti gamocdis dros ganicdis SfoTvas, romelic Zalian uSlis
xels maT akademiur moswrebas. kolejis konsultantma wamoayena hipoTeza, rom
damamSvidebeli musikis mosmena gamocdisas Seamcirebs am SfoTvas. am hipoTezis
Sesamowmeblad, Terapevtma airCia 34 studenti damwyeb studentTa didi
jgufidan, vinc xasiaTdeboda Zlieri SfoTviT gamocdis dros. studentebi
SemTxveviTad iyvnen gadanawilebulni sakontrolo an eqsperimentul jgufSi,
sadac TiToeul jgufSi cdis pirTa toli raodenoba iyo warmodgenili.
gamocdis win sakontrolo jgufis cdis pirebi atarebdnen erT saaTs wynar
dasasvenebel oTaxSi, sadac isini kiTxulobdnen Jurnalebs. eqsperimentuli
jgufis cdis pirebi ki erT saaTs gamocdis win usmendnen damamSvidebel
musikas. gamocdis qulis saSualo sakontrolo jgufisaTvis iyo 73.2(sA1=8.4) da
eqsperimentuli jgufisaTvis ki 75.6(sA2=8.7). gamoiyeneT t kriteriumi, raTa
upasuxoT Semdeg kiTxvas: gansxvavdebian Tu ara jgufebi mniSvnelovnad
erTmaneTisagan?
a.gamoiyeneT orxmrivi kriteriumi da mniSvnelovnobis 0.05 done.
b. aRwereT Sedegebi am Tavis erT-erT qveTavSi mocemuli magaliTis
Sesabamisad, romelic exeboda t kriteriumis Sedegebis prezentacias.
8. maT, vinc cdilobs sxvebis darwmunebas, surT, rom ndoba moipovon. erT-erTi
hipoTezis Tanaxmad mosaubreebi, romlebic uSualod iqcevian, mimarTaven
Tvalebis kontaqts, iRimebian da simpaTias gamoxataven auditoriis mimarT
aRiqmebian rogorc ufro sando, vidre isini, vinc ar mimarTavs qcevis aseT
formebs. davuSvaT, am hipoTezis testirebisaTvis fsiqologma Seqmna 13 cdis
pirisagan Semdgari ori ekvivalenturi jgufi da TiToeuli jgufi ismenda
63 damarwmunebel mokle saubars. im pirebisaTvis, romlebic ar imyofebodnen
uSualo qcevis jgufSi (A1), mosaubre ar mimarTavda raime uSualo qcevas
saubrisas. xolo uSualo qcevis jgufSi (A2) mosaubre mimarTavda Tvalebis
kontaqts, iRimeboda da gamoxatavda auditoriis mimarT simpaTias. saubris
dasrulebis Semdeg, cdis pirebi afasebdnen mosaubres sandoobis skalaze,
romlis qulebic icvleboda 1-dan (sruliad arasando) 9-mde (Zalian sando).
davuSvaT, rom miiRes Semdegi qulebi:
uSualobis piroba (A)
uSualo qcevis gareSe (A1) uSualo qceviT (A2 )
4 7
5 6
3 9
4 8
6 8
7 7
7 6
4 4
3 5
5 8
6 7
3 6
5 8
a. gamoiyeneT t kriteriumi, raTa pasuxi gasceT Semdeg kiTxvebs: moaxdina Tu
ara gavlena sandoobis Sefasebaze mosaubris uSualobam? gamoiyeneT
ornapiriani kriteriumi da 0.05 mniSvnelovnobis done.
b. risi tolia η2 am kvlevaSi? damokidebuli cvladis dispersiis ra wili
SeiZleba mieweros damoukidebeli cvladis donis codnas?
g. ipoveT 95%-iani ndobis intervali ori populaciis saSualoebs Soris
gansxvavebisaTvis.
d. aRwereT Sedegebi am Tavis erT-erT qveTavSi mocemuli magaliTis
Sesabamisad, romelic exeboda t kriteriumis Sedegebis prezentacias.
1
X Tavi. erTfaqtoriani dispersiuli analizi cdis pirTa Soris sqemisaTvis
erTfaqtoriani mravaldoniani sqemis magaliTi jgufebs Soris da jgufSida cvalebadoba
F statistikis miReba
monacemTa dayofa
saSualos kvadratis miReba dayofili monacemebidan
F statistikis gamoTvla
statistikuri hipoTezebis Semowmeba FFrealuriT
MSA da MSSecdomis mniSvnelobaze moqmedi faqtorebi
F statistikiT miRebuli statistikuri gadawyvetilebani
daSvebebi cdis pirTa Soris erTfaqtoriani dispersiuli
analizisaTvis
F-is statistikuri mniSvnelovnobis interpretacia
mravaldonian sqemaSi mravaljeradi Sedarebis kriteriumebi
tukis HSD kriteriumi saboloo SedarebebisaTvis
Carevis efeqtis siZlieris gazomva
eta kvadratis gamoTvla dispersiuli analizisaTvis
tdamoukidebelsa da F Soris damokidebuleba
dispersiuli analizis Sedegebis prezentacia
2
t kriteriumma gagvacno statistikuri hipoTezis SemowmebaSi mravali
mniSvnelovani cneba bihevioristuli mecnierebis kvlevebisaTvis. Tumca, t
kriteriumi mniSvnelovani SezRudvebiT xasiaTdeba, is SeiZleba saSualoTa
Sesadareblad gamoviyenoT mxolod ori jgufisaTvis. magram bihevioristuli
mecnierebis mraval eqsperiments mravaldoniani sqema aqvs. erTfaqtoriani
mravaldoniani eqsperimenti (romelsac, agreTve, uwodeben calmxriv
mravaldonian sqemas) aris eqsperimenti erTi damoukidebeli cvladiT da
damoukidebeli cvladis sami an meti doniT. am sqemebisaTvis statistikuri
hipoTezis Semowmebis mizniT gamoiyeneba statistikuri meTodi, cnobili
rogorc dispersiuli analizi (romlis abreviaturaa ANOVA).
dispersiuli analizi yvelaze farTod gavrcelebuli statistikuri
kriteriumia fsiqologiur kvlevebSi. is SeiZleba gamoyenebuli iyos rogorc
cdis pirTa Soris, ise cdis pirTa Sida sqemebisaTvis da im sqemebisaTvis,
romelSic ori an meti damoukidebeli cvladi icvleba. erTfaqtoriani
sqemisaTvis dispersiuli analizi SeiZleba gamoyenebuli iyos tdamoukidebelis
nacvlad, roca damoukidebeli cvladis mxolod ori donis manipulireba xdeba.
am TavSi warmodgenilia dispersiuli analizi cdis pirTa Soris
erTfaqtoriani mravaldoniani sqemisaTvis, romelSic erTi damoukidebeli
cvladis manipulireba xdeba da am damoukidebeli cvladis TiToeuli done
miwodebulia cdis pirTa gansxvavebuli jgufisaTvis. misi sxva sqemebSi
gamoyeneba warmodgenilia me-11 da me-12 TavebSi.
erTfaqtoriani mravaldoniani sqemis magaliTi
stimulis ra aspeqtebi gansazRvravs, ramdenad kargad vimaxsovrebT Cven mas?
kvlevis erTi hipoTezis, damuSavebis hipoTezis mixedviT, rac ufro
srulyofilia mexsierebaSi warmodgenili stimulis xati, miT ufro ukeTesad
gvamaxsovrdeba is (Craik&Tulving, 1975) kvlevis meore hipoTezis, Zalisxmevis
erTfaqtoriani mravaldoniani sqema (One-factor multilevel design): eqsperimenti erTi damoukidebeli cvladiT da am damoukidebeli cvladis sami an meti doniT.
dispersiuli analizi (Analysis of variance ANOVA): statistikuri kriteriumi, romelic gamoiyeneba mravaldoniani sqemebis analizisaTvis.
3 hipoTezis Tanaxmad, stimulebi, romelTac ufro meti Zalisxmeva dasWirdaT
gadamuSavebisa da damaxsovrebisaTvis, ufro kargad gvaxsendeba, vidre
stimulebi, romelnic nakleb Zalisxmevas moiTxovdnen (Coulter, Coulter&Glover, 1984).
davuSvaT, Cven SevqmeniT Cveulebrivi sagnebis sami gansxvavebuli tipis naxati:
detaluri, eskizi da daumTavrebeli eskizi. damuSavebis hipoTezis mixedviT
detaluri naxatebi yvelaze metad damuSavebas moiTxoven mexsierebaSi da
amrigad, yvelaze kargad unda gvaxsendebodes, xolo eskizebi da dausrulebeli
eskizebi unda iTxovdnen ufro nakleb damuSavebas mexsierebaSi da amitom ufro
cudad unda gvaxsendebodes. Zalisxmevis hipoTeza ki sapirispiro
winaswarmetyvelebas akeTebs. dausrulebeli eskizebi moiTxoven yvelaze met
Zalisxmevas gadamuSavebisa da damaxsovrebis procesSi da amitom ufro kargad
gvaxsendeba. detaluri naxatebi ki moiTxoven yvelaze nakleb Zalisxmevas,
aqedan gamomdinare, isini yvelaze cudad gvaxsendeba.
imis gansazRvrisaTvis, vizualuri stimulebis damaxsovrebis romeli
hipoTezis winaswarmetyvelebaa ukeTesi, davuSvaT, rom Cven gamoviyeneT cdis
pirTa Soris erTfaqtoriani sqema da ganvaxorcieleT damoukidebeli cvladis
_ naxatis tipis manipulacia sam doneze: dausrulebeli eskizi(A1), eskizi(A2) da
detaluri naxatebi(A3). am SemTxvevaSi gavimeoroT me-9 TavSi ganxiluli
procedurebi, romlebic exeboda eqsperimentis sqemasa da Catarebas. Seqmnili
iyo sami ekvivalenturi jgufi, TiToeulSi 5 cdis piri Tanabrad iyo
gadanawilebuli. TiToeuli jgufis cdis pirebs uCvenes naxatebi, romlebzec
gamosaxuli iyo 30 Cveulebrivi sagani. erT jgufs miawodes detaluri naxati,
meore jgufs _ eskizi, xolo mesame jgufs _ dausrulebeli eskizi. TiToeuli
naxatis 4 wm-is ganmavlobaSi daTvalierebis Semdeg, cdis pirebs sTxoves
zepirad gaexsenebinaT rac SeiZleba meti naxati. cxrilSi #10.1 naCvenebia
TiToeuli piris mier gaxsenebuli sagnebis raodenoba, TiToeuli jgufis
saSualo da standartuli gadaxra, qulebis sruli saSualo. sruli saSualo,
romelic aRiniSneba X G-iT, aris 115 qulis saSualo am eqsperimentSi. moqmedebs
Tu ara gaxsenebuli naxatebis raodenobaze naxatis tipi? am kiTxvaze pasuxis
gasacemad, gamoviyenoT cdis pirTa Soris erTfaqtoriani dispersiuli analizi.
sruli saSualo (Grand mean): yvela qulis saSualo eqsperimentSi.
4 cxrili #10.1 gaxsenebuli sagnebis raodenobis hipoTeturi qulebi, rogorc naxatis tipis
funqcia. TiToeuli pirobisaTvis gamoTvlilia saSualo ( X A ) da standartuli
gadaxra (sA ). X G aris 15 qulis sruli saSualo.
naxatis tipi (faqtori A)
dausrulebeli
eskizi(A1)
eskizi(A2) detaluri(A3)
19 14 15
13 13 10
16 16 11
17 12 12
20 10 12
X A 17 13 12 X G=14
sA 2.74 2.24 1.87
jgufebs Soris da jgufSida cvalebadoba
erTfaqtoriani dispersiuli analizi eqsperimentul monacemTa srul
dispersias cdis pirTa Soris sqemisaTvis or nawilad yofs, romlebsac xSirad
uwodeben wyaroebs: (1) dispersia, romelic icvleba damoukidebeli cvladis
sistematur efeqtTan da jgufebis saSualoebs Soris SerCevis SecdomasTan
erTad da (2) dispersia, romelic icvleba mxolod jgufSida Secdomis
cvalebadobiT. miuxedavad amisa, dispersiul analizSi dispersias ewodeba
kvadratebis saSualo. kvadratebis saSualo (abreviatura MS) aris dispersiis
meore saxelwodeba da is dispersiul analizSi gamoiyeneba.
suraTi #10.1 gvexmareba am dispersiebis gaazrebaSi. am suraTis nawili(a)
gviCvenebs yvela 15 qulis sixSireTa ganawilebas cxrilidan #10.1. am qulebis
sruli cvalebadoba aris qulaTa dispersia sruli saSualos anu 14-is garSemo.
nawili (b), (g) da (d) warmogvidgens qulaTa sixSireTa ganawilebebs jgufis
SigniT daumTavrebeli eskizisaTvis, eskizisaTvis da detaluri naxatisaTvis.
kvadratebis saSualo (mean square): dispersiis saxelwodeba, romelic gamoiyeneba dispersiul analizSi.
jgufSida Secdomis dispersia (Within-groups error variance): monacemTa dispersia jgufSi, romelic jgufis saSualos garSemoa gamoTvlili.
5 miaqcieT yuradReba, rom Tumca cdis pirebi TiToeul jgufSi erTnair
pirobebSi imyofebodnen, qulebi jgufis SigniT jgufis saSualos garSemo
icvleba. jgufSida cvalebadoba mocemulia qulaTa dispersiiT maTi Sesatyvisi
jgufis saSualos garSemo. dispersiul analizSi am cvalebadobas ewodeba
jgufSida Secdomis dispersia an ubralod, Secdomis dispersia da MSSecdomis
saSualebiT izomeba. termini Secdoma gamoiyeneba imis aRsaniSnavad, rom jgufis
SigniT yvela qula erTmaneTis toli ar aris, maSinac ki, roca jgufSi cdis
pirs erTi da igive piroba miewodeba.
jgufTa Soris cvalebadoba aisaxeba jgufebis saSualoebis (e.i. 1AX , 2AX , da
3AX ) sruli saSualosagan _ X G-sagan gadaxraSi. jgufebis saSualoebis am
cvalebadobas uwodeben jgufTa Soris dispersias da is MSA-s saSualebiT
izomeba.
suraTi #10.1
(a) 15 qulis sixSireTa ganawileba cxrilidan #10.1. (b) dausrulebeli eskizis
jgufis(A1) qulebis sixSireTa ganawileba (g) eskizis jgufis (A2) qulebis
sixSireTa ganawileba. detaluri naxatis jgufis (A3) qulebis sixSireTa
ganawileba.
suraTi aklia ( a)Total variation in scores: Variation around grand mean - monacemTa
sruli cvalebadoba: monacemTa cvalebadoba sruli saSualos garSemo; frequency
- sixSire; number of items recalled, all groups - gaxsenebuli sagnebis raodenoba, yvela
jgufi;
b) Within groups error variation; variation of scores around group mean - jgufSida Secdomis
cvalebadoba: monacemTa cvalebadoba jgufis saSualos garSemo; frequency-
sixSire, number of items recalled, groupA1 - gaxsenebuli sagnebis raodenoba, A1 jgufi,
Bbetween-groups variation: variation of group mean from grand mean - jgufTa Soris
cvalebadoba: sruli saSualos garSemo jgufis saSualos cvalebadoba;
g) Within groups error variation; variation of scores around group mean - jgufSida Secdomis
cvalebadoba: monacemTa cvalebadoba jgufis saSualos garSemo; frequency-
sixSire, number of items recalled, groupA2 - gaxsenebuli sagnebis raodenoba, A2 jgufi,
jgufTa Soris dispersia (Between-groups variance): dispersia, romelic gamoiTvleba sruli
saSualos garSemo jgufebis saSualoebis cvalebadobis gamoyenebiT.
6
Bbetween-groups variation: variation of group mean from grand mean - jgufTa Soris
cvalebadoba: sruli saSualos garSemo jgufis saSualos cvalebadoba;
Within groups error variation; variation of scores around group mean - jgufSida Secdomis
cvalebadoba: monacemTa cvladebadoba jgusi saSualos garSemo; frequency-
sixSire, number of items recalled, groupA3 - gaxsenebuli sagnebis raodenoba, A3 jgufi,
Bbetween-groups variation: variation of group mean from grand mean - jgufTa Soris
cvalebadoba: sruli saSualos garSemo jgufis saSualos cvalebadoba.
jgufebis saSualoebi eqsperimentSi erTmaneTisagan SeiZleba ori mizeziT
gansxvavdebodes: (1) damoukidebeli cvladis gavleniT da (2) SerCevis Secdomis
gamo. amrigad, MSA asaxavs sistematur cvalebadobas jgufebis saSualoebs
Soris, romelic gamowveulia damoukidebeli cvladis gavleniTa da SerCevis
Secdomis cvalebadobiT. amrigad, ganxilul eqsperimentSi jgufebis
saSualoebi 1AX , 2AX , 3AX gansxvavdeba erTmaneTisagan, Tuki naxatis tipi
gavlenas axdens mis gaxsenebaze, magram isini gansxvavdebian erTmaneTisagan
ubralod SerCevis Secdomis gamoc.
sruli cvalebadobis dayofa jgufTa Soris dispersiad da Secdomis
dispersiad xdeba kriteriumis statistikis saSualebiT, romelsac ronald a.
fiSeris pativsacemad F-s uwodeben (1890-1962). is britaneli statistikosia,
romelmac SeimuSava dispersiuli analizis cnebebi. F statistika aris MSA
Sefardeba MSSecdomasTan da gamoisaxeba, rogorc F
SecdomaMSMSF A=
F statistikis mricxveli aris MSA, romelic zomavs rogorc damoukidebeli
cvladis gavlenas, aseve jgufebis saSualoebs Soris SerCevis Secdomas.
mniSvneli MSSecdoma zomavs mxolod jgufSida qulebis Secdomis cvalebadobas.
Tu damoukidebel cvlads ara aqvs gavlena, maSin jgufebis saSualoebs Soris
cvalebadoba ganpirobebuli iqneba mxolod SerCevis SecdomiT. amitom MSA da
MSSecdoma TiTqmis toli iqneba da F gaxdeba 1.00-is toli. magram Tu
damoukidebel cvlads aqvs gavlena, is gazrdis gansxvavebas SerCevis
saSualoebs Soris imaze metad, rac mosalodneli iqneboda mxolod SerCevis
Secdomis gamo. am SemTxvevaSi, mricxvelis jgufTa Soris dispersia _ MSA, meti
iqneba, vidre mniSvnelis jgufSida dispersia _ MSSecdoma da F Sefardeba iqneba
erTze meti. amrigad, F-is mniSvneloba izrdeba, roca Carevas aqvs gavlena.
7
Sesabamisad, F statistika uzrunvelyofs statistikuri hipoTezis Semowmebis
safuZvels.
F statistikis miReba Semdeg qveTavSi Cven ganvixilavT dispersiebs, romlebic Sedian F SefardebaSi,
raTa uzrunelvyoT ANOVA konceptualuri gaazreba. dispersiuli analizis
Sesruleba realur monacemebze rom mogesurvebinaT, gamosaTvlelad ufro
advil formulas gamoiyenebdiT, romelic mocemulia am Tavis damatebaSi an
kompiuteruli monacemebis analizis programaSi. es gamoTvliTi formula
axals arafers SegmatebT dispersiuli analizis gaazrebaSi, magram
gamoTvlisaTvis advilia.
F statistika moiTxovs monacemebis cvalebadobis or gazomvas, MSA da
MSSecdomis gazomvas. mexuTe TavSi Cven ganvsazRvreT dispersia, anu MS, rogorc
SS/n-1, sadac SS niSnavs kvadratebis jams. mniSvneli, n-1, Seesatyviseba
Tavisuflebis xarisxs, romelic monawileobs dispersiis Sefasebis miRebaSi.
amrigad, dispersiis formula, romelic uzrunvelyofs populaciis dispersiis
Caunacvlebel Sefasebas, aris MS= SS/df. MSA da MSSecdomis misaRebad Cven
Sesabamisi monacemebidan vpoulobT SSA-s da SSSecdomas da Semdeg kvadratebis
TiToeul jams Tavis df-ze vyofT. pirveli nabiji, anu kvadratebis jamebis
miReba iwyeba monacemTa dayofiT.
monacemTa dayofa
dispersiuli analizi yofs eqsperimentis monacemTa srul cvalebadobas
jgufebs Soris da jgufSida dispersiebad cdis pirTa monacemebis martivi
modelis pirobebSi. Tumca zemoT moyvanili magaliTi damoukidebeli cvladis
mxolod sam dones iyenebs, aq moyvanili modeli gamoiyeneba nebismieri
raodenobis doneebis cdis pirTa Soris erTfaqtoriani sqemis dispersiul
analizisaTvis.
cxrilis #10.1 monacemebi simboluradaa warmodgenili cxrilSi #10.2. es
simboloebi msgavsia im simboloebisa, romelic gamoiyeneboda wina paragrafis
magaliTis ganxilvisas. damoukidebeli cvladi ganisazRvreba, rogorc A
faqtori sami doniT A1, A2 da A3. aso a asaxavs A faqtoris doneebis raodenobas.
cxrilebisaTvis #10.1 da #10.2 a=3. cdis pirebis qulebi mocemulia Xij-is
saSulebiT, sadac qveSmiwerili indeqsi i warmoadgens cdis piris ganmsazRvrel
8
A faqtorisa da SerCevis SecdomiT ganpirobebuli cvalebadoba
jgufSida SecdomiT
ganpirobebuli cvalebadoba
ricxvs jgufis SigniT da qveSmiwerili indeqsi j warmoadgens jgufis
ganmsazRvrel ricxvs, A1, A2 da A3. magaliTad, X52 aris A2-jgufis mexuTe cdis
piri. cxrilSi #10.1, am cdis piris qula aris 10, anu X52=10. roca ar aris
aucilebeli ganisazRvros konkretuli cdis piri, ij indeqss vaSorebT da
viyenebT ubralod X-s. SerCevis saSualoebi damoukidebeli cvladis TiToeuli
donisaTvis aRiniSneba ubralod 1AX , 2AX da 3AX -iT an, zogadad, AX -iT. sruli
saSualo aRiniSneba X G-iT. qulaTa raodenoba damoukidebeli cvladis erT
doneze aRiniSneba nA-iT, xolo qulaTa sruli raodenoba eqsperimentSi
warmodgenilia N-iT. cxrilisaTvis #10-1, nA1=5, nA2=5, nA3=5 , N=15, 1AX =17, 2AX =13,
3AX =12 da X G=14. Tumca dispersiuli analizi ar moiTxovs qulaTa tol
raodenobas TiToeul pirobaSi, toli raodenoba ubralod amartivebs saWiro
gamoTvlebs. am qveTavis yvela magaliTi iyenebs qulaTa tol raodenobas
TiToeul pirobaSi.
dispersiuli analizi iRebs qulaTa srul cvalebadobas, anu srul raodenobas
imisa, riTac qula (anu Xij) gansxvavdeba qulaTa sruli saSualosagan (anu X G)
da yofs am srul sxvaobas or nawilad. erTi nawili _ jgufTa Soris
dispersia (e.i. MSA), icvleba damoukidebeli cvladisa da SerCevis Secdomis
gavleniT. meore nawili, romelic zomavs individualuri qulebis Secdomas
maTi jgufis saSualos garSemo, xdeba jgufSida Secdomis dispersia (e.i.
MSSecdoma). es modeli masSi qulaTa asaxvis mizniT SeiZleba aseTnairad
warmovidginoT.
sruli cvalebadoba
10.1
= +
qulaTa terminebiT es toloba SeiZleba Semdegnairad iyos dawerili:
(X- X G) = ( X A- X G) + (X- X A) 10.2
cdis piris eqsperimentuli jgufis cdis piris qulis
qulis gansxvaveba saSualos gansxvaveba gansxvaveba misi jgufis
sruli saSualosagan sruli saSualosagan saSualosagan
sruli cvalebadoba TiToeuli cdis piris qulaSi
a: A faqtoris doneebis raodenoba
nA: qulaTa raodenoba erTfaqtoriani sqemis erT doneze
9
SevamowmoT 10.2 tolobis TiToeuli nawili.
cxrili #10.2
qulaTa asoiTi aRniSvnebi cxrilSi #10.1 moyvanili hipoTeturi eqsperimentidan
damoukidebeli cvladis sami doniT, A faqtoriT. A faqtoris TiToeuli
donisaTvis saSualoebi warmodgenili arian X A da X G.
A faqtori
A1 A2 A3
X11 X12 X13
X21 X22 X23
X31 X32 X33
X41 X42 X43
X51 X52 X53
1AX 2AX 3AX X G
sruli cvalebadoba
qulis gansxvaveba sruli saSualosagan (e.i. X- X G) aris am qulis sruli
cvalebadoba. rogorc 10.2 toloba gviCvenebs, am cvalebadobis nawili
ganpirobebulia damoukidebeli cvladis gavleniT da nawili ki _ Secdomis
cvalebadobiT.
damoukidebeli cvladis gavleniTa da SerCevis SecdomiT ganpirobebuli
cvalebadoba
gansxvaveba jgufis saSualosa da srul saSualos Soris (e.i. X A- X G) icvleba
damoukidebeli cvladis gavleniTa da SerCevis SecdomiT. Sesabamisad, is Seva
MSA-is gamoTvlaSi.
damoukidebeli cvladis efeqti. dispersiuli analizi iyenebs gansxvavebas
TiToeuli jgufis saSualosa da srul saSualos Soris, raTa gaizomos
damoukidebeli cvladis gavlena. es gansxvaveba izrdeba damoukidebeli
cvladis gavlenis gazrdiT. am sakiTxis kargad gaazrebaSi ricxviTi
ilustracia dagvexmareba.
cxrili #10.3 warmoadgens eqsperimentis sam SesaZlo Sedegs, romelic
moicavs mxolod sam qulas TiToeuli sakvlevi jgufisaTvis. am cxrilis (a)
10 nawilSi ar aris warmodgenili Carevis efeqti. TiToeuli jgufis saSualo
sruli saSualos tolia. ufro metic, ori ( X A- X G) sxvaobidan TiToeuli
nulis tolia. Carevis efeqti, +2 emateba A2 qulebs cxrilis (b) nawilSi;
amrigad, arc erTi jgufis saSualo ar aris toli sruli saSualosi. Carevis
es efeqti, romelic +2-is tolia ( X A1- X G) sxvaobisaTvis warmodgenilia
mniSvnelobiT _1 da X A2- X G sxvaobisaTvis ki mniSvnelobiT +1. cxrilis (g)
nawili warmoadgens Carevis efeqts +6 , romelic emateba A2-is qulebs.
dispersiul analizSi es efeqti iyofa -3-is tol mniSvnelobad ( X A1- X G)
sxvaobisaTvis da +3-is tol mniSvnelobad ( X A2- X G) sxvaobisaTvis. miaqcieT
yuradReba, imas, rom es nimuSi gviCvenebs, aqvs Tu ara Carevas efeqti da es
iwvevs jgufis saSualoebis gansxvavebas erTmaneTisagan, is SeiZleba aisaxos
( X A- X G) sxvaobis mniSvnelobaSi, romelic SeiZleba gamoTvlili iyos
eqsperimentSi.
SerCevis Secdoma. garda Carevis efeqtisa, jgufebis saSualoebi, agreTve,
erTmaneTisagan gansxvavdebian SerCevis Secdomis gamo. maSinac ki, Tu
damoukidebel cvlads ara aqvs gavlena, mosalodnelia, rom jgufis saSualo
gansxvavebuli iqneba sruli saSualosagan mxolod SerCevis Secdomis gamo.
amrigad, damoukidebeli cvladis nebismieri efeqti eqsperimentSi warmoiqmneba
SerCevis Secdomis moqmedebis fonze.
jgufSida SecdomebiT gamowveuli cvalebadoba
individualuri qulis gansxvaveba Carevis saSualosagan (e.i. X - X A) Sedis
MSSecdomis gamoTvliT formulaSi. eqsperimentSi is mxolod jgufSida Secdomis
cvalebadobiT icvleba. faqtiurad, es gansxvaveba, romelic me-5 TavSi iyo
gamoyenebuli, rogorc standartuli gadaxrisa da dispersiis mricxveli,
jgufSida qulebs Soris Secdomis cvalebadobas zomavs. am gansxvavebis
ricxobrivi mniSvneloba ar icvleba damoukidebeli cvladis gamo. radgan
damoukidebeli cvladis nebismieri efeqti gulisxmobs TiToeuli cdis piris
qulebis toli sididiT gazrdas an Semcirebas, amitom eqsperimentuli jgufis
saSualo toli sididiT Seicvleba. amrigad, X - X A ucvleli darCeba
damoukidebeli cvladis efeqtisagan damoukideblad. magaliTad, meore cdis
pirisaTvis cxrilSi #10.3 A2-Si (e.i. qula, romelic warmodgenilia X22=13 (a)
svetSi), X - X A gansxvaveba rCeba (a) +1 cxrilis yvela svetisaTvis Carevis
11
efeqtis rogorc arsebobis, aseve ararsebobis SemTxvevaSi. amrigad, X - X A
mniSvneloba asaxavs mxolod qulaTa Soris Secdomis cvalebadobas Carevis
pirobaSi.
cxrili #10.3
cxrilSi naCvenebia, rom eqsperimentSi X A- X G Seesatyviseba damoukidebeli
cvladis efeqts. Carevis saSualoebi warmodgenilia X A-s saSualebiT da srul
saSualo qulaTa TiToeuli simravlisaTvis X G–iT. TiToeuli eqsperimentuli
jgufis saSualo da sruli saSualo naCvenebia qulaTa TiToeuli simravlis
qveviT.
(a) svetis qulebi; ar aRiniSneba Carevis efeqti
(b) svetis qulebi; aRiniSneba Carevis efeqti +2 toli
(c) svetis qulebi; aRiniSneba Carevis efeqti +6 toli
(aA) B(b) (g)
A1 A2 A1 A2 A1 A2
11 12 11 14 11 18
12 13 12 15 12 19
13 11 13 13 13 17
X A
12
12 12 14 12 18
X G 12 13 15
X A1- X G=0 X A1- X G= -1 X A1- X G= _3
X A2- X G=0 X A2- X G= +1 X A2- X G= +3
kvadratebis saSualos miReba monacemTa dayofis saSualebiT
monacemTa nawilebad dayofa, romelic zomavs jgufTa Soris da jgufSida
cvalebadobas, adgens MSA da MSSecdomis miRebis safuZvels, rac saWiroa F
statistikis misaRebad. gaixseneT, rom populaciis dispersiis Sefaseba
gamoiTvleba Semdegi formulis saSualebiT:
1)( 2
2
−−Σ
=n
XXs
12 miaqcieT yuradReba, rom TiToeuli termini 10-2 tolobaSi dispersiis
mricxvels hgavs. Tu s2-is nacvlad MS-s gamoviyenebT, xolo 2)( XX −Σ -is
nacvlad kvadratebis jams (SS), da (n-1)-s ki CavanacvlebT df-iT, es dispersia ase
gamoisaxeba
dfSSMS =
dispersiis es formula amtkicebs, rom Tu 10-2 tolobaSi warmodgenil
TiToeul sxvaobas kvadratSi aviyvanT TiToeuli cdis pirisaTvis da Semdeg
davajamebT yvela cdis pirisaTvis, Sedegi iqneba sxvaobebis kvadratebis jami
anu SS, kvadratebis saSualos mricxveli. 10-2 tolobidan sami aseTi SS termini
SeiZleba iyos miRebuli. termins X- X G-s mivyavarT SSsrulisaken, romelic asaxavs
qulaTa jamur gansxvavebas sruli saSualosagan. X A- X G gansxvavebas mivyavarT
SSA–sken, romelic jgufTaSoris cvalebadobis sazomia, im cvalebadobisa,
romelic ganpirobebulia damoukidebeli cvladisa da SerCevis SecdomiT.
sabolood, X - X A-s mivyavarT SS Secdomisaken, romelic aris jgufSida Secdomis
variaciis sazomi eqsperimentSi.
sami kvadratis jami
ganvixiloT, Tu rogor SeiZleba gamoviyenoT 10-2 toloba kvadratebis jamis
sapovnelad cxrilis #10.1 A1 jgufis pirveli cdis piris qulaTa dayofiT
(e.i.X11), romelic tolia 19-is. am qulisaTvis toloba 10-2 xdeba:
X11 - X G=( X A1- X G)+(X11- X A1).
Tu CavsvamT Sesatyvis ricxviT mniSvnelobebs X11, X A1 da X G , miviRebT
19-14=(17-14)+(19-17),
(+5) = (+3) + (+2).
dayofa gviCvenebs, rom es qula gansxvavdeba sruli saSualosagan (e.i. X11 _ X G=
+5), gansazRvruli sididiT da es gansxvaveba miRebulia im Carevis pirobis gamo,
romelSic imyofeba cdis piri (asaxulia X A1- X G = +3 tolobaSi) da miRebulia
Secdomis arasistematuri gavlenisa da SemTxveviTi faqtorebis gamo (asaxulia
X11 _ X A1= +2 tolobaSi).
SS-is misaRebad, TiToeuli cdis piris qula analogiurad aris
dayofili, rac naCvenebia cxrilSi #10.4 15 qulisaTvis, romelic aRebuli iyo
cxrilidan #10.1. cxrilis #10.4 garegnulma gadatvirTulobam SeiZleba
dagafrTxoT, magram gamoTvlebi aq mxolod mimatebas, gamoklebas, gamravlebas
13 moiTxovs da monacemebi nabij-nabij muSavdeba. dakvirvebiT SeasruleT
TiToeuli nabiji.
cxrili #10.4
kvadratebis jamis miReba cdis pirTa Soris dispersiuli erTfaqtoriani
analizisaTvis. gamoyenebulia qulebi, romelic mocemulia cxrilSi #10.1.
nabiji 1: qulaTa ganawileba 15 cdis pirisaTvis Semdegi tolobis gamoyenebiT
X- X G = ( X A- X G) + ( X - X A)
qulebi
A1-saTvis
X11 19-14 = (17-14) + (19-17)
X21 13-14 = (17-14) + (13-17)
X31 16-14 = (17-14) + (16-17)
X41 17-14 = (17-14) + (17-17)
X51 20-14 = (17-14) + (20-17)
qulebi
A2-saTvis
X12 14-14 = (13-14) + (14-13)
X22 13-14 = (13-14) + (13-13)
X32 16-14 = (13-14) + (16-13)
X42 12-14 = (13-14) + (12-13)
X52 10-14 = (13-14) + (10-13)
qulebi
A3-saTvis
X13 15-14 = (12-14) + (15-12)
X23 10-14 = (12-14) + (10-12)
X33 11-14 = (12-14) + (11-12)
X43 12-14 = (12-14) + (12-12)
X53 12-14 = (12-14) + (12-12)
cxrili #10.4
gagrZeleba:
nabiji 2: pirvel nabijSi mocemul sxvaobaTa povna TiToeuli cdis pirisaTvis ricxviTi
gansxvavebebis misaRebad
X- X G = ( X A- X G) + ( X - X A)
qulebi
A1-saTvis
X11 +5 = +3 + +2
X21 _1 = +3 + _4
X31 +2 = +3 + _1
X41 +3 = +3 + 0
14 X51 +6 = +3 + +3
qulebi
A2-saTvis
X12 0 = _1 + +1
X22 _1 = _1 + 0
X32 +2 = _1 + +3
X42 _2 = _1 + _1
X52 _4 = _1 + _3
qulebi
A3-saTvis
X13 +1 = _2 + +3
X23 _4 = _2 + _2
X33 _3 = _2 + _
X43 _2 = _2 + 0
X53 _2 = _2 + 0
cxrili #10.4
gagrZeleba:
nabiji 3: TiToeuli sxvaobis kvadrati
(X- X G)2 ( X A- X G)2 ( X - X A)2
qulebi
A1-saTvis
X11 25 9 4
X21 1 9 16
X31 4 9 1
X41 9 9 0
X51 36 9 9
qulebi
A2-saTvis
X12 0 1 1
X22 1 1 0
X32 4 1 9
X42 4 1 1
X52 16 1 9
qulebi
A3-saTvis
X13 1 4 9
X23 16 4 4
X33 9 4 1
X43 4 4 0
X53 4 4 0
15 nabiji 4. SeajameT yvela cdis piris sxvaobebis kvadratebis jami TiToeuli
nawilisaTvis
SecdomaSSSSSS
XXXXXX
A
A
n
i
a
jGA
n
i
a
jG
n
i
a
j
AAA
+=+=
−+−=− ΣΣΣΣΣΣ=
=
=
=
=
=
=
=
=
=
=
=
6470134
)()()( 25
1
3
1
25
1
3
1
25
1
3
1
nabiji 1. 15 cdis piris qulebi eqsperimentSi dayofilia 10-2 tolobis
mixedviT.
nabiji 2. TiToeuli qulisaTvis napovnia ricxviTi sxvaobebi.
nabiji 3. TiToeuli dadebiTi da uaryofiTi sxvaoba, romelic napovnia
nabijSi 2 aris kvadratSi ayvanili.
nabiji 4. TiToeuli svetis sxvaobebis kvadratebis mniSvnelobebi
dajamebulia, sul 15 qulis jami. Sedegi aris sami kvadratis jamis miReba:
SSsruli= 134,
SSA=70,
SSSecdoma= 64.
rogorc naCvenebia me-4 nabijSi, SSsruli maTematikurad Semdegi formuliTaa
warmodgenili.
25
1
3
1)( G
n
i
a
jXX
A
−ΣΣ=
=
=
=
ormagi jamis niSani aRniSnavs, rom kvadratebSi ayvanili sxvaobebi [e.i. (X- X G)2]
unda Sejamdes TiToeuli eqsperimentuli pirobis yvela cdis pirisaTvis
(pirveli cdis piridan, i=1-dan, ukanasknel cdis piramde, nA-mde, konkretul
jgufSi) da yvela jgufisaTvis(damoukidebeli cvladis pirveli donidan, j=1-
dan, ukanasknel donemde, a-mde). am magaliTSi TiToeul jgufSi aris 5 qula
(nA=5) da sami jgufi (a=3); amrigad, dajamebis zRvari aris i=1-dan nA=5-mde da
j=1-dan a=3-mde, an
ΣΣ==
5
1
3
1 ij
radgan yovelTvis yvela cdis piris da yvela jgufis Sedegebs vajamebT,
Cveulebriv gamoiyeneba ormagi jamis niSani sazRvrebis gareSe.
16
analogiurad, ASS -is misaRebad, eqsperimentuli jgufis saSualosa da
srul saSualos Soris sxvaobebis kvadratebi yvela cdis pirisaTvis jamdeba
eqsperimentSi. Sesabamisad,
ASS = 2)( GA XX −ΣΣ
sabolood, SSSecdoma miiReba TiToeuli cdis piris qulisa da misi jgufis
saSualos sxvaobebis kvadratebis SejamebiT, anu
SSSecdoma = 2)( AXX −ΣΣ .
mniSvnelovani kavSiri arsebobs kvadratebis jamebs Soris, rac naCvenebia
cxrilis #10.4 qveda nawilSi.
SSsruli = SSA+ SSSecdoma (10-3)
amrigad, eqsperimentis qulebis sruli cvalebadoba, SSsruli, aris sistematuri
cvalebadobis Sedegi, romelic xdeba gansxvavebul eqsperimentul pirobebSi
myof jgufebs Soris, SSA (zogjer aRiniSneba, rogorc SSjgufTaSorisi an SSCarevis)
da im Secdomis cvalebadobis Sedegi, romelic xdeba jgufebs SigniT, SSSecdoma
(zogjer aRiniSneba, rogorc SSjgufSida).
Tavisuflebis xarisxebis povna
saboloo nabiji MSA da MSSecdomis povnaSi aris TiToeuli SS-is gayofa mis df-
ze.
gaixseneT, rom Tavisuflebis xarisxi aris qulaTa raodenoba, romelic
Tavisufalia cvalebadobaSi statistikis gamoTvlisas. am ganmartebidan
advilad SeiZleba ganvsazRvroT Tavisuflebis xarisxi 10-3 tolobis
kvadratebis TiToeuli jamisaTvis.
sruli Tavisuflebis xarisxi. imisaTvis, rom vipovoT SSsruli, unda iyos
cnobili sruli saSualo, X G. radgan kvadratebis sruli jami efuZneba
eqsperimentis TiToeuli qulis gansxvavebas sruli saSualosagan (e.i. GXX − ),
maSin Tavisuflebis xarisxi unda iyos erTiT naklebi qulaTa srul
raodenobaze. magaliTad, cxrilSi #10.1 sruli saSualo 15 qulisaTvis 14-ia.
SSsrulis misaRebad 14 unda gamoakldes am 15 qulidan TiToeuls. am 15
sxvaobis jami unda iyos 0-is toli. amrigad, nebismieri 14 qula 15-dan
Tavisufalia Tavisi cvalebadobisas, magram me-15 qula xdeba fiqsirebuli, Tu
17
X G cnobilia. Sesabamisad SSsrulisaTvis df tolia 14-is. saerTod, sruli
Tavisuflebis xarisxi anu df sruli gasaanalizebel qulaTa raodenobaze erTiT
naklebis ricxvis tolia. is ase Caiwereba:
dfsruli = N-1
sadac N aris qulaTa sruli raodenoba eqsperimentSi.
Tavisuflebis xarisxi ASS -saTvis. ASS gamoiTvleba eqsperimentuli jgufebis (mag.
1AX , 2AX da 3AX ) saSualosa da srul saSualos Soris sxvaobebidan. sami
damoukidebeli cvladis donis mqone kvlevis sqemisaTvis da qulaTa toli
raodenobisaTvis TiToeul eqsperimentul jgufSi, Tu X G cnobilia, maSin
mxolod ori jgufis saSualo SeiZleba Seicvalos. magaliTad, Tu vici, rom
X G=14, 1AX =17, da 3AX =12, rogorc cxrilSi #10.1, maSin 2AX cnobilia. is unda
iyos 13-is toli. 2AX -is nebismieri sxva mniSvneloba ar iqneba X G=14-is
Sesabamisi. Sesabamisad, am magaliTSi SSA-saTvis 2 df arsebobs.
saerTod, df SSA-saTvis an dfA damoukidebeli cvladis doneebis
raodenobaze erTiT naklebi ricxvis tolia, rac Semdegnairad Caiwereba:
df SSA-saTvis = dfA=a-1
sadac a aris A damoukidebeli cvladis doneebis raodenoba.
Tavisuflebis xarisxi SSSecdomisaTvis. SSSecdoma efuZneba TiToeuli jgufis
saSualos, AX -sa da eqsperimentuli jgufebis SigniT yvela cdis piris qulebs
Soris sxvaobas. amitom, damoukidebeli cvladis TiToeuli donisaTvis AX -s
gansazRvris Semdeg, mxolod nA-1 qula aris Tavisufali cvalebadobaSi am
donis SigniT. cxrilSi #10.1, Tu AX cnobilia, am xuTi qulidan mxolod oTxi
aris Tavisufali Tavis cvalebadobaSi TiToeuli eqsperimentuli pirobis A1, A2
da A3-is SigniT. radgan damoukidebeli cvladis sami done gvaqvs da 4 qula
Tavisufalia cvalebadobisas TiToeuli donis SigniT, df SSSecdomisaTvis 12-is
tolia. (4 df A1-saTvis, plus 4 df A2-saTvis , plus 4 df A3-saTvis, rac tolia 12-
is).
saerTod, sadac a aris damoukidebeli cvladis doneebis raodenoba da
arsebobs nA qula damoukidebeli cvladis TiToeul doneze, df SSSecdomisaTvis
18
an dfSecdoma=a(nA-1). radgan (a)(n) qulaTa sruli raodenobis (N) tolia, dfSecdoma
SeiZleba, agreTve gamoisaxos, rogorc N-s gamoklebuli a, anu,
df SSSecdomisaTvis = dfSecdoma = N-a.
Tavisuflebis xarisxis aditiuroba. Tavisuflebis xarisxi garkveuli azriT
aditiuria, iseve rogorc misi Sesatyvisi kvadratebis jamis mniSvnelobebi.
amrigad,
dfsruli=dfA+ dfSecdoma 10-4
es damokidebuleba SenarCunebulia magaliTSic, dfsruli=14, dfA=2, dfSecdoma=12.
kvadratebis saSualos povna SS da df saSualebiT
F statistika aris ori MSA da MSSecdomis kvadratebis saSualos Sefardeba,. ori
saWiro MS mniSvneloba miRebulia SS -dan, romelic, Tavis mxriv, miRebulia 10-
3 tolobaSi da df mniSvnelobidan, romelic dakavSirebulia TiToeul SS -Tan.
konkretulad, kvadratebis saSualo aris
MSAA
A
dfSS
=
da
MSSecdoma=Secdoma
Secdoma
dfΣεχδχμα
SS
Tumca SesaZlebelia MSsrulis miReba SS srulis gayofiT dfsrulze, es mniSvneloba
ar uzrunvelyofs sasargeblo informacias dispersiul analizSi da
Cveulebriv ar gamoiTvleba. cxrilis #10.1 qulebisaTvis, Cven vipoveT SS A=70
da SS Secdoma=64 (ix. cxrili #10.4), agreTve, dfA=2 da dfSecdoma=12, amitom,
MSA 352
70===
A
A
dfSS
da
MSSecdoma= 333.51264
df==
Secdoma
SecdomaSS
F statistikis gamoTvla
am Tavis dasawyisSi F statistika gansazRvruli iyo, rogorc
F=SecdomaMS
MS A
19
cxrilSi #10.1 mocemuli qulebisaTvis: MSA=35 da MSSecdoma=5.333, amrigad
56.6
563.6
=
==
realuri
realuri
FF
an5.333
35
F
F
sadac Frealuri gamoyenebulia F-is im mniSvnelobis aRsaniSnavad, romelic
realurad aris miRebuli gaanalizebuli qulebidan. miaqcieT yuradReba, rom
MSA da MSSecdoma efuZneba kvadratebis jamis mniSvnelobebs, romelic
yovelTvis unda iyos dadebiTi, F agreTve, unda iyos dadebiTi. F-is uaryofiTi
mniSvneloba gviCvenebs, rom gamoTvlebSi Secdomaa daSvebuli.
dispersiuli analizis ricxviTi mniSvneloba xSirad warmodgenilia
cxrilis saxiT, romelic gansazRvravs cvalebadobis wyaroebs; SS , df da MS
mniSvnelobebs da Frealuris mniSvnelobas. cxrili #10.5 gviCvenebs, Tu rogor
aris organizebuli aseTi sistematizebuli cxrili im ricxviTi mniSvnelobebis
gamoyenebiT, romelic miRebulia cxrilis #10.1 monacemebidan. F-is mniSvneloba
ori aseuliTaa gadaweuli sistematizebul cxrilSi, agreTve SS da MS
mniSvnelobebic gadaweulia ori an meti aseuliT, rac saWiroa damrgvalebis
Secdomis minimumamde miyvanisaTvis. cxrili #10.6 ajamebs SS , df da MS-isaTvis
ganmartebiT formulas.
F statistikis miReba da misi mniSvnelobis gamoTvla xSirad iwvevs
mniSvnelovan kamaTs. ufro xSirad praqtikaSi misi mniSvneloba gamoiTvleba
kompiuteruli programiT an gamoTvliTi formuliT. Cveni mizani ar aris
gamoTvla gavamartivoT, aramed unda vuCvenoT rogor gamoiyvaneba F
statistika qulaTa erTobliobidan. Cven ganvixilavT, rogor unda avxsnaT da
rogor unda gamoviyenoT es informacia momavalSi.
cxrili #10.5
dispersiuli analizis ricxviTi Sejameba cxrilSi #10.1 mocemuli
qulebisaTvis
wyaro SS df MS Fa
naxatis tipi (A) 70 2 35 6.56
Secdoma 64 12 5.333
sul 134 14
*p< 0.05 a F-is statistikurad mniSvnelovani gansxvaveba, romelic naCvenebia albaTobis
20
doniT sqolioSi sistematizebul cxrilSi, e.i., *p< 0.05
cxrili #10.6
cdis pirTa Soris erTfaqtoriani dispersiuli analizis ganmartebiTi
formulis Sejameba
wyaro SS dfa MS F
faqtori A 2)( GA XX −ΣΣ a-1 SS A/ dfA
MSA /MSSecdoma
Secdoma 2)( AXX −ΣΣ N-a SS Secdoma/
dfSecdoma
sul 2)( GXX −ΣΣ N-1 ar gamoiTvleba
aa= A faqtoris doneebis raodenoba, N= qulaTa srul raodenobas.
SeamowmeT Tqveni codna X-1
1. gansazRvreT: a, jgufTaSoris dispersia, dfA, dfSecdoma, dfsruli, faqtori A, MSA
MSSecdoma, N, nA, erTfaqtoriani mravaldoniani sqema, SS A, SS Secdoma, SS sruli,
jgufSida dispersia, GXX − , GA XX − , AXX − .
2. rogoria t kriteriumis SezRuduloba eqsperimentuli monacemebis
analizisaTvis?
3. romeli kriteriumis statistika gamoiyeneba dispersiuli analizis dros?
4. dawereT F statistikis formula.
5. romel or wyarod yofs erTfaqtoriani dispersiuli analizi qulaTa srul
cvalebadobas eqsperimentSi?
6. eqsperimentatori iyenebs cdis pirTa Soris erTfaqtorian eqsperiments A A
faqtoris oTxi donisaTvis: A1, A2, A3 da A4 . aq iyo 11 gansxvavebuli cdis piri
TiToeul doneze. risi toli iqneba a, nA1 da N mniSvnelobebi?
7. dawereT sruli toloba qulaTa dayofisaTvis, romelic miRebulia cdis
pirTa Soris erTfaqtorian eqsperimentSi.
8. axseniT, ratom icvleba GA XX − mniSvneloba damoukidebeli cvladis
efeqtTan erTad eqsperimentSi.
9. axseniT, ratom icvleba AXX − mniSvneloba Secdomis cvalebadobasTan erTad
eqsperimentSi.
21
10. daasruleT Semdegi tolobebi cdis pirTa Soris erTfaqtoriani
dispersiuli analizisaTvis.
a. 2)( GXX −ΣΣ =
b. SS sruli=
g. dfA=
d. dfSecdoma=
e. MSA=
v. MSSecdoma=
z. F=
11. risi tolia SS srulis mniSvneloba, Tu SS A=50 da SS Secdoma=100?
12. risi tolia SS A mniSvneloba, Tu SS sruli=75 da SS Secdoma=50?
13. risi tolia dfsruli, Tu dfA=4 da dfSecdoma=75?
14. risi tolia MSA, TuU SS A=100 da dfA=4?
15. risi tolia MSSecdoma, Tu SS Secdoma=760 da dfSecdoma=76?
16. bihevioristi mecnierebi iyeneben cdis pirTa Soris erTfaqtorian A sqemas A
faqtoris oTxi doniT. TiToeul doneze aris 11 cdis piri. risi tolia dfA,
dfSecdoma, dfsruli ?
17. CaatareT eqsperimenti damoukidebeli cvladis ori doniT, sami cdis piriT
TiToeul jgufSi da miiReT Semdegi qulebi:
A faqtori
A1 A2
30 34
32 36
28 32
dayaviT es qulebi cxrilSi #10.4 naCvenebi midgomis Sesabamisad.
ipoveTY SS sruli, SS A, SS Secdoma da df TiToeuli SS -saTvis. Semdeg daasruleT
Semajamebeli cxrilis analizi.
18. CaatareT eqsperimenti damoukidebeli cvladis sami doniT, sami cdis piriT
TiToeul jgufSi da miiReT Semdegi qulebi:
A faqtori
A1 A2 A3
30 34 37
22 32 36 35
28 32 33
dayaviT es qulebi cxrilSi #10.4 naCvenebi midgomis Sesabamisad.
ipoveTY SS sruli, SS A, SS Secdoma da df TiToeuli SS -saTvis. Semdeg daasruleT
Semajamebeli ricxviTi cxrilis analizi.
19. Semdegi ori cxrili aris dausrulebeli Semajamebeli cxrili cdis pirTa
Soris erTfaqtoriani dispersiuli analizisaTvis. davuSvaT, cdis pirTa
raodenoba damoukidebeli cvladis TiToeul doneze tolia. SeavseT
gamotovebuli mniSvnelobebi TiToeuli cxrilSi SS da df Soris kavSiris
gamoyenebiT, romelic mocemulia 10-3 da 10-4 tolobebSi, da cxrilSi #10.6
warmodgenil formulaSi. Semdeg upasuxeT am SekiTxvebs TiToeuli
cxrilisaTvis.
a. damoukidebeli cvladis ramdeni done unda iyos gamoyenebuli?
b. ramdeni cdis piris monacemi unda iyos gazomili TiToeul eqsperimentul
jgufSi?
g. ramdeni cdis piri monawileobs kvlevaSi?
cxrili #1
wyaro SS df MS F
faqtori
A
50 1 _ _
Secdoma 260 26 _
sul _ _
cxrili #2
wyaro SS df MS F
faqtori
A
_ 3 6 _
Secdoma _ 76 _
sul 170 79
23
statistikuri hipoTezis Semowmeba F-iT
Frealuris mniSvneloba gvaZlevs SesaZleblobas gadavwyvitoT mniSvnelovnad
gansxvavdeba Tu ara eqsperimentuli pirobebis saSualoebi erTmaneTisagan. imis
gasagebad, rogor xdeba am daskvnis gamotana, aucilebelia mokled
mimovixiloT is faqtorebi, romlebic gavlenas axdenen MSA da MSSecdomaze da
kavSirze MSA da MSSecdomas Soris.
MSA da MSSecdomis mniSvnelobaze moqmedi faqtorebi
MSA
MSA-s mniSvneloba mocemulia formuliT
,MSAA
A
dfSS
=
sadac
ASS = 2)( GA XX −ΣΣ
rogorc qulis dayofis Sesaxeb ganxilvaSia naCvenebi, nebismier eqsperimentSi
cvalebadobis ori wyaro moqmedebs, jgufis saSualosa da srul saSualos
Soris sxvaobis xarisxze da amrigad, SSA-s mniSvnelobaze nebismier
eqsperimentSi: (1) damoukidebeli cvladis gavlena da (2) SerCevis Secdoma.
aqedan MSA, agreTve, icvleba damoukidebeli cvladis gavleniT da SerCevis
SecdomiT. radgan MSA izrdeba damoukidebeli cvladis gavlenis zrdasTan
erTad, is gamoiyeneba rogorc sistematuri dispersiis sazomi, romelic
damoukidebeli cvladis mieraa Seqmnili.
MSSecdoma
MSSecdoma miRebulia Semdegi formulidan
SecdomadfXX A
2)( −ΣΣ
da zomavs mxolod jgufSida Secdomis cvalebadobas eqsperimentSi. is ar aris
damoukidebeli cvladis gavlenis qveS. rogorc ufro adre iyo aRniSnuli, es
24 imis gamo xdeba, rom qulebSi nebismier sistematur cvlilebas Tan axlavs
Carevis efeqtiT ganpirobebuli jgufis saSualos Sesatyvisi zrda an kleba.
amrigad, es gansxvaveba cdis piris qulasa da jgufis saSualos Soris,
)( AXX − ar icvleba Carevis efeqtiT. amitom, MSSecdoma zomavs eqsperimentSi
mxolod jgufSida Secdomis cvalebadobas qulebSi. faqtorebi, romlebic
moqmedeben TiToeul kvadratebis saSualoze cxrilSi #10.7 Sejamebuli.
cxrili #10.7
faqtorebi, romlebis moqmedeben MSA da MSSecdomaze eqsperimentSi.
kvadratebis saSualo ganicdis gavlenas Semdegi mizezebiT
MSA damoukidebeli cvladis gavlena
SerCevis Secdoma
MSSecdoma jgufSida Secdomis cvalebadoba
MSA da MSSecdomis kavSiri
Caunacvlebeli Sefasebebi. orive MSA da MSSecdomebi populaciis qulaTa
dispersiis, σ2-is Caunacvlebeli Sefasebebia. isini Caunacvlebelia, radgan
TiToeuli MS-is misaRebad kvadratebis jami iyofa Tavisuflebis xarisxebze da
ara qulaTa realur raodenobaze, rogorc es xdeba dispersiis gamoTvlisas.
rogorc Caunacvlebeli Sefaseba, MS ar unda iyos sistematurad mcire da arc
meti σ2-ze, roca mxolod Secdomis cvalebadobaa pasuxismgebeli monacemTa
cvalebadobaze, romlis safuZvelze xdeba Sefaseba.
damoukidebeli Sefasebebi. MSA da MSSecdomebi aris, agreTve, populaciis
dispersiis damoukidebeli Sefasebebi. damoukidebeli Sefaseba niSnavs, rom
nebismieri saSualos kvadratis mniSvneloba SeiZleba Seicvalos iseTnairad,
rom masze ar imoqmedebs meores mniSvneloba. es sakiTxi warmodgenili iyo im
ganxilvaSi, romelic exeboda SS mniSvnelobaze moqmed faqtorebs. MSA da
MSSecdomebis am maxasiaTeblebs mivyavarT F mniSvnelobis Semdeg mosalodnel
Sedegebamde.
• F-is mosalodneli mniSvneloba, roca damoukidebel cvlads ara aqvs
gavlena: Tu damoukidebel cvlads ara aqvs gavlena, maSin
eqsperimentSi mxolod Secdomis cvalebadobaa mosalodneli. am dros
MSA da MSSecdoma Seafaseben mxolod Secdomis cvalebadobas da
25
amrigad, iqneba TiTqmis toli. Sesabamisad, F-is mniSvneloba iqneba
TiTqmis 1-is toli.
• F-is mosalodneli mniSvneloba, roca damoukidebel cvlads aqvs
gavlena: roca damoukidebeli cvlads aqvs gavlena, maSin MSA
Seesatyviseba sistematur cvalebadobas, romelSic wvlili Seaqvs
rogorc damoukidebel cvlads, ise SerCevis Secdomas. Sedegad, MSA
ufro didi iqneba, vidre MSSecdoma da Frealuri 1-ze meti iqneba. es
damokidebuleba statistikuri hipoTezis SemowmebaSi F-is gamoyenebis
saSualebas gvaZlevs.
F statistikiT miRebuli statistikuri gadawyvetilebebi
F-is statistikur Semowmebas mosdevs CvenTvis cnobili Semdegi nabijebi:
• nulovani hipoTezis, H0-isa da alternatiuli hipoTezis H1-is
formulireba;
• H0-is WeSmaritebis daSvebis pirobiT F SerCeviTi ganawilebis miReba.
es ganawileba mocemulia cxrilSi #a-3;
• mniSvnelovnobis donis arCeva;
• F kritikuli mniSvnelobis (Fkritikulis) dadgena F-is SerCeviTi
ganawilebidan, romelic mocemulia cxrilSi #a-3;
• uaryofis areebi ganlagebulni arian F-is SerCeviT ganawilebaze;
• F Sefardeba, romelic ganisazRvreba rogorc Frealuri, gamoTvlilia
SerCevis monacemebidan;
• H0–is uaryofis Tu miRebis gadawyvetileba keTdeba imis safuZvelze,
xvdeba Tu ara Frealuri uaryofis areSi.
statistikuri hipoTezebi
nulovani hipoTeza. eqsperimentisaTvis damoukidebeli cvladis sami doniT,
nulovani hipoTezis dispersiuli analiziT Semowmeba niSnavs imas, rom im
populaciebs, romlebidanac iyo aRebuli sami SerCeva, aqvs erTi da igive
saSualoebi da iwereba Semdegnairad:
H0: µ A1=µA2=µA3
es nulovani hipoTeza warmoadgens situacias, romelic im SemTxvevaSi Cndeba,
Tu damoukidebel cvlads ara aqvs gavlena. Tu H0 WeSmaritia eqsperimentSi,
26 maSin jgufebis saSuloebs Soris nebismieri gansxvaveba mxolod SerCevis
SecdomiTaa ganpirobebuli.
populaciis saSualoebis raodenoba, romelic gansazRvrulia nulovan
hipoTezaSi, yovelTvis Seesatyviseba damoukidebeli cvladis doneebis
raodenobas. Tu eqsperimenti moicavs damoukidebeli cvladis 5 dones, maSin
Semdegnairad iwereba:
H 0: µ A1=µA2=µA3=µA4=µA5
alternatiuli hipoTeza. alternatiuli hipoTeza H1, aris
H1: yvela µ A ar aris erTmaneTis toli,
populaciis saSualoebis im raodenobis miuxedavad, romelic CarTulia
nulovan hipoTezaSi. alternatiuli hipoTeza ayalibebs iseT situacias,
romelic maSin warmoiqmneba, roca damoukidebel cvlads aqvs gavlena.
F SerCeviTi ganawileba
H0-is uaryofis Tu miRebis gadawyvetileba damokidebulia imaze, Tu ramdenad
iSviaTi an naklebad mosalodneli iqneba Frealuris mniSvneloba, Tu H0 ar aris
WeSmariti. statistikis SerCeviTi ganawileba gvawvdis statistikis
mniSvnelobis albaTobas, roca H0 WeSmaritia. amrigad, Frealuris garkveuli
mniSvnelobis miRebis albaToba, roca H0 WeSmaritia, gansazRvrulia F
statistikis SerCeviTi ganawilebiT, magram ara mxolod F-is SerCeviTi
ganawilebiT, Tavad es ganawileba damokidebulia damoukidebeli cvladis
doneebis raodenobaze da qulaTa raodenobaze TiToeul eqsperimentul jgufSi.
kerZod, F SerCeviTi ganawileba icvleba Tavisuflebis xarisxis cvlilebasTan
erTad F statistikis mricxvelisaTvis (e.i. dfA) da mniSvnelisaTvis (e.i.
dfSecdoma).
rogorc nimuSi, F SerCeviTi ganawileba magaliTis saxiT moyvanil
eqsperimentSi 2 df-is toli mricxveliTa (e.i. 2 df dfA-saTvis) da 12-is toli
Tavisuflebis xarisxis mniSvneliT (e.i. 12 df dfSecdomisaTvis) naCvenebia suraTze
#10.2. es ganawileba dadebiTad asimetriulia da misi umciresi mniSvneloba
nulia; F iqneba nulis toli, Tu SerCevis saSualoebi erTmaneTis toli iqneba
da MSA mniSvneloba _ nulis toli. F-is yvelaze mosalodneli mniSvneloba
daaxloebiT 1-ia, Tu H0 WeSmaritia, maSin MSA unda iyos daaxloebiT
MSSecdomis toli. ar arsebobs zeda zRvari im mniSvnelobebisaTvis, romelsac F-
27
ma SeuZlia miaRwios. F-ma SeiZleba miaRwios nebismier mniSvnelobas 0-s da
dadebiT usasrulobas Soris. radgan F mniSvnelobis miRebis albaToba 0-sa da
dadebiT usasrulobas Soris 1-is tolia, ganawilebis qveS mdebare arec erTis
tolia.
mniSvnelovnobis donis arCeva
mniSvnelovnobis done aris albaTobis mniSvneloba, romelic iZleva nulovani
hipoTezis uaryofis pirobas statistikur kriteriumSi. rogorc ganvixileT,
mniSvnelovnobis doned bihevioristuli mecnierebis kvlevaSi Cveulebriv
aRebulia an α=0.05 an α=0.01. uaryofis aris mdebareoba 0.05 mniSvnelovnobis
donisaTvis naCvenebia suraTze #10.2.
suraTi #10.2
SerCeviTi Teoriuli ganawileba 2 da 12 Tavisuflebis xarisxisaTvis. naCvenebia
uaryofis are α=0.05-saTvis. Fkritikuli cxrili #a-3-dan aris 3.89.
suraTi aklia (relative frequency - SefardebiTi sixSire Rejection region - uaryofis are, value of F_ F-is mniSvneloba, critical value _ kritikuli mniSvneloba)
uaryofis aris mdebareobis gansazRvra F-isaTvis
uaryofis are warmoadgens F-is iseT mniSvnelobebs, romelTa albaToba tolia
an naklebia α-ze, Tu H0 WeSmaritia. amrigad, uaryofis are gansazRvravs
Frealuris iseT mniSvnelobebs, romlebic akmayofileben H0-is uaryofis pirobas.
uaryofis are FrealurisaTvis yovelTvis F SerCeviTi ganawilebis marjvena kudSia,
rogorc suraTzea #10.2 naCvenebi. SegviZlia vnaxoT, ratom aris uaryofis are
ganlagebuli aq, Tu gavixsenebT, rom Frealuris mniSvneloba damoukidebeli
cvladis gavleniT icvleba.
Tu damoukidebel cvlads ara aqvs gavlena da aqedan gamomdinare, H0
WeSmaritia, maSin mosalodnelia, rom MSA da MSSecdoma TiTqmis erTmaneTis
toli iqneba; amis Sedegad, Frealuri TiTqmis erTis toli unda iyos. Tu
damoukidebel cvlads aqvs gavlena da amitom H0 ar aris WeSmariti, maSin MSA
iqneba MSSecdomaze meti da Frealuri gaxdeba erTze meti. imisaTvis, rom uarvyoT
nulovani hipoTeza, Frealuri unda iyos sakmarisad meti erTze, ase rom aseTi
Frealuris albaToba nulovani hipoTezis WeSmaritebis pirobebSi arCeul alfa
28
doneze toli an naklebi iqneba. amitom F-is uaryofis are yovelTvis mdebareobs
F-is ufro did mniSvnelobebs Soris SerCevis ganawilebis marjvena napirze.
F-is kritikuli mniSvnelobebi, romlebic (aRiniSneba, rogorc Fkritikuli)
gansazRvraven uaryofis aris qveda sazRvrebs α=0.05-saTvis da 0.01-saTvis
warmodgenilni aris cxril #a-3-is danarTSi. cxrili #a-3a nawili 0.05
mniSvnelovnobis donisaTvis warmodgenilia cxrilSi #10.8. am cxrilSi F
Sefardebis mricxvelis Tavisuflebis xarisxebi (e.i. dfA) mocemulia cxrilis
zeda striqonSi. mniSvnelis Tavisuflebis xarisxebi (e.i. dfSecdoma) ki mocemulia
cxrilis marcxena svetSi.
Fkritikulis mniSvnelobis sapovnelad zemoT xsenebuli magaliTisaTvis,
romlis mricxvelis Tavisuflebis xarisxi 2-ia da mniSvnelis _ ki 12, unda
vipovoT sveti, romlis Tavisuflebis xarisxi 2-ia da Semdeg vpoulobT
striqons 12 df-isaTvis. es sveti da striqoni gadaikveTeba 3.88-is tol
mniSvnelobasTan (cxrilSi gamuqebulia). es mniSvneloba aris Fkritikuli
da gansazRvravs uaryofis ares FrealurisTvis, Tavisuflebis xarisxebiT 2 da 12,
rogorc naCvenebia suraTze #10.2. Fkritikulis nebismieri mniSvneloba
2 da 12 Tavisuflebis xarisxiT da mniSvnelobiT, romelic 3.88-is tolia an
metia, uaryofis areSi mdebareobs. F-is mniSvnelobebi, romlebic 3.88-is toli an
metia, xdeba 100 eqsperimentidan 5 an ufro naklebjer, roca H0 WeSmaritia.
gadawyvetilebebi statistikuri hipoTezebis Sesaxeb
Frealuris mniSvnelobebi uzrunvelyofs statistikuri hipoTezebis Sesaxeb
gadawyvetilebebis gamotanis safuZvels. Tu Frealuri xvdeba uaryofis areSi,
maSin H0 iqneba uaryofili da Tu H0 iqneba uaryofili, maSin H1 unda iyos
miRebuli. SerCevis saSualoebs Soris statistikurad mniSvnelovani
gansxvavebaa.
cxrili #10.8
Fkritikulis mniSvnelobebi α=0.05-saTvis.U
mniSvnelis
Tavisuflebis
xarisxi
mricxvelis Tavisuflebis xarisxi
1 2 3 4 5
1 161.4 199.5 215.7 224.6 230.2
2 18.51 19 19.16 19.25 19.3
3 10.13 9.55 9.28 9.12 9.01
29 4 7.71 6.94 6.59 6.39 6.26
5 6.61 5.79 5.41 5.19 5.05
6 5.99 5.14 4.76 4.53 4.39
7 5.59 4.74 4.35 4.12 3.97
8 5.32 4.46 4.07 3.84 3.69
9 5.12 4.26 3.86 3.63 3.48
10 4.96 4.1 3.71 3.48 3.33
11 4.84 3.98 3.59 3.36 3.2
12 4.75 3.89 3.49 3.26 3.11
es cxrili aris mxolod mTliani cxrilis nawili, romelic warmodgenilia danarTSi
cxrilSi# A a-3.
Tu Frealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da ver miviRebT
H1-s. SerCevis saSualoebs Soris gansxvavebebi ar aris statistikurad
mniSvnelovani da ubralod aramniSvnelovania.
statistikurad mniSvnelovani gansxvaveba. statistikurad mniSvnelovani
gansxvavebis ilustraciisaTvis ganvixiloT zemoT moyvanili eqsperimentis
magaliTi, romelic naxatebis aRdgenas exeboda. am eqsperimentSi xdeboda sami
tipis naxatis manipulireba imisaTvis, rom dagvedgina, moqmedebs Tu ara naxatis
tipi masSi mocemuli elementebis gaxsenebaze. qvemoT mocemulia gaxsenebuli
elementebis raodenobis saSualo, rogorc naxatis tipis funqcia.
naxatis tipi (A)
dausrulebeli
eskizi (A1 )
eskizi ( A2) detaluri
( A3)
AX 17 13 12
cxrilSi #10.1 mocemul qulaTa dispersiuli analizis Sedegad miRebulia
Frealuri (2.12)=6.56 (ix.cxrili #10.5). roca df tolia 2 da 12, Fkritikuli 0.05
doneze aris 3.88. Frealuri aris meti Fkritikulze da amrigad, uaryofis areSi
xvdeba. Sesabamisad, nulovani hipoTeza unda uarvyoT
H0: µ A1=µA2=µA3
da miviRoT alternatiuli
H1: yvela µ A ar aris erTmaneTis toli
30 davaskveniT, rom sami SerCevis saSualo ar aris erTi da imave populaciidan.
damoukidebeli cvladi moqmedebs gaxsenebuli elementebis raodenobaze.
miaqcieT yuradReba, rom arsebobs mravali sxvadasxva gza, imisaTvis, rom
hipoTeza
yvela H1: µ A ar aris erTmaneTis toli iyos WeSmariti. alternatiuli hipoTeza,
magaliTad, WeSmariti iqneba, Tu µ A1 iqneba meti µA2-ze an µA3-ze, µA2 da µA3
iqnebian tolebi, an Tu µA1 iqneba meti µA2-ze da µA2 ki meti µA3-ze. romeli
damokidebuleba gvaCvenebs saSualoTa Soris mimarTulebas am SemTxvevaSi?
mxolod dispersiuli analizi ver gagvcems am kiTxvaze pasuxs garkveviT, H0-is
uaryofisaTvis, roca arsebobs sami an meti eqsperimentuli jgufi
eqsperimentSi. ubralod is gvaZlevs saSualebas davaskvnaT, rom arsebobs sul
cota erTi statistikurad mniSvnelovani gansxvaveba saSualoTa Soris.
(saSualoTa Soris konkretuli mniSvnelovani gansxvavebebis sapovnelad
aucilebelia gamoviyenoT mravaljeradi SedarebisKkriteriumi). am kriteriums
mogvianebiT ganvixilavT am Tavis Semdgom qveTavSi. gadawyvetilebis miRebis
procedura da Sesatyvisi daskvnebi statistikurad mniSvnelovani
gansxvavebisaTvis Sejamebulia cxrilSi #10.9, marcxena svetSi.
aramniSvnelovani gansxvaveba. Tu ver uarvyofT H0-s da ver miviRebT H1-s, maSin
Cven ar gveqneba imis mtkicebis saSualeba, rom nulovani hipoTeza ar aris
WeSmariti. aseTi eqsperimentis mniSvneloba isaa, rom damoukidebel cvlads ara
aqvs gavlena. saSualoebs Soris nebismieri raodenobrivi gansxvaveba miewereba
SerCevis Secdomas. es gadawyvetilebani da daskvnebi Sejamebulia cxril #10.9,
marjvena svetSi.
cxrili #10.9
cdis pirTa Soris erTfaqtoriani sqemis dispersiuli analizis gamoyenebiT
statistikuri hipoTezebis Semowmebis Sesaxeb gadawyvetilebebisa da daskvnebis
Sejameba. gamoyenebuli iyo 0.05 mniSvnelovnobis done.
31
daSvebebi cdis pirTa Soris erTfaqtoriani dispersiuli
analizisaTvis
t kriteriumis analogiurad, cdis pirTa Soris dispersiuli analizi efuZneba
sam daSvebas eqsperimentSi miRebul monacemTa Sesaxeb:
1. monacemebi SerCevaSi SemTxveviTadaa miRebuli populaciis monacemebidan
da TiToeuli monacemi damoukidebelia nebismieri sxva monacemisagan.
2. monacemebi populaciis SerCevaSi normalurad arian ganawilebulni.
3. qulaTa dispersiebi populaciebSi tolia: σ2A1=σ2A2
es daSvebebi mniSvnelovania, radgan F SerCeviTi ganawileba da amitom
Fkritikulis mniSvnelobebi, romelic mocemulia cxrilSi #a-3, im
Tu Frealuri xvdeba
uaryofis areSi α=0.05-saTvis, maSin:
Tu Frealuri ar xvdeba
uaryofis areSi, roca α=0.05, maSin:
Frealuris miRebis albaToba naklebia an
toli 0.05, an p≤0.05
H0 uaryofilia
H1 miRebulia. SerCevis yvela saSualo
ar aris aRebuli erTi da imave
populaciidan.
sul cota, erTi gansxvaveba mainc aris
saSualoebs Soris statistikurad
mniSvnelovani 0.05 doneze.
miiReba gadawyvetileba imis Sesaxeb, rom
SerCevis saSualoebs Soris gansxvaveba
SeiZleba ganpirobebuli iyos sxva
mizeziTac, garda SerCevis Secdomisa.
kargad Catarebul eqsperimentSi
gansxvaveba miewereba damoukidebel
cvlads.
Tu damoukidebel cvlads sami an meti
done aqvs, saWiroa mravaljeradi
Sedarebis kriteriumebi.
Frealuris miRebis albaToba metia an
toli 0.05, an p>0.05
H0 ar aris uaryofili
H1 ar aris miRebuli
ar aris statistikurad
mniSvnelovani gansxvavebani SerCevis
saSualoebs Soris 0.05 doneze.
miiReba gadawyvetileba imis Sesaxeb,
rom SerCevis Secdoma yvelaze
damajerebelad xsnis gansxvavebas
SerCevis saSualoebs Soris. Aar
arsebobs damamtkicebeli sabuTi,
rom damoukidebeli cvladi axdens
gavlenas.
ar aris saWiro mravaljeradi
Sedarebis kriteriumebi.
32 populaciebidanaa Tavmoyrili, romlebic akmayofileben am daSvebebs. imisaTvis,
rom cdis pirTa Soris dispersiuli analizi gamoviyenoT, TiToeuli monacemi
eqsperimentSi unda iyos damoukidebeli nebismieri sxva monacemisagan. amis
Sedegad, yovel cdis pirs SeiZleba miewodos damoukidebeli cvladis mxolod
erTi done da yovelma cdis pirma SeiZleba miiRos monawileoba eqsperimentSi
mxolod erTi monacemiT. Tu cdis pirs miewodeba damoukidebeli cvladis
erTze meti done eqsperimentSi da ori an meti monacemi iqneba gaanalizebuli
TiToeuli cdis pirisaTvis, maSin cdis pirTa Sida dispersiuli analizi unda
gamoviyenoT. es dispersiuli analizi ganxilulia
me-12 TavSi.
zogierTma eqsperimentma SeiZleba ver daakmayofilos meore da mesame
moTxovna. meore (monacemTa normaluri ganawileba) da mesame (dispersiaTa
toloba) daSvebis darRvevam SeiZleba Secvalos Fkritikulis mniSvnelobis
miRebis albaToba da amrigad, imoqmedos pirveli gvaris Secdomis daSvebis
albaTobaze. rogorc mosalodnelia, am debulebebis darRvevebs minimaluri
gavlena eqnebaT pirveli gvaris Secdomis daSvebis albaTobaze, roca Semdegi
pirobebi iqneba dakmayofilebuli:
• TiToeul jgufSi cdis pirebis raodenoba erTi da igivea;
• monacemTa ganawilebis forma TiToeuli jgufisaTvis TiTqmis erTi
da igivea da ganawilebebi arc Zalian wawvetebulia da arc Zalian
gaSlili;
• mniSvnelovnobis doned miRebulia 0.05.
SeamowmeT Tqveni codna X-2
1. gansazRvreT: Fkritikuli, Frealuri, uaryofis are.
2. ra faqtorebi moqmedeben MSA-is mniSvnelobaze eqsperimentSi?
3. ganmarteT, ratom izrdeba MSA-is mniSvneloba, Tuki damoukidebel cvlads
aqvs gavlena eqsperimentSi?
4. ra faqtorebi moqmedeben MSSecdomis mniSvnelobaze eqsperimentSi?
5. axseniT, ratom ar icvleba MSSecdomis mniSvneloba, Tu damoukidebel
cvlads gavlena aqvs eqsperimentSi?
6. Frealuris ra mniSvnelobebia mosalodneli, Tu damoukidebel cvlads ara
aqvs gavlena eqsperimentSi?
33
7. ra mouva Frealuris mniSvnelobas, Tu damoukidebel cvlads eqneba gavlena
eqsperimentSi?
8. dawereT statistikuri hipoTezebi F kriteriumisaTvis:
a. ori damoukidebeli jgufisaTvis
b. oTxi damoukidebeli jgufisaTvis
g. eqvsi damoukidebeli jgufisaTvis
9. romel situacias Seesabameba eqsperimentSi dispersiuli analizis H0?
10. romel situacias Seesabameba eqsperimentSi dispersiuli analizis H1?
11. risi tolia F mniSvnelobis qveda zRvari, Tu H0 WeSmaritia?
12. risi tolia F mniSvnelobis zeda zRvari, Tu H0 WeSmaritia?
13. savarjiSoSi mocemulia dfA, dfSecdomisa da Frealuri cdis pirTa Soris
erTfaqtoriani ramdenime hipoTeturi eqsperimentisaTvis. mniSvnelobaTa
TiToeuli simravlisaTvis miiReT Fkritikulis mniSvneloba 0.05 mniSvnelovnobis
donisaTvis cxrilidan #a-3a. Semdeg aRniSneT, xvdeba Tu ara Frealuri uaryofis
areSi da ra gadawyvetilebas miiRebdiT H0 -isa da H1-is mimarTebaSi. Tu ar aris
mocemuli zusti df cxrilSi dfSecdomisaTvis, maSin gamoiyeneT misi Mmomdevno
mniSvneloba.
eqsperimenti dfA dfSecdoma Frealuri
1 1 10 5.12
2 2 24 3.29
3 1 24 4.26
4 2 90 3.24
5 2 12 3.94
6 2 15 3.6
7 4 45 1.93
8 3 20 1.46
9 6 63 2.3
10 4 95 2.13
14. CamoTvaleT daSvebebi, romlebic safuZvlad udevs cdis pirTa Soris
erTfaqtoriani dispersiuli analizis gamoyenebas.
15. ra pirobebSi eqneba populaciis monacemebis normaluri ganawilebidan
gadaxras minimaluri gavlena pirveli gvaris Secdomaze cdis pirTa Soris
erTfaqtoriani dispersiuli analizisaTvis?
34
F-is statistikuri mniSvnelovnobis interpretacia mravaldonian
eqpsperimentSi
mravaljeradi Sedarebis kriteriumebi
adre aRvniSneT, rom roca eqsperimentSi gamoiyeneba damoukidebeli cvladis
sami da meti done, H0-is uaryofisas, vaskvniT, rom arsebobs sul cota erTi
statistikurad mniSvnelovani gansxvaveba saSualoebs Soris. magram analizi ar
gansazRvravs, konkretulad romeli saSualoebi gansxvavdebian erTmaneTisagan.
imis gadasawyvetad, saSualoTa romeli wyvilebi gansxvavdebian erTmaneTisagan
mniSvnelovnad, unda gamoviyenoT mravaljeradi Sedarebis kriteriumebi.
mravaljeradi Sedarebis kriteriumebi statistikuri kriteriumebia, romlebic
gamoiyeneba wyvilTaSoris SedarebebSi imis dasadgenad, romeli saSualoebi
gansxvavdeba mniSvnelovnad erTmaneTisagan erTfaqtorian mravaldonian sqemaSi.
roca mocemuli gvaqvs sami SerCevis saSualo eqsperimentidan, romelic
exeboda naxatis tipis gavlenas detalebis gaxsenebaze, SegviZlia gavakeToT
sami Sedareba or saSualos Soris:
1AX SevadaroT 2AX ,
1AX SevadaroT 3AX ,
2AX SevadaroT 3AX .
radgan Sedarebebidan TiToeuli moicavs or saSualos, maT wyvilTaSoris
Sedarebebs uwodeben. mas Semdeg, rac statistikur Semowmebas am sami wyvilis
SedarebiT SevasrulebT, SegveZleba davadginoT, Tu saSualoTa romeli
wyvilebi gansxvavdeba erTmaneTisagan mniSvnelovnad.
am wyvilTa Sedarebebis ganxorcielebis erT-erTi gza iqneboda
tdamoukidebelis gamoyeneba sami t kriteriumis dasadgenad, TiTo t kriteriumisa
TiToeuli SedarebisaTvis. am sami Semowmebis Sedegidan Cven SegviZlia
ganvsazRvroT, romeli saSualoebi gansxvavdebian erTmaneTisagan mniSvnelovnad.
es sakiTxi ar aris ase advilad gadasawyveti SedarebebSi pirveli gvaris
mravaljeradi Sedarebis kriteriumebi (Multiple comparison tests): statistikuri kriteriumebi, romlebic gamoiyeneba wyvilTaSoris SedarebebSi imis dasadgenad, Tu romeli saSuloebi gansxvavdeba mniSvnelovnad erTmaneTisagan erTfaqtorian mravaldonian sqemaSi.
wyvilTaSoris Sedarebebi (Pairwise comparisons): statistikuri Sedarebebi, romlebic erTmaneTs adareben or saSualos.
35 Secdomis daSvebis albaTobis gamo. gavixsenoT, rom pirveli gvaris Secdoma
xdeba maSin, roca H0 aris realurad WeSmariti, magram uaryofilia
statistikuri hipoTezis
Semowmebisas. imisaTvis, raTa SeinarCunon pirveli gvaris Secdomis daSvebis
albaTobis dabali done, mkvlevrebi Cveulebriv adgenen 0.05 an 0.01
mniSvnelovnobis doneebs. mTliani dispersiuli analizisaTvis ganxilul
magaliTSi (romelic moxseniebulia cxrilSi #10.5), pirveli gvaris Secdomis
albaToba α-s, an 0.05-is tolia. magram, roca mravaljerad Sedarebebs vatarebT,
mkvlevari atarebs mraval statistikur Semowmebas, romelTa raodenoba
damokidebulia Sedarebebis raodenobaze. ganvixiloT, magaliTad, tdamoukidebelis
gamoTvla TiToeuli wyvilTa Soris SedarebebisaTvis ganxilul eqsperimentSi.
TiToeuli t kriteriumisaTvis pirveli gvaris Secdomis albaToba aris α-s
toli. aqedan gamomdinare, sul cota, erTi pirveli gvaris Secdomis daSvebis
albaToba sam Sedarebas Soris daaxloebiT 0.14-ia, rac miuReblad maRali
mniSvnelobaa. Secdomis xarisxi eqsperimentSi aris sul cota erTi pirveli
gvaris Secdomis daSvebis albaToba mainc Sesrulebul SedarebebSi. Secdomis
xarisxi eqsperimentSi Zalian swrafad izrdeba Sedarebebis ricxvis zrdasTan
erTad. magaliTad, ganvixiloT t kriteriumis gamoyeneba 5 wyvilTaSorisi
SedarebisaTvis α=0.05 TiToeuli SedarebisaTvis. Tu H0 aris WeSmariti, maSin
sul cota erTi pirveli gvaris Secdomis albaToba 5 SedarebaSi 0.23-is tolia.
Secdomis es xarisxi aSkarad Zalian maRalia mkvlevarTa umetesobisaTvis.
Secdomis xarisxi gasakontroleblad Seqmnilia t kriteriumis alternativa
mravaljeradi SedarebebisaTvis. Cven ganvixilavT erT konkretul kriteriums
mravaljeradi Sedarebebis erTi tipisaTvis, saboloo SedarebebisaTvis.
saboloo Sedarebebi axorcieleben yvela SesaZlo wyvilTa Sedarebebs, mas
Semdeg rac statistikurad mniSvnelovani Frealuri dadasturdeba sruli
dispersiuli analizisaTvis. termini Post hoc (saboloo Sedareba) laTinuri
terminia da niSnavs amis Semdeg an am faqtis Semdeg. misi gamoyeneba aq niSnavs,
saboloo Sedarebebi (Post hoc comparisons): statistikuri kriteriumebi, romlebic asruleben yvela SesaZlo wyvilTa Sedarebebs, mas Semdeg, rac statistikurad
mniSvnelovani Frealuri dadasturdeba sruli dispersiuli analizisaTvis.
Secdomis xarisxi eqsperimentSi (Error rate in an experiment) aris sul cota erTi pirveli gvaris Secdomis daSvebis albaToba mainc statistikur SedarebebSi, romelic Catarebulia eqsperimentSi.
36
rom Sedareba Sesrulebuli iqneba mas Semdeg, rac dadasturdeba, rom Frealuri
aris statistikurad mniSvnelovani. Tu sruli dispersiuli analizis Frealuri
aramniSvnelovania da H0 ar aris uaryofili, monacemTa Semdgom analizs azri
ara aqvs da saboloo Sedarebebi ar Catardeba. sxvadasxva miznebisaTvis
Sedarebis mravali kriteriumi arsebobs. amjerad, ganvixilavT mxolod tukis
(Tukey) HSD kriteriums.
tukis namdvilad mniSvnelovani sxvaobis (Tukey HSD) kriteriumi
saboloo SedarebebisaTvis
tukis namdvilad mniSvnelovani sxvaobis kriteriumi (Tukey HSD – honestly significant
difference), uzrunvelyofs kritikul sxvaobas (abreviatura CD), rac warmoadgens
minimalur sxvaobas ori Carevis saSualos Soris, romelic statistikurad
mniSvnelovania arCeul α doneze. realurad miRebuli gansxvavebis absoluturi
mniSvneloba saSualoebis wyvilTaSoris SedarebebSi tukis CD-iTaa
Sedarebuli. Tu saSualoTa Soris realurad miRebuli sxvaobis absoluturi
mniSvneloba CD-s toli an metia, maSin SerCevis saSualoebi mniSvnelovnad
gansxvavdeba da ganixileba rogorc sxvadasxva populaciebis warmomadgenlebi.
CD tukis namdvilad mniSvnelovani sxvaobis kriteriumi moiZebneba
Semdegi formulidan
AnMS
qCD Secdoma= ,
sadac
q aris studentizebuli rangobrivi statistikis ricxviTi mniSvneloba,
romelic damokidebulia (1) arCeul α doneze, (2) damoukidebeli
cvladis doneebis raodenobaze da (3) df-ze MSSecdomisaTvis
dispersiul analizSi. q-s mniSvnelobebi mocemulia cxrilSi #a-
4, roca α=0.01 da α=0.05 damoukidebeli cvladis 10-ze meti
donisaTvis.
MSSecdoma aris Secdomis aRmniSvneli termini, romelic aRebulia sruli
dispersiuli analizidan.
kritikuli gansxvaveba (CD Critical difference): aris or saSualos Soris raodenobrivi gansxvaveba, romelic statistikurad mniSvnelovania.
37
nA aris monacemTa raodenoba TiToeul eqsperimentul pirobaSi.
imisaTvis, rom gamoviyenoT es formula, nA mniSvnelobebi
TiToeuli eqsperimentuli pirobisaTvis toli unda iyos.
tukis kritikuli sxvaobis miReba
ganvixiloT tukis namdvilad mniSvnelovani CD sxvaobis miReba im magaliTSi
moyvanili qulebis mixedviT, romelic mocemulia cxrilSi #10.1 da im qulebis
dispersiuli analizis mixedviT, romelic Sejamebulia cxrilSi #10.5.
Frealuri, 6.56 statistikurad mniSvnelovania 0.05 doneze; amrigad, unda uarvyoT
H0:µA1=µA2=µA3 da miviRoT H1: yvela µA ar aris erTmaneTis toli. . sruli
dispersiuli analizi gviCvenebs, rom arsebobs sul cota erTi mniSvnelovani
gansxvaveba im sam saSualos Soris, romlebsac erTmaneTs adareben. imisaTvis,
rom vipovoT konkretulad romeli saSualoebi gansxvavdeba erTmaneTisagan,
unda gamoviTvaloT tukis namdvilad mniSvnelovani kritikuli sxvaoba (HSD
CD).
dispersiuli analizidan, romelic Sejamebulia cxrilSi #10.5,
MSSecdoma=5.333. monacemTa raodenoba TiToeul eqsperimentul jgufSi 5-ia.
amrigad, nA=5. q-s mniSvneloba miRebulia cxrilidan #a-4. es cxrili moiTxovs
(1) damoukidebeli cvladis, raodenobis, a-s, codnas (am magaliTSi a=3) da (2)
dfSecdomis codnas dispersiuli analizidan (am SemTxvevaSi, 12). 0.05 doneze, a=3
da 12 dfSecdomisaTvis, q=3.77. CavsvaT es mniSvnelobebi formulaSi CD-saTvis da
miviRebT:
.9.35333.577.3
MS
==
=An
qCD Secdoma
tukis kritikuli sxvaobis interpretacia
tukis kriteriumisaTvis am magaliTSi CD=3.9-s. gansxvavebis absoluturi
mniSvneloba (roca + an _ niSani ar miiReba mxedvelobaSi) or eqsperimentul
saSualos Soris cxrilSi #10.1, romelic 3.9-is tolia an metia statistikurad
mniSvnelovania 0.05 doneze. sami wyvilTaSorisi Sedareba, TiToeuli Sedarebis
absoluturi mniSvneloba, statistikuri hipoTeza da gadawyvetileba TiToeuli
hipoTezisaTvis naCvenebia cxrilSi #10.10.
38 miaqcieT yuradReba, rom ori wyvilTaSoris Sedareba aris meti 3.9-ze
absoluturi mniSvnelobiT: 1AX da 2AX Soris (17-13=4) da 1AX da 3AX Soris (17-
12=5). TiToeuli es Sedareba avlens statistikurad mniSvnelovan gansxvavebas
Sesadarebel saSualoebs Soris. 2AX -sa da 3AX -s Soris Sedarebis absoluturi
mniSvneloba (13-12=1) naklebia 3.9-ze. amrigad, gansxvaveba eskizsa da detalur
naxats Soris aramniSvnelovania. aqedan gamomdinare, tukis kriteriums
mivyavarT daskvnamde, rom dausrulebeli eskizis gaxseneba( 1AX ) mniSvnelovnad
ufro kargia, vidre eskizis ( 2AX ) an detaluri naxatis( 3AX ), magram eskizis ( 2AX )
gaxseneba ar gansxvavdeba detaluri naxatisagan ( 3AX ).
es Sedegi mexsierebaSi informaciis gadamuSavebis Zalisxmevis hipoTezis
dadasturebaa, romelic warmodgenili iyo am Tavis dasawyisSi. dausrulebeli
eskizebi, romlebic yvelaze met Zalisxmevas moiTxoven gadamuSavebis procesSi,
yvelaze kargad gaaxsendebaT. es Sedegi ar adasturebs gadamuSavebis hipoTezas
romlis mixedviTac detaluri naxati gaaxsendebodaT.
cxrili #10.10
tukis HSD kritikuli sxvaobis gamoyeneba wyvilTa Soris Sedarebebisas
ganxilul eqsperimentSi
Sedarebebi Sedarebis
absoluturi
mniSvneloba
statistikuri
hipoTezebi
gadawyvetileba
1AX da 2AX
Soris (17-13)
4 H0:µ A1=µA2
H0:µ A1≠µA2
uaryofilia
miRebulia
1AX da 3AX
Soris (17-12)
5 H0:µ A1=µA3
H0:µ A1≠µA3
uaryofilia
miRebulia
2AX da 3AX
Soris (13-12)
1 H0:µ A2=µA3
H0:µ A2≠µA3
ver uarvyofT
ar aris miRebuli
A1 = dausrulebeli eskizi, A2 = eskizi, A3= detaluri naxati
ori SemTxveva, roca saboloo Sedarebis kriteriumi ar aris aucilebeli
arsebobs ori magaliTi, roca erTfaqtoriani dispersiuli analizis
gamoyenebisas saboloo kriteriumi ar aris saWiro. pirveli, maSin, roca
39
Frealuri dispersiuli analizisaTvis aramniSvnelovania. aramniSvnelovani
Frealuri gviCvenebs, rom ar arsebobs mniSvnelovani gansxvavebebi saSualoTa
Soris arc erTi wyvilisaTvis analizSi. amitom ar aris saWiro saboloo
Sedareba am gansxvavebebis dasadgenad.
meore magaliTi, roca arsebobs damoukidebeli cvladis mxolod ori
done. Tu Frealuri aris mniSvnelovani am analizisaTvis, maSin H0:µA1=µA2 aris
uaryofili da yvela : µ A ar aris erTmaneTis toli iqneba miRebuli. aSkaraa,
rom erTaderTi saSualoebi, romlebic erTmaneTisagan gansxvavdebian 1AX da 2AX -
ia. amrigad, am SemTxvevaSi ar aris saWiro am saSualoebis Semdgomi Semowmeba.
Carevis efeqtis siZlieris gazomva
t kriteriumis analogiurad dispersiuli analizi gviCvenebs, aqvs Tu ara
damoukidebel cvlads gavlena damokidebul cvladze eqsperimentSi, magram is
ar gviCvenebs gavlenis sidides. Frealuri statistikuad mniSvnelovani
gansxvavebisas gavlenis siZlieris sazomad SeiZleba kvlavac iyos gamoyenebuli
η2.
eta kvadratis gamoTvla dispersiuli analizisaTvis
cdis pirTa Soris erTfaqtoriani dispersiuli analizisaTvis η2 SeiZleba
miRebuli iyos ori alternatiuli formulidan. Tu kvadratebis jamebis
mniSvnelobebi cnobilia, η2 mocemuli iqneba Semdegi saxiT
η2sruliSSSS A=
roca dispersiuli analizisaTvis Frealuri da df cnobilia, η2 SeiZleba miRebuli
iyos Semdegi formulidan:
Secdomarealuri
realuri
df))(())((2
+=
FdfFdf
A
Aη
η2 gamoTvlis magaliTi cdis pirTa Soris erTfaqtoriani dispersiuli
analizisaTvis
40
η2 SeiZleba gamoviTvaloT nimuSis saxiT moyvanili amocanisaTvis an SS
mniSvnelobidan, an dispersiuli analizis Frealurisa da df mniSvnelobebidan,
romelic Sejamebulia cxrilSi #10.5.
η2 SS mniSvnelobidan
η2sruliSSSS A=
52.013470
=
=
η2 Frealurisa da df mniSvnelobebidan
Secdomarealuri
realuri
df))(())((2
+=
FdfFdf
A
Aη
52.012.2512.13
12)56.6)(2()56.6)(2(
=
=
+=
es η2 gviCvenebs, rom monacemTa srul dispersiaSi xdeba 52-procentiani kleba,
roca damoukidebeli cvladis doneebis saSualoebi gamoiyeneba qulaTa
winaswarmetyvelebisaTvis, sruli saSualos rogorc prognozirebuli qulis
mixedviT. es damoukidebeli cvladi axsnis gaxsenebuli qulebis dispersiis did
nawils.
tdamoukidebelsa da F Soris damokidebuleba
tdamoukidebelsa da cdis pirTa Soris erTfaqtorian dispersiul analizs
mivyavarT igive statistikuri gadawyvetilebebisaken, roca gasaanalizebelia
eqsperimentuli monacemebi damoukidebeli cvladis ori doniT. Tu uarvyofT
H0-s ormxrivi SedarebisaTvis 0.05 doneze, maSin unda uarvyoT, agreTve, H0
dispersiuli analizisaTvis, romelic igive monacemebiTaa Sesrulebuli. amis
mizezi isaa, rom t2 =F an t=√F es damokidebuleba, agreTve, SenarCunebulia
41
cxrilSi mocemuli t-sa da F kritikuli mniSvnelobebisaTvis, roca xdeba ori
SerCevis saSualoebis analizi erTi da imave mniSvnelovnobis doneze.
magaliTad, α=0.05-saTvis, Fkritikulis 1, 8 df 5.32-ia. tkritikulis mniSvneloba 8 df da
α=0.05 ormxriv kriteriumSi aris 2.306 da is 5.32-is kvadratuli fesvis tolia.
amrigad, (2.306)2=5.32 da t2=F. t-sa da F-s Soris am damokidebulebis gamo, orive
kriteriumi SeiZleba iyos gamoyenebuli ori damoukidebeli SerCevis
saSualoebis SedarebisaTvis. kriteriumebi uzrunvelyofen identur Sedegebs da
arc erTs ar SeiZleba mivaniWoT upiratesoba. dispersiuli analizis
upiratesoba maSin Cans, roca sam an met eqsperimentul pirobas vadarebT
erTmaneTs.
amocana #10-1
cdis pirTa Soris erTfaqtoriani dispersiuli analizis gamoyeneba tukis
HSD kriteriumiT
amocana
ganwyobilebaze damokidebuli mexsierebis fenomeni niSnavs, rom gaxseneba
umjobesdeba, roca subieqts masalis gaxsenebisas igive ganwyobileba aqvs, rac
misi Seswavlis dros (Bower, 1981). davuSvaT, rom ganwyobilebaze damokidebuli
mexsierebis fenomenis Seswavlisas Tqven SeqmeniT cdis pirTa xuTi
ekvivalenturi jgufi, TiToeulSi 10-10 cdis piriT. TiToeul jgufs Tqven
umuSavebT ganwyobilebas gansxvavebuli emociebis gamomxatveli debulebebis
wakiTxvis saSualebiT, romlebic qmnian an mxiarul, an sevdian, an neitralur
ganwyobilebas. mas Semdeg, rac ganwyobileba gaCndeba, cdis pirebi 15
Cveulebrivi arsebiTi saxelis sias swavloben. 24 saaTis Semdeg cdis pirebi
brundebian, Tqven kvlav umuSavebT ganwyobilebas da sTxovT cdis pirebs
gaixsenon adre naswavli arsebiTi saxelebis sia. gaxsenebis am seriaSi
ganwyobileba igive an gansxvavebuli unda iyos im ganwyobilebisagan, romlis
pirobebSic xdeboda swavla. ganwyobileba, romelic Seqmnili iyo swavlis
dros da gaxsenebis dros 5 jgufisaTvis mocemulia Semdeg cxrilSi:
jgufis ganwyobileba
A1 A2 A3 A4 A5
swavlis dros mxiaruli sevdiani neitraluri mxiaruli sevdiani
42 Seqmnili
ganwyobileba
gaxsenebis
dros
Seqmnili
ganwyobileba
mxiaruli sevdiani neitraluri sevdiani mxiaruli
ganwyobilebaze damokidebuli mexsierebis Teoria winaswarmetyvelebs, rom
jgufebi, romelTac aqvT swavlisa da gaxsenebis kongruentuli mdgomareoba
(e.i. jgufebi A1, A2, A3) unda ixsenebdnen ufro met arsebiT saxels, vidre
jgufebi, romelTac hqondaT gansxvavebuli ganwyobilebebi swavlisa da
gaxsenebis dros (e.i. jgufebi A4 da A5). davuSvaT, rom Tqvens mier miRebuli
Sedegebi _ gaxsenebuli punqtebis raodenoba naCvenebia cxrilSi #10.11. romeli
jgufis saSualoebi gansxvavdebian mniSvnelovnad erTmaneTisagan 0.05 doneze?
amoxsna
gamoyenebulia cdis pirTa Soris erTfaqtoriani sqema damoukidebeli cvladis,
ganwyobilebis jgufis 5 doniT. monacemTa analizSi pirveli nabiji aris
monacemebis cdis pirTa Sorisi erTfaqtoriani dispersiuli analizis Catareba.
Tu Frealuri statistikurad mniSvnelovania, maSin SesaZlebelia gamoviyenoT
tukis namdvilad mniSvnelovani sxvaobis kriteriumi imisaTvis, raTa
davadginoT statistikurad mniSvnelovani wyvilTaSorisi gansxvavebebi
eqsperimentul saSualoebs Soris.
saWiro statistika: F= MSA / MSSecdoma
cxrili #10.11
sworad gaxsenebuli arsebiTi saxelebis raodenoba rogorc ganwyobilebis
jgufis funqcia
ganwyobilebis jgufi
A1 A2 A3 A4 A5
8 14 9 5 6
12 13 12 9 9
12 6 7 6 8
7 9 7 8 8
10 12 11 4 5
43 9 10 6 6 4
13 8 10 10 5
8 7 8 3 7
14 11 13 7 7
11 9 10 3 3
AX 10.4 9.9 9.3 6.1 6.2
sA 2.37 2.6 2.31 2.42 1.93
daSvebebi:
1. cdis pirebi namdvilad SemTxveviTad arian SerCeuli;
2. gaxsenebuli qulebi normaluradaa populaciaSi ganawilebuli;
3. obieqtebis gaxsenebis dispersiebi populaciebSi erTmaneTis tolia.
statistikuri hipoTezebi:
H0: µ A1=µA2=µA3=µA4=µA5
H1: yvela µ A ar aris erTmaneTis toli.
mniSvnelovnobis done: α=0.05
df: dfA= a-1=5-1=4
dfSecdoma= N_a=50-5=45
dfsruli = N-1=50-1=49
F kritikuli mniSvnelobebi: Fkritikuli (4.45)=2.61. 45 df mniSvneloba
mricxvelisaTvis ar aris warmodgenili cxrilSi #a-3a. maSin unda gamoviyenoT
df-is momdevno dabali mniSvneloba, romelic mocemulia cxrilSi dfSecdomisaTvis,
40.
uaryofis are: Frealuris mniSvneloba tolia an metia 2.61-ze.
gamoTvla: dispersiuli analizi SeiZleba gamoTvlili iyos im gamoTvliTi
formulis gamoyenebiT, romelic mocemulia Tavis damatebaSi, an kompiuteruli
programis gamoyenebiT. am Tavis damatebis formula gamoyenebuli iyo
dispersiuli analizis Semajamebeli Semdegi cxrilis misaRebad.
wyaro SS df MS F ganwyobilebis 171.88 4 42.97 7.86*
44 jgufi(A)
Secdoma 245.9 45 5.464
sul 417.78 49
*p<0.05
gadawyvetileba: radgan Frealuri=7.86 xvdeba uaryofis areSi, unda uarvyoT
H0: µ A1=µA2=µA3=µA4=µA5 da miviRoT H1: yvela µ A ar aris erTmaneTis toli. H0-is
uaryofa gviCvenebs, rom arsebobs statistikurad mniSvnelovani erTi
gansxvaveba mainc SerCevis saSualoebs Soris. statistikurad mniSvnelovani
wyvilTaSoris gansxvavebebis dasadgenad, unda gamoviyenoT tukis namdvilad
mniSvnelovani sxvaobis kriteriumi.
tukis kriteriumi: ricxviTi mniSvnelobebi, romelic saWiroa am
kriteriumisaTvis:
• q miRebulia cxrilidan #a-4. gvaqvs damoukidebeli cvladis 5 done
da dispersiuli analizis Semajamebeli cxrilSi mocemulia, rom df
MSSecdomisaTvis tolia 45-is. radgan cxrilSi #a-4 ar aris mocemuli 45
df, unda gamoviyenoT 40 df _ cxrilSi mocemuli momdevno dabali df
MSSecdomisaTvis. amrigad, q=4.04 mniSvnelovnobis 0.05 donisaTvis.
• dispersiuli analizidan miRebuli MSSecdoma 5.464-ia.
• nA, qulebis raodenoba TiToeuli Sesadarebeli saSualosaTvis aris
10.
CavsvaT es ricxviTi mniSvnelobebi formulaSi:
An
MSqCD Secdoma=
=
sityva310464.504.4
=
=
gansxvaveba or saSualos Soris Tavisi absoluturi mniSvnelobiT tolia an
metia 3 sityvaze statistikurad sarwmunoa mniSvnelovnobis 0.05 doniT.
wyvilTaSorisi Sedarebebi da maTi absoluturi mniSvnelobebi, Semowmebuli
statistikuri hipoTezebi da gadawyvetilebebi, romelic miiReba statistikuri
hipoTezebis Sesaxeb TiToeuli SedarebisaTvis, mocemulia cxrilSi #10.12.
45
Sedarebebis absoluturi mniSvnelobebi 1AX -sa da 4AX -s Soris; 1AX -sa da
5AX -s Soris; 2AX -sa da 4AX -s Soris, 2AX -sa da 5AX -s Soris, 3AX -sa da 4AX -s,
3AX -sa da 5AX -s Soris aRemateba CD-s, romelic 3 sityvis tolia. TiToeuli
gansxvaveba statistikurad mniSvnelovania. darCenili Sedarebebis absoluturi
mniSvnelobebi 1AX -sa da 2AX -s Soris, 1AX -sa da 3AX -s Soris, 2AX -sa da 3AX -s
Soris, 4AX -sa da 5AX -s Soris kritikul sxvaobaze naklebia. es Sedarebebi
aramniSvnelovania da SerCevis saSualoebs Soris miRebuli gansxvavebebi
saukeTesod aixsneba SerCevis SecdomiT.
cxrili #10.12
tukis namdvilad mniSvnelovani sxvaobis kritikuli sxvaobis gamoyeneba
wyvilTaSori SedarebebisaTvis ganwyobilebis 5 jgufisaTvis.
Sedarebebi Sedarebis
absoluturi
mniSvneloba
statistikuri
hipoTezebi
gadawyvetileba
1AX da 2AX
Soris (10.4-9.9)
0.5 H0:µ A1=µA2
H0:µ A1≠µA2
ver uarvyofT
ar aris miRebuli
1AX da 3AX
Soris (10.4-9.3)
1.1 H0:µ A1=µA3
H0:µ A1≠µA3
ver uarvyofT
ar aris miRebuli
1AX da 4AX
Soris (10.4-6.1)
4.3 H0:µ A1=µA4
H0:µ A1≠µA4
uaryofilia
miRebulia
1AX da 5AX
Soris(10.4-6.2)
4.2 H0:µ A1=µA5
H0:µ A1≠µA5
uaryofilia
miRebulia
2AX da 3AX
Soris(9.9-9.3)
0.6 H0:µ A2=µA3
H0:µ A2≠µA3
ver uarvyofT
ar aris miRebuli
2AX da 4AX
Soris (9.9-6.1)
3.8 H0:µ A2=µA4
H0:µ A2≠µA4
uaryofilia
miRebulia
2AX da 5AX
Soris (9.9-6.2)
3.7 H0:µ A2=µA5
H0:µ A2≠µA5
uaryofilia
miRebulia
3AX da 4AX
Soris (9.3-6.1)
3.2 H0:µ A3=µA4
H0:µ A3≠µA4
uaryofilia
miRebulia
3AX da 5AX
Soris(9.3-6.2)
3.1 H0:µ3=µA5
H0:µ A3≠µA5
uaryofilia
miRebulia
46
4AX da 5AX
Soris (6.1-6.2)
0.1 H0:µ A4=µA5
H0:µ A4≠µA5
ver uarvyofT
ar aris miRebuli
A1 = mxiaruli / mxiaruli jgufi, A A2 = sevdiani/sevdiani jgufi , A3 =neitraluri
/ neitraluri jgufi, A4 =mxiaruli/ sevdiani jgufi, da A5 =sevdiani/ mxiaruli
jgufi.
efeqtis Zala: η2=171.88/417.78=0.41
daskvna: tukis kriteriums mivyavarT daskvnamde, rom ganwyobilebis jgufebi,
romlebsac hqondaT daswavlisa da gaxsenebis kongruentuli pirobebi (e.i.
mxiaruli / mxiaruli jgufi, A A2 = sevdiani/sevdiani jgufi, A3 =neitraluri /
neitraluri jgufi) gaxsenebis mxriv erTmaneTisagan mniSvnelovnad ar
gansxvavdebian, Tumca TiToeul am jgufSi cdis pirebi gacilebiT met sityvas
ixseneben, vidre romelime arakongruentuli ganwyobilebis jgufSi
(mxiaruli/sevdiani, da sevdiani/ mxiaruli jgufebi). garda amisa,
mxiaruli/sevdiani da sevdiani/mxiaruli jgufebi erTmaneTisagan mniSvnelovnad
ar gansxvavdebodnen. Sesabamisad, dadgenili kavSiri populaciis saSualoebs
Soris aris
H0: µ A1=µA2=µA3 > (µA4==µA5)
η2 gviCvenebs, rom ganwyobilebis jgufis codna winaswarmetyvelebis srul
dispersias 41 procentiT amcirebs.
magaliTi #10-2
cdis pirTa Soris erTfaqtoriani dispersiuli analizis gamoyeneba:
aramniSvnelovani Frealuri
amocana
moqmedebs Tu ara alkoholis miReba pirovnebis gaxsnilobis xarisxze
socialuri gavlenis efeqtebis mimarT? gustafsonma (Gustafson, 1987) ivarauda,
rom moqmedebs. am hipoTezis Sesamowmeblad, man cdis pirebs Sesafaseblad misca
sxvadasxva sigrZis monakveTebi, romelTa sigrZe icvleboda 5-dan 100
santimetramde. cdis pirebs werilobiT atyobinebdnen TavianT Sedegebs da
sTxovdnen meored SeefasebinaT imave monakveTis sigrZeebi. TiToeuli cdis
piri afasebda 75 monakveTis sigrZes. 75-dan 60 Sefasebis SemTxvevaSi maT
47 uwerdnen, rom pirveli Sefaseba ar iyo swori. iTvleboda im SesworebaTa
raodenoba, romelsac TiToeuli cdis piri akeTebda, roca meorejer afasebda
monakveTebs. eqsperimentSi cdis pirTa sami jgufi iyo gamoyenebuli.
sakontrolo jgufs asmevdnen forToxlis wvens eqsperimentis dawyebidan 15
wuTiT adre. meore jgufi iyo placebo sakontrolo jgufi, romelic
elodeboda, rom alkohols miiRebda. am jgufis cdis pirebs, agreTve,
awodebdnen forToxlis wvens sigrZeTa Sefasebamde, magram maT arwmunebdnen,
rom sasmeli Seicavs alkohols. mesame jgufis cdis pirebs miewodaT davalebis
dawyebamde 15 wuTiT adre forToxlis wvenSi gaxsnili 0.8 mililitri
alkoholi sxeulis erT kilogram wonaze. gustanfsonma ivarauda, rom cdis
pirebi alkoholisa da placebo jgufebSi ufro advilad daeqvemdebarebian
werilobiT miwodebul ukukavSirs da amitom TavianT meore Sefasebebs ufro
xSirad Secvlian, vidre sakontrolo jgufis cdis pirebi. davuSvaT Sefasebebis
raodenoba icvleba jgufis TiToeuli 13 cdis pirisaTvis, romelTa monacemebi
mocemulia cxrilSi #10.13. cvlis Tu ara alkoholi Sefasebebis raodenobas?
amoxsna: gamoyenebulia cdis pirTa Soris erTfaqtoriani sqema damoukidebeli
cvladis sami doniT, Carevis jgufiT. cdis pirTa Soris erTfaqtoriani
dispersiuli analizi am monacemebis damuSavebis Sesatyvisi statistikuri
kriteriumia. gamoviyenoT 0.05 mniSvnelovnobis done.
saWiro statistika: FF= MSA / MSSecdoma
daSvebebi:
1. cdis pirebi namdvilad SemTxveviTad arian SerCeulebi;
2. monakveTis sigrZeTa SefasebebSi daSvebuli cvlilebebi normalurad aris
populaciaSi ganawilebuli;
3. monakveTis sigrZeTa SefasebebSi daSvebuli cvlilebebis dispersiebi
tolia SerCeul populaciebSi.
statistikuri hipoTezebi:
H0: µ A1=µA2=µA3
H1: yvela µ A ar aris erTmaneTis toli
mniSvnelovnobis done: α=0.05
df: dfA=a-1=3-1=2
48
dfSecdoma= N_a=39-3=36
dfsruli = N-1=39-1=38
F-is kritikuli mniSvnelobebi: Fkritikuli (2.36)=3.32. Fkritikulis mniSvneloba 2, 36
df-saTvis ar aris warmodgenili cxrilSi #a-3a. maSin unda gamoviyenoT F
kritikuli 2, 30 df-isaTvis.
uaryofis are: Frealuris mniSvneloba tolia an metia 3.32-is.
gamoTvla: dispersiuli analizi SeiZleba gamoTvlili iyos im gamoTvliTi
formulis gamoyenebiT, romelic mocemulia Tavis damatebaSi. am analizis
Sejameba Semdegi saxiTaa mocemuli:
wyaro SS df MS F ganwyobilebis
jgufi(A)
389.898 2 194.949 2.24
Secdoma 3137.846 36 87.162
sul 3527.744 38
gadawyvetileba: Frealuri=2.24 Fkritikuli=3.32-ze naklebia da ar xvdeba uaryofis
areSi; amrigad, H0-s ver uarvyofT da H1 miRebuli ar iqneba.
cxrili #10.13
Secvlili Sefasebebis raodenoba, rogorc Carevis jgufis funqcia.
Carevis jgufi(A)
sakontrolo
(A1)
placebo
(A2)
alkoholi
(A3)
18 17 31
2 19 27
11 26 16
3 4 24
26 18 41
18 23 17
9 31 12
54 35 32
17 11 16
49 21 8 19
14 29 35
19 25 26
33 38 17
AX 16.54 21.85 24.08
sA 8.83 10.29 8.81
daskvna: realuri gansxvavebebi saSualoebs Soris aramniSvnelovania; SerCevis
saSualoebi 16.5, 21.8 da 24.1 miekuTvneba erTi da imave populaciis SerCevebs. ar
arsebobs damoukidebeli cvladis gavlenis mtkicebis safuZveli. cdis pirebi,
romelTac miawodes alkoholuri sasmeli, monakveTebis sigrZis TavianT
Sefasebebs ar cvlian ufro xSirad, vidre cdis pirebi, romelTac miawodes
placebo da forToxlis wveni. miRebuli gansxvaveba SerCevis saSualoebs Soris
gamowveulia mxolod SerCevis SecdomiT. vinaidan Frealuri iyo
aramniSvnelovani, ar aris mravaljeradi Sedarebis kriteriumebis an kavSiris
siZlieris gazomvis saWiroeba.
am magaliTis Sedegebi gviCvenebs statistikuri hipoTezis Semowmebis
aucileblobas, raTa ganisazRvros, gansxvavdeba Tu ara jgufebis saSualoebi
ufro metad, vidre mxolod SemTxveviTobis gamo iqneboda. mxolod
daTvalierebiT Cans, rom saSualoebi mniSvnelovnad gansxvavdebian
erTmaneTisagan. magram jgufebSi qulaTa Soris gansxvavebac didia, rac iwvevs
Secdomis cvalebadobis mniSvnelovnad gazrdas. es gamoixateba standartuli
gadaxrebis SedarebiT did mniSvnelobebSi. roca Secdomis cvalebadobis
mniSvneloba didia, maSin Cven SeiZleba velodeT did gansxvavebas jgufis
saSualoebs Soris, rac ganpirobebuli iqneba ubralod SerCevis SecdomiT,
rogorc zemoT ganxilul magaliTSi.
amocana #10-3
cdis pirTa Soris erTfaqtoriani dispersiuli analizi damoukidebeli
cvladis ori doniT
amocana
ganvixiloT me-9 Tavis me-4 kiTxva, romelSic gavecaniT morinagas gadaxrilobis
iluzias. is naCvenebia (a) svetSi suraTze #10.3. rogorc naCvenebi iyo me-4
50 kiTxvaSi, figura warmoadgens iluzias, radgan misi kuTxeebis wveroebi erT
vertikalur xazze ganlagebulad ar aRiqmeba. SenarCundeba Tu ara iluzia, Tu
figura rkaluri iqneba, ise rogorc naCvenebia suraTze #10.3(b)? am kiTxvaze
pasuxis gasacemad, davuSvaT, rom fsiqologebma Seqmnes ori ekvivalenturi
jgufi - TiToeulSi 10 cdis piri. cdis pirebs A1 jgufidan uCvenebdnen
kuTxeebian figurebs da A2 jgufidan _ naxevarrkalovan figurebs, rogorc
naCvenebia suraTze #10.3. damokidebul cvlads warmoadgenda stimulebis aRqmis
dros daSvebuli Secdomebi. is gazomes orive jgufisaTvis da davuSvaT, miiRes
cxrilSi #10.14 mocemuli Secdomis qulebi. romelic mocemuli iyo cxrilSi
damokidebulia Tu ara aRqmuli Secdomis sidide figuris tipze?
suraTi #10.3
morinagas gadaxrilobis iluzia.
(a) figura uCvenes A1 jgufs;
(b) figura uCvenes A2 jgufs.
cxrili #10.14
perceptuli Secdoma, (milimetrebSi), rogorc aRqmuli figuris funqcia
figuris tipi (A)
kuTxovani wriuli
2 1.6
1.8 1.7
2 1.8
2.1 1.6
2.2 1.5
1.8 1.8
1.9 1.7
1.8 1.8
1.9 1.7
1.7 1.6
AX 1.92 1.68
sA 0.15 0.10
51 amoxsna: imis sapovnelad, mniSvnelovnad gansxvavdebian Tu ara saSualoebi, Cven
gamoviyenebT cdis pirTa Soris erTfaqtorian dispersiul analizs. (radgan
mxolod ori jgufi monawileobda, tdamoukidebeli, agreTve, Sesatyvisi
statistikuri kriteriumia). gamoviyenoT 0.05 mniSvnelovnobis done.
saWiro statistika: FF= MSA / MSSecdoma
daSvebebi:
1. cdis pirebi namdvilad SemTxveviTad arian SerCeulebi;
2. iluziis aRqmisas daSvebuli Secdomebi normalurad aris populaciaSi
ganawilebuli;
3. SerCeul populaciebSi aRqmis Secdomebis qulebis dispersiebi tolia.
statistikuri hipoTezebi:
H0: µ A1=µA2
H1: yvela µ A ar aris erTmaneTis toli
mniSvnelovnobis done: α=0.05
df: dfA=a-1=2-1=1
dfSecdoma= N_a=20-2=18
dfsruli = N-1=20-1=19
F-is kritikuli mniSvnelobebi: Fkritikuli (1.18)=4.41
uaryofis are: Frealuris mniSvneloba tolia an metia 4.41-ze
gamoTvla: gamoiyeneT gamoTvliTi formula, romelic mocemulia am Tavis
damatebaSi da saboloo analizi Sejamebuli saxiT mocemulia:
wyaro SS df MS F ganwyobilebis
jgufi(A)
0.288 1 0.288 16.62
Secdoma 0.312 18 0.01733
sruli 0.6 19
gadawyvetileba: Frealuri xvdeba uaryofis areSi; amrigad H0 unda uarvyoT da
H1 miviRoT.
52
efeqtis Zala: η2=0.288/0.6=0.48
daskvna: saSualoebi 1.92mm da 1.68 mm mniSvnelovnad gansxvavdebian
erTmaneTisagan 0.05 mniSvnelovnobis doneze. 1.92 mm-is toli saSualo
mniSvnelovnad aRemateba saSualos, romelic tolia 1.68 mm-is. radgan ori
jgufi iyo ekvivalenturi, sanam damoukidebeli cvladi Semovidoda da
damatebiTi cvladebic iyo gakontrolebuli, jgufebs Soris realurad
miRebuli gansxvaveba miewereboda damoukidebeli cvladis gavlenas. cdis
pirebi ufro did iluzias xedaven kuTxovani stimulebis aRqmisas, vidre
wriuli stimulebis aRqmisas. tukis namdvilad mniSvnelovani sxvaobis
kriteriumi ar aris saWiro, radgan saSualos mxolod ori mniSvnelobaa
Sedarebuli. η2 =0.48 gviCvenebs damoukidebeli cvladis did gavlenas, rac xsnis
damokidebuli cvladis dispersiis 48%-s. Tu tdamoukidebels gamoviyenebT
dispersiuli analizis nacvlad, trealuri(18)=4.077. tdamoukidebelis es mniSvneloba
Frealurisagan kvadratuli fesvis tolia (1.18)=16.62. es t statistikurad
mniSvnelovania 0.05 doneze da igive daskvnamde mivyavarT, rasac dispersiul
analizs. es Sedegi, agreTve, gviCvenebs statistikuri hipoTezis Semowmebis
mniSvnelobas monacemTa analizisas. erTi SexedviT, jgufis saSualoebi
erTmaneTisagan mniSvnelovnad ar gansxvavdeba, magram yuradReba miaqcieT imas,
rom jgufis SigniT cdis pirebs Soris cvalebadobac SedarebiT mcirea.
jgufSida Secdomis mcire cvalebadobisas, jgufebs Soris mcire gansxvavebac
ki SeiZleba statistikurad mniSvnelovani iyos.
dispersiuli analizis Sedegebis prezentacia
amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvanelos (1994)
stili moiTxovs dispersiuli analizis Sedegebis prezentacia mocemuli iyos
jgufebis saSualoebi, standartuli gadaxrebi, arCeuli alfas mniSvneloba, F-
is realurad miRebuli mniSvneloba da misi Tavisuflebis xarisxi, MSSecdoma,
da realurad miRebuli F-is albaToba, Tu H0 WeSmaritia. qvemoT mocemulia
Sedegebis prezentaciis es stili erT-erT magaliTze, romelic exeboda naxatis
tipis gavlenas gaxsenebuli detalebis raodenobaze. dispersiuli analizi am
amocanisaTvis warmodgenilia cxrilSi #10.5. Tumca, Cven viyenebT Semajamebel
cxrils dispersiuli analizisaTvis, aseTi cxrili ar SeiZleba moTavsebuli
iyos Jurnalis statiaSi beWdvis maRali fasis gamo. Tu Semajamebeli cxrili
53 ar aris mocemuli, dispersiuli analizi teqstSi Sejamebulia ise, rogorc
qvemoT aris naCvenebi.
sworad gaxsenebuuli detalebis raodenoba iyo 17 dausrulebeli
eskizisaTvis(SD=2.74), 13 eskizisaTvis(SD=2.24) da 12 detaluri
naxatisaTvis(SD=1.87). alfa tolia 0.05-sas cdis pirTa Soris
erTfaqtorianma dispersiulma analizma gviCvena, naxatis tipis
mniSvnelovani gavlena: F(2.12)=6.56, MSE=5.333, p<0.05. saboloo Sedarebam
tukis namdvilad mniSvnelovani sxvaobis kriteriumis gamoyenebiT(CD=3.9,
α=0.05) gviCvena, rom mniSvnelovnad meti iyo gaxsenebuli obieqtebis
raodenoba dausrulebel naxatSi, vidre eskizSi an detalur naxatSi,
magram ar arsebobda mniSvnelovani gansxvaveba gaxsenebuli obieqtebis
raodenobis TvalsazrisiT eskizsa da detalur naxatSi. eta kvadrati am
qulebisaTvis 0.52 iyo.
am Sedegebis prezentaciisas:
α=0.05 gviCvenebs am kriteriumisaTvis arCeul mniSvnelovnobis dones.
F(2, 12) gansazRvravs testis statistikas rogorc F-s. amitom
monacemTa analizisaTvis gamoyenebulia dispersiuli analizi.
es F aris Frealuri, Tumca indeqsi(qveSminaweri) `realuri”
SeiZleba ar axldes. Frealuri df mricxvelisaTvis (e.i. 2) da
mniSvnelisaTvis, (e.i. 12) naCvenebia frCxilebSi. am df-idan Tqven
SegiZliaT gansazRvroT eqsperimentSi monawile jgufebisa da
cdis pirebis raodenoba Semdegnairad: df mricxvelisaTvis
tolia a-1-s. am moxsnebaSi dfA=2, amrigad, a=3. df
mniSvnelisaTvis tolia df Secdomisa, romelic tolia N-
a. am magaliTSi dfSecdoma= 12 da a=3; amitom 12=N-3 da cdis
pirebis sruli raodenoba eqsperimentSi tolia 15-s.
=6.56 gvaZlevs Frealuris mniSvnelobas (ara Fkritikulisa, romelsac
vpoulobT cxrilSi #a-3). Tu Frealuri naklebia 1-ze, es
SeiZleba warmodgenili iyos martivad, rogorc F<1.
p<0.05 gviCvenebs Semdegs:
a) Frealuris albaToba H0 WeSmaritebis pirobebSi aris naklebi
an toli 0.05. es mniSvneloba aris Frealuris albaToba an ufro
54
eqstremaluri Frealuris mniSvnelobis albaToba, roca H0 aris
WeSmariti.
b) H0: µ A1=µA2 ==µA3 uaryofilia
H1: yvela µ A ar aris erTmaneTis toli miRebulia
g) sul cota, erTi gansxvaveba mainc arsebobs jgufebis
saSualoebs Soris, romelic statistikurad mniSvnelovania.
d) saSualoTa Soris realurad miRebuli gansxvaveba
ganpirobebulia ara SerCevis SecdomiT, aramed sxva mizeziT.
e) Tu p>0.05 iqneba SedegebSi warmodgenili, maSin metobis
niSani aRniSnavs imas, rom H0 ar aris uaryofili da ori
SerCevis saSualo erTamaneTisagan mniSvnelovnad ar
gansxvavdeba mniSvnelovnobis 0.05 doneze. Aam SemTxvevaSi
davaskvniT, rom naxatis tipi ar axdens gavlenas gaxsenebuli
detalebis raodenobaze.
MSE=5.333 gvaZlevs MSSecdomis mniSvnelobas FrealurisaTvis. es
mniSvneloba aris jgufSida Secdomis cvalebadobis sazomi
monacemebSi.
CD=3.9 gvaZlevs kritikul gansxvavebas wyvilTaSoris SedarebebSi
tukis
namdvilad mniSvnelovani sxvaobis kriteriumis gamoyenebiT. am
SemTxvevaSi statistikurad mniSvnelovani wyvilTaSoris
Sedarebebia aRwerili.
eta kvadrati
=0.52 η2 mniSvneloba gviCvenebs, rom am eqsperimentSi gaxsenebuli
masalis qulebis dispersiis 52 procenti aixsneba naxatis tipiT.
zogjer gamoiyeneba kavSiris siZlieris sazomi, ω2 (omega
kvadrati).
ω2-is mniSvneloba iqneba odnav mcire, vidre η2.
SeamowmeT Tqveni codna X-3
1. paragrafSi SeamowmeT Tqveni codna IX-3, pirveli amocanis me-9 kiTxva
iyo eqsperimenti, romelic molodinis gavlenas. zomavda drois Sualedis
Sefasebaze. gaixseneT, rom sakontrolo jgufis cdis pirebs ar mouwiaT
55 lodini eqsperimentis dawyebamde, maSin roca eqsperimentuli jgufis cdis
pirebs uTxres, rom eqsperimentatoris asistenti daigvianebda. miuxedavad amisa,
Sualedi orive jgufisaTvis 5 wuTi iyo. mas Semdeg, rac iwyeboda eqsperimenti,
cdis pirebs sTxovdnen SeefasebinaT drois Sualedis xangrZlivoba. davuSvaT,
Tqven gaimeoreT es eqsperimenti 22 cdis pirze da miiReT 5-wuTiani Sualedis
Semdegi Sefasebebi (wuTebSi):
jgufi
sakontrolo eqsperimentuli
6 8
7 9
9 11
7 10
6 9
8 8
5 7
9 6
6 10
8 10
7 9
ipoveT saSualo da standartuli gadaxra TiToeuli jgufisaTvis, Semdeg
gamoiTvaleT dispersiuli analizi da upasuxeT Semdeg kiTxvebs:
a. CamoayalibeT statistikuri hipoTezebi F kriteriumisaTvis am
monacemebze dayrdnobiT.
b. risi tolia Frealuris mniSvneloba?
g. risi tolia df FrealurisaTvis?
d. risi tolia Fkritikuli 0.05 doneze?
e. risi tolia uaryofis are FrealurisaTvis?
v xvdeba Tu ara Frealuri uaryofis areSi?
z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb?
uaryofT, Tu ver uaryofT H0-s? miiRebT Tu ar miiRebT H1-s?
T. jgufis saSualoebi erTi da igive Tu gansxvavebuli populaciebidan
arian?
56 i. aris Tu ara saSualoTa Soris gansxvaveba statistikurad
mniSvnelovani?
k. gansxvavdebian Tu ara jgufebi im drois Sefasebebis mixedviT,
romelsac isini lodinSi atarebdnen? Tu gansxvavdebian, ra
mimarTulebiT gansxvavdebian?
l. saWiroa Tu ara mravaljeradi Sedarebis kriteriumebi am
eqsperimentSi? axseniT Tqveni pasuxi.
m. risi tolia η2 am eqsperimentSi?
n. ramdenad kargad xsnis damoukidebeli cvladi am eqsperimentSi
dispersias?
2. sadehma, ravivma da gruberma (Sadeh, Raviv, Gruber, 2000) ikvlies Zilis
Taviseburebebi me-2, me-4 da me-6 klasel bavSvebSi. davuSvaT, rom Tqvenc
ikvlieT es sakiTxi da meore, meoTxe da meeqvse klaseli gogonebis
jgufebisaTvis aRmoaCineT Zilis Semdegi periodebi wuTebSi. TiToeul jgufSi
iyo 14 bavSvi. Zilis periodi ganisazRvreboda, rogorc drois raodenoba
daZinebidan diliT gaRviZebamde.
klasi
me-2 me-4 me-6
580 497 525
525 515 506
562 543 475
590 478 493
575 567 537
603 532 532
594 517 480
521 510 501
536 534 472
612 556 463
514 511 521
57
544 523 517
573 572 477
592 574 515
ipoveT TiToeuli jgufisaTvis saSualo da standartuli gadaxra, Semdeg
gamoiTvaleT dispersiuli analizi am qulebis gamoyenebiT da upasuxeT Semdeg
kiTxvebs:
a. CamoayalibeT statistikuri hipoTezebi F kriteriumisaTvis am monacemebze
dayrdnobiT.
b. risi tolia Frealuris mniSvneloba?
g. risi tolia df FrealurisaTvis?
d. risi tolia Fkritikuli 0.05 doneze?
e. risi tolia uaryofis are FrealurisaTvis?
v. xvdeba Tu ara Frealuri uaryofis areSi?
z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb? uaryofT
Tu ver uaryofT H0-s? miiRebT Tu ar miiRebT H1-s?
T. ra SegiZlia daaskvnaT am qulebis safuZvelze sruli dispersiuli
analizidan?
i. saWiroa Tu ara mravaljeradi Sedarebis kriteriumiebi am eqsperimentSi?
daasabuTeT Tqveni pasuxi.
k. risi tolia tukis CD mniSvneloba?
l. romeli wyvilTaSoris gansxvaveba aris statistikurad mniSvnelovani?
m. aRwereT, rogor gansxvavdebian jgufebi Zilis periodSi?
n. risi tolia η2 am eqsperimentSi?
o. ramdenad kargad xsnis damoukidebeli cvladi am eqsperimentSi
dispersias?
3. Semdegi savarjiSoebi warmoadgens cdis pirTa Soris erTfaqtoriani
dispersiuli analizis Sedegebis prezentaciis nimuSs. upasuxeT Semdeg
SekiTxvebs TiToeuli angariSisaTvis.
a. damoukidebeli cvladis ramdeni done iyo gamoyenebuli am eqsperimentSi?
b. ramdeni cdis piri monawileobda am eqsperimentSi?
g. risi tolia Frealuris mniSvneloba?
d. risi tolia Fkritikulis mniSvneloba mniSvnelovnobis 0.05 doneze?
58
e. uaryofT Tu ver uaryofT H0-s?
v. miiRebT Tu ar miiRebT H1-s?
z. arsebobs Tu ara, sul cota, erTi statistikurad mniSvnelovani gansxvaveba
saSualoTa Soris?
T. Tu arsebobs, romeli saSualoebi gansxvavdebian mniSvnelovnad
erTmaneTisagan?
3.1. nacnobi leqsis mosmenisas SemCneuli intonaciebis saSualo raodenoba
aris 79.9. xolo ucnobi leqsis mosmenisas SemCneulisa _ 73. alfa tolia 0.05-
saTvis cdispirTa Soris erTfaqtoriani dispersiuli analizi gviCvenebs, rom
saSualoebi mniSvnelovnad gansxvavdebian erTmaneTisagan, F(1.6)=14.15, MSE=5.74,
p<0.01
3.2. roca alfa 0.05-is tolia, ar aris aRmoCenili gansxvaveba gaxsenebuli
sityvebis, rogorc mexsierebaSi Senaxvis intervalis funqciis raodenobaSi,
F(3,36)=1.13 MSE=14.342, p>0.05.
3.3 beWdvis Secdomebis saSualo raodenoba 18.5 iyo im jgufisaTvis, vinc
usmenda klasikur musikas, 29.9 _ vinc usmenda mZime roks da 17.5 _ vinc saerTod
ar usmenda musikas _ sakontrolo jgufisaTvis. alfa tolia 0.05-isa cdis
pirTa Soris erTfaqtoriani dispersiuli analizi gviCvenebs musikis tipis
mniSvnelovan gavlenas, F(2.36)=33.94, MSE=18.261, p<0.001. saboloo Sedarebebi tukis
namdvilad mniSvnelovani sxvaobis kriteriumis gamoyenebiT gviCvenebs
mniSvnelovnad ufro met beWdviT Secdomebs mZime rokis musikis pirobebSi,
vidre musikis gareSe beWdvis an klasikuri musikis mosmenis SemTxvevaSi.
musikis gareSe beWdvisa da klasikuri musikis pirobebi erTmaneTisagan
mniSvnelovnad ar gansxvavdeba.
Sejameba
• erTfaqtoriani mravaldoniani sqema aris eqsperimenti erTi
damoukidebeli cvladiT da am damoukidebeli cvladis sami an meti
doniT.
• cdis pirTa Soris erTfaqtoriani dispersiuli analizi
gamoyenebulia erTi faqtoris mravaldoniani sqemis monacemTa
gasaanalizeblad.
• F statistika mocemulia F= MSA / MSSecdomis saxiT.
• MS aris dispersia da is napovnia SS/df.
59
• MSA icvleba damoukidebeli cvladisa da SerCevis Secdomis gavleniT.
• MSSecdoma icvleba jgufSida Secdomis cvalebadobiT.
• Tu damokidebel cvlads ara aqvs gavlena, maSin MSA da MSSecdoma
afasebs mxolod Secdomis cvalebadobas monacemebSi, da F unda iyos
daaxloebiT 1-is toli.
• roca damoukidebel cvlads aqvs gavlena, maSin MSA mniSvneloba
izrdeba, magram MSSecdomis mniSvnelobaze es ar axdens gavlenas. F-is
mniSvneloba xdeba 1-ze meti.
• F statistikis gamoyenebas statistikur kriteriumSi mohyveba
statistikuri hipoTezebis formulirebis Cveulebrivi nabijebi,
mniSvnelovnobis donis dadgena, uaryofis areebis gansazRvra,
Frealuris gamoTvla da gadawyvetilebebi, romlebic statistikur
hipoTezebs exeba.
• statistikurad mniSvnelovani Frealuri, roca damoukidebeli cvladis
sami an meti donis manipulireba xdeba, moiTxovs mravaljeradi
Sedarebis kriteriums, raTa dispersiuli analizi gagrZeldes.
• sruli dispersiuli analizisaTvis Frealuris statistikuri
mniSvnelovnobis dadasturebis Semdeg mravaljeradi Sedarebis
kriteriumebi gamoiyeneba imis dasadgenad, TuU romeli saSualoebi
gansxvavdebian mniSvnelovnad erTmaneTisagan erTfaqtorian
mravaldonian sqemaSi.
• tukis kriteriumi gamoiyeneba saSualoTa simravlis yvela SesaZlo
wyvilTa SedarebisaTvis saboloo SedarebebSi. tukis kriteriumi
pirveli gvaris Secdomis daSvebis albaTobas inarCunebs α-s tolad
an masze naklebad yvela wyvilTaSoris SedarebebSi.
• η2 SeiZleba gamoviyenoT Carevis efeqtis sazomad.
ZiriTadi terminebi da simboloebi
a
dispersiuli analizi (ANOVA)
jgufTa Soris dispersia (MSA )
kritikuli gansxvaveba (CD)
df SS A-saTvis (dfA)
60
df SS Secdom-isaTvis (dfSecdoma)
Secdomis xarisxi eqsperimentSi
eta kvadrati (η2)
Fkritikuli
Frealuri
A faqtori
sruli saSualo (XG)
saSualos kvadrati (MS)
mravaljeradi Sedarebis kriteriumebi
N
nA
erTfaqtoriani mravaldoniani sqema
wyvilTaSoris Sedareba
saboloo Sedareba
q
SS A
SS Secdoma
SS sruli
sruli Tavisuflebis xarisxi (dfsruli)
tukis HSD kriteriumi
jgufSida Secdomis dispersia (MSSecdoma)
AXX −
GXX −
GA XX −
Tavis danarTi: cdis pirTa Soris erTfaqtoriani dispersiuli
analizis gamoTvla specialuri gamoTvliTi formulis
saSualebiT
dispersiuli analizis gamoTvliTi formula efuZneba qulaTa jams
eqsperimentul jgufSi (da ara jgufis saSualos) da qulaTa srul jams
61 eqsperimentSi ( da ara srul saSualos). Tu gavixsenebT im aRniSvnebs,
romelTac cxrilSi #10.2 gavecaniT, eqsperimenti damoukidebeli cvladis A-s
sami doniTa da 5 cdis piriT TiToeul doneze SeiZleba Semdegnairad iyos
warmodgenili:
A faqtori
A1 A2 A3
X11 X12 X13
X21 X22 X23
X31 X32 X33
X41 X42 X43
X51 X52 X53
TA1 TA2 TA3 G(sruli jami)
sadac Xij=cdis pirTa qulebi
TA=qulaTa jami damoukidebeli cvladis donisaTvis
G=qulaTa sruli jami
nA=qulaTa raodenoba damoukidebeli cvladis doneze
a=damoukidebeli cvladis doneebis raodenoba
N=qulaTa sruli raodenoba
am terminebis gamoyenebiT sami ricxviTi mniSvnelobis dadgena xdeba:
[1] =∑∑ 2ijX yvela qulaTa kvadratebis jami
[2] =A
A
nT∑ 2
TiToeuli jgufis qulaTa jamis kvadratebis jami, gayofili
jgufSi qulaTa raodenobaze
[3] =N
G 2
sruli jamis kvadrati, gayofili qulaTa srul raodenobaze
am ricxviTi mniSvnelobebis gamoyenebiT miRebuli dispersiuli analizi
Semdegnairad gamoiyureba:
62 wyaro SS df MS F faqtori (A) [2] _ [3] a-1
A
A
dfSS
SecdomaMS
MS A
Secdoma [1] _ [2] N_a
Secdoma
Secdoma
dfSS
sul [1] _ [3] N-1 ar
gamoiTvleba
am gamoTvlebis ilustraciisaTvis gamoviyenebT cxrilSi #10.1 magaliTis
saxiT mocemul qulebs, romlis dispersiuli analizi Sejamebulia cxrilSi
#10.5.
faqtori A
A1 A2 A3
19 14 15
13 13 10
16 16 11
17 12 12
20 10 12
TA 85 65 60
G=85+65+60=210
nA=5, a=3, da N=15
ricxviTi gamoTvlebis Sedegad miiReba Semdegi mniSvnelobebi
[1] = 192 + . . . + 202 + 142 + . . . + 102 + 152 + . . . + 122= 3074
[2] = 3010505.15
5606585 222
==++
[3] = 294015
1.4415
2102
==
maSin
SSA= [2]-[3]=3010-2940=70
dfA= a-1=3-1=2
MSA= 352
70=
SS Secdoma= [1]_[2]=3074_3010=64
dfSecdoma = N-a=15-3=12
63
MSSecdoma 333.51264
==
SS sruli = [1] _ [3] = 3074 _ 2940 = 134
dfsruli = N -1=15-1=14
Frealuri = 56.6333.535
==SecdomaMS
MSA
dispersiuli analizis Sejameba Semdegnairad xdeba:
wyaro SS df MS F
faqtori (A) 70 2 35 6.56*
Secdoma 64 12 5.333
sul 134 14
*p< 0.05
am gamoTvliTi midgomiT miRebuli mniSvnelobebi, cxrilSi #10.5 asaxuli im
mniSvnelobebis identuria, romlebic miRebulia dispersiuli analizis
ganmartebiTi formulis saSualebiT.
kiTxvebi
1. Tanamedrove sazogadoebaSi musika yvelgan ismis da fsiqologebi swavloben
mis gavlenas adadmianis mier sxvadasxva amocanis Sesrulebaze. magaliTad,
davuSvaT, eqsperimentatorma Seqmna 8 cdis pirisagan Semdgari sami
ekvivalenturi jgufi da TiToeul pirovnebas sTxova waekiTxaT gazeTis mokle
statiis koreqtura. A A1 jgufis monawileebi am amocanas musikis Tanxlebis
gareSe asrulebdnen, A2 jgufis cdis pirebi irCevdnen Zvel popularul
melodias, xolo A3 jgufis cdis pirebi asrulebdnen amocanas mZime rokis
TanxlebiT. damokidebuli cvladi iyo aRmoCenili Secdomebis raodenoba 50
SesaZleblidan. davuSvaT, Semdegi qulebi iyo miRebuli:
Carevis jgufi
musikis
gareSe
A1
Zveli
popularuli
melodia
mZime roki
A3
64 A2
40 34 26
41 39 24
39 38 19
36 40 23
35 34 18
32 35 21
31 29 23
34 36 29
a. TiToeuli jgufisaTvis ipoveT saSualo da standartuli gadaxra. Semdeg
gaaanalizeT qulebi cdis pirTa Soris erTfaqtoriani analizis safuZvelze da
upasuxeT Semdeg kiTxvas: axdens Tu ara musika gavlenas cdis piris mier
aRmoCenil Secdomebze? gamoiyeneT 0.05 mniSvnelovnebis done. Tu saWiroa,
gamoiyeneT tukis kriteriumi mravaljeradi arCevanisaTvis.
b. risi tolia η2 am esqperimentSi? ra xarisxiT SeiZleba viwinaswarmetyveloT
damokidebuli cvladi, Tu gaviTvaliswineT, rom viciT damoukidebeli cvladis
done?
g. am eqsperimentis Sedegebi im saxiT aRwereT, rogoric naCvenebia qveTavSi
dispersiuli analizis Sedegebis prezentacia
2. skolis maswavlebeli mravalmxrivi niWiT unda xasiaTdebodes, maT Soris
niWiT, rom gamoumuSavos bavSvebs sxvadasxva fizikuri unari, romelic saWiroa
sportis garkveuli saxeobebisaTvis da sxva aqtivobebisaTvis. imisaTvis, rom
ganvsazRvroT, Tu ramdenad kargad SeuZliaT daeuflon maswavleblebi am
xerxebs videofilmebidan, morisonma da rivma (Morrison and Reeve, 1986) Seqmnes
ekvivalenturi jgufi da uCvenes videofilmebi gansxvavebuli instruqciebiT.
pirveli jgufi sakontrolo iyo da firs ar uyurebda. meore jgufma naxa
firi, sadac fexburTelebi asrulebdnen dartymebs da mesame jgufma naxa firi
srolis, daWeris, dartymis unarebze. amis Semdeg cdis pirebs uCvenebdnen meore
videofirs, romelSic iyo demonstrirebuli srolis, daWeris da dartymis
unarebi da sTxovdnen ganesazRvraT, Tu romeli iyo am moqmedebebidan sworad
Sesrulebeuli. davuSvaT Semdegi qulebi miiRes 9 cdis pirisagan TiToeuli
instruqciis pirobebSi:
65
instruqciis piroba
videofiris gareSe (A1) videofiri fexburTelze
(A2)
videofiri srolis,
daWeris da dartymis
demonstrirebiT (A3)
50 53 61
46 48 58
39 37 49
52 49 56
44 46 45
51 48 66
41 37 58
36 41 64
48 40 57
a. ipoveT saSualo da standartuli gadaxra TiToeuli jgufisaTvis, Semdeg
gaaanalizeT qulebi cdis pirTa Soris erTfaqtoriani dispersiuli analiziT,
raTa upasuxoT Semdeg kiTxvebs: SeiZleba instruqciis pirobam moaxdinos
gavlena cdis piris unarze sworad Seafasos srolis, daWeris da dartymis
moqmedebebis Sesruleba? gamoiyeneT 0.05 mniSvnelovnobis done. Tu saWiroa,
gamoiyeneT mravaljeradi Sedarebis tukis namdvilad mniSvnelovani sxvaobis
kriteriumi.
b. risi tolia η2 am eqsperimentSi? damokidebuli cvladis dispersiis ra wili
SeiZleba aixsnas damoukidebeli cvladis donis codniT?
g. aRwereT am eqsperimentis Sedegebi am Tavis erT-erT qveTavSi _
“dispersiuli analizis Sedegebis prezentacia” _ aRwerili stilis mixedviT.
3. mecxre Tavis kiTxvebis amocana #1 exeboda xmauris fiziologiur gavlenas.
aRweril eqsperimentSi, cdis pirebi isxdnen wynar oTaxSi 30 wuTis
ganmavlobaSi, an isxdenen oTaxSi da usmendnen transportis xmaurs pikis
saaTSi 30 wuTis ganmavlobaSi. 30 wuTis gasvlis Semdeg cdis pirebs gauzomes
sistoluri sisxlis wneva da miiRes Semdegi monacemebi( milimetrebSi)
jgufi
66 uxmauro xmauriani
106 141
117 136
124 124
129 139
115 121
131 119
121 147
115 128
128 115
136 134
127 140
a. ipoveT saSualo da standartuli gadaxra TiToeuli jgufisaTvis, Semdeg
gaaanalizeT qulebi cdis pirTa Soris erTfaqtoriani dispersiuli analiziT,
raTa upasuxoT, moqmedebs Tu ara xmauri sistolur sisxlis wnevaze?
gamoiyeneT 0.05 mniSvnelovnobis done.
b. SeadareT Frealuris mniSvneloba trealuris mniSvnelobas, romelic miRebuli
iyo mecxre TavSi mocemul kiTxvebSi (amocana #1). miiReT Tu ara mosalodneli
kavSiri t-sa da F Soris?
4. eqsperimentatorma gamoiyena cdis pirTa Soris erTfaqtoriani sqema
damoukidebeli cvladis 6 doniT, A1, A2, A3, A4, A5 da A6 da 7 cdis piriT
TiToeul doneze.
a. risi tolia a, nA, da N mniSvnelobebi am eqsperimentSi?
b. risi tolia dfA, dfSecdomisa da dfsrulis mniSvnelobebi am eqsperimentis
dispersiuli analizisaTvis?
5. Semdegi cxrili aris dispersiuli analizis dausrulebeli cxrili.
davuSvaT cdis pirTa toli raodenobaa damoukidebeli cvladis TiToeul
doneze. SS , df da MS Soris arsebuli kavSirebis gamoyenebiT SeavseT
TiToeuli cxrilis carieli ujrebi saWiro mniSvnelobebiT.
a. damoukidebeli cvladis ramdeni donis manipulireba xdeba?
b. ramdeni cdis piri iyo Semowmebuli TiToeul pirobaSi?
g. risi tolia cdis pirTa sruli raodenoba am eqsperimentSi?
d. aris Tu ara Frealuris mniSvneloba statistikurad mniSvnelovani 0.05 doneze?
67
e. ra gadawyvetilebas iRebT H0-sa da H1-isaTvis TiToeuli analizis
SemTxvevaSi?
cxrili #1
wyaro SS df MS F
faqtori (A) 50 2 _ _
Secdoma 270 27 _
sul _ _
cxrili #2
wyaro SS df MS F
faqtori (A) 60 _ 12 _
Secdoma _ 72 _
sul 276 77
cxrili #3
wyaro SS df MS F
faqtori (A) _ _ _ 3
Secdoma 320 _ 5
sul 365 67
6. aris Tu ara H0-is uaryofa da H1-is miReba aucilebeli dispersiul
analizSi, raTa uzrunvelyoT kvlevis hipoTezis mxardaWera? axseniT Tqveni
pasuxi.
7. Tqven gamoiTvaleT tdamoukidebeli or jgufze da trealuri(18)=2.000. Tu qulebs
gaaanalizebT dispersiuli analizis Frealuris ra mniSvneloba iqneba napovni?
aRniSneT df FrealurisaTvis.
8. Tqven SeasruleT dispersiuli analizi or damoukidebel jgufze da
Frealurze(1, 22)=9. Tu monacemebs gaaanalizebdiT tdamoukidebeliT, trealuris ra
mniSvnelobas ipovidiT? aRniSneT df trealurisaTvis.
68 9. axseniT, ratom ar SeuZlia statistikur kriteriums daadginos ori jgufis
erTmaneTisagan mniSvnelovnad gansxvavebis mizezi?
10. fsiqologi dainteresebuli iyo 4 pirobis erTmaneTTan SedarebiT
mowevisaTvis Tavis danebebis programaSi. man Seqmna 4 jgufi 17 cdis
pirisaTvis, TiToeuli SemTxveviTi wesiT iyo ganawilebuli da Seuqmna
TiToeul jgufs gansxavebuli mkurnalobis piroba. damoukidebuli cvladi
izomeboda sigaretebis mowevaTa Soris drois saSualoTi mkurnalobidan 5
kviris Semdeg. mkurnalobis jgufebi da saSualo dro sigaretebis mowevas
Soris(wuTebSi) mocemulia cxrilSi #1, xolo dispersiuli analizi am
monacemebisaTvis warmodgenilia cxrilSi #2.
cxrili #1
Carevis pirobebi da saSualo dro sigaretebis mowevas Soris (wuTebSi)
Carevis piroba saSualo dro
A1 sakontrolo jgufi, ar aris Careva 24
A2 qcevis modifikacia 38
A3 sigaretis anonimuri klubebi 34
A4 sigaretis Cvevis sawinaaRmdego saReWi rezina 28
cxrili #2
dispersiuli analizis Semajamebeli cxrili #1 sigaretis mowevebs Soris
saSualo droisaTvis
wyaro SS df MS F
faqtori (A) 1972 3 657.333 21.64*
Secdoma 1944 64 30.375
sul 3916 67
*p < 0.01
tukis namdvilad mniSvnelovani sxvaobis Sedegad gamoTvlili CD=5 wuTi 0.05
mniSvnelovnobis doneze.
a. risi tolia Carevis pirobebis gavlena im drois raodenobaze, romelic
arsebobs sigaretis moweva Soris?
b. aRwereT am eqsperimentis Sedegebi am Tavis erT-erT qveTavSi _
“dispersiuli analizis Sedegebis prezentacia” _ aRwerili stilis mixedviT.
69
Tavi XI. cdis pirTa Soris orfaqtoriani sqema da dispersiuli analizi cdis pirTa Soris 2x2 sqemis magaliTi faqtoruli sqemis Sedegad miRebuli informacia ujredis saSualoebi mTavari efeqtis saSualoebi damoukidebeli cvladebis urTierTqmedeba cdis pirTa Soris orfaqtoriani sqemis dispersiuli analizi monacemTa dayofa dayofili monacemebisagan gadaxrebis kvadratebis saSualos miReba F statistikis gamoTvla statistikuri hipoTezis Semowmeba cdis pirTa Soris orfaqtorian dispersiul analizSi
faqtorebi, romlebic gavlenas axdenen gadaxrebis kvadratebis saSualoebze dispersiul faqtorul analizSi F statistikaze damyarebuli statistikuri gadawyvetilebebis miReba cdis pirTa Soris orfaqtoriani dispersiuli analizis daSvebebi
dispersiis 2x2 faqtoriani dispersiuli analizis axsna damoukidebeli cvladis martivi efeqti Sejameba: dispersiuli analizis axsna eqsperimentis magaliTze
Carevis efeqtis siZlieris gazomva dispersiul faqtorul analizSi
dispersiuli faqtoruli analizis Sedegebis prezentacia
`mravali qceva ganisazRvreba ori an meti erTdroulad moqmedi damoukidebeli cvladis mier. magaliTad, xmauris (damoukidebeli cvladi) gavlena wakiTxulis gaazrebaze (damokidebuli cvladi) gansxvavdeba pirovnebis introvertuli da eqstravertuli tipebisaTvis (meore damoukidebeli cvladi) (Standing, Lynn, & Moxness, 1990). an, Tu sxva magaliTs ganvixilavT, arsebiTi saxelebis wyvilebis (damokidebuli cvladi) damaxsovreba damokidebulia im adamianis warmosaxvis unarze (erTi damoukidebeli cvladi), vinc imaxsovrebs sias, suraTebis tipebze, romelic miewoda adamians (meore damoukidebeli cvladi), da damaxsovrebis drois intervalis xangrZlivobaze (mesame damoukidebeli cvladi) (O’Brien & Wolford, 1982). erTfaqtoriani sqema ar iZleva SesaZleblobas, erTdroulad moxdes erTze meti damoukidebeli cvladiT manipulireba; amitom, imisaTvis, rom Seiswavlon ori an meti damoukidebeli cvladis kombinaciis gavlena, gamoiyeneba faqtoruli analizi. faqtoruli sqema: kvlevis iseTi sqema, romelSic erTdroulad icvleba ori an meti damoukidebeli cvladi.
faqtoruli sqema aris kvlevis iseTi sqema, romelSic erTdroulad icvleba ori an meti damoukidebeli cvladi. yvelaze martiv faqtorul sqemaSi cvalebadobs ori damoukidebeli cvladi da TiToeul damoukidebel cvlads ori mniSvnelobis (donis) miReba SeuZlia. am sqemas xSirad 2x2 (“ori orze”) sqemas uwodeben. pirveli ‘2’ aRniSnavs, rom arsebobs pirveli damoukidebeli cvladis (romelic ganisazRvreba, rogorc A faqtori) ori mniSvneloba (ori done), xolo meore ‘2’ ki aRniSnavs meore damoukidebeli cvladis (romelic ganisazRvreba, rogorc B faqtori) mniSvnelobebis (doneebis) raodenobas. 2x2x2 sqema warmoadgens samfaqtorian sqemas, sadac faqtorebia A, B da C, da TiToeuli maTgani or mniSvnelobas (dones) iRebs. arsebobs faqtoruli analizis mravali SesaZlo varianti, rogoricaa 2x4 sqema (A faqtoris ori done, B faqtoris 4 done), 3x3x2 sqema (A da B faqtorebis ori done, C faqtoris ori done) da 2x2x2x2 sqema (oTxi faqtori, A, B, C da D, TiToeuls aqvs ori done). faqtoruli analizi aseve SesaZlebelia iyos srulad cdis pirebs Soris, srulad cdis pirebs SigniT an am oris kombinacia, rac Sedegad Sereul sqemas gvaZlevs. am TavSi Cven ganvixilavT mxolod cdis pirebs Soris orfaqtorian sqemas (romelsac aseve uwodeben ormxriv sqemas). ujredi an Carevis piroba: faqtorul eqsperimentSi TiToeuli damoukidebeli cvladis erTi donisagan warmoqmnili kombinacia.
cdis pirebs Soris 2x2 sqema mosaxerxebelia faqtoruli sqemebis statistikuri monacemebis analizis wardgenisaTvis. cxrili # 11.1 warmoadgens or damoukidebel cvlads an faqtors, romlebic aRniSnulia, rogorc A da B. maTgan TiToeuli iRebs or mniSvnelobas (dones): faqtori A iRebsAA1 da A2 mniSvnelobebs da faqtori B iRebs B1 da B2 mniSvnelobebs. zogadad, arsebobs A faqtoris a mniSvneloba da B faqtoris b mniSvneloba. am magaliTSi a=2 da b=2. ori damoukidebeli cvladis kombinacia cxrilSi qmnis oTx pirobas, romlebsac ujredebs uwodeben. TiToeuli ujredi anu piroba aris kombinacia, romelic iqmneba
TiToeuli damoukidebeli cvladis erTi mniSvnelobiT, magaliTad, A1B1, A1B2, A2B1 da A2B2 ujredebi. Cveulebriv, TiToeul ujredSi SemTxveviTad nawildeba cdis pirebis toli raodenoba. amdenad, Tu cdis pirebs Soris 2x2 sqemaSi gamoyenebulia 20 cdis piri da TiToeul ujredSi iqneba xuTi cdis piri. yoveli cdis piri mxolod erT ujredSi xvdeba. cxrili # 11.1 cdis pirebs Soris 2x2 sqemis gegma. A faqtoris doneebis kombinacia B faqtoris doneebTan qmnis pirobas anu ujreds.
faqtori A
A1 A2
B1 faqtori B B2
A1B1 piroba (A1B1 ujredi)
A2B1 piroba (A2B1 ujredi)
A1B2 piroba (A1B2 ujredi)
A2B2 piroba (A2B2 ujredi)
cdis pirebs Soris 2x2 sqemis magaliTi zogadad adamianebs ar uyvarT sxva adamianebisTvis cudi siaxleebis Setyobineba. am fenomens MUM efeqts uwodeben. am efeqtis asaxsnelad SemoTavazebuli iyo ramdenime hipoTeza. pirovnuli diskomfortis hipoTezis mixedviT, cudi siaxlis momtani adamiani pirovnul diskomforts grZnobs am siaxlis Setyobinebisas da Tavs aridebs am diskomforts imiT, rom ar yveba am siaxles. meore hipoTeza – TviTprezentaciis hipoTeza – amtkicebs, rom individebi ar grZnoben diskomforts cudi siaxlis Setyobinebisas. piriqiT, cudi siaxlis Setyobinebis Seusrulebloba warmoadgens publikaze gaTvlil mcdelobas, SeinarCunon dadebiTi socialuri imiji. bolos da bolos, vis surs, cudi siaxleebis mimtanad iyos cnobili? am hipoTezebis Sesamowmeblad bondma da andersonma (Bond & Anderson, 1987) gamoiyenes cdis pirebs Soris 2x2 sqema. ori damoukidebeli cvladi iyo cdis piris arseboba im adamianis mxedvelobis areSi, visTvisac unda eambnaT cudi ambebi (cdis piris yofna mxedvelobis areSi, faqtori A) da axali ambebis tipi (siaxlis tipi, faqtori B). TiToeuli damoukidebeli cvladi iRebda or mniSvnelobas. cdis piris mxedvelobis areSi yofnis mxriv, igi an ar imyofeboda (mniSvneloba A1) an imyofeboda (mniSvneloba A2) im piris mxedvelobis areSi, visTvisac unda emcnoT axali ambebi. siaxlis tipi iyo an kargi (mniSvneloba B1), an cudi (mniSvneloba B2). eqsmerimentis procedura Semdegi iyo: cdis pirs unda Caetarebina mravlobiTi pasuxis mqone inteleqtis testi sxva pirisaTvis. es piri eqsperimentatoris TanaSemwe iyo, magram cdis pirma ar icoda amis Sesaxeb. cdis piri da TanaSemwe imyofebodnen sxvadasxva oTaxebSi, romelTa Soris calmxrivi SuSa iyo CamontaJebuli. im pirobaSi, romelSic cdis piri ar Canda (mniSvneloba A1), mas eubnebodnen, rom TanaSemwes ar SeeZlo misi danaxva SuSaSi. im pirobaSi, romelSic cdis piri Canda (mniSvneloba A2), mas eubnebodnen, rom TanaSemwes SeeZlo misi
danaxva SuSaSi. siaxlis tipis cvla xdeboda TanaSemwes mier testis warmatebuli Tu warumatebeli SesrulebiT. kargi siaxlis dros (mniSvneloba B1), cdis piri TanaSemwes awvdida informacias, rom misi Sedegi testis Sedegebis saukeTeso 20 procentSi xvdeboda. cudi siaxlis pirobaSi (mniSvneloba B1), cdis piri TanaSemwes awvdida informacias, rom misi Sedegi testis Sedegebis yvelaze dabal 20 procentSi xvdeboda. damokidebuli cvladi iyo cdis piris meryeobis periodi (latenturi dro) testis Sedegis Setyobinebisas. es periodi warmoadgenda dros testis dasrulebidan im momentamde, rodesac cdis piri atyobinebda TanaSemwes Tavis Sedegs. MUM efeqtis Sesaxeb arsebuli ori hipoTeza eqsperimentis gansxvavebul Sedegebs winaswarmetyvelebda. Tu MUM efeqts safuZvlad udevs cdis pirebis pirovnuli diskomforti, maSin cudi siaxlis Setyobinebas ufro didi dro dasWirdeboda, vidre kargi siaxlisa, imisda miuxedavad, aris Tu ara cdis piri TanaSemwes mxedvelobis areSi. sxva sityvebiT, kargi (mniSvneloba B1) da cudi (mniSvneloba B2) siaxleebis pirobebs Soris gansxvaveba ar unda iyos damokidebuli imaze, xedavs Tu ara TanaSemwe cdis pirs. meore mxriv, Tu MUM efeqti ubralod publikaze gaTvlili qmedebaa, maSin latenturi dro cudi siaxlis miwodebisas damokidebuli unda iyos imaze, xedavs Tu ara TanaSemwe cdis pirs. es niSnavs, rom unda moxdes damoukidebeli cvladebis urTierTqmedeba. rodesac xdeba kargi siaxlis Setyobineba, ar unda iyos gansxvaveba latentur droSi im pirobis miuxedavad, xedavs Tu ara TanaSemwe cdis pirs. xolo rodesac cudi siaxlis Setyobineba xdeba, latenturi dro im pirobaSi, rodesac cdis piri ar imyofeba mxedvelobis areSi, unda mniSvnelovnad maRali iyos, vidre latenturi dro im pirobaSi, rodesac cdis piri imyofeba mxedvelobis areSi. Sesabamisad, rodesac cdis piri ar imyofeba mxedvelobis areSi, ar unda iyos gansxvaveba latentur droSi kargi da cudi siaxleebis Setyobinebisas. xolo rodesac cdis piri imyofeba mxedvelobis areSi, cudi ambebis Setyobinebis latenturi dro unda ufro didi iyos, vidre kargi ambebisa. davuSvaT, xuTi mamrobiTi sqesis cdis piri SemTxveviTad gadanawilda TiToeul ujredSi da miviReT latenturi drois monacemebi (wamebSi), romlebic warmodgenilia cxrilSi # 11.2. im SemTxvevaSi, Tu Sedegebi saerTod romelime winaswarmetyvelebas Seesabameba, romel hipoTezas miekuTvneba es winaswarmetyveleba - pirovnuli diskomfortis hipoTezas, Tu publikaze gaTvlili qmedebis hipoTezas? cxrili # 11.2 pasuxis hipoTeturi latenturi dro (wamebSi), rogorc cdis piris mxedvelobis areSi yofnisa da siaxlis tipis funqcia 20 cdis pirisaTvis sanimuSo cdis pirTa Soris 2x2 eqsperimentSi.
cdis piris mxedvelobis areSi
yofnis piroba (faqtori A) ar Cans (A1) Cans (A1)
siaxlis tipi (faqtori B)
kargi (B1)
97.0 90.0 80.0 107.0 80.0
70.0 87.0 81.0 95.0 90.0
cudi (B2)
68.0 87.0 92.0 80.0 84.0
114.0 96.0 127.0 110.0 115.0
faqtoruli analizis Sedegad miRebuli informacia cxrilis # 11.2 monacemebi (qulebi) sqematuradaa warmodgenili cxrilSi # 11.3. cdis piris monacemi aRniSnulia Xijk, sadac qveda indeqsebi migvaniSneben Semdeg informaciaze: i = ujredSi cdis piris nomeri j = A cvladis done k = B cvladis done cxrili # 11.3 cdis pirebs Soris 2x2 sqemaSi monacemTa sqematuri ganawileba, sadac TiToeul damoukidebel cvlads ori done aqvs da yovel ujredSi xuTi cdis piria.
faqtori A A1 A2
B1 faqtori B B2
X111 X211 X311 X411 X511
X121 X221 X321 X421 X521
X112 X212 X312 X412 X512
X122 X222 X322 X422 X522
magaliTad, X311 aris mesame cdis piris monacemi A1B1 pirobaSi (80.0 wami cxrilSi # 11.2), X421 aris meoTxe cdis piris monacemi A2B1 pirobaSi (95.0 wami), X211 aris meore cdis piris monacemi A1B2 pirobaSi (87 wami), X522 aris mexuTe cdis piris monacemi A2B2 pirobaSi (115 wami) da a.S. rodesac ar
aris aucilebeli konkretuli cdis piris aRniSvna, Cven gamovtovebT ijk indeqsebs da ubralod viyenebT X-s. TiToeuli ujredisaTvis arsebobs nAB cdis pirebis raodenoba, xolo N aris cdis pirebis sruli raodenoba eqsperimentSi. Cvens magaliTSi nAB = 5 da N = 20. monacemebis analizis dros pirveli nabiji centraluri tendenciis sazomis, SerCevis saSualos gamoTvla. cxrilSi # 11.4 naCvenebia ujredisa da mTavari efeqtis saSualoebi, romelTa gamoTvlac SesaZlebelia magaliTis monacemebidan. cxrili # 11.4 ujredisa da mTavari efeqtis saSualoebi cdis pirebs Soris 2x2 sqemaSi. _______________________________________________________________________________
a. ujredis saSualo faqtori A
B1 faqtori B B2
A1 A2
XA1B1 X A2B1 X A1B2 X A2B2
(b) mTavari efeqtis saSualo A faqtorisaTvis,X A1 daX A2
(g) mTavari efeqtis saSualo B faqtorisaTvis, X B1 daX B2
ujredis saSualo: ujredSi nAB qulebis saSualo faqtorul sqemaSi.
faqtori A
B1 faqtori B B1
A1 A2
1AX 2AX
faqtori A A1 A2
B1 faqtori B B2
X B1 X B2
yuradReba ar unda mivaqcioT qulebis klasifikacias B faqtoris mixedviT, raTa miiRoT mTavari efeqtis saSualo A faqtorisaTvis. gansxvavebaX A1-sa daX A2-s Soris aris A faqtoris mTavari efeqti.
yuradReba ar unda mivaqcioT qulebis klasifikacias A faqtoris mixedviT, raTa miiRoT mTavari efeqtis saSualo B faqtorisaTvis. gansxvavebaX B1-sa daX B2-s Soris aris B faqtoris mTavari efeqti.
ujredis saSualo, romelic simbolurad aRiniSneba 11BAX , 21BAX , 12BAX , da
22BAX , an, zogadad, rogorc ABX , warmoadgens ujredSi nAB qulebis saSualos (ixileT cxrili # 11.4a). ujredis saSualo aRniSnavs cdis pirebis tipur Sesrulebas am pirobaSi. ujredebis saSualoebis ricxviTi mniSvnelobebi cxrilisaTvis # 11.2 warmodgenilia cxrilSi # 11.5. standartuli gadaxrebi warmodgenilia frCxilebSi. faqtorul sqemaSi standartuli gadaxra, rogorc wesi, gamoiTvleba mxolod ujredis saSualosaTvis. cxrili # 11.5 ujredisa da mTavari efeqtis saSualoebi qulebisaTvis cxrilidan # 11.2. ricxviTi mniSvnelobebi aris pasuxis latenturi dro wamebSi. ujredis saSualos standartuli gadaxrebi mocemulia frCxilebSi.
cdis piris yofna mxedvelobis areSi (faqtori A)
ar Cans (A1) Cans (A2) mTavari efeqtis saSualo siaxlis
tipisaTvis ( BX )
kargi (B1) siaxlis tipi (faqtori B) cudi (B2) mTavari efeqtis saSualo cdis piris yofnisaTvis mxedvelobis
areSi ( AX )
90.8 84.6 87.7 (11.56) (9.61) 82.2 112.4 97.3 (9.07) (11.15) 86.5 98.5 sruli saSualo 92.5
mTavari efeqtis saSualo: faqtorul analizSi damoukidebeli cvladis erTi donisaTvis yvela cdis piris saSualo, meore damoukidebeli cvladis doneebis mixedviT klasifikaciis ignorirebiT.
mTavari efeqtis saSualo aRniSnavs yvela cdis piris tipur Sesrulebas damoukidebeli cvladis erTi donisaTvis, meore damoukidebeli cvladis doneebis mixedviT klasifikaciis ignorirebiT. amdenad, arsebobs mTavari efeqtis saSualoebis ori erToblioba – erTi A faqtorisaTvis, xolo meore – B faqtorisaTvis.
A faqtoris mTavari efeqti: gansxvaveba 1AX -sa da 2AX -s Soris, gamoxatuli 1AX - 2AX .
A faqtori
mTavari efeqtis saSualo A faqtorisaTvis aRiniSneba 1AX -iTa da 2AX -iT. es saSualoebi, romelTac xandaxan svetebis saSualoebs uwodeben, miiReba yvela cdis pirisaTvis rogorc A1, aseve A2 doneebis Sesabamisad, imisda miuxedavad, Tu ra mniSvnelobebi miiRes maT B faqtoris mixedviT. B faqtoris doneebis mixedviT monacemebis ignorireba, raTa miRebul iqnes A faqtoris mTavari efeqtis saSualo, naCvenebia cxrilSi # 11.4b. magaliTad, mTavari efeqtis saSualo A faqtorisaTvis, romelic aris cdis piris yofna mxedvelobis areSi cxrilSi # 11.2, damyarebulia im cdis pirebis 10 monacemze, romlebic an imyofebian, an ar imyofebian eqsperimentatoris TanaSemwes mxedvelobis areSi. siaxlis tipi (faqtori B) ignorirebulia, rodesac viTvliT A faqtoris mTavari efeqtis
saSualos. 1AX -isa da 2AX -s ricxviTi mniSvnelobebi warmodgenilia
cxrilSi # 11.5. A faqtoris mTavari efeqti aris gansxvaveba 1AX -sa da
2AX -s Soris. Tu A faqtors aqvs gavlena damokidebul cvladze, maSin
1AX da 2AX gansxvavebuli iqneba erTmaneTisagan da 1AX - 2AX mniSvneloba ufro didi gaxdeba.
B faqtoris mTavari efeqti: gansxvaveba 1BX -sa da 2BX -s Soris, gamoxatuli 1BX - 1BX .
B faqtori igive logika gamoiyeneba B faqtorisaTvis mTavari efeqtis saSualos gamoTvlisas. mTavari efeqtis saSualo B faqtorisaTvis (romelsac
xandaxan uwodeben striqonebis saSualos) aRiniSneba 1BX -iTa da 2BX -iT da miiReba A faqtoris monacemebis ignorirebiT, rogorc es naCvenebia cxrilSi # 11.4g. B faqtoris, Cvens magaliTSi siaxlis tipis, mTavari efeqtis saSualos gamoTvlisaTvis xdeba cdis pirebis kategorizaciis ignorireba imis mixedviT, imyofebian Tu ara isini mxedvelobis areSi (A faqtori). 1BX -isa da 2BX -s ricxviTi mniSvnelobebi aseve warmodgenilia
11.5 cxrilSi. B faqtoris mTavari efeqti aris sxvaoba 1BX -sa da 2BX -s
Soris. Tu B faqtors aqvs gavlena damokidebul cvladze, maSin 1BX da
2BX gansxvavebuli iqneba erTmaneTisagan da 1BX - 2BX mniSvneloba ufro didi gaxdeba. sruli saSualo
sruli saSualo, romelic aRiniSneba GX , aris cxrilis yvela monacemis saSualo. cxrilis # 11.2 monacemebisaTvis sruli saSualo – 92.5 wami – aris yvela 20 qulis saSualo. Cveulebriv, sruli saSualo ar aris xolme warmodgenili aRweriT statistikaSi, magram am TavSi saWiroa misi ganxilva, raTa movaxdinoT dispersiuli analizi am sqemisaTvis. damoukidebeli cvladebis urTierTqmedeba urTierTqmedeba: situacia faqtorul sqemaSi, rodesac erTi damoukidebeli cvladis efeqti damokidebulia masTan kombinaciaSi myofi meore damoukidebeli cvladis mniSvnelobaze.
faqtorul sqemaSi urTierTqmedeba xdeba, rodesac erTi damoukidebeli cvladi (mag., faqtori A) damokidebulia meore damoukidebeli cvladis mniSvnelobaze (mag., an B1 an B2), cvladisa, romelTanac aris kombinaciaSi. magaliTad, Tu urTierTqmedeba xdeba 2x2 sqemaSi, maSin A faqtoris efeqti, gansxvaveba qcevaSi A1 da A2 pirobebs Soris, damokidebulia B faqtoris mniSvnelobaze. amis msgavsad, Tu xdeba urTierTqmedeba, B faqtoris efeqti, gansxvaveba qcevaSi B1 da B2 pirobebs Soris, damokidebulia A faqtoris mniSvnelobaze. urTierTqmedeba xSirad aRiniSneba AxB, rac ikiTxeba ase: “A-sa da B-s urTierTqmedeba”. publikaze gaTvlili qcevis hipoTezis winaswarmetyvelebis mixedviT, Cvens sanimuSo eqsperimentSi urTierTqmedeba unda momxdariyo. am hipoTezam ivarauda, rom kargi siaxlis tipisaTvis ar iqneboda gansxvaveba pasuxis latentur droSi mxedvelobis areSi yofnis da mxedvelobis areSi aryofnis pirobebis dros (anu ar iqneboda gansxvaveba A1B1 da A2B1 ujredebis saSualoebs Soris). xolo cudi siaxlis tipisaTvis hipoTezam iwinaswarmetyvela mniSvnelovnad grZeli latenturi dro mxedvelobis areSi yofnis pirobisaTvis (A2B2 ujredis saSualo) mxedvelobis areSi aryofnis pirobasTan (A1B2 ujredis saSualo) SedarebiT. miaqcieT yuradReba, rom urTierTqmedebis arseboba gamoiTvleba ujredebis saSualoebs Soris gansxvavebebis SedarebiT, da ara mTavari efeqtebis saSualoebis gansxvavebebis SedarebiT. SeamowmeT Tqveni codna XI.1
1. gansazRvreT: ujredi, ujredis saSualo, svetis saSualo, faqtori, faqtoruli sqema, sruli saSualo, urTierTqmedeba, damoukidebeli cvladis mniSvneloba (done), A faqtoris mTavari efeqti, B faqtoris mTavari efeqti, mTavari efeqtis
saSualo, striqonis saSualo, piroba,XAB,XA,XB,XG . 2. Semdegi faqtoruli sqemebidan TiToeulisaTvis gansazRvreT, ramdeni
damoukidebeli cvladi cvalebadobs da miuTiTeT doneebis raodenoba TiToeuli faqtorisaTvis. Semdeg, im daSvebiT, rom gvaqvs cdis pirebs Soris sqema, miuTiTeT, ramdeni cdis piri iqneboda saWiro, Tu yovel pirobaSi monawileoba 10 cdis pirma unda miiRos.
3. sqemis tipi: (a) 3x3, (b) 3x2, (g) 2x3, (d) 6x2, (e) 4x4, (v) 2x2x2, (z) 2x4x3, (T) 3x3x2. gamoTvaleT ujredebis saSualoebi, mTavari efeqtis saSualoebi da sruli saSualo qvemoT moyvanili TiToeuli cxrilis monacemebisaTvis:
cxrili 1 faqtori A
A1 A2
B1 faqtori B B2
10 17 13 23 8 17 17 21 12 19 20 24 19 27 13 21 10 16 20 19 22 30
17 24 cxrili 2 faqtori A
A1 A2
B1 faqtori B B2
23 41 13 22 28 27 11 32 19 26 20 24 18 39 29 28 33 31 18 26 29 34 36 39 47 44 27 35
cdis pirebs Soris orfaqtoriani sqemis dispersiuli analizi faqtoruli sqemis statistikuri analizi moicavs hipoTezebis Semowmebas, raTa ganisazRvros, mniSvnelovnad gansxvavdeba Tu ara mTavari efeqtis saSualoebi TiToeuli damoukidebeli cvladisaTvis da dadgindes, statistikurad mniSvnelovania Tu ara damoukidebel cvladebs Soris urTierTqmedeba. es Semowmeba xdeba kvlav dispersiuli analizis meSveobiT, romelic monacemebis srul cvalebadobas dayofs cvalebadobis iseT wyaroebad, romlebic erTmaneTisagan damoukidebelia. cdis pirebs Soris erTfaqtoriani dispersiuli analizi (romelic me-10 TavSi ganvixileT) srul cvalebadobas or damoukidebel wyarod yofs: erTi, romelic damoukidebeli cvladis (anu A faqtoris) efeqtis gamo xdeba, da meore, romelic jgufebs SigniT cvalebadobis gamo. cdis pirebs Soris orfaqtorian sqemaSi qulebis sruli cvalebadoba iyofa jgufebs Soris sam wyarod da jgufebs SigniT erT wyarod. jgufebs Soris cvalebadobis wyaroebi Semdegia: cvalebadoba A faqtoris efeqtis gamo cvalebadoba B faqtoris efeqtis gamo
cvalebadoba A da B faqtorebs Soris urTierTqmedebis gamo, romelic simbolurad aRiniSneba AxB
jgufebs SigniT cvalebadobis wyaro aris ujredebs SigniT SemTxveviTi cvalebadoba. Semdeg dispersiuli analizi iyenebs am danawilebas, raTa warmoqmnas kvadratebis oTxi saSualo:
MSA, romelic icvleba A faqtoris efeqtTan erTad MSB, romelic icvleba B faqtoris efeqtTan erTad MSAxB, romelic icvleba A da B faqtorebis urTierTqmedebasTan erTad MSSecdoma, romelic icvleba ujredebs SigniT monacemebis cvalebadobasTan erTad
am etapis Semdeg gamoiTvleba sami F statistika, ori TiToeuli damoukidebeli cvladisaTvis da erTi – urTierTqmedebisaTvis. es Semdegi gziT xdeba: cvalebadobis wyaro F statistika -----------------------------------------------------------------------------------
faqtori A MSA/ MSSecdoma faqtori B MSB/ MSSecdoma urTierTqmedeba AxB MSAxB/ MSSecdoma
----------------------------------------------------------------------------------- TiToeuli am sami F statistikidan gamoiyeneba statistikuri hipoTezebis SemowmebisaTvis, raTa dadgenil iqnes, statistikurad mniSvnelovania Tu ara cvalebadobis is wyaro, romelsac Seesabameba F statistika. dispersiuli faqtoruli analizis konceptualuri gaazrebisaTvis Semdeg qveTavSi ganvixilavT, rogor miiReba gadaxrebis kvadratebis saSualo, romelic Sedis am F statistikebSi. Tumca, Tu Tqven dagWirdebodaT dispersiuli faqtoruli analizis Catareba monacemTa realur bazaze, Tqven gamoiyenebdiT, gamoTvlis TvalsazrisiT, ufro advil formulebs im gamosaTvleli formulebidan, romlebic moyvanilia am Tavis damatebaSi an mimarTavdiT monacemTa analizis kompiuterul programas. monacemTa dayofa dispersiul analizSi sami F statistikisaTvis gadaxrebis kvadratebis saSualoebi gamoiTvleba eqsperimentSi monacemebis sruli cvalebadobis iseT nawilebad dayofiT, romlebic icvleba A faqtorTan (jgufebs Soris), B faqtorTan (jgufebs Soris), AxB urTierTqmedebasTan (jgufebs Soris) da SemTxveviT cvalebadobasTan (jgufebs SigniT) erTad. qulis wardgenis es modeli cdis pirebs Soris orfaqtorian sqemaSi SeiZleba Semdegnairad gamoisaxos: (11.1) = + + +
sruli cvalebadoba cdis piris monacemebSi
cvalebadoba A faqtoris efeqtis gamo plus SerCevis Secdoma
cvalebadoba B faqtoris efeqtis gamo plus SerCevis Secdoma
cvalebadoba A da B faqtorebis urTierTqme-debis gamo plus SerCevis Secdoma
cvalebadoba ujredebs SigniT Secdomis gamo
monacemebis terminebiT es gantoleba SeiZleba Caiweros Semdegnairad:
(X - GX ) = ( AX - GX ) + ( BX - GX ) + ( ABX - AX - BX + GX ) + (X - ABX ). (11.2) Cven ganvixilavT 11.2 gantolebis TiToeul nawils. sruli cvalebadoba me-10 TavSi warmodgenili dispersiis erTfaqtoriani analizis msgavsad,
monacemis gansxvaveba sruli saSualosagan (anu X - GX ) warmoadgens am qulis srul cvalebadobas. rogorc gviCvenebs 11.2 gantoleba, am cvalebadobis nawili xdeba TiToeuli damoukidebeli cvladis efeqtisa da damoukidebeli cvladebis urTierTqmedebis gamo, xolo nawili ki – SemTxveviTi cvalebadobis gamo. cvalebadoba A faqtoris efeqtisa da SerCevis Secdomis gamo A faqtoris mTavari efeqtis saSualosa da srul saSualos Soris
gansxvaveba (anu AX - GX ) icvleba A faqtoris efeqtsa da A faqtoris mTavari efeqtis saSualoebs Soris SerCevis SecdomasTan erTad. misi gamoyeneba orfaqtorian sqemaSi misi cdis pirebs Soris erTfaqtorian sqemaSi gamoyenebis identuria, raTa gaizomos damoukidebeli cvladis gavlena. mosalodnelia, rom A faqtoris nebismieri efeqti yvela cdis pirze Tanabrad moaxdens gavlenas am faqtoris yvela doneze. amdenad, A faqtoris efeqti cvlis mTavari efeqtis saSualoebis mniSvnelobas A faqtorisaTvis. es mTavari efeqtis saSualoebi Semdeg gansxvavebuli
iqneba sruli saSualosagan. aqedan gamomdinare, gansxvaveba AX - GX asaxavs A faqtoris nebismier efeqts (pirobis efeqts A faqtorisaTvis 11.1 gantolebaSi). nebismieri pirobis efeqtis garda, mTavari efeqtis saSualoebi A faqtorisaTvis aseve gansxvavebuli iqneba erTmaneTisagan
SerCevis Secdomis gamo. amitom, AX - GX sxvaobis mniSvneloba asaxavs rogorc A faqtoris pirobis efeqts, aseve SerCevis Secdomas, romelic
gavlenas axdens AX -s mniSvnelobebze. Sesabamisad, is monawileobs MSA.s gamoTvlaSi. cvalebadoba B faqtoris efeqtisa da SerCevis Secdomis gamo B faqtoris mTavari efeqtis saSualosa da srul saSualos Soris
gansxvaveba (anu BX - GX ) icvleba B faqtoris efeqtsa da B faqtoris
cdis piris qulis gan-sxvaveba sruli saSualosagan
A faqtoris mTavari efeqtis saSualos gansxvaveba sruli saSualosagan
B faqtoris mTavari efeqtis saSualos gansxvaveba sruli saSualosagan
urTierTqmedeba: ujredis saSualos gansxvaveba sruli saSualosagan mas Semdeg, rac A da B faqtorebis mTavari efeqtebi gauqmebulia
cdis piris qulis gansxvaveba misi ujredis saSualosagan
mTavari efeqtis saSualoebs Soris SerCevis SecdomasTan erTad. iseve, rogorc A faqtoris SemTxvevaSi, mosalodnelia, rom B faqtoris nebismieri efeqti gavlenas moaxdens yvela individze am faqtoris yvela doneze da amdenad Secvlis mTavari efeqtis saSualo mniSvnelobas B faqtorisaTvis, romelic gansxvavebuli gaxdeba sruli saSualosagan.
amitom, gansxvaveba BX - GX asaxavs B faqtoris nebismier efeqts (pirobis efeqts B faqtorisaTvis 11.1 gantolebaSi). garda amisa, mTavari efeqtis saSualoebi B faqtorisaTvis aseve gansxvavebuli iqneba erTmaneTisagan
SerCevis Secdomis gamo. amitom, BX - GX gansxvaveba asaxavs rogorc B faqtoris pirobis efeqts, aseve SerCevis Secdomas, romelic gavlenas
axdens BX -s mniSvnelobebze da, Sesabamisad, monawileobs MSB-s gamoTvlaSi. cvalebadoba A da B faqtorebis urTierTqmedebis gamo A da B faqtorebis urTierTqmedeba aris sruli saSualosagan ujredebis saSualoebis darCenili gansxvaveba mas Semdeg, rac TiToeuli damoukidebeli cvladis efeqti gauqmebulia. simbolurad es debuleba SeiZleba Semdegnairad warmovadginoT:
A x B = ( ABX - GX ) - ( AX - GX ) - ( BX - GX ) = - - gantoleba gviCvenebs, rom urTierTqmedeba aris ujredebis saSualoebis cvalebadoba mas Semdeg, rac TiToeuli damoukidebeli cvladis mTavari efeqti gauqmebulia. ujredebis saSualoebis es darCenili cvalebadoba xdeba damoukidebel cvladebs Soris urTierTqmedebisa da ujredebis saSualoebs Soris nebismieri SerCevis Secdomis gamo. gantolebaSi moyvanili gamoklebis Catarebis Sedegad viRebT urTierTqmedebis gamartivebul gamosaxulebas:
ABX - AX - BX + GX
urTierTqmedeba am saxiT aris warmodgenili 11.2 gantolebaSi. gantolebis es nawili monawileobs MSAxB-is gamoTvlaSi. cvalebadoba ujredebs SigniT Secdomis gamo
monacemis gansxvaveba misi ujredis saSualosagan (anu X - ABX ) asaxavs mxolod eqsperimentSi ujredebs SigniT SemTxveviT cvalebadobas. amiT gamowveuli Secdoma aris is, rac rCeba monacemis miRma mas Semdeg, rac orive faqtoris mTavari efeqti da maTi urTierTqmedeba gaTvaliswinebul iqna. rogorc dispersiuli erTfaqtoriani analizis dros, Secdoma asaxavs pirovnebis monacemis unikalurobas individebis jgufSi, romelTaganac yvela erTsa da imave pirobaSi iyo. is monawileobs MSSecdomis gamoTvlaSi.
urTierTqmedeba A.sa da B-s Soris
ujredis saSualos gansxvaveba sruli saSualosagan
A faqtoris mTavari efeqti
B faqtoris mTavari efeqti
dayofili monacemebidan gadaxrebis kvadratebis saSualos miReba rogorc ganvixileT me-10 TavSi, dayofis gantoleba gvaZlevs safuZvels, rom miviRoT SS da df, romlebic qmnian kvadratebis saSualos F SefardebisaTvis. pirveli nabijia SS-is miReba cdis pirebis monacemebidan. kvadratebis jamebis miReba me-10 TavSi ganxiluli midgomis msgavsad, kvadratebis jamebi (SS) miiReba TiToeuli qulis dayofiT 11.2 gantolebis mixedviT. am dayofis ilustraciisaTvis Cven gamoviyenebT pirveli cdis piris monacems pirobaSi “ar aris mxedvelobis areSi/kargi siaxle” (anu X111) 11.2 cxrilidan, romelic aris 97.0. am monacemisaTvis 11.2 gantoleba iqneba
(X111 - GX ) = ( 1AX - GX ) + ( 1BX - GX ) + ( 11BAX - 1AX - 1BX + GX ) + (X111 -
11BAX ) ricxviTi mniSvnelobebis CasmiT viRebT 97.0–92.5 = (86.5 – 92.5) + (87.7 – 92.5) + (90.8 – 86.5 – 87.7 + 92.5) + (97.0 – 90.8) anu, miviRebT +4.5 = (-6.0) + (-4.8) + (+9.1) + (+6.2) es dayofa gvaCvenebs, rom es monacemi gansxvavdeba sruli saSualosagan,
radganac adgili aqvs A faqtoris sistematur efeqts (aRniSnulia 1AX - GX = - 6.0 ), B faqtoris sistematur efeqts (aRniSnulia 1BX - GX = - 4.8), A
da B faqtorebis urTierTqmedebas (aRniSnulia 11BAX - 1AX - 1BX + GX = 9.1) da ujredebs SigniT SemTxveviTi cvalebadobis arasistematur gavlenebs
(aRniSnulia X111 - 11BAX = +6.2). cxrilis # 11.2 yvela sxva monacemis msgavsi dayofa naCvenebia cxrilis # 11.6 pirvel nabijSi. cxrilis danarCen nabijebSi am dayofili monacemebidan gamoTvlilia saSualoebis jamebi. yuradRebiT misdieT TiToeuls am nabijebidan. nabiji 1. eqsperimentSi monawile yvela 20 cdis piris monacemi
dayofilia 11.2 gantolebis mixedviT. nabiji 2. TiToeuli monacemisaTvis miRebulia ricxviTi
sxvaobebi. nabiji 3. me-2 nabijSi miRebuli yoveli dadebiTi da uaryofiTi
sxvaoba ayvanilia kvadratSi. nabiji 4. eqsperimentis yvela monacemis kvadratSi ayvanili
sxvaobebis mniSvnelobebi dajamebulia. es dajameba miiRweva ocive mniSvnelobis mimatebiT TiToeul svetSi. Sedegad viRebT kvadratebis xuT jams: SSsruli = 4567.000,
SSA = 720.000, SSB = 460.800, SSAxB = 1656.200, SSSecdoma = 1730.000.
SSsruli misi komponentebis jamia, amitom SSsruli = SSA + SSB + SSAxB + SSSecdoma. (11.3) cxrili # 11.6 kvadratebis jamis miReba orfaqtoriani dispersiuli analizisaTvis cxrilis # 11.2 monacemebze dayrdnobiT.
nabiji 1: 20 cdis piris qulebis dayofa Semdegi gantolebis gamoyenebiT:
(X - GX ) = ( AX - GX ) + ( BX - GX ) + ( ABX - AX - BX + GX ) + (X - ABX ) __________________________________________________________________ 97.0 - 92.5 = (86.5 - 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (97.0 - 90.8) 90.0 – 92.5 = (86.5 – 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (90.0 - 90.8) monacemebi 80.0 - 92.5 = (86.5 - 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (80.0 - 90.8) A1B1-Si 107.0 - 92.5 =(86.5 - 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (107.0 - 90.8) 80.0 - 92.5 = (86.5 - 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (80.0 - 90.8) 70.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (70.0 - 84.6) 87.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (87.0 - 84.6) monacemebi 81.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (81.0 - 84.6) A2B1-Si 95.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (95.0 - 84.6) 90.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (90.0 - 84.6)
68.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (68.0 - 82.2) 87.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (87.0 - 82.2) monacemebi 92.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (92.0 - 82.2) A1B2-Si 80.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (80.0 - 82.2) 84.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (84.0 - 82.2)
114.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (114.0 -112.4) 96.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (96.0 - 112.4) monacemebi 127.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (127.0 -112.4) A2B2-Si 110.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (110.0 -112.4) 115.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (115.0 -112.4)
nabiji 2: CavataroT pirvel nabijSi miTiTebuli gamokleba, raTa miviRoT ricxviTi sxvaobebi TiToeuli cdis pirisaTvis. + 4.5 = - 6.0 + - 4.8 + + 9.1 + + 6.2 - 2.5 = - 6.0 + - 4.8 + + 9.1 + - 0.8 monacemebi - 12.5 = - 6.0 + - 4.8 + + 9.1 + - 10.8 A1B1-Si +14.5 = - 6.0 + - 4.8 + + 9.1 + + 16.2 _ 12.5 = - 6.0 + - 4.8 + + 9.1 + - 10.8 - 22.5 = + 6.0 + - 4.8 + - 9.1 + - 14.6 - 5.5 = + 6.0 + - 4.8 + - 9.1 + + 2.4 monacemebi - 11.5 = + 6.0 + - 4.8 + - 9.1 + - 3.6 A2B1-Si + 2.5 = + 6.0 + - 4.8 + - 9.1 + + 10.4 - 2.5 = + 6.0 + - 4.8 + - 9.1 + + 5.4
- 24.5 = _ 6.0 + + 4.8 + - 9.1 + - 14.2
- 5.5 = _ 6.0 + + 4.8 + - 9.1 + + 4.8 monacemebi - 0.5 = _ 6.0 + + 4.8 + - 9.1 + + 9.8 A1B2-Si - 12.5 = _ 6.0 + + 4.8 + - 9.1 + - 2.2 - 8.5 = _ 6.0 + + 4.8 + - 9.1 + + 1.8
+ 21.5 = + 6.0 + + 4.8 + + 9.1 + + 1.6 + 3.5 = + 6.0 + + 4.8 + + 9.1 + - 16.4 monacemebi + 34.5 = + 6.0 + + 4.8 + + 9.1 + + 14.6 A2B2-Si + 17.5 = + 6.0 + + 4.8 + + 9.1 + _ 2.4 + 22.5 = + 6.0 + + 4.8 + + 9.1 + + 2.6
cxrili # 11.6 (gagrZeleba) nabiji 3: kvadratSi aiyvaneT TiToeuli sxvaoba:
(X - GX )2 ( AX - GX )2 ( BX - GX )2 ( ABX - AX - BX + GX )2 (X -
ABX )2 -----________________________________________________________________ 20.25 36.0 23.04 82.81 38.44 6.25 36.0 23.04 82.81 0.64 monacemebi 156..25 36.0 23.04 82.81 116.64 A1B1-Si 210.25 36.0 23.04 82.81 262.44 156..25 36.0 23.04 82.81 116.64 506.25 36.0 23.04 82.81 213.16 30.25 36.0 23.04 82.81 5.76 monacemebi 132.25 36.0 23.04 82.81 12.96 A2B1-Si 6.25 36.0 23.04 82.81 108.16 6.25 36.0 23.04 82.81 29.16
600..25 36.0 23.04 82.81 201.64 30.25 36.0 23.04 82.81 23.04 monacemebi 0.25 36.0 23.04 82.81 96.04 A1B2-Si 156..25 36.0 23.04 82.81 4.84 72.25 36.0 23.04 82.81 3.24
462.25 36.0 23.04 82.81 2.56 12.25 36.0 23.04 82.81 268.96 monacemebi 1190..25 36.0 23.04 82.81 213.16 A2B2-Si 306.25 36.0 23.04 82.81 5.76 506.25 36.0 23.04 82.81 6.76
nabiji 4: daajameT kvadratSi ayvanili sxvaobebi TiToeuli dayofisaTvis yvela cdis piris mixedviT. 4567.000 = 720.000 + 460.800 + 1656.200 + 1730.000
SSsruli SSA SSB SSAxB SSSecdoma
es gantoleba gvaCvenebs, rom eqsperimentSi monacemebis sruli cvalebadoba (SSsruli) im cvalebadobis Sedegia, romelic oTxi damoukidebeli wyaros gamo xdeba: A faqtori, B faqtori, A da B faqtorebis urTierTqmedeba da ujredebs SigniT Secdoma. Tavisuflebis xarisxis gamoTvla
kvadratebis saSualo (MS) miiReba TiToeuli SS-is gayofiT Tavis Tavisuflebis xarisxze. gaixseneT, rom Tavisuflebis xarisxebi ganisazRvreba, rogorc monacemebis (qulebis) raodenoba, romelic Tavisuflad icvleba statistikis gamoTvlisas. gamoviyenoT es gansazRvreba 11.3 gantolebaSi TiToeuli SS-isaTvis df–is gamosaTvlelad. Tavisuflebis sruli xarisxi. SSsruli -s gamosaTvlelad unda vicodeT
sruli saSualo GX . radganac kvadratebis sruli jami damokidebulia
eqsperimentSi yoveli monacemis sxvaobaze sruli saSualosagan (anu X -
GX ), amdenad monacemebis srul raodenobaze erTiT naklebs SeuZlia Tavisufali cvlileba SSsruli -s gamoTvlisas. amitom Tavisuflebis sruli xarisxi = dfsruli = N – 1, sadac N aris eqsperimentSi monacemTa sruli raodenoba. rodesac N = 20, dfsruli = 20 – 1 = 19.
Tavisuflebis xarisxi SSA.-isaTvis. SSA gamoiTvleba GX –sagan A faqtoris mTavari efeqtis saSualoebis gamoklebiT (anu Cvens magaliTSi
1AX da 2AX ). faqtorul sqemaSi, sadac A faqtors ori done aqvs, Tu viciT GX , maSin A faqtoris mxolod erT mTavari efeqtis saSualos SeuZlia Tavisuflad cvlileba. zogadad, df SSA.-isaTvis (dfA) udris erTiT naklebs, vidre A faqtoris doneebi, anu Tavisuflebis xarisxi SSA.-isaTvis = dfA = a – 1, sadac a aris A damoukidebeli cvladis doneebis raodenoba. Cvens magaliTSi, rodesac a = 2, amitom dfA = 2 – 1 = 1.
Tavisuflebis xarisxi SSB-isaTvis. SSB gamoiTvleba GX –sagan B faqtoris mTavari efeqtis saSualoebis gamoklebiT (anu Cvens magaliTSi 1BX da
2BX ). faqtorul sqemaSi, sadac B faqtors ori done aqvs, Tu viciT GX , maSin B faqtoris mxolod erT mTavari efeqtis saSualos SeuZlia Tavisuflad cvlileba. zogadad, df SSB-isaTvis (dfB) udris erTiT naklebs, vidre B faqtoris doneebi, anu Tavisuflebis xarisxi SSB-isaTvis = dfB = b – 1, sadac b aris B damoukidebeli cvladis doneebis raodenoba. Cvens magaliTSi, rodesac b = 2, amitom dfB = 2 – 1 = 1. Tavisuflebis xarisxi SSAxB-isaTvis. SSAxB gamoiTvleba TiToeuli ujredis saSualosagan Sesabamisi mTavari efeqtis saSualoebis gamoklebiT. 2x2 sqemisaTvis, Tu cnobilia TiToeuli mTavari efeqtis saSualo, maSin mxolod erT ujredis saSualos SeuZlia Tavisufali cvlileba. rodesac viciT erTi ujredis saSualo da A da B faqtorebis mTavari efeqtebis saSualoebi, SegviZlia vipovoT danarCeni sami ujredis saSualoebi. zogadad, Tavisuflebis xarisxi SSAxB-isaTvis = dfAxB = (a - 1) (b – 1)
2x2 sqemaSi a = 2 da b = 2, amitom dfAxB = (2 - 1) (2 – 1) = 1
Tavisuflebis xarisxi SSSecdomisaTvis. SSSecdoma gamoiTvleba yoveli ABX -is gamoklebiT ujredis yvela monacemidan, eqsperimentis yvela ujredisaTvis. amdenad, yoveli pirobisaTvis, mas Semdeg, rac napovnia
ABX , mxolod nAB – 1 monacemebs SeuZliaT Tavisufali cvalebadoba am ujredSi. magaliTad, cxrilSi # 11.2 xuTi monacemidan mxolod oTxs SeuZlia Tavisufali calebadoba TiToeul ujredSi, Tu cnobilia
TiTeuli ABX . radganac arsebobs oTxi ujredi da TiToeulSi Tavisufali cvlileba oTx qulas SeuZlia, SSSecdomisaTvis arsebobs 16 df. zogadad, rodesac a aris A faqtoris doneebis raodenoba, b – B faqtoris doneebis raodenoba, xolo nAB ki qulebis raodenoba TiToeuli ujredis SigniT, Tavisuflebis xarisxi SSSecdomisaTvis = dfSecdoma = ab (nAB - 1) Tavisuflebis xarisxis aditiuroba. faqtorul analizSi Tavisuflebis xarisxebi imave gziT jamdeba, rogorc erTfaqtorian dispersiul analizSi. amdenad dfsruli = dfA + dfB + dfAxB + dfSecdoma (11.4) es damokidebuleba sruldeba Cvens magaliTSic, radgan dfsruli =19, dfA =1, dfB =1, dfAxB =1, dfSecdoma =16. MS-is miReba SS-is da df-is saSualebiT. 11.3 gantolebaSi TiToeuli SS-is gayofa mis Sesabamis df-ze warmoqmnis kvadratebis oTx saSualos, romlebic saWiroa cdis pirebs Soris orfaqtoriani dispersiuli analizisaTvis:
MSA = A
A
dfSS -
MSB = B
B
dfSS -
MSAxB = AxB
AxB
dfSS
MSSecdoma = Secdoma
Secdoma
dfSS
Cven ar vangariSobT MSsruls, radganac is ar gamoiyeneba am analizSi romelime F Sefardebis misaRebad. cxrilis # 11.2 monacemebisaTvis SSA=720.000, SSB=460.800, SSAxB=1656.200 da SSSecdoma =1730.000. ufro metic, dfA=1, dfB=1, dfAxB=1 da dfSecdoma =16. Sesabamisad,
MSA = 1000.720 = 720.000
MSB = 1800.460 = 460.800
MSAxB = 1
200.1656 = 1656.200
MSSecdoma = 16
000.1730 = 108.125
F statistikis gamoTvla F Sefardeba iqmneba kvadratebis ori saSualos SefardebiT, erTi MS-isa, romelic icvleba damoukidebeli cvladis efeqtTan erTad da aseve SerCevis SecdomasTan erTad, da meore MS_isa, romelic icvleba mxolod Secdomis cvalebadobasTan erTad eqsperimentSi. orfaqtorian cdis pirebs Soris dispersiul analizSi, MSA icvleba A faqtorTan erTad, MSB – B faqtorTan erTad, MSAxB – A da B faqtorebis urTierTqmedebasTan erTad da MSSecdoma ki – mxolod ujredebs SigniT SemTxveviT cvalebadobasTan erTad. ase rom, viRebT sami saxis F statistikas. maTgan TiToeuli asaxavs cvalebadobis erT sistematur wyaros eqsperimentSi: cvalebadobis wyaro F -------------------------------------------------------------------------------------------
faqtori A SecdomaMS
MS A
faqtori B SecdomaMS
MSB
A da B urTierTqmedeba SecdomaMS
MS AxB
------------------------------------------------------------------------------------ gadaxrebis kvadratebis saSualoebis ricxviTi mniSvnelobebis CasmiT viRebT Semdeg F statistikebs: cvalebadobis wyaro F -------------------------------------------------------------------------------------------------
faqtori A 125.108000.720 = 6.66
faqtori B 125.108800.460
= 4.26
A da B urTierTqmedeba 125.108200.1656 = 15.32
--------------------------------------------------------------------------------------------------- mokled ganvixiloT, Tu rogor gamoviyenoT da rogori axsna mivceT TiToeul F-s. cxrilSi # 11.7 moyvanilia cvalebadobis wyaroebi cdis pirebs Soris orfaqtorian dispersiul analizSi da formulebi TiToeuli SS-isTvis, df-isTvis, MS-isTvis da F statistikisaTvis. cxrilis # 11.2 monacemebis dispersiuli analizis Sejamebis Sedegebi mocemulia cxrilSi # 11.8. cxrili # 11.7 cdis pirebs Soris orfaqtoriani dispersiuli analizSi gamoyenebuli ganmartebiTi formulebis Sejameba. ------------------------------------------------------------------------------------------------------------------------------------------------ wyaro SS dfa MS F -----------------------------------------------------------------------------------------------------------------------------------------------------
A faqtori ‡ ∑ ∑ ∑ ( AX - GX )2 A.1 A
A
dfSS
SecdomaMS
MSA
B faqtori ∑ ∑ ∑ ( BX - GX )2 b-1 B
B
dfSS
SecdomaMS
MSB
A da B
urTierTqmedeba ∑ ∑ ∑ ( ABX - AX - BX + GX )2 (A.1)(b-1) AxB
AxB
dfSS
SecdomaMS
MS AxB
Secdoma ∑ ∑ ∑ (X - ABX )2 ab(nAB-1) Secdoma
Secdoma
dfSS
sul ∑ ∑ ∑ (X - GX )2 N-1 ar gamoiTvleba
-------------------------------------------------------------------------------------------------------------------------------------------------------
aa = A faqtoris doneebis raodenoba; b = B faqtoris doneebis raodenoba; nAB = qulebis raodenoba TiToeul ujredSi; N = qulebis sruli raodenoba. -------------------------------------------------------------------------------------------------------------------------------------------------------
cxrili # 11.8 cxrilis # 11.2 hipoTeturi qulebis dispersiuli analizis gamoTvlis Sedegebi.
wyaro SS df MS F
subieqtis yofna mxedvelobis areSi (A) 720.000 1 720.000 6.66* siaxlis tipi (B) 460.800 1 460.800 4.26 A x B 1656.200 1 1656.200 15.32** Secdoma 1730.000 16 1656.200 sul 4567.000 19
* p < .05 ** p < .01
SeamowmeT Tqveni codna XI.2 1. gansazRvreT monacemebis (qulebis) cvalebadobis wyaroebi cdis pirebs
Soris orfaqtorian sqemaSi. 2. dawereT gantoleba, romlis mixedviTac xdeba qulebis dayofa dispersiis
orfaqtorian cdis pirebs Soris analizSi.
3. axseniT, ratom icvleba AX - GX mniSvneloba A faqtoris efeqtTan erTad faqtorul eqsperimentSi.
4. axseniT, ratom icvleba BX - GX mniSvneloba B faqtoris efeqtTan erTad faqtorul eqsperimentSi.
5. axseniT, ratom icvleba ABX - AX - BX + GX mniSvneloba A da B faqtorebs Soris urTierTqmedebis efeqtTan erTad faqtorul eqsperimentSi.
6. axseniT, ratom icvleba X - ABX mniSvneloba mxolod ujredebs SigniT SemTxveviT cvalebadobasTan erTad faqtorul eqsperimentSi.
7. daasruleT Semdegi tolobebi cdis pirebs Soris orfaqtoriani dispersiuli analizisaTvis: a. dfA = b. dfB = g. dfAxB = d. dfSecdoma = e. dfsruli = v. MSA = z. MSB = T. MSAxB = i. MSSecdoma =
8. ras udris SSsrulis mniSvneloba, Tu SSA =40.00, SSB =70.00, SSAxB =30.00 da SSSecdoma =100.00?
9. ras udris SSA–is mniSvneloba, Tu SSsruli = 224.00, SSB =14.00, SSAxB =64.00 da SSSecdoma =101.00?
10. ras udris SSAxB–is mniSvneloba, Tu SSsruli = 302.00, SSA =75.00, SSB =19.00 da SSSecdoma =153.00?
11. ras udris SSSecdomis mniSvneloba, Tu SSsruli = 98.00, SSA =14.00, SSB =7.00 da SSAxB =17.00?
12. ipoveT dfA, dfB, dfAxB, dfSecdoma da dfsruli 2x3 cdis pirebs Soris dispersiuli analizisaTvis, sadac TiToeul ujredSi eqvsi cdis piria.
13. ipoveT dfA, dfB, dfAxB, dfSecdoma da dfsruli 3x2 cdis pirebs Soris dispersiuli analizisaTvis, sadac TiToeul pirobaSi 13 qulaa.
14. ras udris dfsrulis mniSvneloba, Tu dfA=3, dfB=1, dfAxB=3 da dfSecdoma = 72? 15. gamoTvaleT gadaxrebis kvadratebis saSualoebis ricxviTi mniSvnelobebi
Semdegi SS-isa da df-isaTvis: SSA =14.00, SSB =26.00, SSAxB =44.00, SSSecdoma = 480.00, dfA=1, dfB=2, dfAxB=2 da dfSecdoma = 60.
16. davuSvaT, Tqven CaatareT 2x2 faqtoruli eqsperimenti, sadac TiToeul ujredSi sami cdis piria da miiReT Semdegi qulebi:
faqtori A
A1 A2
B1 faqtori B B2
30 34 32 36 28 32 28 38 30 34 26 36
a. dayaviT es qulebi cxrilSi # 11.6 naCvenebi midgomis mixedviT. gamoTvaleT SSsruli, SSA, SSB, SSAxB, SSSecdoma da Sesabamisi Tavisuflebis xarisxebi yoveli dayofisaTvis. Semdeg SeqmeniT ricxviTi Sejamebis cxrili MS-isa da F-is mniSvnelobebiT. b. ratom aris SSB nulis toli am magaliTSi? 17. cxrilebi # 1 da 2 cdis pirebs Soris orfaqtoriani dispersiuli analizis dausrulebeli cxrilebia. dauSviT, rom TiToeul ujredSi qulebis toli raodenobebia. TiToeul cxrilSi SeavseT gamotovebuli mniSvnelobebi SS-isa da df-is cxrilebSi # 11.3 da 11.4 mocemuli kavSiris gamoyenebiT. Semdeg TiToeuli cxrilisaTvis upasuxeT Semdeg kiTxvebs:
a. A faqtoris ramdeni done iyo gamoyenebuli? b. B faqtoris ramdeni done iyo gamoyenebuli? g. ramdeni cdis piri gaizoma TiToeul ujredSi? d. ramdeni cdis piri monawileobda kvlevaSi?
cxrili # 1
wyaro SS df MS F A faqtori ____ 2 16.00 ------ B faqtori 24.00 1 --------- 3.00 A x B -------- ------ 12.00 ------- Secdoma --------- ------ --------- sul 560.00 65 ------------------------------------------------------------------------------------------------------------------------------------------------------- cxrili # 2
wyaro SS df MS F A faqtori ____ 2 --------- 4.50 B faqtori 24.00 4 --------- 1.50 A x B 160.00 ------ --------- 5.00
Secdoma --------- ------ 4.00 sul --------- 224 -------------------------------------------------------------------------------------------------------------------------------------------------------
statistikuri hipoTezis Semowmeba cdis pirTa Soris orfaqtorian dispersiul analizSi imisaTvis, rom gavigoT, Tu rogor xdeba statistikuri hipoTezebis Semowmeba F statistikis gamoyenebiT, romelic miiReba dispersiuli faqtoruli analizis Sedegad, unda vicodeT, Tu ra axdens gavlenas MSA.ze, MSB-ze, MSAxB-sa da MSSecdomaze. faqtorebi, romlebic gavlenas axdenen gadaxrebis kvadratebis saSualoebze dispersiul faqtorul analizSi faqtorul analizSi yovel MS-ze gavlenas axdens cvalebadobis gansxvavebuli wyaroebi. Tu damokidebul cvladze arc erT damoukidebel cvlads ar aqvs gavlena da aseve ar arsebobs urTierTqmedeba damoukidebel cvladebs Soris, maSin yoveli MS asaxavs mxolod SemTxveviT cvalebadobas. aseT pirobebSi oTxive kvadratebis saSualo daaxloebiT erTi da imave mniSvnelobis unda iyos da aCvenebdes mxolod SerCevis Secdomis gavlenas. magram, Tu romelime an orive damoukidebel cvlads aqvs mTavari efeqti an Tu isini urTierTqmedeben, maSin efeqti zrdis Sesabamisi MS-is mniSvnelobas. magaliTad, Tu A faqtori qmnis mTavar efeqts, maSin MSA gaizrdeba MSSecdomasTan SedarebiT, magram es mTavari efeqti ar moaxdens gavlenas MSB-ze, MSAxB-ze da MSSecdomaze. msgavsi situacia iqneba, Tu B faqtors aqvs mTavari efeqti. Tu B faqtori qmnis mTavar efeqts, maSin MSB gaizrdeba MSSecdoma -sTan SedarebiT, magram es mTavari efeqti ar moaxdens gavlenas MSA.ze, MSAxB-ze da MSSecdoma -ze. Tu damoukidebeli cvladebi urTierTqmedebaSia, maSin izrdeba MSAxB, magram MSA.ze, MSB-ze da MSSecdoma -ze gavlena ar aris. MSSecdoma -ze gavlenas ar axdens arc mTavari efeqtebi da arc damoukidebeli cvladebis urTierTqmedeba. is asaxavs mxolod ujredebs SigniT SemTxveviT cvalebadobas eqsperimentSi. amdenad, TiToeuli MS-is mniSvneloba damoukidebelia sxva kvadratebis saSualoebis mniSvnelobebisagan. kvadratebis oTxi saSualo da cvalebadobis wyaroebi, romlebic maTze axdenen gavlenas, SeiZleba Semdegnairad Sejamdes: kvadratebis saSualo masze gavlenas axdens:
MSA A faqtoris gamo arsebuli sistematuri cvalebadoba SerCevis Secdoma MSB B faqtoris gamo arsebuli sistematuri cvalebadoba SerCevis Secdoma MSAxB A da B faqtorebis interaqciis gamo arsebuli
sistematuri cvalebadoba SerCevis Secdoma MSSecdoma ujredebs SigniT SemTxveviT cvalebadoba
imisaTvis, rom yvelaferi zemoT Tqmuli davukavSiroT sam F Sefardebas cxrilidan # 11.8, miaqcieT yuradReba, rom yoveli F Sefardeba iyenebs MSSecdomas, rogorc mniSvnels da damoukidebeli cvladis an urTierTqmedebis Sesabamis MS-s, rogorc mricxvels. Tu konkretuli F Sefardebis gamoTvlaSi MS-ze gavlenis mqone damoukidebel cvlads ara aqvs efeqti, maSin es F Sefardeba daaxloebiT 1-s unda udrides, radganac rogorc mricxvelis, aseve mniSvnelis MS zomavs mxolod SemTxveviT cvalebadobas eqsperimentSi. magram Tu damoukidebel cvlads aqvs efeqti, maSin MS-is mniSvneloba am cvladisaTvis izrdeba da F-is mniSvneloba am faqtorisaTvis 1.0-ze meti xdeba. es dasabuTeba me-10 TavSi dispersiuli erTfaqtoriani analizisaTvis moyvanili dasabuTebis identuria.
F statistikaze damyarebuli statistikuri gadawyvetilebebis miReba axla, rodesac viciT, Tu rogor axdens gavlenas faqtorebi F Sefardebis TiToeul mniSvnelobaze, Cven mzad varT, gamoviyenoT TiToeuli F hipoTezebis Semowmebisas. saWiro nabijebi igivea, rac me-10 TavSi.
a. TiTeuli F SefardebisaTvis yalibdeba nulovani hipoTeza, H0, da alternatiuli hipoTeza, H1;
b. im daSvebiT, rom H0 sworia, F-is TiTeuli mniSvnelobisaTvis gamoiTvleba F-is SerCeviTi ganawileba. es ganawileba mocemulia cxrilSi # A.3;
g. xdeba mniSvnelovnobis donis SerCeva; d. F-is kritikuli done, romelic aRiniSneba Fkritikuli,
TiToeuli F SefardebisaTvis F–is SerCeviTi ganawilebidan gamoiTvleba, romelic mocemulia cxrilSi # A.3;
e. F–is SerCeviT ganawilebaSi TiToeuli F-isaTvis gamoiyofa uaryofis are; v. SerCevis monacemebidan gamoiTvleba sami F-is sami mniSvneloba,
romlebic aRiniSneba, rogorc Frealuri; z. gadawyvetileba imis Taobaze, uarvyoT Tu ara H0 TiToeuli
FrealurisaTvis, miiReba imaze dayrdnobiT, xvdeba Tu ara Frealuri uaryofis areSi.
statistikuri hipoTezebi cdis pirebs Soris orfaqtoriani dispersiuli analiziT mowmdeba sami gansxvavebuli nulovani hipoTeza, TiTo yoveli Frealuri-saTvis. 2x2 faqtorul dispersiul analizSi nulovani (H0) da alternatiuli (H1) hipoTezebi yoveli F-isaTvis Semdegia: F statistika statistikuri hipoTezebi
A faqtorisaTvis H0: µ A1 = µ A2
H1: µ A.ebi ar aris toli
B faqtorisaTvis H0: µ B1 = µ B2
H1: µ B-ebi ar aris toli
A x B H0: (µAB - µ A - µ B + µ G) = 0 H1: yvela (µAB - µ A - µ B + µ G) ar udris 0 mTavari efeqtebi. A da B faqtorebis mTavari efeqtebis nulovani da alternatiuli hipoTezebi formiT identuria erTfaqtoriani analizis statistikuri hipoTezebisa. damoukidebeli cvladis mTavari efeqtis SemTxvevaSi dispersiuli faqtoruli analizi monacemebs ganixilavs ise, TiTqos isini miRebulia erTfaqtoriani sqemidan, rogorc naCvenebia cxrilSi # 11.4. populaciis saSualoebis raodenoba, romelic ganisazRvreba H0-Si, rogorc A, aseve B faqtorisaTvis, Seesabameba eqsperimentSi am faqtoris doneebis raodenobas. magaliTad, Tu eqsperimenti moicavs A faqtoris sam dones, maSin H0 A faqtorisaTvis ase gamoisaxeba:
H0: µ A1 = µ A2 = µ A3, xolo H1 ki ase:
H1: yvela µ A.-ebi ar aris erTmaneTis toli. nulovani hipoTeza TiToeuli faqtorisaTvis aris situacia, romelic arsebobs, Tu damoukidebel cvlads ar aqvs efeqti. Tu nulovani hipoTeza sworia, maSin nebismieri aRmoCenili gansxvaveba mTavari efeqtebis Sesabamis saSualoebs Soris mxolod SerCevis Secdomis gamoa. alternatiuli hipoTeza amtkicebs, rom populaciis saSualoebi ar aris erTmaneTis toli. es situacia arsebobs maSin, Tu damoukidebel cvlads aqvs efeqti. urTierTqmedeba. urTierTqmedebis nulovani hipoTeza amtkicebs, rom Tu
populaciaSi urTierTqmedeba ar xdeba, ujredebis saSualoebis, µAB, gansxvaveba sruli saSualosagan, µG, nulis toli iqneba, mas Semdeg, rac mas gamovaklebT TiToeuli damoukidebeli cvladis mTavar efeqts (anu
(µ A - µ G) da (µ B - µ G)). sxva sityvebiT, es H0 amtkicebs, rom Tu damoukidebel cvladebs Soris urTierTqmedeba ar arsebobs, damoukidebeli cvladebis mTavari efeqtebidan SesaZlebelia zustad viwinaswarmetyveloT TiToeuli ujredis saSualos mniSvneloba. alternatiuli hipoTeza amtkicebs, rom es ase ar aris da SeuZlebelia ujredebis saSualoebis cvalebadobis winaswarmetyveleba mxolod mTavari efeqtebidan. F–is mniSvnelobaze damyarebuli gadawyvetilebebis miReba hipoTezebis samive tipis Sesaxeb gadawyvetilebebis miReba Frealuris mniSvnelobebze dayrdnobiT msgavsia im procesisa, romelic mimdinareobs
erTfaqtoriani dispersiuli analizis dros. α-s mniSvneloba, romelic gansazRvravs uaryofis ares, arCeulia analizis Catarebamde. H0
hipoTezis pirobebSi F -is SerCeviTi ganawileba ganisazRvreba samive F
SefardebisaTvis da uaryofis are arsebobs samive SerCevis ganawilebaSi. es nabiji moiTxovs Fkritikulis sami mniSvnelobis moZebnas Sesabamisi mricxveli da mniSvneli df-ebiT A.3a an b cxrilSi, imis mixedviT, Tu ra
aris arCeuli α-s mniSvneloba. Semdeg, TiToeuli FrealurisaTvis, Tu Frealuri udris an metia, vidre misi Sesabamisi Fkritikulis mniSvneloba, H0
am FrealurisaTvis uaryofilia da miRebulia H1. Frealuris am mniSvnelobaSi Semaval saSualoebs Soris sxvaoba statistikurad mniSvnelovania. Tu Frealuri naklebia, vidre misi Sesabamisi Fkritikulis mniSvneloba, maSin gadawyvetileba isaa, rom cvalebadobis am wyarosaTvis ver uarvyofT H0–s da ver miviRebT H1–s. Frealuris am mniSvnelobaSi Semaval saSualoebs Soris gansxvaveba ar aris statistikurad mniSvnelovani. orfaqtorian dispersiul analizSi TiToeuli FrealurisaTvis arsebuli gadawyvetilebebis ori jgufi Sejamebulia cxrilSi # 11.9. gadawyvetilebis miRebis ilustracias movaxdenT cxrilSi # 11.2 sanimuSo monacemebis dispersiuli analizis meSveobiT, rac Sejamebulia cxrilSi # 11.8. am cxrilSi, TiToeul Frealurs aqvs 1 df mricxvelad da 16 df mniSvnelad. amitom TiToeul Frealurs aqvs Fkritikulis erTi da igive mniSvneloba da, Sesabamisad, erTi da igive uaryofis are. es mimarTeba aucileblad ar aris aseTi faqtoruli analizis yvela SemTxvevaSi. TiToeuli damoukidebeli cladis doneebis raodenobis mixedviT, SesaZlebelia, rom TiToeul Frealurs mricxvelad hqondes gansxvavebuli df da, Sesabamisad, gansxvavebuli uaryofis are. cxrili # 11.9 dispersiis orfaqtorian analizSi statistikuri gadawyvetilebebis Sejameba -------------------------------------------------------------------------------------------------------------------------------------------------------
F Sefardeba:
Frealuri-s mniSvneloba
statistikuri gadawyvetileba
A faqtorisaTvis
Fkritikulze naklebi ver uarvyofT H0-s ver viRebT H1-s
A faqtoris mTavari efeqti ar aris mniSvnelovani.
Fkritikulis toli an masze meti
uarvyofT H0-s viRebT H1-s
A faqtoris mTavari efeqti statistikurad mniSvnelovania.
B faqtorisaTvis
Fkritikulze naklebi ver uarvyofT H0-s ver viRebT H1-s
B faqtoris mTavari efeqti ar aris mniSvnelovani.
Fkritikulis toli an masze meti
uarvyofT H0-s viRebT H1-s
B faqtoris mTavari efeqti statistikurad mniSvnelovania.
A x B
Fkritikulze naklebi ver uarvyofT H0-s ver viRebT H1-s
A da B faqtorebis urTierTqmedeba ar aris mniSvnelovani.
Fkritikulis toli an masze meti
uarvyofT H0-s viRebT H1-s
A da B faqtorebis urTierTqmedeba statistikurad mniSvnelovania.
rodesac α=.05, Fkritikuli 1,16 df-iT aris 4.49 (aRebulia cxrilidan # A.3a). aqedan gamomdinare, cxrilSi # 11.8 TiToeuli FrealurisaTvis uaryofis are Sedgeba Frealuris mniSvnelobebisagan, romlebic udris an metia 4.49-ze. dispersiis am analizidan Semdeg gadawyvetilebebs viRebT: wyaro: Frealuri xvdeba
uaryofis areSi
statistikuri gadawyvetileba
A faqtorisaTvis 6.66 ki uarvyofT H0-s viRebT H1-s
A faqtoris mTavari efeqti statistikurad mniSvnelovania.
B faqtorisaTvis 4.26 Aara ver uarvyofT H0-s ver viRebT H1-s
B faqtoris mTavari efeqti ar aris mniSvnelovani.
A x B 15.32 ki uarvyofT H0-s viRebT H1-s
A da B faqtorebis urTierTqmedeba statistikurad mniSvnelovania.
cdis pirTa Soris orfaqtoriani dispersiuli analizis daSvebebi cdis pirTa Soris dispersiuli faqtoruli analizi damyarebulia imave sam daSvebaze eqsperimentSi miRebuli monacemebis (qulebis) Sesaxeb, rac cdis pirTa Soris erTfaqtorian analizSi:
1. monacemebi TiToeul SerCevaSi SemTxveviTad aris aRebuli populaciis monacemebidan da TiToeuli qula damoukidebelia yvela sxva qulisagan;
2. im populaciaSi, saidanac aris aRebuli SerCeva, monacemebi normaluradaa ganawilebuli;
3. monacemebis dispersiebi populaciebSi tolia. rogorc aRvniSneT, erTfaqtorian analizTan dakavSirebiT, Tu gvsurs, gamoviyenoT cdis pirTa Soris dispersiuli analizi, unda Sesruldes pirveli daSveba monacemebis damoukideblobis Sesaxeb. TiTeul cdis pirs mxolod erTi qulis mocema SeuZlia eqsperimentSi. Tumca, ar aris aucilebeli, eqsperimentis monacemebSi Sesruldes meore da mesame daSvebebi. mosalodnelia, rom normaluridan gadaxra da dispersiebis
tolobis daSvebis darRveva minimalur gavlenas moaxdens I gvaris Secdomis daSvebis albaTobaze, Tu sruldeba Semdegi pirobebi:
1. cdis pirebis raodenoba yovel ujredSi tolia; 2. TiToeul pirobaSi qulebis ganawilebis forma daaxloebiT
erTnairia da ganawilebebi arc Zalian wamaxvilebulia, arc Zalian brtyeli;
3. mniSvnelovnobis done aris .05.
SeamowmeT Tqveni codna XI.3 a. gansazRvreT faqtorebi, romlebsac aqvs gavlena yovel MS-ze cdis pirTa Soris
dispersiul faqtorul analizSi. b. axseniT, Tu ratom izrdeba MSA.-is mniSvneloba, Tu A faqtors aqvs efeqti
faqtorul eqsperimentSi. g. axseniT, Tu ratom izrdeba MSB-is mniSvneloba, Tu B faqtors aqvs efeqti faqtorul eqsperimentSi. d. axseniT, Tu ratom izrdeba MSAxB-is mniSvneloba, Tu A da B faqtorebi urTierTqmedeben faqtorul eqsperimentSi. e. axseniT, Tu ratom ar aqvs gavlena MSSecdomis mniSvnelobaze arc A da B faqtorebs, arc maT urTierTqmedebas faqtorul eqsperimentSi. v. ra aris mosalodneli, rom daemarTos Frealuris mniSvnelobas A faqtorisaTvis, Tu A faqtors aqvs efeqti eqsperimentSi? z. ra aris mosalodneli, rom daemarTos Frealuris mniSvnelobas AxB-saTvis, Tu damoukidebeli cvladebi urTierTqmedeben eqsperimentSi? T. gansazRvreT statistikuri Semowmebis nabijebi dispersiul faqtorul analizSi. i. dawereT statistikuri hipoTezebi SemdegisaTvis:
c. 2x4 cdis pirTa Soris dispersiuli analizi d. 3x2 cdis pirTa Soris dispersiuli analizi b. 3x3 cdis pirTa Soris dispersiuli analizi
k. am savarjiSoSi mocemulia ramdenime cdis pirTa Soris orfaqtoriani sqemis
dispersiuli analizis Sejamebis FrealurisTvis cxrilebi. TiToeuli
ipoveT Fkritikuli 0.5 mniSvnelovnobis donisaTvis cxrilidan # A.3a da miuTiTeT, xvdeba Tu ara Frealuri uaryofis areSi. Semdeg daafiqsireT Tqveni gadawyvetileba H0-Tan da H1-Tan dakavSirebiT. Tu dfSecdomisaTvis cxrilSi ar aris mocemuli zusti df, gamoiyeneT cxrilSi qvemoT miTiTebuli momdevno mniSvneloba.
cxrili 1 cxrili 2 wyaro df Frealuri
A faqtori 1 4.63
B faqtori 3 2.11
A x B 3 3.97 Secdoma 56 sul 63 cxrili 3
wyaro df Frealuri
A faqtori 3 1.42
B faqtori 2 3.51
A x B 6 2.37 Secdoma 144 sul 155
2x2 faqtoruli dispersiuli analizis interpretacia faqtoruli sqemis Sedegebis axsnis Sesaxeb diskusias daviwyebT im nabijebiT, romlebic saWiroa statistikurad mniSvnelovani urTierTqmedebis axsnisaTvis. Semdeg ki avxsniT statistikurad mniSvnelovan mTavar efeqtebs. damoukidebeli cvladis martivi efeqti damoukidebeli cvladis martivi efeqti: faqtorul sqemaSi erTi damoukidebeli cvladis efeqti meore damoukidebeli cvladis mxolod erTi donis dros.
ori damoukidebeli cvladis urTierTqmedeba xdeba, rodesac erTi damoukidebeli cvladis efeqti damokidebulia masTan kombinaciaSi myofi meore damoukidebeli cvladis doneze. Sesabamisad, dispersiul analizSi statistikurad mniSvnelovani urTierTqmedebis interpretacia xdeba eqsperimentSi ujredebis saSualoebs Soris sxvaobebis SedarebiT. ujredebis saSualoebs Soris sxvaobebi gviCveneben damoukidebeli cvladebis martiv efeqtebs. damoukidebeli cvladis martivi efeqti (romelsac xandaxan martiv mTavar efeqtsac uwodeben) faqtorul sqemaSi aris am damoukidebeli cvladis efeqti meore damoukidebeli cvladis mxolod erTi donis dros. cxrilSi # 11.10 moyvanilia oTxi martivi efeqti, iseve, rogorc ori mTavari efeqti 2x2 sqemaSi. TiToeuli damoukidebeli cvladis martivi efeqti gansazRvrulia marTkuTxedebSi ujredebs Soris. 2x2 sqemaSi mxolod ori martivi efeqti arsebobs TiToeuli damoukidebeli cvladisaTvis.
wyaro df Frealuri
A faqtori 2 4.12
B faqtori 4 2.83
A x B 8 1.87 Secdoma 150 sul 164
cxrili # 11.10 martivi da mTavari efeqtebi 2x2 cdis pirebs Soris dispersiuli analizSi. A faqtori
A1 A2
mTavari efeqtis saSualoebi B faqtorisaTvis
B1 B faqtori B2
11BAX 12BAX
21BAX 22BAX
1BX
2BX
mTavari efeqtis saSualoebi A faqtorisaTvis
1AX 2AX
A faqtoris martivi efeqti A faqtoris martivi efeqti B1 doneze 11BAX da 12BAX ujredebis
saSualoebs Soris gansxvavebiTaa mocemuli (anu 11BAX - 12BAX ). es gansxvaveba asaxavs A faqtoris efeqts B faqtoris B1 doneze. amis
msgavsad, A faqtoris martivi efeqti B faqtoris B2 doneze 21BAX da
22BAX ujredebis saSualoebs Soris gansxvavebiTaa mocemuli (anu 21BAX - 22BAX ). es gansxvaveba asaxavs A faqtoris efeqts B2 doneze. martivi
efeqtebis ricxviTi mniSvnelobebis gamoTvlisas mniSvnelovania, SevinarCunoT Sedarebebis mimarTulebis SeTanxmebuloba da efeqtis niSani (+ an _). B faqtoris martivi efeqti B faqtoris martivi efeqti A1 doneze 11BAX da 21BAX ujredebis
saSualoebs Soris gansxvavebiTaa mocemuli (anu 11BAX - 21BAX ), xolo B faqtoris martivi efeqti A2 doneze 12BAX da 22BAX ujredebis saSualoebs
Soris gansxvavebiTaa mocemuli (anu 12BAX - 22BAX ). TiToeuli martivi efeqti miuTiTebs B faqtoris efeqts A faqtoris mxolod erT doneze. martivi da mTavari efeqtebis ricxviTi magaliTi. cxrilis # 11.5 saSualoebisagan martivi da mTavari efeqtebis ricxviTi mniSvnelobebis sxvaobebi mocemulia cxrilSi # 11.11. saSualoebs Soris mTavari efeqtis da martivi efeqtis gansxvavebebi naCvenebia marTkuTxedebSi. magaliTad, kargi siaxlis mosmenisas cdis piris mxedvelobis areSi yofnis/ar
A-s martivi efeqti B1-
ze: 11BAX - 12BAX
A-s martivi efeqti B2-
ze: 21BAX - 22BAX
A-s martivi efeqti B2-ze:
11BAX - 21BAX
A-s martivi efeqti B2-ze:
12BAX - 22BAX
B faqtoris mTavari efeqti:
1BX - 2BX
A faqtoris mTavari efeqti:
1AX - 2AX
yofnis martivi efeqti (A faqtoris martivi efeqti B1 doneze) aris 11BAX - 12BAX , rac udris 90.8-84.6, anu +6.2 wams. amis msgavsad, siaxlis tipis martivi efeqti im pirobisaTvis, rodesac cdis piri ar imyofeba
mxedvelobis areSi (B faqtoris martivi efeqti A1 doneze) aris 11BAX - 21BAX , rac udris 90.8-82.2, anu +8.6 wams.
cxrili # 11.11 pasuxis latenturi drois (wamebSi) mTavari efeqtisa da martivi efeqtis sxvaobebi cdis piris mxedvelobis areSi yofnisa da siaxlis tipisaTvis sanimuSo eqsperimentSi.
subieqtis yofna mxedvelobis areSi (A)
Cans (A1) ar Cans (A2)
mTavari efeqtis saSualoebi siaxlis tipisaTvis (B)
kargi (B1)
siaxlis tipi (B) cudi (B2)
90.8 84.6
82.2 112.4
87.7
97.3
mTavari efeqtis saSualoebi subieqtis mxedvelobis areSi yofnisaTvis (A)
86.5 98.5
martivi efeqtebis axsna, rodesac vlindeba statistikurad mniSvnelovani urTierTqmedeba Tu dispersiuli analizisas xdeba damoukidebeli cvladebis statistikurad mniSvnelovani urTierTqmedeba, maSin faqtoris ori martivi efeqti ar aris erTmaneTis toli an am faqtoris mTavari efeqtis toli. amitom, Tu xdeba statitikurad mniSvnelovani urTierTqmedeba, A faqtoris martivi efeqti B1 doneze, 11BAX - 12BAX , ar udris A faqtoris martiv efeqts B2 doneze, 21BAX - 22BAX . garda amisa, A faqtoris arc erTi martivi efeqti ar udris A faqtoris mTavar efeqts, romelic
ganisazRvreba 1AX - 2AX . Tu am mimarTebas mivusadagebT magaliTis Sedegebs cxrilSi # 11.11, aRmovaCenT, rom kargi siaxlis mosmenisas cdis piris mxedvelobis areSi yofnis martivi efeqti (efeqtia +6.2 wami) ar udris cudi siaxlis mosmenisaTvis cdis piris mxedvelobis areSi yofnis martiv efeqts (efeqtia –30.2 wami), da arc erTi martivi efeqti ar udris cdis piris mxedvelobaSi yofnis mTavar efeqts (efeqtia –12.0 wami). msgavsi urTierToba axasiaTebs B faqtoris martiv efeqtebsac. Tu xdeba statistikurad mniSvnelovani urTierTqmedeba, maSin B faqtoris martivi
+6.2
+8.6 -27.8
-30.2
-12.0
-9.6
efeqti A1 doneze, 11BAX - 21BAX , ar udris B faqtoris martiv efeqts A2
doneze, 12BAX - 22BAX ; da arc erTi martivi efeqti ar udris B faqtoris mTavar efeqts, romelic ganisazRvreba 1BX - 2BX . Tu am mimarTebas mivusadagebT magaliTis Sedegebs cxrilSi # 11.11, naTelia, rom siaxlis tipis martivi efeqti im pirobaSi, rodesac cdis piri ar imyofeba mxedvelobis areSi (efeqtia +8.6 wami) ar udris siaxlis tipis martiv efeqts im pirobaSi, rodesac cdis piri imyofeba mxedvelobis areSi (efeqtia –27.8 wami), da arc erTi efeqti ar udris siaxlis tipis mTavar efeqts (efeqtia –9.6 wami). mravlobiTi Sedarebis kriteriumebi statistikurad mniSvnelovani urTierTqmedebisaTvis. martivi efeqtebis TiToeuli Sedareba cxrilebSi # 11.10 da 11.11 warmoadgens wyvilTaSoris Sedarebas. erTmaneTs vadarebT ori ujredis saSualos. imis gansasazRvrad, mniSvnelovnad gansxvavdeba Tu ara erTmaneTisagan ori ujredis saSualo, gamoiyeneba mravlobiTi Sedarebis kriteriumi, raTa dadgenil iqnes kritikuli gansxvaveba (CD) martivi efeqtisaTvis. Tu martivi efeqtis realuri mniSvneloba tolia an aRemateba CD–s, maSin ujredebis saSualoebs Soris gansxvaveba statistikurad mniSvnelovania. aqac, martivi efeqtebis saboloo SedarebisaTvis xSirad gamoiyeneba tukis (Tukey) HSD kriteriumi. am kriteriumisaTvis CD ase gamoiTvleba:
CD = qABn
MSSecdoma
q-s mniSvneloba gamoiTvleba cxrilidan # A.4 da damokidebulia martivi efeqtis Casatarebeli Sedarebebis raodenobaze da dfSecdomaze. radgan cxrili # A.4 Sedegil iqna erTfaqtoriani sqemisaTvis, Cven cxrilSi viyenebT Semdeg gardaqmnas:
cxrilidan # A.4 gamoiyeneT q, romelic
moTavsebulia svetSi damoukidebeli sqemis tipi cvladis _____ doneebisaTvis
______________________________________________________ 2 x 2 3 2 x 3 5 3 x 2 5 3 x 3 7 2 x 4 8 4 x 3 8 4 x 4 10
______________________________________________________ MSSecdoma aris MSSecdomis mniSvneloba, gamoTvlili dispersiis faqtorul analizSi da nAB aris qulebis raodenoba TiToeul ujredSi. am gamoTvlis ilustraciisaTvis 2x2 eqsperimentSi cdis piris mxedvelobis areSi yofnisa da siaxlis tipis efeqtebisaTvis, viyenebT svets a=3 cxrilidan # A.4, raTa vipovoT q. radganac MSSecdomis df aris
16 (cxrilidan # 11.8) da α = .05, q-s mniSvneloba (3, 16) aris 3.65. cxrilSi # 11.8 MSSecdoma =108.125 da nAB, qulebis raodenoba, romlebic Sedis TiToeuli ujredSi, aris 5. aqedan gamomdinare, tukis CD-s ricxviTi mniSvneloba aris
CD = 3.655125.108 = 17.0 wami, damrgvalebuli erT meaTedamde.
nebismieri martivi efeqti cxrilidan # 11.11, romelic absoluturi mniSvnelobiT tolia an aRemateba 17.0-s, statistikurad mniSvnelovania .05 doneze. martivi efeqtebi cdis piris mxedvelobis areSi yofnisaTvis cudi siaxlis mosmenis pirobaSi (anu - 30.2 wami) da siaxlis tipisaTvis cdis piris mxedvelobis areSi yofnis pirobaSi (anu -27.8 wami) aRemateba absoluturi mniSvnelobiT CD-s, romelic aris 17.0 wami. es martivi efeqtebi gviCvenebs statistikurad mniSvnelovan gansxvavebebs ujredebis saSualoebs Soris. martivi efeqtebi cdis piris mxedvelobis areSi yofnisTvis kargi siaxlis mosmenis pirobaSi (anu + 6.2 wami) da siaxlis tipisaTvis cdis piris mxedvelobis areSi aryofnis pirobaSi (anu + 8.6 wami) absoluturi mniSvnelobiT naklebia, vidre CD, romelic 17.0 wamia. es martivi efeqtebi ar aris mniSvnelovani da monawile ujredebis saSualoebi gansxvavdeba mxolod SerCevis Secdomis gamo. mTavari efeqtebis axsna, rodesac xdeba urTierTqmedeba rodesac xdeba statistikurad mniSvnelovani urTierTqmedeba, damoukidebeli cvladebis mTavari efeqtebi SeiZleba ver gamoviyenoT raime mniSvnelobis mqone axsnisaTvis. es xdeba imis gamo, rom damoukidebeli cvladis mTavari efeqti aris am cvladis martivi efeqtebis saSualo. magaliTad, yvela ujredSi cdis pirebis toli raodenobis arsebobisas, cdis piris mxedvelobis areSi yofnis mTavari efeqti (-12.0 wami) cxrilSi # 11.11 aris kargi da cudi siaxleebisTvis cdis piris mxedvelobis areSi yofnis martivi efeqtebis saSualo (anu [(+6.2) + (-30.2)]/2 = _12.0 wami). msgavsi damokidebulebaa siaxlis tipis mTavari efeqtisaTvis, -9.6 wami. es aris saSualo martivi efeqtebisa siaxlis tipisaTvis cdis piris mxedvelobis areSi aryofnis pirobaSi, +8.6 wami, da cdis piris mxedvelobis areSi yofnis pirobaSi, -27.8 wami. Sesabamisad, rodesac xdeba urTierTqmedeba, ise, rom damoukidebeli cvladis martivi efeqtebi ar udris erTmaneTs, am damoukidebeli cvladis mTavari efeqti SeZleba zustad ar warmoadgendes cvladis efeqts. aseTi mTavari efeqtebi aris urTierTqmedebis artefaqtebi da ar iZlevian raime mniSvnelobis mqone informacias eqsperimentis Sedegebis Sesaxeb. Sesabamisad, artefaqturi mTavari efeqti aris mTavari efeqti, romelic ar iZleva mniSvnelobis mqone informacias damoukidebeli cvladis efeqtis Sesaxeb, radgan is Tavs iCens mxolod realuri urTierTqmedebis specifikuri paternis gamo. es aris am cvladis martivi efeqtebis saxis xelovnuri Sedegi. aqedan gamomdinare, faqtorul sqemaSi mTavari efeqtebis axsnasTan dakavSirebiT SemoRebulia Tavis dasazRvevi mniSvnelovani SeniSvna:
a. Tu faqtorul sqemaSi xdeba statistikurad mniSvnelovani urTierTqmedeba, maSin rogorc A, aseve B Bfaqtorebis mTavari efeqtebi SeiZleba iyos artefaqturi da SeiZleba ar
warmoadgendnen mniSvnelobis mqone Sedegebs am damoukidebeli cvladis efeqtis Sesaxeb.
b. Tu urTierTqmedeba faqtorul sqemaSi aramniSvnelovania, maSin rogorc A, aseve B faqtorebis mTavari efeqtebi warmoadgenen mniSvnelobis mqone Sedegebs am damoukidebeli cvladis efeqtis Sesaxeb.
artefaqturi mTavari efeqti: mTavari efeqti, romelic ar iZleva mniSvnelobis mqone informacias damoukidebeli cvladis efeqtis Sesaxeb faqtorul sqemaSi.
dispersiuli analizSi statistikurad mniSvnelovani urTierTqmedeba ar niSnavs imas, rom mTavari efeqtebi iyos artefaqturi, magram is aris gafrTxileba, rom Semowmdes nebismieri mniSvnelovani mTavari efeqti, raTa gamovlindes, iZleva Tu ara is mniSvnelobis mqone Sedegebs damoukidebeli cvladis efeqtis Sesaxeb. martivi mTavari efeqtebis axsna, rodesac ar xdeba statistikurad mniSvnelovani urTierTqmedeba Tu damoukidebel cvladebs Soris ar aris statistikurad mniSvnelovani urTierTqmedeba, maSin erTi damoukidebeli cvladis efeqti ar aris damokidebuli meore damoukidebeli cvladis doneze. amitom, Tu urTierTqmedeba ar xdeba, A faqtoris martivi efeqti B1 donisaTvis B2 donisaTvis A faqtoris martivi efeqtis toli iqneba. nebismieri realuri gansxvaveba am martiv efeqtebs Soris mxolod SerCevis SecdomiT aris gamowvuli. ufro metic, Tu A faqtoris martivi efeqtebi erTmaneTis tolia, isini aseve A faqtoris mTavari efeqtis tolia, da A faqtoris mTavari efeqti am damoukidebeli cvladis efeqtis saukeTeso aRweras iZleva. msgavsi urTierToba aRiniSneba B faqtoris martivi efeqtebisaTvisac. Tu urTierTqmedeba ar xdeba, maSin B faqtoris martivi efeqti A1 doneze A2 doneze B faqtoris martivi efeqtis toli iqneba. nebismieri realuri gansxvaveba am martiv efeqtebs Soris mxolod SerCevis SecdomiT aris gamowvuli. B faqtoris martivi efeqtebi aseve B faqtoris mTavari efeqtis tolia, da B faqtoris mTavari efeqti am damoukidebeli cvladis efeqtis saukeTeso aRweras iZleva. es mimarTebebi Sejamebulia cxrilis # 11.12 (b) nawilSi. cxrili # 11.12 urTierTqmedeba 2x2 faqtorul dispersiul analizSi da martivi efeqtebis da mTavari efeqtebis axsna.
(a) Tu Frealuri AxB urTierTqmedebisaTvis xvdeba uaryofis areSi, maSin:
1. urTierTqmedebis F Sefardeba statistikurad mniSvnelovania; 2. erTi damoukidebeli cvladis efeqti damokidebulia meore damoukidebeli
cvladis doneze;
3. A faqtoris martivi efeqti B1 doneze (anu 11BAX – 12BAX ) ar udris A
faqtoris martiv efeqts B2 doneze (anu 21BAX – 22BAX ). A faqtoris martiv efeqtebs Soris realuri sxvaoba gamowveulia damoukidebeli cvladebis urTierTqmedebiT da ara mxolod SerCevis SecdomiT;
4. B faqtoris martivi efeqti A1 doneze (anu 11BAX – 21BAX ) ar udris B
faqtoris martiv efeqts A2 doneze (anu 12BAX – 22BAX ). B faqtoris martiv
efeqtebs Soris realuri sxvaoba gamowveulia damoukidebeli cvladebis urTierTqmedebiT da ara mxolod SerCevis SecdomiT;
5. A faqtoris martivi efeqtebi ar udris A faqtoris mTavar efeqts; 6. B faqtoris martivi efeqtebi ar udris B faqtoris mTavar efeqts; 7. TiToeuli martivi efeqti unda Sedardes tukis CD-s, raTa vipovoT, Tu
ujredis romeli saSualoebi gansxvavdeba erTmaneTisagan mniSvnelovnad; 8. rogorc A, aseve B faqtoris mTavari efeqtebi SeiZleba artefaqturi iyos da
SeiZleba ar warmoadgendes mniSvnelobis mqone Sedegebs damoukidebeli cvladis efeqtis Sesaxeb.
(b) Tu Frealuri AxB urTierTqmedebisaTvis ar xvdeba uaryofis areSi, maSin:
1. urTierTqmedebis F Sefardeba ar aris statistikurad mniSvnelovani; 2. erTi damoukidebeli cvladis efeqti ar aris damokidebuli meore damoukidebeli
cvladis doneze; 3. A da B faqtorebis mTavari efeqtebi warmoadgenen mniSvnelobis mqone Sedegebs
eqsperimentis Sedegebis Sesaxeb;
4. A faqtoris martivi efeqti B1 doneze (anu 11BAX – 12BAX ) mniSvnelovnad ar
gansxvavdeba A faqtoris martivi efeqtisagan B2 doneze (anu 21BAX – 22BAX ). A faqtoris martiv efeqtebs Soris realuri gansxvaveba gamowveulia SerCevis SecdomiT;
5. B faqtoris martivi efeqti A1 doneze (anu 11BAX – 21BAX ) mniSvnelovnad ar
gansxvavdeba B faqtoris martivi efeqtisagan A2 doneze (anu 12BAX – 22BAX ). B faqtoris martiv efeqtebs Soris realuri gansxvaveba gamowveulia SerCevis SecdomiT;
6. A faqtoris martivi efeqtebi ar gansxvavdeba A faqtoris mTavari efeqtisagan. A faqtoris mTavari efeqti warmoadgens A faqtoris efeqtis saukeTeso aRweras;
7. B faqtoris martivi efeqtebi ar gansxvavdeba B faqtoris mTavari efeqtisagan. B faqtoris mTavari efeqti warmoadgens B faqtoris efeqtis saukeTeso aRweras;
8. ar aris saWiro martivi efeqtebis gaanalizeba mravlobiTi Sedarebis kriteriumiT.
Sejameba: dispersiuli analizis axsna sanimuSo eqsperimentSi faqtoruli eqsperimentis Sedegebis gageba moiTxovs TiToeuli mTavari efeqtis da urTierTqmedebis ganxilvas da maT axsnas dispersiul analizSi miRebuli gadawyvetilebebis gaTvaliswinebiT. rodesac xdeba statistikurad mniSvnelovani urTierTqmedeba, rogorc es mocemuli eqsperimentis magaliTSia (ix. cxrili # 11.8), Sedegebis axsna urTierTqmedebiT unda daiwyos. cdis piris mxedvelobis areSi yofnisa da siaxlis tipis urTierTqmedebis axsna Frealuris mniSvneloba urTierTqmedebisaTvis cxrilidan # 11.8 = 15.32 metia, vidre Fkritikulis mniSvneloba 4.49 da amdenad xvdeba uaryofis areSi. aqedan gamomdinare, H0 A da B faqtorebis urTierTqmedebisaTvis, H0: yvela (µAB – µ A – µ B + µ G) = 0 uaryofilia da
H1: yvela (µAB – µ A – µ B + µ G) ar udris 0-s miRebulia. am urTierTqmedebis interpretaciisaTvis Tavdapirvelad viyenebT tukis HSD kritikul sxvaobas cxrilis # 11.11 martivi efeqtebisaTvis. tukis CD aris 17.0 wami; amitom nebismieri martivi efeqti cxrilSi # 11.11, romelic absoluturi mniSvnelobiT tolia an metia 17.0 wamze, statistikurad mniSvnelovania .05 doneze.
Tu davubrundebiT cxrils # 11.11, jer ganvixilavT A faqtoris – cdis piris mxedvelobis areSi yofnis martiv efeqtebs. A faqtoris martivi efeqti B1 doneze, sxvaoba +6.2 wami, naklebia CD-ze, romelic 17.0 wamia; es martivi efeqti aramniSvnelovania. cdis piris mxedvelobis areSi yofnas ar aqvs efeqti, rodesac kargi siaxlis Setyobineba xdeba. sxvaoba “Cans” da “ar Cans” ujredebis saSualoebs Soris, rodesac kargi siaxlis Setyobineba xdeba (Sesabamisad, 90.8 da 84.6 wamebi), mxolod SerCevis Secdomis gamoa miRebuli. A faqtoris martivi efeqti B2 doneze, sxvaoba –30.2 wami, absoluturi mniSvnelobiT metia, vidre CD, romelic 17.0 wamia. es martivi efeqti statistikurad mniSvnelovania. cudi siaxlis Setyobineba mniSvnelovnad ufro nela xdeba, rodesac cdis piri mxedvelobis areSi imyofeba (112.4 wami), vidre rodesac cdis piri mxedvelobis areSi ar imyofeba (82.2 wami). axla SevudgeT B faqtoris – siaxlis tipis – martiv efeqtebs. B faqtoris martivi efeqti A1 doneze, sxvaoba +8.6 wami, absoluturi mniSvnelobiT naklebia CD-ze, romelic 17.0 wamia; es martivi efeqti aramniSvnelovania. rodesac cdis piri ar imyofeba mxedvelobis areSi, kargi da cudi siaxleebis Setyobineba erTnairi siswrafiT xdeba. sxvaoba “kargi siaxlisa” da “cudi siaxlis” ujredebis saSualoebs Soris (Sesabamisad, 90.8 da 82.2. wami) mxolod SerCevis Secdomis gamoa miRebuli. B faqtoris martivi efeqti A2 doneze, sxvaoba –27.8 wami, absoluturi mniSvnelobiT metia CD-ze, romelic 17.0 wamia; amdenad, es martivi efeqti mniSvnelovani sxvaobaa. rodesac cdis piri imyofeba mxedvelobis areSi, cudi siaxlis Setyobineba mniSvnelovnad ufro nela (112.4 wami) xdeba, vidre kargi siaxlis Setyobineba (84.6 wami). am analizs mivyavarT cdis piris mxedvelobis areSi yofnasa da siaxlis tips Soris urTierTqmedebis Sedegebis Semdgom Sejamebamde:
cdis piris mxedvelobis areSi yofnis efeqti damokidebulia Sesatyobinebeli siaxlis tipze;
a. kargi siaxlis SemTxvevaSi sxvaoba latentur droSi mxedvelobis areSi myof da armyof cdis pirebs Soris (Sesabamisad, 90.8 da 84.6 wami) ar iyo mniSvnelovani;
b. cudi siaxlis SemTxvevaSi im cdis pirebis latenturi dro, romlebic iyvnen mxedvelobis areSi (112.4 wami) mniSvnelovnad meti iyo im cdis pirebis latentur droze, romlebic ar imyofebodnen mxedvelobis areSi (82.2 wami). siaxlis tipis efeqti damokidebulia cdis piris mxedvelobis
areSi yofnaze; a. im cdis pirebisTvis, romlebic ar imyofebodnen mxedvelobis areSi,
ar iyo mniSvnelovani sxvaoba kargsa da cud siaxleebs Soris; aseTi cdis pirebi karg da cud siaxleebs ekvivalenturi latenturi drois Semdeg atyobinebdnen (Sesabamisad, 90.8 da 82.2 wami);
b. is cdis pirebi, romlebic imyofebian mxedvelobis areSi, mniSvnelovnad did dros andomeben cudi siaxlis Setyobinebas, vidre kargis Setyobinebas, Sesabamisad, 112.4 da 84.6 wami.
es Sedegebi aSkarad Seesabameba TviTprezentaciis hipoTezis winaswarmetyvelebas, rom MUM efeqti warmoadgens mxolod publikaze
gaTvlil qmedebas, romelic dadebiTi socialuri imijis SenarCunebazea mimarTuli. eqsperimentis Sedegebis grafikuli warmodgena. faqtoruli sqemebis gamoqveynebul SedegebSi ujredebis saSualoebi grafikulad xSirad suraTis saxiTaa warmodgenili, da ara # 11.5 msgavsi cxrilis saxiT, radganac suraTi xSirad urTierTqmedebis ufro naTel suraTs iZleva. ujredebis saSualoebi cxrilidan # 11.5 naCvenebia suraTze # 11.1. am suraTze gansazRvrulia TiToeuli ujredis saSualo, magram gamoqveynebul statiebSi ujredebis saSualoebi warmodgenilia xolme mxolod iseTi simboloTi, rogoricaa Savi an TeTri wreebi. iqve suraTze aRniSnulia, Tu ras gamosaxavs es wreebi. rodesac grafiks vagebT, erTi damoukidebeli cvladi gamoisaxeba abscisaze (anu horizontalur RerZze), xolod meore ki – ordinataze (anu vertikalur RerZze). romeli damoukidebeli cvladi gamoisaxeba abscisaze? zogadad, miRebuli wesis mixedviT, abscisaze aigeba is damoukidebeli cvladi, romelic bunebiT ufro raodenobrivia an romelsac ufro uwyveti ganzomileba udevs safuZvlad. Cvens magaliTSi, aseTi cvladia B faqtori, siaxlis tipi, radganac siaxlis tips SeuZlia uwyvetad cvalebadoba kargi-cudi ganzomilebis gaswvriv; zogi siaxle SeiZleba Zalian kargi iyos, zogi – Zalian cudi, xolo zogi siaxle am or ukiduresobas Soris aris. amitom siaxlis tipis cvladi moTavsda abscisaze. meore damoukidebeli cvladi – cdis piris mxedvelobis areSi yofna – naklebad uwyvetad cvalebadobs; ar arsebobs uwyveti gradacia mxedvelobis areSi yofnasa da aryofnas Soris. Sesabamisad, es damoukidebeli cvladi grafikze ori funqciiT gamosaxeba. uwyveti xaziT SeerTebuli Savi wreebi gamosaxavs cdis piris mxedvelobis areSi aryofnis pirobas (A1), xolo wyvetili xaziT SeerTebuli TeTri wreebi - cdis piris mxedvelobis areSi yofnis pirobas (A2). damokidebuli cvladi gamosaxulia ordinataze. radganac yvelaze mokle latenturi
dro aris 82.2 wami, RerZi dayofilia (=) da pirveli aRniSnuli
mniSvneloba aris 80 wami. suraTi # 11.1 pasuxis latenturi dro (wamebSi) sanimuSo eqsperimentSi, rogorc siaxlis tipis (kargi siaxle - cudi siaxle) da cdis piris mxedvelobis areSi yofnis (Cans – ar Cans) funqcia. am suraTze agebuli cxrilis # 11.5 ujredebis saSualoebi.
80
85
90
95
100
105
110
115
1 2
siaxlis tipi (faqtori B)
Set
yobinebis
lat
enturi dro
(wamebi)
ar Cans Cans
suraTi # 11.1 gvaCvenebs gansxvavebul martiv efeqtebs saTiTaod orive damoukidebeli cvladisaTvis. rodesac xdeba statistikurad mniSvnelovani urTierTqmedeba, erTi damoukidebeli cvladis efeqti damokidebulia meore damoukidebeli cvladis doneze, im cvladis, romelTanac is kombinirebulia. urTierTqmedebis es saxe, romelic gamoiTqmeba “damokidebulia”, naTeli xdeba am suraTidan. Tu Tqven gkiTxavdnen: “rogor moqmedebs cdis piris mxedvelobis areSi yofna pasuxis latentur droze?”, Tqveni pasuxi iqneboda: “es damokidebulia im siaxlis tipze, romlis Setyobinebac xdeba. kargi siaxlisaTvis cdis piris mxedvelobis areSi yofnis da mxedvelobis areSi aryofnis pirobebs Soris aramniSvnelovani sxvaoba arsebobs. magram cudi siaxlisaTvis cdis piris mxedvelobis areSi yofnis dros pasuxis latenturi dro mniSvnelovnad metia, vidre cdis piris mxedvelobis areSi aryofnis dros.” amis msgavsad, Tu Tqven gkiTxavdnen: “axdens Tu ara gavlenas siaxlis tipi pasuxis latenturobaze?”, Tqveni pasuxi aseTi iqneboda: “es damokidebulia imaze, Tu rogoria cdis piris mxevelobis areSi yofnis piroba. rodesac cdis piri ar imyofeba mxedvelobis areSi, kargi da cudi siaxleebis Setyobinebis latenturoba aramniSvnelovnad aris gansxvavebuli. xolo rodesac cdis piri imyofeba mxedvelobis areSi, cudi siaxlis Setyobinebis latenturi dro mniSvnelovnad metia, vidre kargi siaxlisa.” monacemTa aRmniSvnelo wertilebis damakavSirebeli xazebi suraTze # 11.1 ar aris paraleluri, rac statistikurad mniSvnelovani urTierTqmedebis amsaxveli suraTis ganmasxvavebeli maxasiaTebelia. radganac urTierTqmedeba gulisxmobs, rom damoukidebeli cvladis martivi efeqtebi ar udris mis mTavar efeqts, statistikurad mniSvnelovani urTierTqmedebis agebis Sedegad araparaleluri xazebi unda miviRoT. magram ar aris aucilebeli, rom xazebma erTmaneTi
ar Cans/kargi siaxle ujredis saSualo
Cans/cudi siaxle ujredis saSualo
ar Cans/cudi siaxle ujredis saSualo
Cans/kargi siaxle ujredis saSualo
B1 B2
cdis piris mxedvelobaSi yofna (faqtori A)
gadakveTon, rogorc es xdeba suraTze # 11.1; es moxdeba mxolod maSin, rodesac damoukidebeli cvladis martiv efeqtebs urTierTsawinaaRmdego niSnebi aqvT, rogorc es am magaliTSi moxda. praqtikaSi iSviaTad xdeba, rom nebismieri ori xazi suraTze zustad paraleluri iyos. SerCevis Secdomis gamo xazebi ar iqneba zustad paraleluri, maSinac ki, rodesac ar arsebobs statistikurad mniSvnelovani urTierTqmedeba. amdenad, mxolod vizualuri Semowmeba ar aris sakmarisi imisaTvis, rom ganisazRvros, moxda Tu ara urTierTqmedeba. urTierTqmedeba xdeba monacemebSi mxolod maSin, rodesac dispersiul analizSi urTierTqmedebisaTvis aRmoCenilia statistikurad mniSvnelovani Frealuri. A damoukidebeli cvladis – cdis piris mxedvelobis areSi yofnis - mTavari efeqtis axsna. cxrilSi # 11.8 Frealuris mniSvneloba A faqtoris mTavari efeqtisaTvis Frealuri (1.16) = 6.66 metia, vidre Fkritikuli (1.16) = 4.49. es Frealuri xvdeba
uaryofis areSi da “ 210 : AAH µµ = ” cdis piris mxedvelobis areSi
yofnisaTvis uaryofilia, xolo “ AH µ:1 -ebi ar udris erTmaneTs” ki –
miRebuli. es gadawyvetileba gulisxmobs, rom cdis piris mxedvelobis areSi yofnis mTavari efeqti (sxvaoba –12.0 wami mTavari efeqtebis saSualoebs - 86.5 da 98.5 wamebs Soris cdis piris mxedvelobis areSi aryofnis da yofnis dros, Sesabamisad) statistikurad mniSvnelovania. es mTavari efeqti miuTiTebs, rom siaxlis Setyobineba ufro swrafad xdeba, rodesac cdis piri ar Cans, vidre roca cdis piri Cans. magram es mTavari efeqti artefaqturia, da kavSiri, romelsac is gamoxatavs, ar aris rogorc kargi, aseve cudi siaxlisaTvis. rogorc aRmoCnda urTierTqmedebis Seswavlis Sedegad, piroba, rodesac cdis piri Cans, mniSvnelovnad xargrZliv latentur dros iwvevs mxolod maSin, rodesac siaxle cudia. kargi siaxlisaTvis, latenturi droebi cdis piris mxedvelobis areSi yofnis da aryofnis pirobebSi mniSvnelovnad ar gansxvavdeba. amdenad, miuxedavad imisa, rom cdis piris mxedvelobis areSi yofnis mTavari efeqti ufro did latentur dros gviCvenebs im pirobisaTvis, rodesac cdis piri Cans, martivi efeqtebis analizi gviCvenebs, rom es mimarTeba marTebulia mxolod cudi siaxlis Setyobinebis dros; is ar aris marTebuli kargi siaxlisaTvis. amdenad, cdis piris mxedvelobis areSi yofnis mTavari efeqti ar gvawvdis mniSvnelovan informacias eqsperimentis SedegebTan dakavSirebiT; is artefaqturia. B damoukidebeli cvladis – siaxlis tipis - mTavari efeqtis axsna Frealuris mniSvneloba B faqtorisTvis – siaxlis tipisaTvis – aris 4.26. Frealuris es mniSvneloba naklebia, vidre Fkritikuli (4.49) da, aqedan
gamomdinare, “ 210 : AAH µµ = ” ar aris uaryofili da “ AH µ:1 -ebi ar udris
erTmaneTs” ar aris miRebuli. es gadawyvetileba niSnavs, rom siaxlis tipis mTavari efeqti (sxvaoba –9.6 wami 87.7 da 97.3 wamebs Soris kargi da cudi siaxlisTvis, Sesabamisad) aramniSvnelovania. mxolod am aramniSvnelovani mTavari efeqtidan davaskvnidiT, rom siaxlis tipi ar axdens gavlenas cdis piris pasuxis latentur droze; mTavari efeqtis 9.6-wamiani sxvaoba kargi da cudi siaxlis pirobebisaTvis SerCevis
Secdomis gamo iqna miRebuli. magram zustad aRwers Tu ara es daskvna siaxlis tipis efeqts? aqac martivi efeqtebis Seswavla gviCvenebs, rom es ase ar aris; siaxlis tips aqvs efeqti, magram efeqti damokidebulia cdis piris mxedvelobis areSi yofnaze. rodesac cdis piri ar aris eqsperimentatoris TanaSemwes mxedvelobis areSi, kargsa da cud siaxles Soris ar arsebobs mniSvnelovani gansxvaveba pasuxis latenturobaSi (anu sxvaoba 90.8 da 82.2 wamebs Soris SerCevis Secdomis gamo miviReT); magram rodesac cdis piri aris eqsperimentatoris TanaSemwes
mxedvelobis areSi, cudi siaxlis ( X A2B2 = 112.4 wams) Setyobineba
mniSvnelovnad ufro nela xdeba, vidre kargi siaxlisa ( X A2B1 = 84.6 wams). statistikurad mniSvnelovani urTierTqmedebis gamo ar aris saWiro siaxlis tipis aramniSvnelovani efeqtis axsna. pirovnuli diskomfortis hipoTezam iwinaswarmetyvela, rom siaxlis tipisaTvis iarsebebs statistikurad mniSvnelovani da mniSvnelobis mqone mTavari efeqti. cudi siaxlis Setyobinebas ufro didi dro unda sWirdebodes, vidre kargi siaxlis Setyobinebas, miuxedavad imisa, aris Tu ara cdis piri TamaSemwes mxedvelobis areSi. am magaliTSi siaxlis tipis mTavari efeqtis Sedegi aSkarad ar Seesabameba pirovnuli diskomfortis hipoTezis winaswarmetyvelebas. urTierTqmedebis saxeebi kavSiri ujredis saSualoebs Soris sanimuSo eqsperimentSi mxolod erT-erTia im mravali kavSiridan, romlis Sedegad statistikurad mniSvnelovan urTierTqmedebas viRebT. dispersiul analizSi statistikurad mniSvnelovani urTierTqmedeba gviCvenebs mxolod imas, rom damoukidebeli cvladis martiv efeqtsa da mTavar efeqts Soris sxvaobebi ar iqneba erTmaneTis toli. mxolod es ar iZleva raime Semdgom informacias im mravali gzis Sesaxeb, romlebiTac es Sedegi SeiZleba migveRo. urTierTqmedebis zusti buneba SeiZleba ganisazRvros mxolod ujredebis saSualoebs Soris kavSiris frTxili analizis gziT.
Carevis efeqtis siZlieris gazomva dispersiul faqtorul analizSi rogorc es iyo erTfaqtoriani dispersiuli analizis dros, statistikurad mniSvnelovani efeqti faqtorul sqemaSi SeiZleba
gaizomos 2η -iT. radganac arsebobs ori mTavari efeqtis da
urTierTqmedebis kriteriumebi, 2η -is sam gansxvavebul mniSvnelobas
viRebT:
2η = sruliSSSS A A faqtoris statistikurad mniSvnelovani mTavari
efeqtisaTvis.
2η = sruliSSSSB B faqtoris statistikurad mniSvnelovani mTavari
efeqtisaTvis.
2η = sruliSS
SS AxB A da B faqtorebis statistikurad mniSvnelovani
urTierTqmedebisaTvis.
vaCvenoT 2η -is gamoyeneba magaliTSi, roca mxolod urTierTqmedebam
mogvca mniSvnelobis mqone informacia; am SemTxvevaSi 2η
urTierTqmedebisaTvis, damyarebuli SS mniSvnelobebze cxrilidan # 11.8, iqneba:
2η = sruliSS
SS AxB = 000.4567200.1656 = .36
es mniSvneloba gviCvenebs, rom pasuxebis latenturobis dispersiis 36 procenti aixsneba cdis piris mxedvelobis areSi yofnisa da siaxlis tipis urTierTqmedebiT. SeamowmeT Tqveni codna XI.4
1. gansazRvreT: artefaqturi mTavari efeqti, mTavari efeqti, martivi efeqti. 2. Semdegi cxrilebi da suraTebi # 11.2 da 11.3 warmoadgenen eqsperimentSi miRebul
ujredebis saSualoebs. saSualoebis TiToeuli erTobliobisaTvis mocemulia dispersiuli Sejamebuli analizi. Sejamebul analizSi statistikurad
mniSvnelovani Frealuri aRniSnulia p < .05. statistikurad aramniSvnelovani
Frealuri aRniSnulia p > .05. rodesac xdeba mniSvnelovani urTierTqmedeba,
tukis HSD CD martivi efeqtebisaTvis mocemulia dispersiis analizis Sejamebis boloSi. davuSvaT, rom A faqtori aris swavlebis meTodi, romelsac aqvs ori done: kompiuterizebuli swavleba (A1) da arakompiuterizebuli swavleba (A2). faqtori B aris saswavlo masala – maTematikuri (B1) da socialuri (B2) Sinaarsis. damokidebuli cvladi aris testSi miRebuli qula. axseniT saSualoebis TiToeuli erToblioba, ipoveT martivi da mTavari efeqtebi TiToeuli damoukidebeli cvladisaTvis da daukavSireT es efeqtebi Sejamebul dispersiul analizs. Semdeg TiToeuli magaliTisaTvis upasuxeT Semdeg kiTxvebs:
a. mniSvnelovnad gansxvavdebian Tu ara erTmaneTisagan A faqtoris mTavari efeqtis saSualoebi?
b. mniSvnelovnad gansxvavdebian Tu ara erTmaneTisagan B faqtoris mTavari efeqtis saSualoebi?
g. xdeba Tu ara statistikurad mniSvnelovani urTierTqmedeba A da B faqtorebs Soris?
d. rogor moqmedebs swavlebis meTodis (A faqtori) efeqti testSi miRebul qulebze?
e. rogor moqmedebs masalis tipis (B faqtori) efeqti testSi miRebul qulebze?
v. iZlevian Tu ara mTavari efeqtebi gamosadeg informacias eqsperimentis Sedegebis Sesaxeb?
suraTi # 11.2 A da B faqtorebis manipulaciis Semcveli eqsperimentis Sedegi
76
80
84
88
92
96
100
1 2
A faqtori
qula
B1 B2
suraTi # 11.3 A da B faqtorebis manipulaciis Semcveli eqsperimentis Sedegi
667074788286909498
1 2
A faqtori
qula
B1 B2
cxrili 1 A faqtori
A1 A2 dispersiuli analizis Sedegebis Sejameba
} B faqtori
dispersiuli analizis Sedegebis Sejameba A faqtori p > .05 B faqtori p < .05 AxB p < .05
A1 A2
} B faqtori
dispersiuli analizis Sedegebis Sejameba A faqtori p < .05 B faqtori p > .05 AxB p < .05
A1 A2
B1 B faqtori B2
91 85 82 87
A faqtori p < .05. B faqtori p > .05. A x B p > .05.
cxrili 2 A faqtori
A1 A2 dispersiuli analizis Sedegebis Sejameba
B1 B faqtori B2
89 90 81 80
A faqtori p > .05. B faqtori p < .05. A x B p > .05.
cxrili 3 A faqtori
A1 A2 dispersiuli analizis Sedegebis Sejameba
B1 B faqtori B2
85 94 78 86
A faqtori p < .05. B faqtori p < .05. A x B p > .05.
cxrili 4 A faqtori
A1 A2 dispersiuli analizis Sedegebis Sejameba
B1 B faqtori B2
88 75 76 90
A faqtori p > .05. B faqtori p > .05. A x B p < .05.
cxrili 5 A faqtori
A1 A2
dispersiuli analizis Sedegebis Sejameba
B1 B faqtori B2
87 88 74 80
A faqtori p < .05. B faqtori p < .05. A x B p < .05.
cxrili 6 A faqtori
A1 A2 dispersiuli analizis Sedegebis Sejameba
B1 B faqtori B2
90 80 81 82
A faqtori p < .05. B faqtori p < .05. A x B p < .05.
magaliTi # 11.1 2x2 sqemis analizi statistikurad mniSvnelovani urTierTqmedebiT da aramniSvnelovani mTavari efeqtiT amocana mxedveloba da smena ori damoukidebeli SegrZnebaa, da mainc maT Soris SesaZlebelia arsebobdes raRac saerTo gancdis TvalsazrisiT. vizualur stimulebs zogjer axasiaTeben, rogorc JRerads, xolo xmebs - rogorc naTels. amis safuZvelze lindnerma da hainanma (Lindner & Hynan, 1987) wamoayenes hipoTeza, rom musikis tipi, romelsac adamiani usmenda, moaxdenda gavlenas am adamianis mier abstraqtuli naxatis aRqmaze. am hipoTezis Sesamowmeblad maT Caatares cdis pirTa Soris 2x2 sqemis eqsperimenti. isini cvlidnen musikis tips, romelsac cdis piri usmenda: es iyo an arastruqturirebuli avangarduli an struqturirebuli minimalisturi musika (A faqtori). aseve icvleboda cdis piris sqesi: mamrobiTi an mdedrobiTi (B faqtori). musikis mosmenisas yoveli cdis piri aTvalierebda rva abstraqtul naxats. naxatebis daTvalierebis Semdeg igi afasebda maT ramdenime 7-quliani Sefasebis skalaze. erTi skala Sedgenili iyo zedsarTavi saxelebis wyvilebisagan: daZabuli (1) - Tavisufali (7), rigiduli (1) - fleqsiburi (7), Seviwrovebuli (1) - gafarToebuli (7). davuSvaT, rom qulebi miRebulia am zedsarTavi saxelebis samive erTobliobis SejamebiT, ise, rom Sefasebis qulebi SeiZleba icvlebodes 3-dan 21-mde. davuSvaT, rom 24 cdis pirisagan miRebulia Semdegi hipoTeturi qulebi: musikis tipi (A)
avangarduli (A1) minimalisturi (A2)
mamrobiTi (B1) cdis pirTa sqesi (B) mdedrobiTi (B2)
4 16 5 14 7 17 6 19 10 16 4 20 12 5 15 9 18 7 13 6 16 10 16 5
axdens Tu ara musikis tipi gavlenas naxatebis Sefasebaze, gansxvavdebian Tu ara mamakacebi da qalebi micemuli Sefasebebis mixedviT, da arsebobs Tu ara urTierTqmedeba musikis tipisa da cdis piris sqess Soris? amoxsna
gamoyenebuli sqema warmoadgens cdis pirTa Soris 2x2 sqemas. pirveli nabiji aris cdis pirTa Soris orfaqtoriani dispersiuli analizis Catareba imis gansasazRvrad, arsebobs Tu ara statistikurad mniSvnelovani mTavari efeqtebi an statistikurad mniSvnelovani urTierTqmedeba. gamosayenebeli daSvebebi:
1. cdis pirebi populaciisagan SemTxveviTad da damoukideblad iqna SerCeuli;
2. Sefasebebis Secdomebi populaciaSi normaluradaa ganawilebuli;
3. populaciis SerCevebSi Sefasebis qulebis dispersiebi tolia.
statistikuri hipoTezebi:
A faqtori H0: µ A1 = µ A2
H1: µ A.ebi ar aris toli
B faqtori H0: µ B1 = µ B2
H1: µ B-ebi ar aris toli
AxB H0: yvela (µAB - µ A - µ B + µ G) = 0 H1: yvela (µAB - µ A - µ B + µ G) ar udris 0
mniSvnelovnobis done: α = .05. df: dfA = a - 1 = 2 - 1 = 1 dfB = b - 1 = 2 - 1 = 1 dfAxB = (a - 1) (b - 1) = (2 - 1)(2 - 1) = 1 dfSecdoma = ab(nAB - 1) = (2)(2)(6 - 1) = 20 dfsruli = N - 1 = 24 - 1 = 23 F-is kritikuli mniSvneloba: TiToeuli Frealuris aqvs 1 df mricxvelad da 20 df mniSvnelad; amitom Fkritikuli (1.20) = 4.35 [cxrilidan # A.3a]. uaryofis are: 4.35-ze toli an meti Frealuri-s mniSvnelobebi. gamoTvla: dispersiuli analizi SeiZleba gamoTvlil iqnes qulebidan im gamoTvliTi formulebis gamoyenebiT, romlebic mocemulia am Tavis damatebaSi an kompiuteruli programis gamoyenebiT. Cven gamoviyeneT formulebi am Tavis damatebidan, raTa migveRo saboloo cxrili: wyaro df SS MS F musika (A) sqesi (B) AxB Secdoma sul
1 1 1 20 23
13.50 1.50 541.50 96.00 652.50
13.50 1.50 541.50 4.80
2.81 0.31 112.81*
* p < .01 gadawyvetilebebi: am analizis statistikuri hipoTezebisaTvis misaRebia sami gadawyvetileba, erTi yoveli FrealurisTvis.
• musikis tipi (A faqtori): Frealuri = 2.81. es Frealuri ar xvdeba uaryofis areSi; H0-is uaryofis safuZveli ar aris da ar viRebT H1-s.
• sqesi (B faqtori): Frealuri = 0.31. es Frealuri ar xvdeba uaryofis areSi; H0-is uaryofis safuZveli ar aris da ar viRebT H1-s.
• musikis tipisa da sqesis urTierTqmedeba (AxB): Frealuri = 112.81. es Frealuri xvdeba uaryofis areSi; uarvyofT H0-s da viRebT H1-s.
efeqtis siZliere: η2 statistikurad mniSvnelovani urTierTqmedebisaTvis udris SSAxB/SSsruli = 541.50/652.50 = .83. axsna da daskvnebi: statistikurad mniSvnelovani urTierTqmedebis asaxsnelad, viRebT tukis HSD CD-s martivi efeqtebisaTvis da Semdeg vadarebT martiv efeqtebs am CD-s. am kriteriumisTvis saWiro ricxviTi mniSvnelobebi Semdegia:
• q aRebulia cxrilidan # A.4. 2x2 sqemis gardaqmnis matricis gamoyenebiT, romelic TandarTulia am cxrilTan, q-s mniSvneloba miRebulia, rodesac a=3 da MSSecdomis df aris 20. es mniSvnelobaa
3.58, rodesac α = .05. • dispersiuli analizis Sedegad miRebuli MSSecdoma udris 4.80.
• nAB, qulebis raodenoba yovel ujredSi, aris 6. CavsvaT es ricxviTi mniSvnelobebi formulaSi:
CD = qABn
MSSecdoma
= 3.58 680.4
= 3.2 Sefasebis wertili nebismieri martivi efeqti, romlis absoluturi mniSvneloba tolia an metia 3.2 Sefasebis wertilze, statistikurad mniSvnelovania .05 doneze. cxrili # 11.13 gviCvenebs ujredis da mTavari efeqtis saSualoebs da martiv da mTavar efeqtebs. am cxrilSi yoveli martivi efeqti absoluturi mniSvnelobiT metia CD-ze, romelic aris 3.2 da amdenad statistikurad mniSvnelovania. Tumca, aRsaniSnavia, rom TiToeuli damoukidebeli cvladisaTvis martiv efeqtebs sapirispiro niSnebi aqvT. musikis tipis efeqti aSkarad damokidebulia cdis piris sqesze. mamakacebi, rodesac isini avangardul musikas usmendnen, naxatebs Cveulebriv 6.0 qulas aniWebdnen (daZabuli, rigiduli, Seviwrovebuli), xolo rodesac isini minimalistur musikas usmendnen, naxatebs bevrad ufro maRali quliT afasebdnen - 17.0 (Tavisufali, fleqsiburi, gafarTovebuli). qalebis SemTxvevaSi Sefasebis es paterni Sebrunebulia. rodesac isini avangardul musikas usmendnen, qalebi naxatebs 15.0 quliT afasebdnen, maSin, rodesac minimalisturi musikis mosmenisas bevrad nakleb qulas aniWebdnen - 7.0. martivi efeqtebi musikis tipisaTvis (-11.0
mamakacebisaTvis da +8.0 qalebisaTvis) ar aris erTmaneTis toli, aseve ar aris musikis aramniSvnelovani mTavari efeqtis - -1.5 - tolic. cxrili # 11.13 naxatebis SefasebebSi mTavari efeqtebisa da martivi efeqtebis sxvaobebi, rogorc musikis tipisa da cdis piris sqesis funqcia.
musikis tipi (A)
avangarduli (A1) minimalisturi (A2)
mTavari efeqtis saSualoebi sqesisaTvis (B)
mamrobiTi (B1)
cdis piris sqesi (B) mdedrobiTi (B2)
6.0 17.0
15.0 7.0
11.5
11.0
mTavari efeqtis saSualoebi musikis tipisaTvis (A)
10.5 12.0
amis msgavsad, naxatebis Sefasebisas gansxvaveba qalebsa da mamakacebs Soris damokidebulia musikis tipze. qalebi, romlebic avangardul
musikas usmenen ( X 21BA = 15.0), naxatebs afaseben, rogorc mniSvnelovnad
ufro Tavisufals, fleqsibursa da gafarTovebuls, vidre mamakacebi
( X 11BA = 6.0), romlebic usmenen avangardul musikas. magram mamakacebi,
romlebic usmenen minimalistur musikas ( X 12BA = 17.0), naxatebs am
ganzomilebaSi mniSvnelovnad maRali qulebiT afaseben, vidre qalebi
( X 22BA = 7.0). amgvarad, martivi efeqtebi sqesisaTvis (-9.0 avangarduli
musikisaTvis da +10.0 minimalisturi musikisaTvis) ar aris arc erTmaneTis da arc cdis pirTa sqesis aramniSvnelovani mTavari efeqtis - +0.5 - toli. mTavari efeqtebi ar iZleva gamosadeg informacias am eqsperimentis Sedegebis Sesaxeb. mxolod mTavari efeqtebidan Cven davaskvnidiT, rom musikis tipisaTvis efeqti ar arsebobs da rom mamakacebi da qalebi ar gansxvavdebian TavianTi Sefasebebis mxriv. magram urTierTqmedebis analizi gviCvenebs, rom musikis tips aqvs efeqti micemul Sefasebebze, magram efeqtis mimarTuleba damokidebulia sqesze. ufro metic, qalebi da mamakacebi gansxvavdebian Sefasebebis mxriv, magram gansxvaveba
-11.0
-9.0 +10.0
+8.0
-1.5
+0.5
damokidebulia musikis tipze. µ2 gviCvenebs, rom Sefasebebis dispersiis 83 procents ganapirobebs urTierTqmedeba. qulebis winaswarmetyveleba ujredis saSualodan, da ara sruli saSualodan, am SerCevaSi srul dispersias 83 procentiT amcirebs.
magaliTi # 11.2 2x2 sqemis analizi statistikurad mniSvnelovani urTierTqmedebiT da mTavari efeqtiT amocana sityva-gasaRebis meTodi warmoadgens mexsierebis gaZlierebis teqnikas, rodesac adamiani iyenebs vizualur xats, raTa warmoqmnas asociacia or erTeuls Soris. am meTodis gamoyenebis erT-erTi magaliTia meore enis leqsikonis Seswavla. davuSvaT, rom Tqven gTxoves, SeiswavloT ucxo franguli sityvebis sia da am sityvebis inglisuri ekvivalentebi, rogoricaa, magaliTad, franguli sityva boue (talaxi) da misi inglisuri ekvivalenti mud (talaxi). aq sityva-gasaRebi SeiZleba iyos boot (Ceqma), inglisuri sityva, romelic franguli boue-s msgavsia JReradobiT. am teqnikis gamoyenebiT, rodesac Tqven gesmiT sityva boue, Tqven fiqrobT boot-ze da Semdeg warmoidgenT adamians, romelsac acvia Ceqmebi da midis talaxSi. bevrma kvlevam aCvena, rom sityva-gasaRebis teqnika aumjobesebs daswavlas da meore enis sityvis dauyovnebel gaxsenebas (Wang, Thomas, & Ouellette, 1992). magram Riad rCeba kiTxva, Tu ramden xans grZeldeba es gaumjobesebuli gaxseneba. vangma da misma kolegebma (Wang et al. 1992) aRmoaCines, rom miuxedavad imisa, rom sityva-gasaRebis teqnikam, sityvebis martivi gameorebis gziT meqanikur SeswavlasTan SedarebiT, gaaumjobesa mexsierebaSi Senaxva da dauyovnebeli gaxseneba, cdis pirebs didwilad sityva-gasaRebic aviwydebodaT erTkviriani Senaxvis intervalis Semdeg.
davuSvaT, Tqven gainteresebT, marTebulia Tu ara es kavSiri zrdasruli adamianebisaTvis iseve, rogorc kolejis studentebisaTvis, rac Seamowmes vangma da misma kolegebma. Sesabamisad, Tqven gamoiyenebT cdis pirTa Soris 2x2 sqemas, sadac Seswavlis sxvadasxva meTodi - an meqanikuri, an sityva-gasaRebiT - iqneba faqtori A da Senaxvis intervali - dauyovnebeli an erTkviriani aRdgena - iqneba faqtori B. Tqven SemTxveviTad gadaanawileT rva frangulis armcodne zrdasruli adamiani TiToeul pirobaSi. cdis pirebs, romlebzec meqanikuri Seswavlis meTodi gamoiyeneboda sTxoves, ubralod mravaljer gaemeorebinaT franguli sityvebi da maTi inglisuri ekvivalentebi damaxsovrebis mizniT. sityva-gasaRebis meTodis gamoyenebisas cdis pirebs aswavles sityva-gasaRebis teqnika da misces sityva-gasaRebebi yoveli frangul-inglisuri wyvilisaTvis. dauyovnebeli aRdgenis pirobaSi inglisuri ekvivalentebis gaxseneba mowmdeboda 5 wuTSi siis Seswavlis Semdeg. gadavadebuli aRdgenis pirobaSi cdis pirebs sTxovdnen, gaexsenebinaT inglisuri ekvivalentebi Seswavlidan erTi kviris Semdeg. sityvebis sia 25 sityvas Seicavda, amitom cdis piris gaxsenebis qula SeiZleboda Secvliliyo 0-dan (verc erTi sityva ver gaixsena) 25-mde (srulyofilad gaixsena). dauSviT, rom 32 cdis pirisagan miRebulia gaxsenebis hipoTeturi qulebi. gaaanalizeT da aRwereT am eqsperimentis Sedegebi.
daswavlis piroba (A) meqanikuri (A1) gasaRebi sityviT (A2)
dauyovnebeli aRdgena (B1) Senaxvis intervali (B) gadavadebuli aRdgena (B2)
12 9 11 19 16 14 13 15
25 22 19 23 17 24 20 21
6 7 3 11 5 10 5 8
9 4 7 11 5 8 12 6
amoxsna. eqsperimentis statistikuri gaanalizebisaTvis saWiroa cdis pirTa Soris dispersiuli orfaqtoriani analizi. viyenebT mniSvnelovnobis dones .05. daSvebebi:
1. cdis pirebis SerCeva moxda SemTxveviTad da populaciisagan damoukideblad;
2. gaxsenebis qulebi populaciaSi normaluradaa ganawilebuli; 3. gaxsenebis qulebis dispersiebi SerCeul populaciaSi tolia.
statistikuri hipoTezebi:
A faqtori H0: µ A1 = µ A2
H1: µ A.ebi ar aris toli
B faqtori H0: µ B1 = µ B2
H1: µ B-ebi ar aris toli
AxB H0: yvela (µAB - µ A - µ B + µ G) = 0 H1: yvela (µAB - µ A - µ B + µ G) ar udris 0
mniSvnelovnobis done: α = .05. df: dfA = a - 1 = 2 - 1 = 1 dfB = b - 1 = 2 - 1 = 1 dfAxB = (a - 1) (b - 1) = (2 - 1)(2 - 1) = 1 dfSecdoma = ab(nAB - 1) = (2)(2)(8 - 1) = 28 dfsruli = N - 1 = 32 - 1 = 31 F-is kritikuli mniSvneloba: Fkritikuli (1,28) = 4.20 TiToeuli Frealuris mniSvnelobisaTvis.
uaryofis are: Frealuris mniSvnelobebi, romlebic tolia an metia 4.20-ze. gamoTvla: Semdegi Sejamebis cxrilis misaRebad gamoyenebul iqna formulebi Tavis damatebidan: wyaro SS df MS F* Seswavlis piroba (A) Senaxvis intervali (B) AxB Secdoma sul
148.781 830.281 94.531 224.125 1297.718
1 1 1 28 31
148.781 830.281 94.531 8.004
18.59* 103.73* 11.81*
* p < .01 gadawyvetilebebi:
• daswavlis piroba (A faqtori): Frealuri = 18.59. es Frealuri xvdeba uaryofis areSi; uarvyofT H0-s da viRebT H1-s.
• Senaxvis intervali (B faqtori): Frealuri = 103.73. es Frealuri xvdeba uaryofis areSi; uarvyofT H0-s da viRebT H1-s.
• daswavlis pirobisa da Senaxvis intervalis urTierTqmedeba (AxB): Frealuri = 11.81. es Frealuri xvdeba uaryofis areSi; uarvyofT H0-s da viRebT H1-s.
efeqtis siZliere:
Seswavlis piroba: η2 = SSA/SSsruli = 148.781/1297.718 = .11.
Senaxvis intervali: η2 = SSB/SSsruli = 830.281/1297.718 = .64. Seswavlis piroba x Senaxvis intervali:
η2 = SSAxB/SSsruli = 94.531/1297.718 = .07. axsna da daskvnebi: Tavdapirvelad urTierTqmedebas avxsniT tukis HSD CD-s moZebniT martivi efeqtebisaTvis da Semdeg martivi efeqtebis am CD-sTan SedarebiT. am kriteriumisaTvis saWiro ricxviTi mniSvnelobebi Semdegia:
• q aRebulia cxrilidan # A.4, rodesac a=3 da MSSecdomis df aris 28. radgan MSSecdomis 28 df ar aris moyvanili cxrilSi, gamoviyeneT
momdevno df-is, 24-is q. es mniSvnelobaa 3.53, rodesac α = .05. • MSSecdoma aris 8.004.
• nAB aris 8. CavsvaT es ricxviTi mniSvnelobebi formulaSi:
CD = qABn
MSSecdoma
= 3.53 8004.8
= 3.5 sityva, damrgvalebuli erT meaTedamde.
nebismieri martivi efeqti, romelic absoluturi mniSvnelobiT tolia an metia 3.5 sityvisa, statistikurad mniSvnelovania .05 doneze. cxrilSi # 11.14 warmodgenilia ujredisa da mTavari efeqtis saSualoebi da martivi da mTavari efeqtebi am qulebisaTvis. Tavdapirvelad avxsniT urTierTqmedebas TiToeuli damoukidebeli cvladisaTvis da Semdeg am cvladis mTavar efeqts. cxrili # 11.14 sworad aRdgenili sityvebis raodenobebis mTavari efeqtisa da martivi efeqtis gansxvavebebi, rogorc daswavlis pirobisa da Senaxvis intervalis funqcia. Seswavlis piroba (A)
meqnikuri (A1) gasaRebi sityviT (A2)
mTavari efeqtis saSualoebi Senaxvis intervalisaTvis (B)
dauyovnebeli aRdgena (B1)
Senaxvis intervali (B) gadavadebuli aRdgena (B2)
13.6 21.4
6.9 7.8
17.5
7.3
mTavari efeqtis saSualoebi Seswvalis pirobisaTvis (A)
10.2 14.6
Seswavlis piroba: Seswavlis pirobis martivi efeqti dauyovnebeli aRdgenisaTvis, -7.8, absoluturi mniSvnelobiT metia, vidre CD - 3.5 da statistikurad mniSvnelovania. dauyovnebeli aRdgenisas gasaRebi sityvis
pirobas ( X A2B1 = 21.4 sityva) Sedegad moyveba mniSvnelovnad ufro didi
gaxseneba, vidre meqanikur daswavlas ( X A1B1 = 13.6 sityva). xolo
gadavadebuli aRdgenisas Seswavlis pirobis martivi efeqti, -0.9, absoluturi mniSvnelobiT naklebia, vidre CD - 3.5. es martivi efeqti
aramniSvnelovania. amdenad, gadavadebuli aRdgenisas meqanikuri (X A1B2 =
6.9 sityva) da gasaRebi sityvis ( X A2B2 = 7.8 sityva) pirobebi mniSvnelovnad
ar gansxvavdeba. Seswavlis pirobas ar aqvs efeqti gaxsenebaze gadavadebuli aRdgenisas. Frealuri (1.28) = 18.59 Seswavlis pirobis mTavari efeqtisaTvis statistikurad mniSvnelovania da migviTiTebs, rom
-7.8
+6.7 +13.6
-0.9
-4.4
+10.2
Seswavlis pirobis mTavari efeqtis saSualoebi ( X A1 = 10.2 da X A2 = 14.6) mniSvnelovnad gansxvavdebian erTmaneTisagan. magram es mniSvnelovani mTavari efeqti ar gvawvdis mniSvnelobis mqone informacias eqsperimentis Sedegis Sesaxeb. es mTavari efeqti miuTiTebs, rom cdis pirebi ufro met sityvebs ixseneben gasaRebi sityvis pirobaSi, magram Seswavlis pirobis martivi efeqtebis analizma daadgina, rom es kavSiri arsebobs mxolod dauyovnebeli aRdgenis intervalis dros; is ar aris marTebuli gadavadebuli aRdgenis intervalis dros. aqedan gamomdinare, Seswavlis pirobis mTavari efeqti ar iZleva Seswavlis pirobis efeqtis swor aRweras. amis gamo Cven mas ugulvebelvyofT eqsperimentis Sedegebis aRwerisas. Senaxvis intervali: mniSvnelovani urTierTqmedeba aseve migvaniSnebs imaze, rom Senaxvis intervalis efeqti damokidebulia Seswavlis pirobaze. Seswavlis pirobis orive donisaTvis Senaxvis intervalis martivi efeqtebi (+6.7 meqanikuri SeswavlisaTvis da +13.6 gasaRebi sityviT SeswavlisaTvis) absoluturi mniSvnelobiT metia, vidre CD = 3.5. es statistikurad mniSvnelovani martivi efeqtebi migviTiTebs, rom rogorc meqanikuri, aseve gasaRebi sityviT Seswavlis SemTxvevaSi gadavadebuli aRdgena Sedegad mniSvnelovnad nakleb gaxsenebas iZleva, vidre dauyovnebeli aRdgena. magram martivi efeqtebis Seswavla gviCvenebs, rom Senaxvis Semcireba dauyovneblidan gadavadebuli aRdgenisken ufro metia gasaRebi sityviT Seswavlis pirobaSi (13.6 sityva), vidre meqanikuri daswavlisas (6.7 sityva). Senaxvis intervalis statistikurad mniSvnelovani mTavari efeqti [anu Frealuri (1.28) = 103.73, p < .05] gviCvenebs, rom Senaxvis intervalis mTavari efeqtis saSualoebi
( 1BX = 17.5 da 2BX = 7.3) mniSvnelovnad gansxvavdebian. mxolod am statistikurad mniSvnelovani mTavari efeqtidan vaskvniT, rom dauyovnebel aRdgenas ufro karg Senaxvamde mivyavarT, vidre gadavadebul aRdgenas. es daskvna sworia, radgan es kavSiri erTnairad marTebulia rogorc meqanikuri, aseve gasaRebi sityviT Seswavlis pirobebisaTvis. aqedan gamomdinare, Senaxvis intervalis mTavari efeqti mniSvnelobis mqonea da gamosadeg informacias gvawvdis eqsperimentis Sedegebis Sesaxeb. Tumca, am mTavari efeqtis axsnas Tan unda mohyves imis aRiareba, rom urTierTqmedeba migviTiTebs imaze, rom gansxvavebis xarisxi dauyovnebel da gadavadebul aRdgenas Soris Seswavlis pirobazea damokidebuli.
η2 Senaxvis intervalis mTavari efeqtisaTvis .64 migviTiTebs, rom Senaxvis intervali gaxsenebis qulebis dispersiis did nawilzea
pasuxismgebeli. η2 urTierTqmedebisaTvis, .07, gviCvenebs, rom urTierTqmedeba pasuxismgebelia gaxsenebis qulebis dispersiis ufro mcire nawilze. radganac Seswavlis pirobis mTavari efeqti artefaqturia, am faqtorisaTvis ar viyenebT eta kvadrats.
dispersiuli faqtoruli analizis Sedegebis prezentacia faqtoruli eqsperimentis Sedegebis wardgenis stili igivea, rac erTfaqtorian sqemaSi, romelic aRwerilia me-10 TavSi. Sedegebis wardgenis magaliTs moviyvanT sanimuSo amocanis Sedegebis gamoyenebiT,
sadac ganxiluli iyo cdis piris mxedvelobis areSi yofnis (cdis piri imyofeba an ar imyofeba mxedvelobis areSi) da siaxlis tipis (kargi an cudi siaxle) efeqtebi am siaxlis Setyobinebis latenturobaze. am eqsperimentis ujredis saSualoebi warmodgenilia cxrilSi # 11.5, xolo dispersiuli analizi Sejamebulia cxrilSi # 11.8. cxrilebis dabeWdvis siZviris gamo eqsperimentis Sedegebi, ujredebis saSualoebisa da statistikuri analizis CaTvliT, xSirad teqstis formatiT aris xolme warmodgenili, rogoc es aq aris naCvenebi. yvela statistikuri kriteriumisaTvis gamoyenebuli iyo alfas .05 done. cdis piris mxedvelobaSi yofnasa da Sesatyobinebeli siaxlis tips Soris arsebobda mniSvnelovani urTierTqmedeba, F(1,16) = 15.32, MSE = 108.125, p < .01. tukis HSD kriteriumi (kritikuli sxvaoba = 17.0, rodesac α = .05) gamoviyeneT, raTa Segvemowmebina martivi efeqtebis statistikuri mniSvnelovnoba. rodesac kargi siaxlis Setyobineba xdeboda, cdis piris mxedvelobis areSi yofnis efeqti aramniSvnelovani iyo (M = 90.8, SD = 11.56, rodesac cdis piri ar Cans, da M = 84.6, SD = 9.61, rodesac cdis piri Cans). xolo rodesac cudi siaxlis Setyobineba xdeboda, cdis piris mxedvelobis areSi yofnis pirobis latenturoba (M = 112.4, SD = 11.15) mniSvnelovnad ufro xangrZlivi iyo, vidre cdis piris mxedvelobis areSi aryofnis pirobisa (M = 82.2, SD = 9.07). sxvaoba latenturobaSi kargsa da cud siaxleebs Soris cdis piris mxedvelobis areSi aryofnis pirobaSi aramniSvnelovani iyo. xolo cdis piris mxedvelobis areSi yofnis pirobisTvis cudi siaxlis Setyobinebis latenturoba mniSvnelovnad xangrZlivi iyo, vidre kargi siaxlisa. urTierTqmedebis eta kvadrati iyo .36. cdis piris mxedvelobis areSi yofnis mTavari efeqti statistikurad mniSvnelovani iyo, F(1,16) = 6.66, p < .05, magram ar iyo mniSvnelobis mqone. siaxlis tipis mTavari efeqti aramniSvnelovani iyo, F(1,16) = 4.26, p < .05. am prezentaciaSi:
α = .05 aRniSnavs mniSvnelovnobis dones, SerCeuls am statistikuri kriteriumebisaTvis.
F(1,16) aRniSnavs kriteriumis statistikas, rogorc F-s. TiTeuli
warmodgenili Frealuris mniSvnelisa da mricxvelis df naCvenebia frCxilebSi. wamodgenilia Frealuris sami mniSvneloba, ori yoveli damoukidebeli cvladisaTvis da erTi - damoukidebeli cvladebis urTierTqmedebisaTvis. Tanmimdevroba, romliTac warmodgenilia Frealurebi, ar aris aucileblad A faqtori, B faqtori da AxB. xSirad eqsperimentatori ufro metad dainteresebulia urTierTqmedebiT da Sedegebis aRwerisas pirvel rigSi Frealuris am mniSvnelobas gamosaxavs, rogorc es naCvenebia magaliTSi.
= 15.32 gvaZlevs Frealuri-s mniSvnelobas (da ara Fkritikulis
mniSvnelobas cxrilidan # 11.3). zogierT SemTxvevaSi, SesaZlebelia warmodgenili iyos dispersiuli analizis Sedegad miRebuli sami Frealuridan mxolod ori. aseT
SemTxvevebSi SegiZliaT dauSvaT, rom im Frealuris mniSvneloba, romelic ar aris warmodgenili, aramniSvnelovania.
p < .01 aRniSnavs, rom a. Frealuris albaToba, Tu Ho WeSmaritia, naklebia an udros
.01. es mniSvneloba aris Frealuris albaToba an Frealuris ufro ukiduresi mniSvneloba, Tu Ho WeSmaritia; is SeiZleba ar
iyos igive, rac SerCeuli α-s mniSvneloba. b. Ho uaryofilia da H1 miRebulia. g. Frealuri miRebul iqna dispersiuli analizis urTierTqmedebis efeqtisaTvis; amdenad, is miuTiTebs, rom A da B faqtorebis urTierTqmedeba statistikurad mniSvnelovania. d. A da B faqtorebis urTierTqmedeba xdeba ara SemTxveviTi Secdomis, aramed sxva mizezis gamo.
MSE = 108.125 iZleva MSSecdomas FrealurisTvis. es mniSvneloba qulebis
ujredebs Soris cvalebadobis sazomia. CD = 17.0 gvaZlevs kritikul sxvaobebs martivi efeqtebis Sesadareblad
tukis HSD kriteriumis gamoyenebiT. Semdeg aRiwereba statistikurad mniSvnelovani martivi efeqtebi.
η2 = .36 gviCvenebs, rom pasuxebis latenturobebis cvalebadobis 36 procenti am eqsperimentSi miRebulia A da B faqtorebis urTierTqmedebis gamo. am magaliTSi urTierTqmedeba iZleva erTaderT mniSvnelovan informacias eqsperimentis Sedegebis
Sesaxeb; amdenad, η2 mocemulia cvalebadobis mxolod am wyarosaTvis.
SeamowmeT Tqveni codna # 11.5 amoxseniT es amocanebi cdis pirTa Soris orfaqtoriani dispersiuli analizis gamoyenebiT. misdieT sanimuSo amocanebSi gamoyenebul formats. yoveli amocanisaTvis gamoiyeneT .05 mniSvnelovnobis done.
1. erTma fsiqologma wamoayena hipoTeza, rom instruqciis tipis efeqti adamianis mier anagramis amoxsnaze damokidebulia am adamianis kontrolis lokusze. am hipoTezis Sesamowmeblad man eqsperimenti Caatara 12 qalze, romlebmac dabali qulebi miiRes kontrolis lokusis skalaze (rac niSnavs kontrolis Sida mdebareobas) da 12 qalze, romlebmac maRali qulebi miiRes am skalaze (rac niSnavs kontrolis gare mdebareobas). TiToeuli jgufidan qalebis naxevari SemTxveviTad ganawilda oridan erT instruqtirebis pirobaSi - unarebis an SemTxveviTobis instruqtirebaSi. unarebis instruqtirebis pirobaSi cdis pirebs uTxres, rom maT mier anagramis warmatebuli amoxsna damokidebuli iyo maT verbalur unarebze da, Sesabamisad, maT kontrols eqvemdebareboda. SemTxveviTobis instruqcia miuTiTebda, rom maT mier am davalebis Sesruleba ar iyo damokidebuli maT verbalur unarebze da maTi kontrolis miRma iyo. Semdeg cdis pirebi xsnidnen anagramebis seriebs da maT mier am davalebis Sesrulebaze daxarjuli dro aRiniSneboda eqsperimentatoris mier. dauSviT, rom miRebul iqna Semdegi droebi (wamebSi), romlebsac TiToeuli cdis piri xarjavda anagramebis amoxsnaze:
kontrolix lokusi (A) Sida (A1) gare (A2)
unari (B1) instruqciebi (B) SemTxveviToba (B2)
175 202 193 186 150 212
115 106 87 93 99 157
125 100 77 101 131 152
218 202 196 244 237 180
a. gamoTvaleT ujredis da mTavari efeqtis saSualoebi. Semdeg gaaanalizeT qulebi,
raTa gansazRvroT (1) arsebobs Tu ara mTavari efeqti romelime damoukidebeli cvladisaTvis da (2) arsebobs Tu ara urTierTqmedeba damoukidebel cvladebs Soris.
b. aRwereT instruqciis tipis efeqti. Tu saWiroa, gamoiyeneT tukis HSD kriteriumi statistikurad mniSvnelovani martivi efeqtebis gansazRvrisaTvis.
g. gaqvT Tu ara SesaZlebloba, mniSvnelobis mqone axsna misceT romelime statistikurad mniSvnelovan mTavar efeqts, Tu isini artefaqturebia?
d. damokidebuli cvladis dispersiis ra nawili SeiZleba mivaweroT damoukidebeli cvladebis urTierTqmedebas?
2. mraval marTlmsajulebaSi potenciur nafic msajulebs winaswar uCveneben films,
romelic acnobs maT iuridiul process da iuridiuli sistemis arss. davuSvaT, rom iurists ainteresebs, daadginos, cvlis Tu ara filmi nafici msajulis qcevas standartul verbalur instruqciasTan SedarebiT. ufro metic, iurists surs, gaigos, SesaZlebelia Tu ara, filmis efeqti damokidebuli iyos im danaSaulis tipze, romelic unda Seafasos msajulma. am kiTxvebze pasuxis gasacemad iuristma Caatara 2x2 cdis pirTa Soris faqtoruli eqsperimenti, sadac TiToeul pirobaSi cxra cdis piri - fiqtiuri msajuli - monawileobda. A faqtori iyo msajulebisaTvis micemuli instruqciis tipi: A1 - normaluri verbaluri instruqciebi, gacemuli sasamarTlos muSakis mier da A2 - instruqciebi filmis saxiT. B faqtori iyo danaSaulis tipi, romlis Sefasebasac sTxovdnen cdis pirs. am cvlads cvlidnen Semdegnairad: cdis pirs akiTxebdnen avtomanqanis moparvis SemTxvevas (B1) an fizikuri Zaladobis gamoyenebiT Tavdasxmis SemTxvevas (B2). Semdeg TiToeul cdis pirs sTxovdnen, miesadagebina patimrobis vada SemTxvevaSi aRwerili braldebulisaTvis. miCenili patimrobis vadebi (TveebSi) mocemulia momdevno cxrilSi:
instruqciis tipi (A)
normaluri (A1) filmis saxiT (A2) manqanis moparva (B1)
15 10 10 14 17 9 8 12 10
13 16 9 11 14 8 10 10 12
danaSaulis tipi (B) Tavdasxma fizikuri Zaladobis gamoyenebiT (B2)
25 20 23 20 19 17 18 22 20
30 33 26 25 30 35 30 28 32
a. gamoTvaleT ujredisa da mTavari efeqtis saSualoebi. Semdeg gaaanalizeT
qulebi, raTa gansazRvroT (1) arsebobs Tu ara mTavari efeqti romelime damoukidebeli cvladisaTvis da (2) arsebobs Tu ara urTierTqmedeba damoukidebel cvladebs Soris.
b. aRwereT instruqciis tipis efeqti. g. aRwereT danaSaulis tipis efeqti. d. SegviZlia Tu ara mniSvnelobis mqone axsna mivceT romelime
statistikurad mniSvnelovan mTavar efeqts, Tu es efeqtebi artefaqturebia? e. damokidebuli cvladis dispersiis ra nawili SeiZleba mivaweroT danaSaulis
tips da damoukidebeli cvladebis urTierTqmedebas? 3. Semdeg savarjiSoSi warmodgenilia cdis pirTa Soris orfaqtoriani dispersiuli analizi. TiToeuli prezentaciisaTvis upasuxeT Semdeg kiTxvebs:
a. A faqtoris ramdeni done iyo am eqsperimentSi? b. B faqtoris ramdeni done iyo am eqsperimentSi? g. ramdeni cdis piri iRebda monawileobas eqsperimentSi? d. ra aris Fkritikulis mniSvneloba .05 mniSvnelovnobis doneze yoveli Frealuri-
sTvis? e. aris Tu ara Frealuris mniSvneloba A faqtorisaTvis statistikurad
mniSvnelovani? v. aris Tu ara Frealuris mniSvneloba B faqtorisaTvis statistikurad
mniSvnelovani?
z. aris Tu ara Frealuris mniSvneloba AxB-saTvis statistikurad mniSvnelovani?
3.1. alfas done .05 gamoyenebul iqna yvela statistikuri kriteriumisaTvis. medikamentis donis (A faqtori) mTavari efeqti statitikurad mniSvnelovani iyo, F(2,36) = 9.43, p < .01, MSE = 17.814. aseve miRebuli iyo mTavari efeqti Senaxvis intervalisaTvis (B faqtori), F(1,36) = 5.60, p < .05. urTierTqmedeba aramniSvnelovani iyo, F(2,36) = 2.46, p > .05. 3.2. alfas done .05 gamoyenebul iqna yvela statistikuri kriteriumisaTvis. miRebul iqna aTletebis sqesisa (A faqtori) da sportis saxeobis (B faqtori) mniSvnelovani urTierTqmedebis gavlena TviTSefasebis qulebze, F (3,88) = 3.28, p < .05, MSE = 42.163. garda amisa, mamakacebma ufro maRali TviTSefasebis qulebi mogvces, vidre qalebma, F (1,88) = 4.82, p < .05. sportis saxeobebs Soris ar aRmoCnda mTavari efeqtebis sxvaobebi, F (3,88) = 1.65, p > .05. 3.3. alfas done .05 gamoyenebul iqna yvela statistikuri kriteriumisaTvis. mniSvnelovani urTierTqmedeba gamovlinda xmauris pirobasa da pirovnebis tips Soris F (1,56) = 5.33, p < .05, MSE = 26.718. mniSvnelovani ar aRmoCnda mTavari efeqtebi arc xmauris pirobis, F (1,56) < 1.0, arc pirovnebis tipisaTvis, F (1,56) = 1.43, p > .05.
Sejameba
• faqtorulia kvlevis iseTi sqemebi, sadac ori an meti damoukidebeli cvladi erTdroulad icvleba.
• faqtoruli sqemis TiToeuli ujredi anu Carevis piroba warmoadgens yoveli damoukidebeli cvladis erTi donisagan warmoqmnil kombinacias.
• mTavari efeqtis saSualo gviCvenebs yvela individis tipur Sesrulebas damoukidebeli cvladis erTi donisaTvis, meore
damoukidebeli cvladis doneebis mixedviT klasifikaciis ignorirebiT.
• faqtorul sqemaSi urTierTqmedeba xdeba, rodesac erTi damoukidebeli cvladis efeqti damokidebulia masTan kombinaciaSi myofi meore damoukidebeli cvladis doneze.
• cdis pirTa Soris orfaqtorian dispersiul analizSi miiReba sami F statistika:
cvalebadobis wyaro F A faqtori B faqtori AxB urTierTqmedeba
MSA/MSSecdoma
MSB/MSSecdoma
MSAxB/MSSecdoma
• statistikuri Semowmebis nabijebi am F SefardebebiT me-10 TavSi ganxiluli nabijebis identuria.
• faqtorul sqemaSi damoukidebeli cvladis martivi efeqti aris am damoukidebeli cvladis efeqti meore damoukidebeli cvladis mxolod erT doneze.
• Tu xdeba damoukidebeli cvladebis statistikurad mniSvnelovani urTierTqmedeba, maSin erTi faqtoris ori martivi efeqti ar udris erTmaneTs an am faqtoris mTavar efeqts.
• artefaqturi mTavari efeqti aris mTavari efeqti, romelic ar SeiZleba mniSvnelovnad aixsnas; is xdeba mxolod realuri urTierTqmedebis specifikuri saxis gamo.
• Tu xdeba mniSvnelovani urTierTqmedeba, saWiroa Semdgomi Semowmeba, raTa statistikurad gaanalizdes martivi efeqtebi. tukis HSD kriteriumi gamoiyeneba saboloo SedarebisaTvis.
• η2 gamoiyeneba, rogorc Carevis efeqtis siZlieris sazomi statistikurad mniSvnelovani mTavari efeqtebisa da urTierTqmedebisaTvis.
__________________________________________________________________ ZiriTadi terminebi da simboloebi
a artefaqturi mTavari efeqti b ujredi
ujredis saSualo ( ABX ) kritikuli sxvaoba (CD) dfA dfB dfAxB dfSecdoma dfsruli A faqtori B faqtori faqtoruli sqema
sruli saSualo ( GX ) urTierTqmedeba (AxB) A faqtoris mTavari efeqti
MSA MSB MSAxB MSSecdoma nAB damoukidebeli cvladis martivi efeqti SSA SSB SSAxB SSSecdoma SSsruli Carevis piroba tukis HSD kriteriumi
AX
BX
B faqtoris mTavari efeqti mTavari efeqtis saSualo
GX
__________________________________________________________________
Tavis damateba cdis pirTa Soris orfaqtoriani dispersiuli analizis gamoTvla formulebis meSveobiT __________________________________________________________________ am damatebaSi mocemulia gamoTvliTi formulebi cdis pirTa Soris orfaqtoriani dispersiuli analizisaTvis, sadac TiToeul ujredSi cdis pirTa Tanabari raodenobaa. amis simboluri gamosaxuleba TiToeuli faqtoris ori donisaTvis da TiToeul ujredSi xuTi qulisaTvis aseTia:
faqtori A
A1 A2
B1
faqtori B B2
X111 X211 X311 TA1B1 X411 X511
X121 X221 X321 TA2B1 X421 X521
TB1
X112 X212 X312 TA2B1 X412 X512
X122 X222 X322 TA2B2 X422 X522
TB2
TA1 TA2 G sadac Xijk = cdis piris qula TA = qulebis sruli jami A faqtoris donisaTvis TB = qulebis sruli jami B faqtoris donisaTvis TAB = qulebis jami ujredSi G = qulebis saerTo sruli jami nAB = qulebis raodenoba ujredSi a = A faqtoris doneebis raodenoba b = B faqtoris doneebis raodenoba
N = qulebis sruli raodenoba am aRniSvnebis gamoyenebiT gamoiTvleba xuTi ricxviTi mniSvneloba:
[1] = ∑∑∑ 2ijkX yvela qulis kvadratebis jami,
( ∑∑∑ aRniSnavs, rom kvadratSi ayvanili qulebi dajamebulia ujredSi yvela qulis mixedviT da Semdeg A da B faqtorebis yvela donis mixedviT),
[2] = AB
AB
nT 2∑∑
TiToeuli ujredis kvadratebis jami, gayofili
ujredSi qulebis raodenobaze,
[3] = AB
A
bnT 2∑
A faqtoris yoveli donis sruli jamebis kvadratebis
jami, gayofili TiToeul srul jamSi Semavali qulebis raodenobaze.
[4] = AB
B
anT 2∑
B faqtoris yoveli donis sruli jamebis kvadratebis
jami, gayofili TiToeul srul jamSi Semavali qulebis raodenobaze.
[5] = N
G 2
sruli saerTo jamis kvadrati, gayofili qulebis srul
raodenobaze. am ricxviTi mniSvnelobebis gamoyenebiT, dispersiuli analizi ase miiReba: cvalebadobis wyaro
SS df MS F
A faqtori B faqtori AxB Secdoma sul
[3] – [5] [4] – [5] [2] – [3] – [4] + [5] [1] – [2] [1] – [5]
a – 1 b – 1 (a – 1)( b – 1)
ab(nAB - 1) N - 1
A
A
dfSS
B
B
dfSS
AxB
AxB
dfSS
Secdoma
Secdoma
dfSS
ar gamoiTvleba
SecdomaMSMS A
SecdomaMSMSB
SecdomaMSMS AxB
am gamoTvlebis ilustraciisaTvis gamoviyenoT monacemebi magaliTidan. es monacemebi moyvanilia cxrilSi # 11.2, xolo maTi dispersiuli analizi Sejamebulia cxrilSi # 11.8. faqtori A
A1 A2
B1
faqtori B B2
97 90 80 TA1B1 = 454 107 80
70 87 81 TA2B1 = 454
95 90
TB1 = 877
68 87 92 TA2B1 = 411
80 84
114 96 127 TA2B2 = 562
110 115
TB2 = 973
TA1 = 865 TA2 = 985 G = 1850 da nAB = 5, a = 2, b = 2 da N = 20. ricxviTi mniSvnelobebia [1] = 972 + . . . + 802 + 702 . . . + 902 + 682 + . . . + 842 + 1142 + . . . + 1152 =
175,692.000
[2] = 2222
5)562()411()423()454( +++=
500.810,869
= 173,962.000
[3] = )5)(2(
)985()865( 22 + =
1000.450,718,1
= 171,845.000
[4] = )5)(2(
)973()877( 22 + =
1000.858,715,1
= 171,585.800
[5] = 20
18502
= 20
00.500,422,3 = 171,125.000
Semdeg SSA = [3] - [5] = 171,845.000 - 171,125.000 = 720.000 dfA = a - 1 = 2 - 1 = 1
MSA = 1000.720 = 720.000
SSA = [4] - [5] = 171,585.800 - 171,125.000 = 460.800 dfB = b - 1 = 2 - 1 = 1
MSB = 1800.460 = 460.800
SSAxB = [2] - [3] - [4] + [5] = 173,962.000 - 171,845.000 - 171,585.800 + 171,125.000 =
1656.200 dfAxB = {a - 1}(b - 1) = (2 - 1) (2 - 1) = 1
MSAxB = 1
1656.200 = 1656.200
SSSecdoma = [1] - [2] = 175,692.000 - 173,962.000 = 1730.000 dfSecdoma = ab(nAB - 1) = (2)(2)(5-1) = 16
MSSecdoma = 16
1730.000 = 108.125
SSsruli = [1] - [5] = 175,692.000 - 171, 125.000 = 4567.000 dfsruli = N - 1 = 20 - 1 = 19 amdenad, Frealuris mniSvnelobebia
A faqtori: Frealuri = SecdomaMS
MSA = 125.108000.720
= 6.66
B faqtori: Frealuri = SecdomaMS
MSB = 125.108800.460
= 4.26
A da B faqtorebis urTierTqmedeba: Frealuri = SecdomaMS
MSAxB
Frealuri = 125.108200.1656
= 15.32
dispersiuli analizis Sejamebaa wyaro SS df MS F cdis piris mxedvelobis areSi yofna (A) siaxlis tipi (B) AxB Secdoma sul
720.000 460.000 1656.200 1730.000 4567.000
1 1 1 16 19
720.000 460.000 1656.200 108.125
6.66* 4.26 15.32**
* p < .05 ** p < .01 am gamoTvliTi midgomis Sedegad miRebuli mniSvnelobebi cxrilSi # 11.8 warmodgenili im mniSvnelobebis identuria, romlebic miviReT dispersiis analizis ganmartebiTi formulebis gamoyenebis Sedegad.
kiTxvebi
mocemul qulebze cdis pirTa Soris orfaqtoriani dispersiuli analizis Catarebis Sedegad upasuxeT sam kiTxvas.
1. rogor icvleba fiziologiuri agznebis done, rodesac Cven socialuri gacvlis procesSi verTvebiT? am kiTxvaze nawilobrivi pasuxis gasacemad velensma (Wellens, 1987) gamoiyena cdis pirTa Soris 2x2 sqema, sadac eqsperimentatori intervius utarebda cdis pirebs. erTi damoukidebeli cvladi (A) iyo eqsperimentatoris mxedvelobiTi kontaqti cdis pirTan intervius dros. aq icvleboda ori done: mxedvelobiTi kontaqtis ararseboba (A1) da mxedvelobiTi kontaqtis damyareba (A2). meore damoukidebeli cvladi (B faqtori) iyo emociiT manipulireba or doneze: dadebiTi (B1) da uaryofiTi (B2). dadebiTi emociis pirobaSi cdis pirebs werilobiTi saxiT aZlevdnen maTze eqsperimentatoris "pirvel STabeWdilebas", romelic Zalian dadebiTi iyo. uaryofiTi emociis pirobaSi maT werilobiTi saxiT aZlevdnen maTze eqsperimentatoris pirvel STabeWdilebas, romelic Zalian uaryofiTi iyo. damokidebuli cvladi iyo cdis piris guliscema, gamoxatuli wuTSi dartymebis raodenobiT, romelsac afiqsirebdnen intervius Catarebis procesSi. dauSviT, rom Tqven gaimeoreT es eqsperimenti TiToeul ujredSi rva cdis pirze da guliscemis Semdegi monacemebi miiReT:
mxedvelobiTi kontaqtis piroba (A)
araseboba (A1) damyareba (A2) dadebiTi (B1) emociis piroba (B) uaryofiTi (B2)
75 77 68 72 83 78 70 65
61 68 71 59 74 66 67 57
79 82 75 77 84 72 78 69
84 79 88 90 75 75 77 85
a. damokidebulia Tu ara mxedvelobiTi kontaqtis efeqti emociis
doneze? Tu ki, gamoiyeneT tukis HSD kriteriumi martivi efeqtebis SedarebisaTvis. aRwereT Tqvens mier aRmoCenili urTierToba.
b. damokidebulia Tu ara emociis pirobis efeqti mxedvelobiTi kontaqtis pirobaze? Tu ki, gamoiyeneT tukis HSD kriteriumi martivi efeqtebis SedarebisaTvis. aRwereT Tqvens mier aRmoCenili urTierToba.
g. SegiZliaT Tu ara, mniSvnelovnad CavTvaloT romelime mTavari efeqti, Tu isini artefaqturia?
d. damokidebuli cvladis cvalebadobis ra nawili xdeba
damoukidebeli cvladebis urTierTqmedebis gamo? e. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic
ilustrirebuli iyo am Tavis seqciaSi "dispersiuli faqtoruli analizis Sedegebis prezentacia".
2. adamianebis umetesobis mier aRiarebuli pirovnebis erT-erTi
ganzomileba aris introversia-eqstraversia. eqstravertebs axasiaTeben, rogorc Ria, socialur da xalisiT aRsavse adamianebs, xolo introvertebs ki, rogorc Caketil da naklebad socialur tipebs. radganac introvertebi ufro metad sakuTar fiqrebsa da ideebisken mimarTulni Canan, SeiZleba vifiqroT, rom introvertebi da eqstravertebi SeiZleba gansxvavbulad reagirebdnen garegan stimulirebaze, rogoricaa xmauri. am sakiTxis ufro Rrmad gamokvlevisaTvis stendingma, linma da moqsnesma (Standing, Lynn and Moxness, 1990) gamoiyenes cdis pirTa Soris 2x2 sqema, sadac cdis pirebi asrulebdnen wakiTxulis Sinaarsis gagebis davalebas da icvleboda pirovnebis tipi (A faqtori) - introvertebi (A1) da eqstravertebi (A2) da xmauris foni (B faqtori) - xmadabali (B1) da xmamaRali (B2). davuSvaT, Tqven CaatareT msgavsi kvleva 12 cdis piriT yovel ujredSi da yovelma cdis pirma Seasrula 15 debulebisgan Semdgari "sworia/arasworia" tipis wakiTxulis Sinaarsis gagebis davaleba. cdis piris qula warmoadgenda sworad gagebuli debulebebis raodenobas. Tqven miiReT Semdegi qulebi:
pirovnebis tipi (A)
introvertuli (A1) eqstravertuli (A2) xmadabali (B1) xmauris foni (B) xmamaRali (B2)
10 8 12 15 10 9 14 13 14 7 12 11
15 14 12 15 11 8 12 8 13 10 9 11
6 8 11 4 5 8 9 10 9 10 7 11
13 8 9 14 13 12 15 11 9 11 12 10
a. ra damokidebuleba arsebobs pirovnebis tipsa da xmauris fonis efeqts Soris? Tu xdeba urTierTqmedeba, martivi efeqtebisTvis gamoiyeneT tukis HSD kriteriumi da aRwereT Tqvens mier dadgenili urTierTqmedeba.
b. ra aris η2 urTierTqmedebisaTvis? g. SegiZliaT Tu ara, mniSvnelovnad axsnaT romelime mTavari efeqti,
Tu isini artefaqturia? d. SegiZliaT Tu ara, mniSvnelovnad CavTvaloT romelime mTavari
efeqti, Tu isini artefaqturia? e. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic
ilustrirebuli iyo am Tavis seqciaSi "dispersiuli faqtoruli analizis Sedegebis prezentacia".
3. romelia kiTxvis damwyebTaTvis fonetikis swavlebis
saukeTeso meTodi? jozefma (Joseph, 2000) Seamowma pirvelklaselebisaTvis fonetikis swavlebis ori axali meTodi - sityvebis yuTis instruqciiT da sityvebis dajgufebis instruqciiT, da aseve tradiciuli fonetikuri meTodis instruqciiT. sityvebis yuTis instruqciis gamoyenebisas bavSvs eZleva oTxkuTxedi, romelic dayofilia seqciebad. TiToeuli seqcia sityvaSi yovel fonemas Seesabameba (fonema metyvelebis yvelaze mcire nawilakia). am instruqciis sawyis fazebze, rodesac sityva warmoiTqmis, bavSvi Jetons aTavsebs oTxkuTxedis im seqciaze, romelic Seesabameba gaJRerebul fonemas. sityvebis dajgufebis instruqcia isaa, rom bavSvebs sTxoven, moaxdinon sityvebis dajgufeba garkveuli zogadi aspeqtebis mixedviT, rogoricaa marTlwera an JReradoba. fonetikis tradiciuli swavlebis dros moswavleebi gundurad kiTxuloben im sityvebs, romelTac marTlweris saerTo tipi aqvT. dauSviT, rom Tqven dainteresdiT fonetikis swavlebis meTodebis efeqtiT da amas garda moisurveT, gagegiT am meTodebze varjiSis raodenobis efeqtebi. ase rom, Tqven cvlidiT fonetikis swavlebis (A faqtori) sam meTods: tradiciuls (A1), sityvebis yuTis (A2) da sityvebis dajgufebis (A3). aseve varjiSis raodenoba (B faqtori) icvleboda ori donis mixedviT: erTi kvira (B1) da eqvsi kvira (B2). cxra pirvelklaseli bavSvi SemTxveviTad gadanawilda TiToeul ujredSi. varjiSis periodis bolos yoveli bavSvi Semowmda marTlweris 30-debulebiani testiT da miRebuli iqna Semdegi qulebi (yoveli qula sworad dawerili sityvebis raodenobaa):
instruqciis meTodi (A) tradiciuli sityvebis yuTi sityvebis dajgufeba (A1) (A2) (A3)
erTi kvira (B1) varjiSis raodenoba (B) eqvsi kvira (B2)
10 14 16 12 14 13 15 11 13
16 17 23 19 20 15 19 18 21
21 24 23 20 23 25 21 26 22
15 18 22 18 21 17 18 16 16
18 24 19 22 21 20 20 22 23
21 19 20 23 24 24 22 25 20
a. damokidebulia Tu ara instruqciis meTodi varjiSis raodenobaze?
Tu ki, gamoiyeneT tukis HSD kriteriumi martivi efeqtebisaTvis. aRwereT Tqvens mier aRmoCenili urTierToba instruqciis meTodsa da varjiSis raodenobas Soris.
b. ra aris η2 urTierTqmedebisaTvis? g. SegiZliaT Tu ara, mniSvnelovnad CavTvaloT romelime mTavari
efeqti Tu isini artefaqturia? d. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic
ilustrirebuli iyo am Tavis seqciaSi "dispersiuli faqtoruli analizis Sedegebis prezentacia".
4. erTma fsiqologma Caatara cdis pirTa Soris 2x2 eqsperimenti rva cdis piriT TiToeul ujredSi.
a. ra aris am sqemisaTvis A.s, b-s, nAB-s da N-is mniSvnelobebi? b. ra aris am sqemisaTvis dfA.s, dfB-s, dfAxB-s, dfSecdomis da dfsrulis
mniSvnelobebi? 5. Semdegi cxrilebi warmoadgens cdis pirTa Soris dispersiuli faqtoruli analizis arasrul Sejamebis cxrilebs. dauSviT, rom yovel ujredSi cdis pirTa toli raodenobebia. gamoiyeneT SS-s, df-sa da MS-s Soris kavSiri, raTa SeavsoT TiToeul cxrilSi gamotovebuli mniSvnelobebi. Semdeg yoveli cxrilisaTvis upasuxeT Semdeg kiTxvebs:
a. A faqtoris ramdeni done iyo? b. B faqtoris ramdeni done iyo? g. ramdeni cdis piri gamoiyenes TiToeul ujredSi? d. ra iyo cdis pirTa sruli raodenoba eqsperimentSi? e. Frealuris romeli mniSvnelobebia statistikurad mniSvnelovani .05 doneze? v. am analizSi rogor gadawyvetilebebs gaakeTebT yoveli nulovani da alternatiuli hipoTezisaTvis?
cxrili # 1 wyaro SS df MS F davalebis tipi (A) xmauris done (B) AxB Secdoma sul
------- 16.00 36.00 ------- -------
3 ------- 6 60 -------
5.00 ------- ------- 2.00
------- ------- -------
cxrili # 2 wyaro SS df MS F treningis xangrZlivoba (A) unaris done (B) AxB Secdoma sul
40.00 ------- 960.00 3200.00 -------
------- 3 ------- 160 179
10.00 40.00 80.00 -------
------- ------- -------
cxrili # 3 wyaro SS df MS F ukukavSiris tipi (A) davalebis sirTule (B) AxB Secdoma sul
200.00 ------- ------- ------- 3300.00
------- ------- 8 ------- 149
50.00 ------- 30.00 20.00-
------- ------- -------
6. dauSviT, rom Tqven sTxoveT cdis pirebs, SeeswavlaT winadadebebis seriebi da Semdeg gaexsenebinaT arsebiTi saxelebi am winadadebebidan. Tqven iyenebdiT xatebis or dones (A faqtori) – Cveulebriv da ucnaur xatebs, da damuSavebis or dones (B faqtori) – dabal da maRal damuSavebas. yovel cdis pirs 30 winadadeba unda eswavla, Sesabamisad, aRsadgeni arsebiTi saxelebis maqsimumi iyo 30. cxrilSi # 1 warmodgenilia ujredis saSualoebi sworad aRdgenili arsebiTi saxelebis raodenobebisaTvis. am monacemebis dispersiuli analizi warmodgenilia cxrilSi # 2. dauSviT, rom gamoyenebuli iyo cdis pirTa Soris sqema xuTi cdis piriT TiTo ujredSi. cxrili # 1 sworad aRdgenili arsebiTi saxelebis saSualo raodenoba, rogorc xatis tipisa da damuSavebis donis funqcia. xatis tipi (A)
Cveulebrivi (A1) ucnauri (A2) dabali (B1) damuSavebis done (B) maRali (B2)
21.4
4.8
10.4
18.8
cxrili # 2 dispersiuli analizis Sejamebis cxrili aRdgenili arsebiTi saxelebis saSualo raodenobisaTvis cxrilidan # 1. wyaro SS df MS F xatis tipi (A) damuSavebis done (B) AxB Secdoma sul
84.050 11.250 781.250 274.000 1150.550
1 1 1 16 19
84.050 11.250 781.250 17.125
4.91* 0.66 45.62**
* p < .05 ** p < .01
tukis HSD kriteriumma gviCvena, rom CD = 6.8 sityvas martivi efeqtebisaTvis.
a. axdens Tu ara xatebis tipi gavlenas arsebiTi saxelebis aRdgenaze?
b. axdens Tu ara damuSavebis done gavlenas arsebiTi saxelebis aRdgenaze?
7. fsiqologi dainteresebuli iyo pilotebis unariT, waikiTxon TviTmfrinavis xelsawyoebis monacemebi Zilis deprivaciis periodis Semdeg. rogorc es naCvenebia naxatze # 11.4, man gamoiyena ori sxvadasxva tipis xelsawyo, romelic pilotebs unda waekiTxaT – analoguri da cifruli. fsiqologma gamoiyena pilotebis SerCeva. maT naxevars unda waekiTxa analoguri xelsawyos monacemebi, xolo meore naxevars ki – cifruli xelsawyos monacemebi. TiToeul jgufSi pilotebis naxevari Semowmda normaluri Ramis Zilis Semdeg, xolo meore naxevari ki – mas Semdeg, rac 24 saaTis ganmavlobaSi ar eZinaT. ase rom, gamoyenebul iqna cdis pirTa 2x2 sqema, sadac xelsawyos tipi - analoguri an cifruli - iyo erTi damoukidebeli cvladi, da Zilis deprivaciis raodenoba - arc erTi an 24 saaTi - iyo meore damoukidebeli cvladi. damokidebuli cvladi iyo 100 cdidan swori wakiTxvis raodenoba. ujredis saSualoebi swori wakiTxvis raodenobisaTvis mocemulia cxrilSi # 1. dispersiuli analizi am monacemebisaTvis warmodgenilia cxrilSi # 2. cxrili # 1 xelsawyos sworad wakiTxvis saSualo raodenobebi, rogorc xelsawyos tipisa da Zilis deprivaciis raodenobis funqcia.
xelsawyos tipi (A)
analoguri (A1) cifruli (A2) arc erTi (B1) Zilis deprivaciis raodenoba (B) 24 saaTi (B2)
89.0
91.0
72.0
81.0
cxrili # 2 dispersiuli analizis Sejameba xelsawyos sworad wakiTxvis saSualo raodenobisaTvis cxrilidan # 1. wyaro SS df MS F xelsawyos tipi (A) Zilis deprivaciis raodenoba (B) AxB Secdoma sul
211.750 1275.750 85.750 433.920 2007.170
1 1 1 24 27
211.750 1275.750 85.750 18.080
11.71* 70.56 4.74**
* p < .05 ** p < .01
tukis HSD kriteriumis gamoyenebiT gamoTvlili CD martivi efeqtebisaTvis udris 5.7 swor wakiTxvas.
a. rogoria xelsawyos tipis efeqti swori wakiTxvis raodenobaze? b. rogoria Zilis deprivaciis raodenobis efeqti sworad wakiTxvis
raodenobaze?
1
Tavi XII. cdis pirTa Sida erTfaqtoriani sqema da dispersiuli analizi cdis pirTa Sida erTfaqtoriani sqemis magaliTi cdis pirTa Sida erTfaqtoriani sqemis dispersiuli analizi monacemTa dayofa da kvadratebis jamebis gamoTvla
Tavisuflebis xarisxebis gamoTvla gadaxrebis kvadratebis saSualos da F statistikis gamoTvla
statistikuri hipoTezis Semowmeba Frealuris meSveobiT Carevis efeqtis siZlieris gazomva cdis pirTa Sida dispersiuli analizis daSvebebi
dispersiuli analizis Sedegebis prezentacia kvlevis sqemis arCeva: cdis pirebs Soris Tu cdis pirebs Sida cdis pirebis saWiro raodenoba damoukidebeli cvladis efeqturoba mravaljeradi Carevis efeqtebi
2
me-9 TavSi warmodgenili iyo cdis pirTa Sida sqemebi da t kriteriumi damokidebuli qulebisTvis. am TavSi grZeldeba cdis pirTa Sida sqemebis ganxilva da warmodgenilia maTi dispersiuli analizi. cdis pirTa Sida sqema: kvlevis sqema, rodesac cdis pirTa erTi jgufi ganicdis damoukidebeli cvladis zemoqmedebas, rac izomeba am cvladis yoveli donis Sesabamisad.
cdis pirTa Sida erTfaqtorian sqemaSi cdis pirTa erTi jgufi ganicdis damoukidebeli cvladis zemoqmedebas, rac izomeba am cvladis yoveli donis Sesabamisad. radgan cdis pirTa Sida sqemaSi cdis pirebi ganmeorebiT izomebian, am sqemas aseve uwodeben ganmeorebiTi gazomvis sqemas an “Careva cdis pirebze” sqemas, rac imaze migviTiTebs, rom TiToeuli cdis piri damoukidebeli cvladis yvela donis mixedviT mowmdeba. cxrilSi # 12.1 sqematuradaa warmodgenili cdis pirTa Sida erTfaqtoriani sqemis monacemebi damoukidebeli cvladis oTxi doniT da xuTi cdis piris monawileobiT. damoukidebeli cvladi gansazRvrulia, rogorc A faqtori A1, A2, A3 da A4 doneebiT. A faqtoris doneebis raodenoba aRniSnulia a-Ti; cxrilisaTvis # 12.1 a = 4. cdis piris monacemi (qula) aRniSnulia Xij-T, sadac indeqsi i migviTiTebs cdis piris aRmniSvnel ricxvs (mag., X1j, X2j, . . . X5j) da indeqsi j migviTiTebs ricxvs, romelic aRniSnavs damoukidebeli cvladis dones. amdenad, X34 aRniSnavs me-3 cdis piris qulas A faqtoris me-4 doneze. rodesac ar aris saWiro konkretuli cdis piris identificireba, gamovtovebT ij indeqsebs da ubralod X-s viyenebT. damoukidebeli cvladis TiToeuli donis
saSualoebi aRiniSneba 1AX , 2AX , 3AX da 4AX an zogadad AX . qulebis raodenoba damoukidebeli cvladis donisaTvis aRiniSneba nA. miaqcieT yuradReba, rom nA aseve aris cdis pirTa raodenoba cdis pirTa Soris sqemaSic. qulebis sruli raodenoba eqsperimentSi aRniSnulia N-iT. cxrilisaTvis # 12.1, nA = 5 da N = 20. cdis piris saSualo A faqtoris yvela doneze aRiniSneba SX . es saSualo gamosaxavs cdis piris tipur Sesrulebas damoukidebeli cvladis yvela doneze.
cdis pirTa Sida erTfaqtoriani sqemis magaliTi cxrili # 12.1 qulebis sqematuri gamosaxuleba cdis pirTa Sida erTfaqtorian sqemaSi damoukidebeli cvladis oTxi doniTa da xuTi cdis piriT. cdis piri A faqtori
________________________________________________ A1 A2 A3 A4
cdis piris saSualo
S1 S2 S3 S4 S5
X11 X21 X31 X41 X51
X12 X22 X32 X42 X52
X13 X23 X33 X43 X53
X14 X24 X34 X44 X54
1SX
2SX
3SX
4SX
5SX
A faqtoris saSualoebi
1AX 2AX 3AX 4AX GX
3
SexedeT suraTs # 12.1. miuxedavad imisa, rom aq mxolod nawilobriv Sevsebuli wreebia warmodgenili, suraTze isini TiTqos kideebs qmnian, romelTa Sorisac samkuTxedia moTavsebuli. am kideebs subieqturi konturebi ewodeba, radganac isini TavisTavad stimulSi ar aris mocemuli. SesaZlebelia Tu ara subieqturi konturebis gamoyeneba aRqmis iluziis gamosawvevad? davuSvaT, gadavwyviteT am sakiTxis gamorkveva poncos (Ponzo) iluziis ramdenime formis gamoyenebiT, rogorc es maierma gaakeTa (Meyer, 1986). suraTze # 12.2a naCvenebia poncos standartuli iluzia. es suraTi iluziaa, radganac ori horizontaluri xazi fizikurad erTi da imave sigrZisaa, Tumca isini tolad ar gamoiyureba. zeda xazi ufro grZeli Cans, vidre qveda. suraTze # 12.2b warmodgenilia am iluziis versia subieqturi konturebiT. versia suraTze # 12.3g gvawvdis saorientacio informacias, magram mTliani wertilebi ar iZleva subieqtur konturebs. dabolos, suraTze # 12.2d mocemulia sakontrolo piroba, sadac iluzia ar unda iyos. davuSvaT, cdis pirs sTxoven, ise Secvalos qveda horizontaluri xazis sigrZe, rom is zeda xazis tolad aRiqmebodes, Semdeg ki davuSvaT, rom zeda xazis sigrZe 14 santimetria (sm). Tu iluzia ar Cndeba, maSin qveda xazi aseve 14 sm unda gaxdes. xolo Tu iluzia Cndeba, maSin zeda xazi ufro grZeli unda aRmoCndes, vidre es sinamdvileSia (anu is 14 sm-ze grZeli unda aRmoCndes) da qveda xazi ise unda Seicvalos, rom aRqmuli sigrZis Sesabamisi iyos. gamotovebulia suraTi 12.1 cdis pirTa Sida erTfaqtoriani eqsperimenti iluziis tipis (A faqtori) oTxi doniTa da xuTi cdis piriT gamoiyeneba imisaTvis, raTa Semowmdes iluziis pirobis efeqti iluziis aRqmul sidideze. yoveli cdis piri aTvalierebs oTxive stimuls, romlebic naCvenebia suraTze # 12.2 da cvlis qveda horizontalur xazs ise, rom is sigrZeSi zeda xazis tolad gamoiyurebodes. santimetrebSi gamosaxuli qveda xazis Secvlili sigrZe aris damokidebuli cvladi. davuSvaT, miiReT monacemebi, romlebic mocemulia cxrilSi # 12.2. imoqmeda Tu ara aRqmuli iluziis sidideze iluziis pirobam? cxrili # 12.2 im xuTi cdis pirisagan miRebuli xazebis aRqmuli toli sigrZeebi (santimetrebSi), romelTac poncos iluziis oTxi versia naxes. cdis piri iluziis piroba (A)
________________________________________________ standartuli subieqturi mTliani sakontrolo konturebi wertilebi (A1 ) (A2 ) (A3) (A4)
cdis piris saSualo
1 2 3 4 5
16.6 16.9 17.1 17.2 16.8
15.8 15.6 16.1 16.3 15.7
14.7 15.1 14.8 15.2 15.3
14.4 13.8 13.9 14.1 14.3
15.38 15.35 15.48 15.70 15.53
AX sA
16.92 0.24
15.90 0.29
15.02 0.26
14.10 0.25
4
iseve, rogorc cdis pirTa Soris sqemebSi, cxrilSi # 12.2 moyvanili monacemebis analizi aRweriTi statistikebis gamoTvliT iwyeba. amdenad, pirobebis saSualoebi da standartuli gadaxrebi naCvenebia cxrilis yoveli svetis boloSi. Tu damoukidebeli cvladi gavlenas axdens iluziis aRqmaze, maSin pirobebis saSualoebi gansxvavebuli unda iyos. magram, rogorc Cven viciT, pirobebis saSualoebs Soris miRebuli nebismieri sxvaoba yovelTvis unda SevadaroT Secdomis cvalebadobas. ramdenad aris SesaZlebeli, rom saSualoebis gansxvaveba mxolod SerCevis Secdomis gamo momxdariyo? saSualoebis ubralo mxedvelobiTi Semowmeba ar gvaZlevs am kiTxvaze pasuxis gacemis SesaZleblobas. rogorc es cdis pirTa Soris sqemisaTvis gavakeTeT, am gadawyvetilebis miRebis problemas statistikuri kriteriumis gamoyenebiT vwyvetT. es aris cdis pirTa Sida erTfaqtoriani dispersiuli analizi.
cdis pirTa Sida erTfaqtoriani sqemis dispersiuli analizi cdis pirTa Sida erTfaqtoriani dispersiuli analizi gamoiyeneba erTfaqtoriani mravaldoniani cdis pirTa Sida sqemebis gaanalizebisaTvis. Tu damoukidebeli cvladis mxolod ori done icvleba, cdis pirTa Soris erTfaqtoriani dispersiuli analizi SeiZleba tdamokidebulis magivrad gamoviyenoT. es analizi eqsperimentSi qulebis srul cvalebadobas sam wyarod yofs:
1. A faqtoris efeqtiT gamowveuli cvalebadoba; 2. cdis pirTa Soris individualuri gansxvavebebis, S faqtoris
efeqtiT gamowveuli cvalebadoba; 3. pirobebisa da cdis pirebis urTierTqmedebiT (AxS) gamowveuli
cvalebadoba. es sami dayofa gamoiyeneba sami kvadratebis saSualos misaRebad:
1. MSA, romelic A faqtoris efeqtTan da Secdomis cvalebadobasTan erTad icvleba;
2. MSS, romelic cdis pirebs Soris gansxvavebebTan erTad icvleba; 3. MSAxS, romelic pirobebis cdis pirebTan urTierTqmedebasTan
erTad icvleba da eqsperimentSi Secdomis cvalebadobis zomas iZleva.
F statistika, romelic statistikuri hipoTezis Semowmebisas gamoiyeneba, aris
F = AxS
A
MSMS
monacemis dayofa da kvadratebis jamebis miReba am seqciaSi ganvixilavT monacemebis dayofas da kvadratebis jamebis miRebas cdis pirTa Sida sqemebisaTvis. ramdenadac aq gamoyenebuli midgoma Zalian hgavs imas, romelic gamoiyeneba cdis pirTa Soris sqemebSi, dayofis wardgena mokle iqneba.
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imis gasagebad, Tu rogor unda daiyos cvalebadobis wyaroebi cdis pirTa Sida sqemebSi, cdis pirTa Sida sqema warmovidginoT, rogorc orfaqtoriani sqema, sadac damoukidebeli cvladi erTi faqtoria (A faqtori), da cdis pirebi ki - meore faqtori (S faqtori, sadac S warmoadgens cdis pirs). radganac TiToeuli cdis piri damoukidebeli cvladis yovel doneze mowmdeba, cdis pirTa Sida erTfaqtoriani sqema SeiZleba ganvixiloT, rogorc AxS faqtoruli sqema A faqtoris a doneebiT da S faqtoris nA doneebiT. ase rom, miviRebT a x nA pirobebs anu ujredebs, da TiToeuli ujredi warmoadgens, konkretul cdis pirTan kombinirebul damoukidebeli cvladis erT dones. cxrilSi # 12.2 A faqtoris oTxi done da xuTi cdis piria; amdenad, eqsperimenti SeiZleba ganvixiloT, rogorc 4 (iluziis pirobis doneebi) x 5 (cdis pirebis raodenoba) faqtoruli sqema, sadac sul 20 gansxvavebuli Careva/cdis piri (AxS) pirobaa. magram TiToeuli AxS pirobisaTvis mxolod erTi qula miiReba. cdis pirTa Sida erTfaqtoriani dispersiuli analizis ganxilva, rogorc orfaqtoriani dispersiuli analizisa, A da S faqtorebisaTvis "mTavari efeqtebis" saSualoebs gvaZlevs. cxrilSi # 12.2 miaqcieT yuradReba, rom saSualoebi mocemulia TiToeuli cdis pirisaTvis, iseve, rogorc damoukidebeli cvladis TiToeuli donisaTvis. TiToeuli cdis piris saSualo aris erTi cdis piris qulebis saSualo iluziis yvela pirobisaTvis. es saSualoebi asaxavs iluziis aRqmis individualur gansxvavebebs eqsperimentSi monawile cdis pirebs Soris. miuxedavad imisa, rom cdis pirebis saSualoebi dispersiuli analizis relevanturia, es saSualoebi, Cveulebriv, ar aris xolme warmodgenili im eqsperimentebis aRweriT statistikaSi, romlebSic cdis pirTa Sida sqema gamoiyeneba. cdis pirTa Sida erTfaqtoriani dispersiuli analizis ganxilva, rogorc orfaqtoriani analizisa, ar cvlis eqsperimentis arss: icvleba mxolod erTi damoukidebeli cvladi da xdeba misi efeqtebis analizi. Tumca, analizis orfaqtorian sqemad warmodgena gvexmareba, gavigoT AxS urTierTqmedebis Secdomis nawilad SemoReba F statistikisaTvis. F statistikis misaRebad cdis pirTa Sida dispersiul analizSi iReben srul raodenobas, romliTac qula gansxvavdeba sruli saSualosagan da am gansxvavebas sam komponentad yofen: = + + . (12.1) dispersiuli analizis ZiriTadi midgomis mixedviT, dayofili qulebis komponentebi ayvanilia kvadratSi da jamdeba yvela cdis piris da Carevis yvela pirobis mixedviT, raTa kvadratebis jami miviRoT: SSsruli = SSA + SSS + SSAxS (12.2)
sruli cvalebadoba cdis piris qulaSi
cvalebadoba A faqtoris gamo plius Secdoma A da S faqtorebis urTierTqme-debis gamo
cvalebadoba cdis pirTa Soris indi-vidualuri gansxvavebe-bis gamo (S faqtori)
cvalebadoba A da S faqtorebis urTierTqme-debis gamo
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es kvadratebis jamebi migviTiTebs, rom eqsperimentSi qulebis sruli cvalebadoba (aRniSnuli SSsruli) aris im cvalebadobis Sedegi, romelic xdeba A faqtoris (aRniSnuli SSA), S faqtoris (aRniSnuli SSS) da A da S faqtorebis urTierTqmedebis gamo. cxrilSi # 12.2 mocemuli qulebisaTvis es procesi Semdeg ricxviT mniSvnelobebs gvaZlevs: SSsruli = 22.926, SSA = 21.830, SSS = 0.314, SSAxS = 0.782. am mniSvnelobebis misaRebad gamyenebulia gamoTvliTi formulebi am Tavis damatebidan. ganvixiloT 12.2 tolobaSi mocemul TiToeuli kvadratebis jami. SSsruli SSsruli warmoadgens eqsperimentSi qulebis srul cvalebadobas. rogorc 12.2 tolobaSia aRniSnuli, am cvalebadobis nawili damoukidebeli cvladis efeqtis, nawili - cdis pirebs Soris individualuri gansxvavebebis, xolo nawili ki - pirobebis da cdis pirTa urTierTqmedebis Sedegia. SSA SSA icvleba damoukidebeli cvladis efeqts da pirobebsa da cdis pirTa urTierTqmedebasTan erTad. rogorc cdis pirTa Soris erTfaqtorian sqemaSi, damoukidebeli cvladis efeqti zrdis SSA -is mniSvnelobas. garda amisa, SSA reagirebs pirobebisa da cdis pirTa nebismier urTierTqmedebaze. cdis pirTa Sida sqemaSi pirobebisa da cdis pirTa urTierTqmedeba ganixileba Secdomis cvalebadobad. SSS SSS icvleba eqsperimentSi cdis pirTa Soris individualur gansxvavebebTan erTad. cdis pirTa Soris am gansxvavebebs aRvniSnavT S faqtoriT. cxrilSi # 12.2 miaqcieT yuradReba, rom me-4 cdis pirs aqvs tendencia, iluziis yvelaze meti raodenoba dainaxos damoukidebeli
cvladis yvela doneze ( 4SX = 15.70 sm), maSin, rodesac me-2 cdis piri
yvelaze nakleb iluzias aRiqvams ( 2SX = 15.35 sm). rac ufro metad gansxvavdeba eqsperimentSi cdis piris saSualoebi erTmaneTisagan, miT ufro didi xdeba SSS. SSAxS pirobebis urTierTqmedeba cdis pirebTan qulis darCenili cvalebadobaa da gavlenas axdens SSAxS-ze. pirobebisa da cdis pirebis es urTierTqmedeba SeiZleba vnaxoT suraTze # 12.3, romelic cxrilis # 12.2 hipoTetur monacemebs gamoxatavs. abscisaze iluziis oTxi tipia warmodgenili. suraTze moTavsebuli funqciebi gamoxataven xuTi cdis pirisagan TiToeulis miRebul oTx qulas. xazis toli aRqmuli sigrZe yvelaze didia yvela cdis pirisaTvis standartuli iluziis pirobaSi. amdenad, aSkaraa iluziis pirobis efeqti xazis aRqmul sigrZeze. Tumca, radganac xuTi funqcia paraleluri ar aris, xdeba pirobebisa da cdis
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pirebis urTierTqmedebac. xazis aRqmul sigrZes Soris sxvaobebi iluziis pirobebs Soris ar aris erTi da igive yvela cdis pirisaTvis. magaliTad, pirveli cdis piris SemTxvevaSi qulebis sxvaoba standartul pirobasa da subieqturi konturis pirobas Soris aris 16.6 - 15.8 = 0.8 sm, xolo meore cdis piris SemTxvevaSi sxvaoba aris 16.9 - 15.6 = 1.3 sm. ase rom, iluziis pirobis efeqti xazis aRqmul sigrZeze cdis pirebzea damokidebuli. SSAxS zomavs pirobebisa da cdis pirebis urTierTqmedebis xarisxs cdis pirTa Sida eqsperimentSi. magram, radganac Carevis TiToeul pirobaSi erTi cdis pirisagan mxolod erT qulas viRebT, pirobebisa da cdis pirebis urTierTqmedebas ver gamovacalkevebT nebismieri Secdomisagan, romelic SeiZleba gavlenas axdendes qulaze. Sesabamisad, SSAxS ganixileba, rogorc Secdomis cvalebadobis sazomi cdis pirTa Sida erTfaqtorian sqemaSi da warmoadgens Secdomis nawilis safuZvels F statistikisaTvis am analizSi. pirobebisa da cdis pirTa urTierTqmedeba (AxS): situacia cdis pirTa Sida erTfaqtorian sqemaSi, rodesac damoukidebeli cvladis efeqti damokidebulia cdis pirze.
Tavisuflebis xarisxebis gamoTvla F statistikisaTvis saWiro gadaxrebis kvadratebis saSualos misaRebad TiToeuli SS iyofa mis Sesabamis Tavisuflebis xarisxze. Tavisuflebis xarisxebis gamoTvlis procesi Zalian hgavs cdis pirTa Soris sqemaSi gamoyenebul gamoTvlas.
Tavisuflebis sruli xarisxi dfsruli = N - 1, sadac N aris qulebis mTeli raodenoba eqsperimentSi. magaliTad, N = 20; amitom dfsruli = 20 - 1 = 19. Tavisuflebis xarisxi SSA-sTvis dfA = a - 1, sadac a aris A damoukidebeli cvladis doneebis raodenoba. Cvens magaliTSi A faqtoris oTxi donisaTvis a = 4; aqedan gamomdinare, dfA = 4 - 1 = 3.
Tavisuflebis xarisxi SSS-sTvis dfS = nA - 1 Cvens magaliTSi xuTi cdis piria; Sesabamisad, nA = 5 da dfS = 5 - 1 = 4.
Tavisuflebis xarisxi SSAxS-isTvis dfAxS = (a - 1)( nA - 1)
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Cvens magaliTSi a = 4 da nA = 5; Sesabamisad, dfAxS = (4 - 1)( 5 - 1) = 12.
Tavisuflebis xarisxis aditiuroba Tavisuflebis xarisxebi aditiuria, ase rom dfsruli = dfA + dfS + dfAxS es mimarTeba Cvens magaliTSic sruldeba, radgan dfsruli = 19, dfA = 3, dfS = 4, da dfAxS = 12. gadaxrebis kvadratebis saSualoebisa da F statistikis gamoTvla kvadratebis sami saSualo miiReba SS-idan 12.2 tolobaSi TiToeuli SS-is mis df-ze gayofiT:
MSA = A
A
dfSS
,
MSS = S
S
dfSS
,
MSAxS = AxS
AxS
dfSS
.
MSsruli ar gamoiTvleba, radganac is ar gamoiyeneba dispersiul analizSi. cxrilis # 12.2 qulebisaTvis SSA = 21.830, SSS = 0.314, SSAxS = 0.782. am kvadratebis saSualoebisaTvis Tavisuflebis xarisxebia dfA = 3, dfS = 4, dfAxS = 12. aqedan gamomdinare,
MSA = 3830.21
= 7.277,
MSS = 4314.0
= 0.078,
MSAxS = 12782.0
= 0.065.
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F statistika cdis pirTa Sida erTfaqtorian dispersiul analizSi aris
F = AxS
A
MSMS .
ase rom, cxrilis # 12.1 qulebisaTvis
Frealuri = 065.0277.7
= 111.95.
es Frealuri aris am analizSi gamosaTvleli erTaderTi F; S faqtorisaTvis F ar gamoiTvleba. cxrilis # 12.2 qulebis dispersiuli analizis ricxviTi Sejameba warmodgenilia cxrilSi # 12.3. gamoTvliTi formulebi am analizisaTvis mocemulia Tavis damatebaSi. cxrili # 12.4 ajamebs cdis pirTa Sida erTfaqtorian dispersiul analizSi cvalebadobis wyaroebs - SS, df, MS da F. cxrili # 12.3 dispersiuli analizis cxrili xazebis hipoTeturi Tanabari aRqmuli sigrZis qulebisaTvis cxrilidan # 12.2. wyaro SS df MS F iluziis tipi (A) cdis pirebi (S) AxS sul
21.830 0.314 0.782 22.926
3 4 12 19
7.277 0.078 0.065
111.95*
* p < .01
cxrili # 12.4 formulebis Sejameba cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis. wyaro SS dfa MS F A faqtori S faqtori AxS sul
SSA SSS SSAxS SSsruli
a - 1 nA - 1 (a - 1)( nA - 1) N - 1
A
A
dfSS
S
S
dfSS
AxS
AxS
dfSS
ar gamoiTvleba
AxS
A
MSMS
aa = A faqtoris doneebis raodenoba; nA = qulebis raodenoba Carevis pirobaSi anu, Sesabamisad, cdis pirebis raodenoba; N - qulebis sruli raodenoba.
SeamowmeT Tqveni codna # 12.1
1. gansazRvreT: AxS urTierTqmedeba, dfA, dfAxS, dfS, dfsruli, A faqtori, S faqtori, MSA, MSAxS, MSS, MSsruli, nA, cdis pirTa Sida erTfaqtoriani sqema, ganmeorebiTi gazomvebis sqema, SSA, SSAxS, SSS, SSsruli, pirobebisa da cdis pirebis urTierTqmedeba.
2. gansazRvreT qulebis cvalebadobis wyaroebi cdis pirTa Sida erTfaqtorian sqemaSi.
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3. daasruleT Semdegi gantolebebi cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis:
a. SSsruli = b. dfA = g. dfS = d. dfAxS = e. MSA = v. MSS = z. MSAxS = T. F =
4. ra aris SSsrulis mniSvneloba, Tu SSA = 74.00, SSS = 29.00 da SSAxS = 96.00? 5. ra aris SSA-is mniSvneloba, Tu SSsruli = 426.00, SSS = 92.00 da SSAxS = 301.00? 6. ra aris SSS -is mniSvneloba, Tu SSsruli = 96.00, SSA = 15.00 da SSAxS = 56.00? 7. ra aris SSAxS -is mniSvneloba, Tu SSsruli = 173.00, SSA = 41.00 da SSS = 36.00? 8. erTma fsiqologma gamoiyena cdis pirTa Sida erTfaqtoriani dispersiuli
analizi A faqtoris xuTi doniT da 13 cdis piriT. ra aris dfA, dfS, dfAxS da dfsrulis mniSvnelobebi?
9. ra aris dfsrulis mniSvneloba cdis pirTa Sida erTfaqtorian dispersiul analizSi, Tu dfA = 2 da dfS = 19?
10. ipoveT gadaxrebis kvadratebis saSualoebis ricxviTi mniSvnelobebi Semdegi SS-isa da df-isaTvis: SSA = 22.00, SSS = 18.00, SSAxS = 36.00, dfA = 2, dfS = 9 da dfAxS = 18.
11. cxrilebi # 1 da 2 warmoadgens arasrul Sejamebis cxrilebs cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis. TiToeul cxrilSi SeavseT gamotovebuli mniSvnelobebi cxrilSi # 12.4 SS-sa da df-s Soris mocemuli mimarTebis gamoyenebiT. Semdeg upasuxeT am kiTxvebs yoveli cxrilisaTvis.
a. damoukidebeli cvladis ramdeni done icvleba? b. ramdeni qulaa miRebuli TiToeul pirobaSi? g. ramdeni cdis piri iRebda monawileobas kvlevaSi?
cxrili # 1 wyaro SS df MS F A faqtori S faqtori AxS sul
48.00 27.00 72.00 -------
4 9 ------- -------
------- ------- -------
-------
cxrili # 2 wyaro SS df MS F A faqtori S faqtori AxS sul
6.00 36.00 ------- 78.00
------- 12 24 -------
------- 3.00 -------
-------
statistikuri hipoTezis Semowmeba Frealuris meSveobiT Frealuri = MSA/MSAxS gamoiyeneba imis gadasawyvetad, mniSvnelovnad gansxvavdeba Tu ara Carevis pirobebis saSualoebi. rogorc SSAxS-is ganxilvisas vnaxeT, F-is mniSvneli, MSAxS, am analizSi Secdomis cvalebadobas warmoadgens. F-is mricxveli, MSA, icvleba damoukidebeli cvladis sistematur efeqtTan erTad da pirobebisa da cdis pirebis urTierTqmedebasTan erTad. Tu damoukidebel cvlads ar aqvs efeqti da pirobebis saSualoebis cvalebadoba mxolod pirobebisa da cdis pirebis urTierTqmedebis gamo xdeba, maSin MSA-is mniSvneloba daaxloebiT igive
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unda iyos, rac MSAxS-is mniSvneloba da Frealuri unda iyos daaxloebiT 1.00. xolo, Tu A faqtors aqvs efeqti da pirobebis saSualoebi gansxvavdeba rogorc am efeqtis, aseve pirobebisa da cdis pirebis urTierTqmedebis gamo, maSin MSA-s ufro didi mniSvneloba unda hqondes, vidre MSAxS-s, da Frealuri 1.00-ze meti xdeba. F Sefardeba MSS-is monawileobiT, romelic individualur gansxvavebebis kvadratis saSualos aRniSnavs, ar gamoiTvleba. miuxedavad imisa, rom MSS icvleba eqsperimentSi cdis pirebs Soris gansxvavebebis xarisxTan erTad, am analizSi ar gamoiTvleba MS, romelic iZleva individebs Soris mosalodneli Secdomis cvalebadobis Sefasebas. aqedan gamomdinare, MSS-is mniSvneloba xSirad ar aris warmodgenili saboloo cxrilSi. sami kvadratebis saSualo da maTze gavlenis mqone cvalebadobis wyaroebi SeiZleba Semdegnairad Sejamdes: kvadratebis saSualo masze gavlenas axdens:
MSA A faqtoriT gamowveuli sistematuri cvalebadoba
pirobebisa da cdis pirebis urTierTqmedeba (anu Secdomis cvalebadoba)
MSS S faqtoriT gamowveuli sistematuri cvalebadoba
MSAxS pirobebisa da cdis pirebis urTierTqmedeba (anu Secdomis cvalebadoba)
statistikuri Semowmeba F-is gamoyenebiT ukve nacnob proceduras warmoadgens:
• yalibdeba nulovani hipoTeza, H0, da alternatiuli hipoTeza, H1;
• im daSvebiT, rom H0 sworia, gamoiTvleba F-is SerCevis ganawileba. es ganawileba mocemulia cxrilSi # A-3;
• xdeba mniSvnelovnobis donis SerCeva;
• F-is kritikuli done, romelic aRiniSneba FkritikuliT, gamoiTvleba F–is SerCeviTi ganawilebidan, romelic mocemulia cxrilSi # A-3;
• F–is SerCeviT ganawilebaSi gamoiyofa uaryofis are;
• SerCevis monacemebidan gamoiTvleba F statistika, romelic aRiniSneba, rogorc Frealuri;
• gadawyvetileba imis Taobaze, uarvyoT Tu ara H0, miiReba imaze dayrdnobiT, xvdeba Tu ara Frealuri uaryofis areSi.
statistikuri hipoTezebi nulovani hipoTeza. nulovani hipoTeza cdis pirTa Sida sqemisaTvis A faqtoris oTxi doniT ase gamoixateba:
H0: µ A1 = µ A2 = µ A3 = µ A3
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es nulovani hipoTeza asaxavs situacias, romelic Cndeba, Tu damoukidebel cvlads ar aqvs efeqti. Tu H0 marTldeba, maSin pirobebis saSualoebs Soris miRebuli sxvaobebi ubralod Secdomis cvalebadobis gamoa. rogorc cdis pirTa Soris sqemaSi, nulovan hipoTezaSi gansazRvruli populaciis saSualoebis raodenoba yovelTvis Seesabameba damoukidebeli cvladis doneebis raodenobas. alternatiuli hipoTeza. alternatiuli hipoTezaa:
H1: µ A-ebi ar aris toli. hipoTezaSi monawile populaciis saSualoebis raodenobis miuxedavad. alternatiuli hipoTeza asaxavs situacias, romelic Cndeba, Tu damoukidebel cvlads aqvs efeqti. F statistikaze damyarebuli gadawyvetilebebis miReba cdis pirTa Sida erTfaqtorian sqemaSi statistikuri gadawyvetilebebis miRebis procesi cdis pirTa Soris erTfaqtorian dispersiul analizSi
gamoyenebuli procesis identuria. α–s mniSvneloba, romelic gansazRvravs uaryofis ares FrealurisaTvis, analizis Catarebamde SeirCeva. Semdeg cxrilSi # A-3a an b moiZebneba Fkritikulis mniSvneloba Sesabamisi mricxveli da mniSvneli df-iT. Tu Frealuri udris an metia Fkritikulze, maSin Frealuri xvdeba uaryofis areSi. nulovani hipoTeza uaryofilia da miRebulia alternatiuli hipoTeza. arsebobs sul mcire erTi statistikurad mniSvnelovani sxvaoba pirobebis saSualoebs Soris. Tu Frealuri naklebia, vidre Fkritikuli, maSin Frealuri ar xvdeba uaryofis areSi; nulovani hipoTeza ar aris uaryofili da alternatiuli hipoTeza ar miiReba. pirobebis saSualoebi mniSvnelovnad ar gansxvavdeba; saSualoebs Soris miRebuli sxvaobebi Secdomis cvalebadobis gamo xdeba. vaskvniT, rom damoukidebel cvlads ar hqonda efeqti. Frealurze damyarebuli gadawyvetilebebi da Semdgomi daskvnebi Sejamebulia cxrilSi # 12.5. cxrili # 12.5 gadawyvetilebebis da daskvnebis Sejameba statitikuri hipoTezebis SemowmebaSi cdis pirTa Sida erTfaqtoriani sqemisaTvis dispersiuli analizis gamoyenebiT. mocemulia mniSvnelovnobis done .05.
Tu Frealuri xvdeba uaryofis areSi α = .05-isaTvis, maSin:
Tu Frealuri ar xvdeba uaryofis areSi α = .05-isaTvis, maSin:
Frealuris albaToba naklebia, an tolia
.05-is, anu p≤ .05.
H0 uaryofilia.
H1 miRebulia. SerCevis yvela saSualo ar aris erTi da imave populaciidan. pirobebis saSuloebs Soris sul mcire erTi sxvaoba statistikurad mniSvnelovania .05 doneze. gadawyvetilia, rom SerCevis Secdomis garda kidev raRac sxva ganapirobebs sxvaobas SerCevis saSualoebs Soris. yvela pirobis dacviT Catarebul eqsperimentSi sxvaoba damoukidebel
Frealuris albaToba metia .05-ze, anu p>.05.
H0 ar aris uaryofili.
H1 ar aris miRebuli. SerCevis saSualoebs Soris ar aris mniSvnelovani sxvaobebi .05 doneze. gadawyvetilia, rom SerCevis Secdoma SerCevis saSualoebs Soris sxvaobis yvelaze metad savaraudo axsnaa. ar arsebobs imis safuZveli, rom damoukidebel
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cvlads miewereba. Tu damoukidebel cvlads sami an meti done aqvs, saWiroa mravlobiTi Sedarebis kriteriumebi.
cvlads efeqti hqonda. mravlobiTi Sedarebis kriteriumebi ar aris saWiro.
gadawyvetilebis procesis ilustraciisaTvis gamoviyenebT cxrilSi # 12.3 Sejamebul analizs cxrilis # 12.2 sanimuSo qulebisaTvis. am analizSi Frealurs aqvs 3 (mricxveli) da 12 (mniSvneli) df. amdenad,
rodesac α = .05, Fkritikuli = 3.49 (miRebulia cxrilidan A-3a.) Frealuri = 111.95 metia Fkritikulze da xvdeba uaryofis areSi. uarvyofT
H0: µ A1 = µ A2 = µ A3 = µ A3 da viRebT
H1: µ A-ebi ar aris toli, es gadawyvetileba migviTiTebs, rom arsebobs sul mcire erTi statistikurad mniSvnelovani sxvaoba cxrilis # 12.2 oTx saSualos Soris, magram is ar aRniSnavs konkretulad, Tu romeli saSualoebi gansxvavdeba erTmaneTisagan. imis dasadgenad, Tu romeli wyvilTaSorisi Sedarebebi gansxvavdeba mniSvnelovnad, gamoiyeneba mravlobiTi Sedarebis kriteriumebi. mravlobiTi Sedarebis kriteriumebi cdis pirTa Sida dispersiuli analizisaTvis mravlobiTi Sedarebis kriteriumebTan dakavSirebuli problemebi da sakiTxebi cdis pirTa Sida dispersiuli analizisaTvis igivea, rac ganvixileT me-10 TavSi cdis pirTa Soris dispersiuli analizisaTvis. saboloo SedarebisaTvis SeiZleba gamoyenebul iqnes tukis HSD kriteriumi kritikuli sxvaobiT
CD = qA
AxS
nMS .
Tu am kriteriums Cvens eqsperiments mivusadagebT:
• MSAxS = 0.065 (miRebulia cxrilidan # 12.3)
• nA – 5. • q .05 doneze damoukidebeli cvladis oTxi donisaTvis da 12 df –
MSAxS-saTvis aris 4.20 (miRebulia cxrilidan # A-4). am mniSvnelobebis CasmiT viRebT:
CD = 4.205065.0
= 4.20 013.0 = 0.48 sm.
14
TiToeuli wyvilTaSorisi SedarebisaTvis pirobebis saSualoebs Soris sxvaoba, romelic absoluturi mniSvnelobiT 0.48 sm-ze metia, statistikurad mniSvnelovania .05 doneze. eqvsi wyvilTaSorisi Sedareba, TiToeuli Sedarebis absoluturi mniSvnelobebi, statistikuri hipoTeza da gadawyvetileba TiToeuli hipoTezisaTvis naCvenebia cxrilSi # 12.6 (sadac A1 = standartuli iluzia, A2 = subieqturi konturebis iluzia, A2 = mTliani wertilebis iluzia, A4 = sakontrolo piroba). TiToeuli sxvaobis absoluturi mniSvneloba CD – 0.48 sm-ze metia; amdenad, TiToeuli Sedareba statistikurad mniSvnelovania .05 doneze. standartuli iluzia iluziis mniSvnelovnad nakleb Sedegs qmnis, mTliani wertilebi – mniSvnelovnad kidev ufro nakleb iluzias da sakontrolo piroba ki – yvelaze nakleb iluzias. Carevis efeqtis siZlieris gazomva eta kvadrati efeqtis siZlieris sazomad SeiZleba gamoviyenoT cdis pirTa Sida erTfaqtorian sqemaSi. gamoyenebuli formula mciredaa Secvlili cdis pirTa Soris sqemasTan SedarebiT:
η2 = AxSA
A
SSSSSS+
miaqcieT yuradReba, rom am formulis mixedviT individualuri gansxvavebebiT gamowveuli cvalebadoba, romelic mocemulia SSS-iT, ar aris CarTuli eqsperimentis srul cvalebadobaSi. cxrili # 12.6 tukis HSD CD-s gamoyeneba wyvilTaSorisi SedarebebisaTvis sanimuSo eqsperimentSi. Sedareba Sedarebis
absoluturi mniSvneloba
statistikuri hipoTezebi
gadawyvetileba
1AX 2AX -Tan
(16.92 – 15.90)
1AX 3AX -Tan
(16.92 – 15.02)
1AX 4AX -Tan
(16.92 – 14.10)
2AX 3AX -Tan
(15.90 – 15.02)
2AX 4AX -Tan
(15.90 – 14.10)
3AX 4AX -Tan
(15.02 – 14.10)
1.02 1.90 2.82 0.88 1.80 0.92
H0: µ A1 = µ A2 H1: µ A1 ≠ µ A2
H0: µ A1 = µ A3 H1: µ A1 ≠ µ A3
H0: µ A1 = µ A4 H1: µ A1 ≠ µ A4
H0: µ A2 = µ A3 H1: µ A2 ≠ µ A3
H0: µ A2 = µ A3 H1: µ A2 ≠ µ A3
H0: µ A3 = µ A4 H1: µ A3 ≠ µ A4
uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia
15
Cveni magaliTisaTvis, SSA = 21.830 da SSAxS = 0.782. ase rom,
η2 = 782.0830.21
830.21+
= .97.
xazis sigrZis gazomvis cvalebadobis 97 procenti xdeba iluziis pirobis gamo, mas Semdeg, rac individualuri gansxvavebebis gamo cvalebadoba gauqmebulia. es damoukidebeli cvladis Zalian Zlieri efeqtia. cdis pirTa Sida dispersiuli analizis daSvebebi daSvebebi, romlebsac emyareba cdis pirTa Sida erTfaqtoriani analizis gamoyeneba, Semdegia:
1. TiToeuli cdis piri mowmdeba damoukidebeli cvladis yovel doneze;
2. monacemebi SerCevaSi SemTxveviTad aris aRebuli populaciidan; 3. monacemTa populaciebi Carevis sxvadasxva pirobisaTvis
normaluradaa ganawilebuli; 4. populaciaSi monacemTa dispersiebi Carevis sxvadasxva pirobisaTvis
erTnairia; 5. cdis piris individualuri gansxvavebebis wili misi qulebisaTvis
erTi da igive rCeba Carevis yvela pirobaSi. pirveli daSveba aucilebelia cdis pirTa Sida dispersiul analizSi monacemebis saTanado statistikuri kriteriumis arsebobisaTvis; am daSvebis ugulvebelyofa dauSvebelia. meore, mesame da meoTxe daSvebebi cdis pirTa Soris dispersiul analizSi arsebuli daSvebebis identuria. xandaxan eqsperimentebSi maTi darRveva xdeba, magram dispersiuli analizi inarCunebs garkveul mdgradobas aseTi darRvevebis mimarT. mexuTe daSvebis mixedviT, adamianis qceva, Carevis efeqtisagan damoukideblad, stabiluria damoukidebeli cvladis yvela doneze. mosalodnelia, rom es daSveba dairRveva bihevioristuli mecnierebis kvlevis mraval SemTxvevaSi. aseTi darRvevis gavlena isaa, rom mniSvnelovnobis donis mixedviT dadgenili mniSvnelobis zeviT izrdeba I gvaris Secdomis daSvebis albaToba. Secdomis am gazrdil SesaZleblobasTan gamklavebis ramdenime midgoma arsebobs. gaiser-grinhausis (Geisser-Greenhouse) koreqcia cvlis Fkritikulis misaRebad gamoyenebul df-s ise, rom Frealuris gamosaTvlelad gamoiyeneba ufro didi Fkritikuli. am koreqciiT, Fkritikuli miiReba mricxvelad 1 df-is da mniSvnelad (nA - 1) df -is gamoyenebiT, Cveulebriv gamoyenebuli a – 1, (a - 1)( nA - 1) df-is nacvlad. gaiser-grinhausis koreqciis Cvens magaliTze misadagebiT, Fkritikuli (1.4) = 7.71. Frealuri (3.12) = 111.95 mainc ufro didia, vidre Fkritikuli da H0 uaryofilia – miRebulia igive gadawyvetileba, rac 3 da 12 df-iani Fkritikulis gamoyenebiT. es da sxva midgomebi, romlebic amcirebs Secdomis SesaZleblobas, Tu darRveulia me-5 daSveba, ganxilulia kepelis naSromSi (Keppel, 1991).
16
alternatiuli midgomaa, gamoviyenoT araparametruli uilkoqsonis niSniani rangebis kriteriumi an fridmanis dispersiuli analizi cdis pirTa Sida dispersiuli analizis magivrad. uilkoqsonis kriteriumi warmodgenilia me-15 TavSi. SeamowmeT Tqveni codna XII.2
1. gansazRvreT TiToeul MS-ze gavlenis mqone faqtorebi cdis pirTa Sida erTfaqtorian dispersiul analizSi.
2. axseniT, Tu ratom izrdeba MSA-s mniSvneloba, Tu A faqtors aqvs efeqti cdis pirTa Sida erTfaqtorian eqsperimentSi.
3. dawereT statistikuri hipoTezebi cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis damoukidebeli cvladis xuTi doniT.
4. gansazRvreT daSvebebi, romlebic safuZvlad udevs cdis pirTa Sida erTfaqtoriani dispersiuli analizis gamoyenebas.
magaliTi # 12.1 cdis pirTa Sida erTfaqtoriani sqemis analizi damoukidebeli cvladis oTxi doniT amocana: mravali davalebis Sesruleba moiTxovs adamianebisagan, fxizlad iyvnen drois xangrZlivi periodis ganmavlobaSi. amis magaliTia sahaero moZraobis kontroliori, romelic radaris ekrans akvirdeba, raTa TviTmfrinavebs Soris saWiro distancia SenarCundes. ase, fsiqologebma Seiswavles sifxizleze moqmedi mravali faqtori. magaliTad, pfendlerma da uidelma (Pfendler & Widdel, 1986) Caatares eqsperimenti, sadac cdis pirebi 2,5 saaTis ganmavlobaSi akvirdebodnen ekrans masze gamosaxuli skaliT. rac ufro meti dro gadioda, cdis pirebs ufro didi dro sWirdebodaT ekranze cvlilebebis dasadgenad. dauSviT, rom Catarda analogiuri kvleva, sadac cdis pirebs 4 saaTis ganmavlobaSi ramdenime skala unda ekontrolebinaT. cdis piris davaleba iyo, mieTiTebina, rodis Seicvleboda monacemi skalaze. damokidebuli cvladi zomavda im drois xangrZlivobas, romelic cdis pirs cvlilebis SesamCnevad sWirdeboda. cvlilebis SemCnevis dro (wamebSi) 13 cdis pirisaTvis oTxsaaTiani periodidan TiToeuli saaTisaTvis mocemulia cxrilSi # 12.7. mniSvnelovnad gaizarda Tu ara SemCnevis dro? amoxsna. sqema warmoadgens cdis pirTa Sida erTfaqtorian sqemas damoukidebeli cvladis – drois periodis - oTxi doniT. monacemTa analizis pirveli nabijia cdis qulebis pirTa Sida erTfaqtoriani dispersiuli analizi. Tu Frealuri statistikurad mniSvnelovania, gamoviyenoT tukis HSD kriteriumi, raTa vipovoT statistikurad mniSvnelovani wyvilTaSorisi sxvaobebi pirobebis saSualoebs Soris. saWiro statistika: F = MSA/MSAxS gamosayenebeli daSvebebi:
1. TiToeuli cdis piri mowmdeba drois yovel periodSi. 2. SerCevis qulebi SemTxveviTad aris aRebuli populaciidan.
17
3. SemCnevis droebis populacia gansxvavebuli drois periodebisaTvis normaluradaa ganawilebuli.
4. SemCnevis droebis populaciis dispersiebi gansxvavebuli drois periodebisaTvis erTnairia.
5. cdis piris individualuri gansxvavebebis wvlili igive rCeba misi qulebisaTvis drois yvela periodisaTvis.
cxrili # 12.7 skalaze cvlilebis SesamCnevad saWiro wamebis saSualo raodenoba oTxsaaTiani drois periodisaTvis 13 cdis pirisaTvis. cdis piri drois periodi (saaTebSi) (A)
________________________________________________ 1 (A1 ) 2 (A2 ) 3 (A3) 4 (A4)
1 2 3 4 5 6 7 8 9 10 11 12 13
6.2 5.9 8.4 7.6 4.1 5.4 6.6 6.1 4.9 8.2 5.7 5.9 6.9
6.7 4.8 8.7 7.8 4.7 5.3 6.7 5.8 5.1 8.6 5.7 6.4 6.6
7.4 6.1 9.9 8.7 5.4 5.9 7.2 6.4 5.2 9.3 6.5 6.9 7.1
7.8 6.9 10.3 8.9 6.6 7.1 7.5 6.7 6.8 10.4 7.2 7.6 7.5
AX sA
6.30 1.24
6.38 1.34
7.08 1.45
7.79 1.28
statistikuri hipoTezebi:
H0: µ A1 = µ A2 = µ A3 = µ A4 H1: µ A-ebi ar aris toli,
mniSvnelovnobis done: α = .05. df: dfA = a – 1 = 4 – 1 = 3 dfS = nA – 1 = 13 – 1 = 12 dfAxS = (a – 1) (nA – 1) = (4 – 1) (13 – 1) = 36 dfsruli = N – 1 = 52 – 1 = 51 F-is kritikuli mniSvneloba: Fkritikuli (3.36) = 2.92. df-is mniSvneloba 36, rogorc mniSvneli, ar aris warmodgenili cxrilSi # A-3a. dfSecdomisaTvis gamoviyeneT kritikuli df-is uaxloesi Semdgomi mniSvneloba cxrilSi – 30. uaryofis are: Frealuris mniSvnelobebi tolia an meti 2.92-ze. gamoTvla: Tavis damatebaSi mocemuli formulebis gamoyenebiT gamoviTvaleT dispersiuli analizi, raTa migveRo dispersiuli analizis Semdegi cxrili:
18
wyaro SS df MS F drois periodi (A) cdis pirebi (S) AxS sul
18.985 81.188 3.727 103.900
3 12 36 51
6.328 6.766 0.104
60.85*
* p < .05
gadawyvetileba: Frealuri = 60.85 xvdeba uaryofis areSi; uarvyofT H0: µ A1
= µ A2 = µ A3 = µ A3 da viRebT H1: µ A-ebi ar aris toli. H0–is uaryofa gviCvenebs, rom arsebobs sul mcire erTi statistikurad mniSvnelovani sxvaoba Carevis pirobebis saSualoebs Soris. statistikurad mniSvnelovani wyvilTaSorisi sxvaobebis sapovnelad viyenebT tukis HSD kriteriums. tukis kriteriumi: am kriteriumisaTvis saWiro ricxviTi mniSvnelobebi Semdegia:
• q miiReba cxrilidan # A-4. gvaqvs damoukidebeli cvladis oTxi done da 36 df MSAxS-isaTvis. radganac 36 df ar aris mocemuli cxrilSi # A-4, viyenebT 30df-s, df-is uaxloes Semdgom mniSvnelobas cxrilSi. ase rom, q = 3.85 .05 mniSvnelovnobis donisaTvis;
• dispersiuli analizidan miRebuli MSAxS aris 0.104;
• nA, qulebis raodenoba TiToeul Sesadarebel saSualoSi, aris 13. formulaSi am ricxviTi mniSvnelobebis CasmiT
CD = qA
AxS
nMS = 3.85
13104.0 = 0.34 wami.
nebismieri sxvaoba, romelic absoluturi mniSvnelobiT 0.34 wamis tolia an metia, statistikurad mniSvnelovania .05 doneze. wyvilTaSorisi Sedarebebi, maTi absoluturi mniSvnelobebi, Semowmebuli statistikuri hipoTezebi da statistikur hipoTezebze damyarebuli gadawyvetilebebi TiToeuli SedarebisaTvis mocemulia cxrilSi # 12.8 (sadac A1 = 1 saaTs, A2 = 2 saaTs, A3 = 3 saaTs da A4 = 4saaTs). efeqtis siZliere:
η2 = AxSA
A
SSSSSS+
= 727.3985.18
985.18+
= .84.
cxrili # 12.8 tukis HSD CD-s gamoyeneba oTxi drois periodis wyvilTaSoris SedarebaSi. Sedareba Sedarebis
absoluturi mniSvneloba
statistikuri hipoTezebi
gadawyvetileba
1AX 2AX -Tan
(6.30 – 6.38)
0.08
H0: µ A1 = µ A2 H1: µ A1 ≠ µ A2
ar aris uaryofili ar aris miRebuli
19
1AX 3AX -Tan
(6.30 – 7.08)
1AX 4AX -Tan
(6.30 – 7.79)
2AX 3AX -Tan
(6.38 – 7.08)
2AX 4AX -Tan
(6.38 – 7.79)
3AX 4AX -Tan
(7.08 – 7.79)
0.78 1.49 0.70 1.41 0.71
H0: µ A1 = µ A3 H1: µ A1 ≠ µ A3
H0: µ A1 = µ A4 H1: µ A1 ≠ µ A4
H0: µ A2 = µ A3 H1: µ A2 ≠ µ A3
H0: µ A2 = µ A3 H1: µ A2 ≠ µ A3
H0: µ A3 = µ A4 H1: µ A3 ≠ µ A4
uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia
daskvna: erTaderTi Sedareba, romelic naklebia CD-ze – 0.34 wamze, aris pirvel da meore saaTebs Soris. es sxvaoba aramniSvnelovania. yvela sxva sxvaoba statistikurad mniSvnelovania. ase rom, Sedarebebma gvaCvena, rom SemCnevis droebSi ar aris zrda davalebis Sesrulebis pirvel da meore saaTebs Soris, magram statistikurad mniSvnelovani zrda aRiniSneba me-2
da me-3 saaTebs Soris da me-3 da me-4 saaTebs Soris. efeqtis siZliere, η2 = .84, gviCvenebs, rom drois periodi pasuxs agebs SemCnevis drois cvalebadobis did nawilze mas Semdeg, rac individualuri gansxvavebebis cvalebadoba eqsperimentSi ugulvebelyofilia.
magaliTi # 12.2 cdis pirTa Sida erTfaqtoriani dispersiuli analizis gamoyeneba damoukidebeli cvladis ori doniT amocana: rodesac studenti leqciis dros amTqnarebs, leqtori amas xSirad mowyenis niSnad aRiqvams. magram aris Tu ara mTqnareba namdvilad amis niSani? am hipoTezis gamosakvlevad provinma da hamernikma (Provine & Hamernik, 1986) Caatares eqsperimenti imis Sesamowmeblad, namdvilad zrdis Tu ara mTqnarebis sixSires uintereso masalis swavla. am eqsperimentSi ori sxvadasxva sesiis dros studentebi 30 wuTis ganmavlobaSi uyurebdnen rok-musikis videoklipebs (A1, saintereso stimuli) da 30 wuTis ganmavlobaSi – vertikalur ferad-zolebian ekrans (A2, uintereso stimuli). aRiricxeboda damTqnarebis raodenoba ori sesiis dros. davuSvaT, es eqsperimenti 11 cdis pirze gavimeoreT da aRvricxeT damTqnarebis raodenoba cxrilSi # 12.9. mocemul TiToeul sesiaSi moqmedebs Tu ara videoklipis tipi mTqnarebis raodenobaze? cxrili # 12.9 mTqnarebis miRebuli raodenoba, rogorc nanaxi videoklipis funqcia 11 cdis pirisaTvis.
20
cdis piri videoklipis tipi (A) _______________________________________ saintereso (A1 ) uintereso (A2)
1 2 3 4 5 6 7 8 9 10 11
5 2 4 3 0 4 7 6 3 1 8
7 1 7 8 2 5 6 9 3 4 9
AX sA
3.9 2.47
5.5 2.77
amoxsna. cdis pirTa Sida erTfaqtoriani dispersiuli analizi Sesabamisi statistikuri kriteriumia am monacemebisaTvis. ramdenadac gamoyenebuli iyo mxolod ori Carevis piroba, tdamokidebuli aseve Sesabamisi statistikuri kriteriumi iqneboda. viyenebT .05 mniSvnelovnobis dones. saWiro statistika: F = MSA/MSAxS gamosayenebeli daSvebebi:
1. TiToeuli cdis piri mowmdeba videoklipis orive tipis pirobaSi;
2. SerCevis monacemebi SemTxveviTad aris aRebuli populaciidan; 3. mTqnarebis monacemebis populacia Carevis gansxvavebuli
pirobisaTvis normaluradaa ganawilebuli; 4. mTqnarebis monacemebis dispersiebi populaciaSi Carevis
gansxvavebuli pirobebisaTvis erTnairia; 5. cdis piris individualuri gansxvavebebis wvlili igive rCeba
misi qulebisaTvis videoklipis orive tipis pirobaSi. statistikuri hipoTezebi:
H0: µ A1 = µ A2 H1: µ A-ebi ar aris toli,
mniSvnelovnobis done: α = .05. df: dfA = a – 1 = 2 – 1 = 1 dfS = nA – 1 = 11 – 1 = 10 dfAxS = (a – 1) (nA – 1) = (2 – 1) (11 – 1) = 10 dfsruli = N – 1 = 22 – 1 = 21 F-is kritikuli mniSvneloba: Fkritikuli (1,10) = 4.96. uaryofis are: Frealuris mniSvnelobebi tolia an meti 4.96-ze.
21
gamoTvla: am Tavis damatebaSi mocemuli formulebis gamoyenebiT, am analizis Sejameba aseTia: wyaro SS df MS F videoklipis tipi (A) cdis pirebi (S) AxS sul
14.728 120.364 17,272 152.364
1 10 10 21
14.728 12.036 1.727
8.53*
* p < .05
gadawyvetileba: Frealuri xvdeba uaryofis areSi; amitom uarvyofT H0-s da viRebT H1-s. efeqtis siZliere:
η2 = 272.17728.14
728.14+
= .46.
daskvna: sxvaoba ori Carevis pirobis saSualos Soris – 3.9 da 5.5 damTqnarebas Soris – statistikurad mniSvnelovania. rodesac saintereso videoklipebs uyurebdnen, cdis pirebi naklebad amTqnarebdnen, vidre uintereso videoklipebis yurebisas. radganac mxolod ori saSualoa CarTuli, mravlobiTi Sedarebis kriteriumi ar
aris saWiro. η2 miuTiTebs, rom damokidebuli cvladis dispersiis 46 procentze pasuxs agebs videoklipis tipi, mas Semdeg, rac eqsperimentidan amoRebulia SSS.
dispersiuli analizis Sedegebis prezentacia iluziis pirobasa da xazis aRqmul sigrZeze (ix. cxrilebi # 12.2 da 12.3) moyvanili magaliTis analizis Sedegebis aRwera, romelic misdevs amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils, statiis Sedegebis seqciaSi ase SeiZleba daiweros:
xazis aRqmuli sigrZeebis saSualoebi iluziis standartuli, subieqturi konturebis, mTliani wertilebis da sakontrolo pirobebisaTvis aRmoCnda 16.92 (SD = 0.24), 15.90 (SD = 0.29), 15.02 (SD = 0.26) da 14.10 (SD = 0.25) sm, Sesabamisad. alfas .05 doneze, cdis pirTa Sida erTfaqtorianma dispersiulma analizma gviCvena, rom saSualoebi mniSvnelovnad gansxvavdeba, F (3,12) = 111.95, MSE = 0.065, p < .01. wyvilTaSorisi Sedarebebisas tukis kriteriumma aCvena, rom iluziis TiToeuli piroba mniSvnelovnad gansxvavdeboda iluziis yoveli sxva pirobisagan, p < .05 (CD = 0.48 sm). eta kvadrati qulebisaTvis iyo .97.
am angariSSi informaciis didi nawili cdis pirTa Soris sqemis msgavsia. mokled ganvixiloT es prezentacia.
SerCevis saSualo SerCevis saSualoebis da standartuli gadaxrebis miRebuli mniSvnelobebi warmodgenilia pirvel winadadebaSi.
22
radganac miRebuli iyo mxolod oTxi saSualo, isini warmodgenilia winadadebis da ara cxrilis an suraTis saxiT.
α = .05 miuTiTebs kriteriumisaTvis SerCeul mniSvnelovnobis dones.
F (3,12) aRniSnavs kriteriumis statistikis saxels,
rogorc F-s. frCxilebSi naCvenebi ricxvebi aRniSnavs Frealuris mricxvelis da mniSvnelos df-ebs (Sesabamisad, 3 da 12).
= 111.95 iZleva Frealuris mniSvnelobas. p < .01 aRniSnavs, rom:
a. Frealuris albaToba, Tu H0 marTldeba, naklebia an tolia .01-is. es mniSvneloba aris Frealuris albaToba an Frealuris kidev ufro ukiduresi mniSvneloba, Tu H0marTalia.
b. H0: µ A1 = µ A2 = µ A3 = µ A4 uaryofilia.
H1: µ A-ebi ar aris toli, miRebulia. g. arsebobs sul mcire erTi statistikurad mniSvnelovani sxvaoba pirobebis saSualoebs Soris.
d. SerCevis Secdomis garda kidev sxva mizezi agebs pasuxs pirobebis saSualoebs Soris sxvaobaze.
MSE = 0.065 gvaZlevs MSAxS mniSvnelobas FrealurisaTvis. es
MS aris Secdomis nawili F SefardebisaTvis; agreTve, is SeiZleba ganisazRvros, rogorc MSAxS.
CD = 0.48 sm gvaZlevs kritikul sxvaobas wyvilTaSorisi
SedarebebisaTvis tukis HSD kriteriumis gamoyenebiT. Semdeg aRiwereba statistikurad mniSvnelovani wyvilTaSorisi Sedarebebi.
η2 = .97 gviCvenebs, rom xazis aRqmuli Tanabari sigrZeebis dispersiis 97 procenti am eqsperimentSi iluziis pirobis gamo xdeba, mas Semdeg, rac individualuri gansxvavebebis gamo arsebuli cvalebadoba gauqmebulia.
kvlevis sqemis arCeva: cdis pirebs Soris Tu cdis pirebs Sida rogor unda gadawyvitos bihevioristma mecnierma, romeli kvlevis sqema gamoiyenos kvlevis hipoTezis Sesafaseblad – cdis pirebs Soris Tu
23
cdis pirebs Sida? sqemis TiToeul tips aqvs upiratesobebi da SezRudvebi. zogadad, am gadawyvetilebis miRebisas sami sakiTxi dgeba xolme: (1) cdis pirTa saWiro raodenoba, (2) damoukidebeli cvladis mosalodneli efeqturoba da (3) Carevis mravaljeradi efeqtebis miRebis albaToba. cdis pirTa saWiro raodenoba cdis pirTa Sida sqemaSi cdis pirs Carevis yovel pirobaSi zomaven; ase rom, aq naklebi cdis pirebia saWiro, vidre cdis pirTa Sida sqemaSi. cxrilSi # 12.2 naCveneb magaliTSi iluziis pirobis oTxi donidan TiToeulisaTvis miRebulia xuTi qula, Sesabamisad, sul 20 qulaa. magram mxolod xuTi cdis piria saWiro am 20 qulis misaRebad. cdis pirTa Soris sqemaSi am 20 qulis misaRebad saWiro iqneboda 20 sxvadasxva cdis piri. ase rom, rodesac cdis pirebis moZieba Znelia an rodesac cdis pirebis moZieba da maTTvis instruqciebis micema bevrad met dros moiTxovs, vidre TviTon eqsperimentuli davalebis Sesruleba, cdis pirTa Sida sqema SeiZleba upiratesi iyos cdis pirTa Soris sqemasTan SedarebiT cdis pirebis gamoyenebis TvalsazrisiT. damoukidebeli cvladis efeqturoba is, Tu ramdenad advilad xdeba eqsperimentSi damoukidebeli cvladis efeqtis dadgena, nawilobriv damokidebulia am cvladis efeqtis siZliereze. cdis pirTa Sida sqema xSirad ufro mgrZnobiarea damoukidebeli cvladis efeqtis aRmoCenis mxriv, vidre cdis pirTa Soris sqema. amdenad, cdis pirTa Sida sqemas SeiZleba upiratesoba mieniWos, rodesac damoukidebeli cvladis mosalodneli efeqti sustia. cdis pirTa Sida sqema metad mgrZnobiarea imis gamo, rom TiToeuli cdis piri iRebs damoukidebeli cvladis yovel dones da xvdeba Carevis yvela pirobaSi. cdis pirTa Sida dispersiuli analizi gamoacalkevebs cvalebadobas cdis pirTa Soris individualuri gansxvavebebis gamo (anu S faqtoris gamo cvalebadobas - MSS), ase rom am ukanasknels ar Seaqvs wvlili F-is Secdomis nawilSi (anu MSAxS). cdis pirTa Soris sqemaSi es cvalebadoba Sedis MSSecdomaSi. ase rom, Secdomis veqtori cdis prTa Sida sqemaSi (anu MSAxS) ufro mcire iqneba, vidre Secdomis veqtori cdis pirTa Soris sqemaSi (anu MSSecdoma). ufro mcire Secdomis veqtori iwvevs Frealuris ufro did mniSvnelobas, ris Sedegadac ufro mosalodnelia, rom H0-s uaryofa miviRoT. ase rom, eqsperimentSi, sadac cdis pirTa Soris didi individualuri gansxvavebebia da damoukidebeli cvladis efeqti ar unda iyos Zlieri, cdis pirTa Sida sqema ufro kargad gamoavlens damoukidebeli cvladis efeqts, vidre cdis pirTa Soris sqema. Carevis mravaljeradi efeqtebi cdis pirTa Sida sqemaSi cdis pirs damoukidebeli cvladis yovel doneze zomaven drois garkveuli periodis ganmavlobaSi. Carevebis es Tanmimdevroba cdis pirTa Sida sqemaSi Carevis mravaljeradi efeqtebis moxdenis SesaZleblobas qmnis; damoukidebeli cvladis donis efeqti SeiZleba damokidebuli iyos mis winamorbed doneebze. Carevis mravaljeradi efeqtebi moicavs rogorc gavarjiSebis, aseve Carevis gadatanis efeqtebs.
24
gavarjiSebis efeqtebi aris Sesrulebis cvlilebebi, romlebic imis gamo xdeba, rom eqsperimentuli davaleba ganmeorebiT sruldeba. am cvlilebebma SeiZleba gamoiwvios an gaumjobesebuli, an gauaresebuli Sesruleba eqsperimentis msvlelobisas. magaliTad, eqsperimentis magaliTSi, sadac izomeba iluziis xarisxi iluziis oTx gansxvavebul pirobaSi, cdis pirebs SeiZleba mobezrdeT eqsperimenti misi msvlelobisas an SeiZleba isini, rac ufro met Sefasebebs akeTeben, miT ufro skrupulozurni xdebian TavianT SefasebebSi. orive SemTxvevaSi cdis piris Sesrulebam SeiZleba Secvalos, moqmedebs Tu ara qcevaze iluziis piroba. Carevis gadatanis efeqtebi aris damoukidebeli cvladis erTi donis specifikuri efeqtebi, romlebic SeiZleba ise iqnen gadatanili, rom gavlena moaxdinon Sesrulebaze, damoukidebeli cvladis Semdgom doneze. magaliTad, amocanaSi, sadac iluziebis aRqma ganixileba, rogorc iluziis pirobis funqcia, Tu cdis pirebi sakontrolo iluzias yovelTvis bolo rigSi xedavdnen, maT SeiZleba aReqvaT iluzia, radgan sakontrolo pirobis win isini xedavdnen iluziis Semqmnel pirobebs. maT aqvT iluziis danaxvis molodini, romelic gadaitaneba sakontrolo pirobaze. Tu isini sakontrolo pirobas pirvel rigSi naxavdnen iluziis danaxvis molodinis gareSe, isini mas ar aRiqvamdnen rogorc iluzias. gavarjiSebis efeqtebi da Carevis gadatanis efeqtebi mniSvnelovania, radgan Tu isini Cndebian da ar arian kontrolirebadi, isini arTuleben eqsperiments. am efeqtebis kontroli Cveulebriv xorcieldeba cdis pirebisaTvis Carevis pirobebis sxvadasxva TanmimdevrobiT miwodebis gziT. ganxiluli sakiTxebidan – cdis pirebis saWiro raodenoba, damoukidebeli cvladis mosalodneli efeqturoba da Carevis mravaljeradi efeqtebis miRebis albaToba – TiToeuli unda miviRoT mxedvelobaSi, rodesac kvlevis sqemas virCevT. cdis pirebis efeqturad gamoyeneba da ufro meti mgrZnobiaroba damoukidebeli cvladis efeqtis dasafiqsireblad cdis pirTa Sida sqemis upiratesobebia. xolo Carevis mravaljeradi efeqtebis miRebis SesaZlebloba am sqemis naklovanebaa. Carevis mravaljeradi efeqtebi ar SeiZleba moxdes cdis pirTa Soris sqemaSi, radganac yoveli cdis piri eqvemdebareba damoukidebeli cvladis mxolod erTi dones. ase rom, rodesac eqsperimentSi mosalodnelia Carevis mravaljeradi efeqtebis miReba da maTi gakontroleba ar aris advili Carevebis Tanmimdevrobis gziT, sasurvelia cdis pirTa Soris sqemis gamoyeneba. radganac Carevis mravaljeradi efeqtebi xdeba maSin, rodesac mravali damoukidebeli cvladi gvaqvs da am efeqtebis gakontroleba ar aris advili cdis pirTa Sida sqemaSi, bihevioristi mecnierebi xSirad arCeven cdis pirTa Soris sqemis gamoyenebas. Carevis mravaljeradi efeqti: cdis pirTa Soris sqemaSi Sesrulebis cvlilebebi, romlebic xdeba imis gamo, rom mowmdeba damoukidebeli cvladis yovel doneze. gavarjiSebis efeqti: Carevis mravaljeradi efeqti, romelic xdeba imis gamo, rom cdis pirebi SeiZleba gavarjiSdnen an daiRalnen eqsperimentis davalebis Sesrulebisas. Carevis gadatanis efeqti: Carevis mravaljeradi efeqti, romelic xdeba maSin, rodesac damoukidebeli cvladis erTi donis erTi efeqti gadaitaneba, gavlenas axdens Sesrulebaze damoukidebeli cvladis meore doneze.
25
SeamowmeT Tqveni codna XII.3 1. gansazRvreT: Carevis mravaljeradi efeqtebi, gavarjiSebis efeqti, Carevis gadatanis efeqti. 2. rodesac adamianebis geografiul adgilmdebareobebze fiqroben, adgilis maT mier identifikaciaze xSirad gavlenas axdens Tanadaqvemdebarebis gadaxra. es niSnavs, rom iseTi geografiuli adgilebi, rogoricaa qalaqebi, moTavsebulia ufro did geografiul arealSi, rogoricaa regioni an qveyana. ufro didi geografiuli arealebi (anu Tanadaqvemdebarebuli adgilebi) SeiZleba gavlenas axdendnen imaze, Tu rogor warmovidgenT maTSi moTavsebul adgilebs. magaliTad, kanada amerikis SeerTebuli Statebisagan CrdiloeTiT aris ganlagebuli, Tumca arsebobs zogierTi kanaduri qalaqi, romelic bevrad ufro samxreTiT mdebareobs, vidre zogierTi aSS-s qalaqi. magram adamianebis umetesoba CaTvlida, rom aSS-s qalaqebi ufro samxreTiTaa, ubralod imis gamo, rom SeerTebuli Statebi kanadis samxreTiT mdebareobs. marTlac, fridmanma da braunma (Friedman and Brown, 2000) aRmoaCines, rom adamianebi Tvlidnen, rom vaSingtonSi mdebare sietli (SeerTebuli Statebi) bevrad ufro samxreTiTaa, vidre niufaundlendSi mdebare sent-jonsi (kanada), Tumca orive qalaqi erTsa da imave ganedzea. davuSvaT, Tqven gainteresebT es sakiTxi da sTxoveT cdis pirebs, daesaxelebinaT sami arseboTad erT ganedze ganlagebuli qalaqis ganedebi. es qalaqebia dalasi, texasis Stati (aSS), tihuana (meqsika) da tripoli (libia). yovel cdis pirebs eZleva instruqcia da utardeba ganedis dadgenis mcire savarjiSo, amis Semdeg ki sTxoven, daasaxelos yoveli qalaqis ganedi. miRebuli iyo Semdegi monacemebi: ' cdis piri
qalaqi (A) dalasi (A1) tihuana (A2) tripoli (A3)
1 2 3 4 5 6 7 8 9 10 11 12 13 14
33 29 35 28 30 40 32 31 30 25 27 36 26 29
18 27 21 17 18 26 25 19 23 20 20 24 22 23
20 24 26 19 27 23 25 24 17 21 20 23 18 20
ipoveT saSualo da standartuli gadaxra Carevis TiToeuli pirobisaTvis. Semdeg am qulebze CaatareT cdis pirTa Sida erTfaqtoriani dispersiuli analizi imis gansasazRvrad, arsebobs Tu ara Tanadaqvemdebarebis gadaxra ganedebis dasaxelebisas da upasuxeT Semdeg kiTxvebs: a. CamoayalibeT F kritetiumis statistikuri hipoTezebi am qulebisaTvis.
b. ra aris Frealuris mniSvneloba? g. ra aris Frealuris df-ebi? d. ra aris Fkritikuli .05 doneze? e. ra aris uaryofis are FrealurisaTvis? v. xvdeba Tu ara Frealuri uaryofis areSi?
z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb? uaryofT Tu ar uaryofT H0-s? miiRebT Tu ar miiRebT H1-s? T. ras daaskvniT, saSualoebi erTi da igive Tu sxvadasxva populaciebidanaa? i. arsebobs Tu ara statistikurad mniSvnelovani sxvaoba sul mcire saSualoebis erT wyvils Soris? k. saWiroa Tu ara am eqsperimentSi mravlobiTi Sedarebis kriteriumebi? daasabuTeT Tqveni pasuxi.
l. ra aris tukis CD-s mniSvneloba? m. romeli wyvilTaSorisi sxvaobebia statistikurad mniSvnelovani?
26
n. ra aris η2 am eqsperimentisaTvis? o. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic ilustrirebuli iyo am Tavis seqciaSi "dispersiuli analizis Sedegebis prezentacia".
savarjiSoebi # 3 da 4 warmoadgens cdis pirTa Sida erTfaqtoriani dispersiuli analizis angariSis nimuSebs. TiToeuli savarjiSosaTvis upasuxeT Semdeg kiTxvebs: a. damoukidebeli cvladis ramdeni done iyo gamoyenebuli eqsperimentSi? b. ramdeni cdis piri iRebda monawileobas eqsperimentSi? g. ra aris Frealuris mniSvneloba? d. ra aris Fkritikulis mniSvneloba .05 mniSvnelovnobis doneze? e. uaryofT Tu ar uaryofT H0-s? v. miiRebT Tu ar miiRebT H1-s?
z. arsebobs Tu ara sul mcire erTi statistikurad mniSvnelovani sxvaoba saSualoebs Soris? T. romeli Carevis pirobebi gansxvavdeba mniSvnelovnad, Tuki aseTebi saerTod arseboben?
1. saSualo reaqciis dro martivi stimulis pirobaSi 396 miliwami (mw) iyo, maSin, rodesac rTuli stimulis pirobaSi - 412 mw. alfas .05 doneze cdis pirTa Sida erTfaqtorianma dispersiulma analizma aCvena saSualoebs Soris mniSvnelovani sxvaoba, F (1,18) = 4.76, MSE = 510.924, p < .05.
2. sworad amocnobili saxeebis saSualo raodenoba 13.9 iyo pirvel cdaSi, 14.6 - meore cdaSi da 14.1 - mesame cdaSi. alfas .05 doneze cdis pirTa Sida erTfaqtorianma dispersiulma analizma ar aCvena sxvaoba cdebSi sworad amocnobis raodenobebs Soris, F (2,22) = 2.81, MSE = 0.555, p > .05.
Sejameba
• cdis pirTa Sida erTfaqtorian sqemaSi cdis pirTa erTi jgufi ganicdis damoukidebeli cvladis yoveli donis zemoqmedebas.
• cdis pirTa Sida erTfaqtoriani dispersiuli analizi gamoiyeneba cdis pirTa Sida erTfaqtoriani mravaldoniani sqemis qulebis gasaanalizeblad.
• cdis pirTa Sida erTfaqtoriani dispersiuli analizidan miiReba sami kvadratebis saSualo: MSA, romelic icvleba damoukidebeli cvladis efeqtTan erTad da Carevis pirobebis da cdis pirTa urTierTqmedeba MSS, romelic icvleba cdis pirTa Soris gansxvavebebTan erTad, da MSAxS, romelic icvleba pirobebis cdis pirebTan urTierTqmedebasTan erTad da iZleva eqsperimentSi Secdomis cvalebadobis sazoms.
• F statistika, romelic gamoiyeneba cdis pirTa Sida erTfaqtorian dispersiul analizSi, aris F = MSA/ MSAxS.
• F statistikis gamoyeneba misdevs statistikuri hipoTezebis Camoyalibebis Cveulebriv nabijebs, rogorebicaa mniSvnelovnobis donis dadgena, uaryofis aris dadgena, Frealuris gamoTvla da statistikur hipoTezebTan dakavSirebuli gadawyvetilebebis miReba.
• rodesac damoukidebel cvlads sami an meti done aqvs, statistikurad mniSvnelovani F moiTxovs mravlobiTi Sedarebis kriteriumis gamoyenebas, raTa dadgindes statistikurad mniSvnelovani wyvilTaSorisi Sedarebebi.
27
• tukis kriteriumi gamoiyeneba yvela SesaZlebeli wyvilTaSorisi saboloo SedarebisaTvis.
• η2 SeiZleba gamoyenebuli iqnes, rogorc Carevis efeqtis siZlieris sazomi.
• cdis pirTa Soris da cdis pirTa Sida sqemebs Soris arCevanis gakeTeba moiTxovs Semdegi sakiTxebis ganxilvas: cdis pirTa saWiro raodenoba, damoukidebeli cvladis mosalodneli efeqturoba, da Carevis mravaljeradi efeqtebis miRebis SesaZlebloba.
------------------------------------------------------------------------------------------------------------------------------ ZiriTadi terminebi da simboloebi
kritikuli sxvaoba (CD) dfA dfS dfAxS dfsruli eta kvadrati (η2) A faqtori S faqtori MSA MSS MSAxS Carevis mravaljeradi efeqtebi
nA gavarjiSebis efeqti ganmeorebiTi gazomvis sqema SSA SSS SSAxS SSsruli Carevis gadatanis efeqti Carevis pirobebis da cdis pirebis urTierTqmedebis sqema Carevis pirobebis da cdis pirebis urTierTqmedeba (AxS) cdis pirTa Sida sqema
__________________________________________________________________ Tavis damateba cdis pirTa Soris orfaqtoriani dispersiuli analizis gamoTvla formulebis meSveobiT __________________________________________________________________ am damatebaSi mocemulia gamoTvliTi formati cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis. eqsperimenti A damoukidebeli cvladis oTxi doniT da xuTi cdis piriT yovel doneze qvemoTaa warmodgenili: cdis piri A faqtori
________________________________________________ A1 A2 A3 A4
S1 S2 S3 S4 S5
X11 X21 X31 X41 X51
X12 X22 X32 X42 X52
X13 X23 X33 X43 X53
X14 X24 X34 X44 X54
1ST
2ST
3ST
4ST
5ST
1AT 2AT 3AT 4AT G
28
sadac Xij = cdis piris qula Ta = qulebis jami damoukidebeli cvladis donisaTvis TS = qulebis jami cdis pirisaTvis G = qulebis saerTo sruli jami nA = qulebis jami damoukidebeli cvladis doneze a = damoukidebeli cvladis doneebis raodenoba N = qulebis sruli raodenoba. am aRniSvnebis gamoyenebiT miiReba oTxi ricxviTi mniSvneloba:
[1] = ∑∑ 2ijX yvela qulis kvadratebis jami
[2] = A
A
nT∑ 2
damoukidebeli cvladis TiToeuli donis kvadratebis
jami, gayofili cdis pirebis raodenobaze.
[3] = aTS∑ 2
TiToeuli cdis piris qulebis kvadratebis jami,
gayofili A faqtoris doneebis raodenobaze.
[4] = NG2
saerTo sruli jamis kvadrati, gayofili qulebis srul
raodenobaze. am ricxviTi mniSvnelobebis gamoyenebiT miiReba Semdegi dispersiuli analizi: wyaro SS dfa MS F A faqtori cdis pirebi (S) AxS sul
[2] – [4] [3] – [4] [1] – [2] – [3] + [4] [1] – [4]
a - 1 nA - 1 (a - 1)( nA - 1) N - 1
A
A
dfSS
S
S
dfSS
AxS
AxS
dfSS
ar gamoiTvleba
AxS
A
MSMS
am gamoTvlebis ilustraciisaTvis gamoviyenoT cxrilSi # 12.2 moyvanili monacemebi, romelTa dispersiuli analizi Sejamebulia cxrilSi # 12.3. cdis piri
A faqtori cdis piris jamebi A1 A2 A3 A4
1 16.6 15.8 14.7 14.4 1ST = 61.50
2 16.9 15.6 15.1 13.8 2ST = 61.40
3 17.1 16.1 14.8 13.9 3ST = 61.90
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4 17.2 16.3 15.2 14.1 4ST = 62.80
5 16.8 15.7 15.3 14.3 5ST = 62.10
AT = 84.60 79.50 75.10 70.50 G = 309.70
nA = 5, a = 4 da N = 20. ricxviTi mniSvnelobebia: [1] = [(16.6)2 + ... + (16.8)2 + (15.8)2 + ... + (15.7)2 + (14.7)2 + ... (15.3)2 + (14.4)2 + ... +
(14.3)2] = 4818.630
[2] = 5
)50.70()10.75()50.79()60.84( 2222 +++ = 4817.534
[3] = 4
)10.62()80.62()90.61()40.61()50.61( 22222 ++++ = 4796.018
[4] = 20
)70.309( 2
= 4795.704
Semdeg SSA = [2] – [4] = 4817.534 – 4795.704 = 21.830 dfA = a – 1 = 4 – 1 = 3
MSA = 3830.21
= 7.277
SSS = [3] – [4] = 4796.018 – 4795.704 = 0.314 dfS = nA – 1 = 5 – 1 = 4
MSS = 4314.0
= 0.078
SSAxS = [1] – [2] – [3] + [4] = 4818.630 - 4817.534 - 4796.018 + 4795.704 = 0.782 dfAxS = (a – 1)(nA – 1) = (4 - 1)(5 – 1) = 12
MSAxS = 12782.0
= 0.065
SSsruli = [1] – [4] = 4817.534 – 4795.704 = 22.926 dfsruli = N – 1 = 20 – 1 = 19
Frealuri = AxS
A
MSMS
= 065.0277.7
= 111.95
dispersiuli analizis Sejamebaa:
wyaro SS df MS F A faqtori cdis pirebi AxS sul
21.830 0.314 0.782 22.926
3 4 12 19
7.277 0.078 0.065
111.95*
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* p < .01
am gamoTvliTi midgomis Sedegad miRebuli mniSvnelobebi identuria im mniSvnelobebisa, romelic warmodgenilia cxrilSi # 12.3 dispersiuli analizis ganmartebiTi formulebidan. __________________________________________________________________
kiTxvebi ________________________________________________________
1. davuSvaT, gainteresebT, moqmedebs Tu ara sityviT (mxiaruli, neitraluri an sevdiani) gamoxatuli xasiaTis tipi imaze, Tu ramdenad kargad xdeba mocemuli siidan sityvis damaxsovreba. SeadgenT eqvsi samxiarulo sityvisagan (magaliTad, mxiaruli, gaciskrovnebuli), eqvsi neitraluri sityvisagan (magaliTad, gamoyvana, gadacema) da eqvsi sevdiani sityvisagan (magaliTad, daRvremili, martosuli) Semdgar sias. Semdeg rva cdis piridan TiToeuli Seiswavlis sias, sanam orjer sruli sizustiT ar gaimeorebs mas. erTi kviris Semdeg TiToeuli cdis piri cdilobs siis gaxsenebas. sworad aRdgenili erTeulebis raodenobebi, rogorc sityvis tipis funqcia, ase gamoiyureba:
cdis piri
sityvis tipi bednieri neitraluri mowyenili
1 2 3 4 5 6 7 8
5 6 4 5 3 6 2 5
4 3 5 3 1 3 2 3
3 4 2 1 2 4 3 1
a. sityvis pirobis TiToeuli tipisaTvis ipoveT saSualo da
standartuli gadaxra. Semdeg gaaanalizeT qulebi cdis pirTa Sida erTfaqtoriani analizis meSveobiT, raTa upasuxoT Semdeg kiTxvebs: moqmedebs Tu ara sityvis tipi aRdgenaze? gamoiyeneT .05 mniSvnelovnobis done. Tu saWiroa, gamoiyeneT tukis HSD kriteriumi mravlobiTi SedarebisaTvis.
b. ra aris damoukidebeli cvladis - sityvis tipis - efeqtis siZliere?
g. aRwereT Sedegebi im stilis gamoyenebiT, romelic ilustrirebuli iyo am Tavis seqciaSi "dispersiuli analizis Sedegebis prezentacia".
2. sanimuSo magaliTSi 10.3 warmoadgenili iyo cdis pirTa Sida sqemis
mixedviT agebuli morinagas (Morinaga) SeuTavseblobis iluziis kvleva [ix. suraTi # 10.3, paneli (a)]. davuSvaT, eqsperimentators surs, icodes, Secvlis Tu ara cdis pirebis mier iluziis aRqmas maTi instruqtireba am iluziis Sesaxeb. Tavdapirvelad eqsperimentatorma sTxova 12 cdis pirs, SeeTavsebinaT (erT xazze ganelagebinaT) kuTxeebis wveroebi da gazoma Secdomis raodenoba yoveli adamianisTvis. mas Semdeg, rac cdis pirebma SeaTavses
31
wveroebi, maT Seatyobines iluziis Sesaxeb da kidev erTxel sTxoves kuTxeebis SeTavseba. miRebuli iqna Secdomis ori gazomva (milimetrebSi) yoveli cdis pirisaTvis:
cdis piri
instruqciis piroba (A) _______________________________________ instruqciamde (A1 ) instruqciis Semdeg (A2)
1 2 3 4 5 6 7 8 9 10 11 12
1.3 2.1 2.5 1.9 3.0 2.5 1.8 2.4 3.1 3.4 2.5 2.9
1.4 2.0 2.7 2.0 3.4 2.8 1.6 2.5 3.3 3.4 2.3 2.8
a. instruqciis TiToeuli pirobisaTvis ipoveT saSualo da
standartuli gadaxra. Semdeg gaaanalizeT qulebi cdis pirTa Sida erTfaqtoriani analizis meSveobiT, raTa upasuxoT Semdeg kiTxvebs: iqonia Tu ara instruqciam gavlena aRqmuli iluziis raodenobaze? gamoiyeneT .05 mniSvnelovnobis done.
b. SeadareT Frealuri me-9 Tavis "kiTxvebidan" meoTxeSi moyvanil
tdamokidebuls. sworia Tu ara realuriF = tdamokidebuli?
g. dispersiuli analizisa da tdamokidebuli kriteriumis gamoyenebis Sedegad erTsa da imave daskvnebs gaakeTebT damoukidebeli cvladis efeqtis Sesaxeb, Tu gansxvavebulebs?
d. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic ilustrirebuli iyo am Tavis seqciaSi "dispersiuli analizis Sedegebis prezentacia".
3. davuSvaT, gegmavT, CaataroT cdis pirTa Sida erTfaqtoriani
eqsperimenti damoukidebeli cvladis xuTi doniTa da 12 cdis piriT.
a. dawereT dispersiuli analiziT Semowmebuli statistikuri hipoTezebi im qulebisaTvis, romlebsac Tqven miiRebdiT.
b. ipoveT df am sqemis dispersiuli analizisaTvis. g. ra aris Fkritikulis mniSvneloba .05 mniSvnelovnobis doneze am
sqemis dispersiuli analizisaTvis? d. davuSvaT, rom Frealuris mniSvneloba ufro didi iyo, vidre
Fkritikulis mniSvneloba. ra gadawyvetilebebs miiRebdiT H0-Tan da H1-Tan dakavSirebiT?
e. davuSvaT, rom am eqsperimentisaTvis Tqven uaryaviT H0 da miiReT H1. ra daskvnas gaakeTebdiT am gadawyvetilebebidan Carevis saSualoebis Sesaxeb?
4. qvemoT moyvanilia dispersiuli analizis arasruli Sejamebis
cxrilebi. SS-s, df-sa da MS-s Soris arsebuli mimarTebebis
32
gamoyenebiT TiToeul cxrilSi SeitaneT gamotovebuli mniSvnelobebi. Semdeg TiToeuli cxrilisaTvis upasuxeT Semdeg kiTxvebs:
a. ramdeni done hqonda damoukidebel cvlads? b. damoukidebeli cvladis TiToeul doneze ramdeni qula iyo
miRebuli? g. ramdeni cdis piri iRebda monawileobas kvlevaSi? d. aris Tu ara Frealuris mniSvneloba statistikurad
mniSvnelovani .05 doneze? e. ra gadawyvetilebebs miiRebT H0-isa da H1-isaTvis yovel
analizSi? cxrili # 1
wyaro SS df MS F A faqtori S faqtori AxS sul
____ 138.00 253.00 ____
____ 23 46 71
11.00 6.00 ____
____
cxrili # 2
wyaro SS df MS F A faqtori S faqtori AxS sul
60.00 133.00 ____ 573.00
____ 19 95 ____
____ 7.00 ____
____
5. ramdenime eqsperimentatorma gamoikvlia Cvilis interesi ucnobi saxeebis mimarT nacnob saxeebTan SedarebiT. 14 Cvilidan TiToeuli aTvalierebda nacnobi saxeebis (magaliTad, dedis da mamis fotoebi), ucnobi saxeebis (magaliTad, CvilisTvis ucnobi mamakacebisa da qalebis fotoebi) da saxis zomis carieli ovalebis fotoebs. eqsperimentatorebi afiqsirebdnen dros, romlis ganmavlobaSic TiToeuli Cvili akvirdeboda saxes. cxrilSi # 1 warmodgenilia stimulis piroba da drois saSualo xangrZlivoba, romelic ixarjeboda stimulze TiToeul pirobaSi. cxrilSi # 2 warmodgenilia am monacemebis dispersiuli analizi.
cxrili # 1 dakvirvebis saSualo dro (wamebSi), rogorc stimulis pirobis funqcia.
stimulis piroba
nacnobi saxe (A1) ucnobi saxe (A2) carieli ovali (A3) 34.2 26.8 18.5
cxrili # 2 dispersiuli analizis Sejameba cxrilSi # 1 warmodgenili dakvirvebis saSualo droebisaTvis.
wyaro SS df MS F stimulis piroba (A) 1727.32 2 863.66 15.99*
33
cdis pirebi (S) AxS sul
2047.37 1404.52 5179.21
13 26 41
157.49 54.02
* p < .01
tukis HSD-m mogvca CD = 6.9 wuTi mniSvnelovnobis .05 doneze. a. ra aris stimulis pirobis efeqti dakvirvebis droze? b. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic
ilustrirebuli iyo am Tavis seqciaSi "dispersiuli analizis Sedegebis prezentacia".
Tavi XIII. korelacia _______________________________________________________________________
1
Tavi XIII korelacia __________________________________________________________
wertilTa gabnevis diagrama dadebiTi kavSiri uaryofiTi kavSiri
pirsonis korelaciis koeficienti korelaciis sazomis miReba: jvaredina namravlebi
r : XYCP -is Sedareba qulebis saerTo cvalebadobasTan pirsonis korelaciis koeficientis alternatiuli formulebi
r -is standartuli qulebis formula r -is gamoTvliTi formula
r -is formulebi _ Sejameba
r -is Tvisebebi kavSiris mimarTuleba kavSiris xarisxi diapazonis SezRudva da r -is mniSvneloba determinaciis koeficienti korelacia da mizezobrioba
pirsonis r koeficientis statistikuri mniSvnelovnobis Semowmeba
populaciis korelacia, ρ aranulovani korelaciis Semowmeba populaciaSi
korelaciuri kvlevis Sedegebis warmodgena spirmenis ranguli korelaciis koeficienti
Sr -is gamoyenebis magaliTi
spirmenis ranguli korelaciis koeficientis statistikuri mniSvneloba
Sr -is Tvisebebi
Tavi XIII. korelacia _______________________________________________________________________
2
korelaciuri kvlevebi (Correlational studies): kvlevebi, romlebSic izomeba ori an meti cvladi maTi korelaciis mimarTulebisa da xarisxis dasadgenad.
mecxredan meTormete Tavebis CaTvliT ganixileboda statistikis
gamoyeneba eqsperimentebSi, romlebSic xdeba damoukidebel cvladze manipulacia da izomeba damokidebuli cvladi. Tumca bevr kvlevaSi mkvlevari mxolod imiT aris dainteresebuli, rom aRmoaCinos, dakavSirebulia Tu ara erTmaneTTan ori cvladi. korelaciuri kvleva gulisxmobs kvlevas, romelSic izomeba ori an ramdenime cvladi, raTa gairkves maTi kavSiris mimarTuleba da xarisxi. am kvlevebSi arc erT cvladze ar xdeba manipulireba da, Sesabamisad, ar arsebobs damoukidebeli da damokidebuli cvladebi. am TavSi gagacnobT korelaciis cnebebs, romlebic saWiroa aseTi kvlevebis gasagebad da statistikas, romelsac korelaciis koeficienti ewodeba. erTmaneTTan korelaciaSi myofi qulebis gamoyeneba imisaTvis, rom erTis mixedviT meore viwinaswarmetyveloT, ganxilulia me-14 TavSi. korelaciuri kvlevis umartivesi SemTxveva gulisxmobs cdis pirTa jgufis TiToeuli wevrisgan ori gansxvavebuli cvladis monacemis miRebas. Semdeg xdeba monacemTa am wyvilebis statistikuri analizi, raTa dadgindes, arsebobs Tu ara raime kavSiri maT Soris. ganvixiloT Semdegi magaliTi:
martooba fsiqologiuri mdgomareobaa, romelsac yvela Cvengani ginicdis odesme. bevrisaTvis martooba xangrZlivi mdgomareobaa da is SeiZleba dakavSirebuli iyos fsiqikuri janmrTelobis problemebTan. (Roberts, Lewinsohn, & Seeley, 1993). davuSvaT, rom gvainteresebs martoobasa da depresias Soris kavSiris kvleva. viRebT 10 adamianisagan Semdgar SerCevas da vzomavT maTi depresiisa da martoobis dones. martoobis gasazomad adamians vTxovT, upasuxos 10 debulebas rogoricaa, magaliTad, ,,Tavs sxvebisgan gariyulad vgrZnob”. SesaZlo pasuxebia: arasodes (0), iSviaTad (1), zogjer (2), xSirad (3). frCxilebSi mocemuli cifrebi gamoxatavs pasuxebisaTvis miniWebul ricxviT mniSvnelobas (wonas). subieqti, romlis pasuxi iqneba ,,arasodes” aTive debulebaze, miiRebs nul qulas martoobis skalaze maSin, roca subieqti, romlis pasuxi iqneba ,,xSirad” aTive debulebaze, miiRebs 30 qulas martoobis skalaze. depresiac aseve 10 debulebiT izomeba, rogoricaa ,,me mowyenili var”. aqac cdis piri upasuxebs: arasodes (1), iSviaTad (2), zogjer (3), xSirad (4). Tumca yuradReba miaqcieT imas, rom am skalaze pasuxebisaTvis miniWebuli ricxviTi mniSvnelobebi gansxvavebulia martoobis skalis pasuxebis ricxviTi mniSvnelobebisagan. yvelaze dabali qulaa 10, romelsac iRebs adamiani, romelic upasuxebs ,,arasodes” aTive debulebaze da umaRlesi qulaa 40, romelsac iRebs adamiani, romelic pasuxobs ,,xSirad” aTive debulebaze. maSasadame, ToToeuli subieqtisagan miviReT ori qula: pirveli - martoobis da meore - depresiis.
davuSvaT, rom miRebuli martoobis qulebi (aRniSnuli rogorc X cvladi) da depresiis qulebi (aRniSnuli roogrc Y cvladi) warmodgenilia cxrilSi #13.1. cvladis X -iT an Y -iT aRniSvna pirobiTia. Tumca zogadad mklevari svams kiTxvas: rogoria Y -is, rogorc X -is funqciis mniSvneloba?
es debuleba gamoxatavs, rom mkvlevari dainteresebulia, erTi cvladis ( X ) mniSvnelobis mixedviT iwinaswarmetyvelos meore cvladis
Tavi XIII. korelacia _______________________________________________________________________
3
(Y ) mniSvneloba. cvladi, romelsac prediqtorad viyenebT, aRiniSneba X -iT, xolo nawinaswarmetyvelebi cvladi - Y -iT. maSasadame, rodesac mkvlevari svams kiTxvas: `ras udris depresiis qula martoobis garkveuli qulis SemTxvevaSi?~, is cdilobs, depresiis done (Y ) iwinaswarmetyvelos martoobis donis ( X ) mixedviT. erTi qulis winaswarmetyvelebisaTvis meore cvladis gamoyenebis detalebi warmodgenilia me-14 TavSi.
qulaTa wyvilebi cxrilSi # 13.1 warmoadgens qulebis organzomilebian (bivariaciul) ganawilebas.
prefiqsi ,,bi” niSnavs ors (2); es qulebi organzomilebiania, radgan
TiToeuli cdis pirisgan miRebulia 2 monacemi (qula). dakavSirebulia Tu ara es qulebi erTmaneTTan? iqneb martoobis maRali qulebi dakavSirebulia depresiis maRal qulebTan, xolo martoobis dabali qulebi dakavSirebulia depresiis dabal qulebTan? an iqneb kavSiri Sebrunebulia: martoobis maRali qulebi dakavSirebulia depresiis dabal qulebTan da martoobis dabali qulebi dakavSirebulia depresiis maRal qulebTan. aseve SesaZlebelia, rom qulebi ar korelirebdes da martoobisa da depresiis qulebi urTierTdamoukidebeli iyos. imisaTvis, rom davadginoT, dakavSirebulia Tu ara qulebi, Tavdapirvelad vagebT monacemebis grafiks, romelsac wertilTa gabnevis diagrama ewodeba; Semdeg am monacemebisTvis viTvliT pirsonis korelaciis koeficients.
cxrili # 13.1 10 cdis piris hipoTeturi qulebi martoobisa da depresiis skalebze
cdis piri
martoobis qula: X
depresiis qula: Y
1 2 3 4 5 6 7 8 9 10
4 27 18 7 30 12 18 23 19 12
16 37 33 23 34 32 24 29 26 26
wertilTa gabnevis diagramebi
wertilTa gabnevis diagrama (Scatterplot): organzomilebiani ganawilebis grafiki, romelSic X cvladi gadazomilia horizontalur RerZze da Y cvladi gadazomilia vertikalur RerZze.
organzomilebiani ganawileba (Bivariate Distribution): ganawileba, romelSic TiToeuli cdis pirisagan miiReba ori monacemi (qula).
Tavi XIII. korelacia _______________________________________________________________________
4
cxrilis #13.1 monacemebis wertilTa gabnevis diagrama warmodgenilia suraTze #13.1. wertilTa gabnevis diagramaze X -iT aRniSnuli cvladi (Cvens magaliTSi martoobis qula) gadazomilia horizontalur RerZze (abscisaze), xolo Y cvladi (Cvens magaliTSi depresiis qula) gadazomilia vertikalur RerZze (ordinataze). cdis piris mier TiToeul skalaze miRebuli qula gafantulobis diagramaze erTi wertiliT gamoisaxeba. magaliTad, grafikze #13.1 pirveli cdis piris qulebi warmodgenilia martoobis Sesabamisi 4 qulis da depresiis 16 qulis gadakveTis wertiliT. imis saCveneblad, Tu rogor aigeba wertilTa gabnevis diagrama, grafikze #13.1 TiToeuli cdis piris Sesabamisi wertilis gverdiT miTiTebulia am cdis piris nomeri. Tumca Cveulebriv, wertilTa gabnevis diagramaze cdis pirebis saxeliT an nomriT aRniSvna ar xdeba.
dadebiTi kavSiri
korelacia (covary): ori cvladi korelirebs, rodesac erTi cvladis cvlileba dakavSirebulia meore cvladis SeTanxmebul cvlilebasTan. wertilTa gabnevis diagrama gviCvenebs, rom monacemebs aqvT korelaciis tendencia: depresiis SedarebiT maRali qulebi dakavSirebulia martoobis SedarebiT maRal qulebTan, xolo depresiis SedarebiT dabali qulebi dakavSirebulia martoobis SedarebiT dabal qulebTan. monacemebi korelirebs (msgavsad varirebs) rodesac erTi qulis cvlilebas Tan axlavs meore qulis SeTanxmebuli cvlileba. grafikze #13.1 monacemebs Soris kavSiri dadebiTia. dadebiTi kavSiris SemTxvevaSi (rasac pirdapiri kavSiric ewodeba), X cvladis mniSvnelobis zrdasTan erTad izrdeba Y cvladis mniSvnelobac.
suraTi #13.1 10 cdis piris martoobisa da depresiis qulebis wertilTa gabnevis diagrama. wertilebis gverdiT miTiTebuli ricxvebi aRniSnavs cdis pirebs. qulebi aRebulia 13.1 cxrilidan.
maSasadame, grafikze #13.1 martoobis qulebis zrdasTan erTad xdeba depresiis qulebis zrdac. radgan orive cvlads vzomavT da arcerTiT ar vaxdenT manipulirebas, Tanabrad marTebulia vTqvaT, rom depresiis qulebis zrdasTan erTad izrdeba martoobis qulebic. magram grafikze #13.1 gamosaxuli kavSiri ar aris sruli. magaliTad, me-9 cdis pirs ufro maRali martoobis qula aqvs, vidre me-10 cdis pirs, magram orives depresiis qulaa 26. analogiurad, me-6 da me-10 cdis pirebis martoobis qulaa 12, magram pirvelis depresiis qulaa 32 da meoris - 26.
sruli dadebiTi kavSiri depresiisa da martoobis qulebs Soris gamosaxulia suraTze @13.2. am SemTxvevaSi wetilTa gabnevis diagrama gvaZlevs wrfes. or cvlads Soris kavSirs, romelic SeiZleba aRiweros wrfiT, wrfivi kavSiri ewodeba.
dadebiTi kavSiri (positive relationship): or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad izrdeba meore cvladis mniSvnelobac.
Tavi XIII. korelacia _______________________________________________________________________
5
wrfivi kavSiri (linear relationship): or cvlads Soris kavSiri, romelic SeiZleba aRiweros wrfiT.
sruli dadebiTi kavSiris SemTxvevaSi, rogoric gamosaxulia suraTze #13.2, X -is TiToeul mniSvnelobas Seesabameba Y -is mxolod erTi mniSvneloba. amgvarad, erTi cvladis mniSvnelobis codna saSualebas gvaZlevs, zustad gavigoT an viwinaswarmetyveloT meore cvladis mniSvneloba. magaliTad, cdis pirs, romelsac aqvs 8 qula martoobis skalaze, eqneba 18 qula depresiis skalaze; an cdis pirma, romelmac miiRo 24 qula depresiis skalaze, miiRo 14 qula martoobis skalaze. sruli kavSiris SemTxvevaSi erTi qulis mixedviT meores winaswarmetyveleba SecdomasTan ar aris dakavSirebuli. Tumca realuri kvlevis monacemebs Soris sruli dadebiTi kavSirebi iSviaTad an arasodes aRiniSneba.
suraTi #13.2 10 cdis piris martoobisa da depresiis qulebs Soris sruli dadebiTi kavSiri.
uaryofiTi kavSiri davuSvaT, rom martoobisa da depresiis miRebul maCveneblebs Soris kavSiri naCvenebia suraTze #13.3. es grafiki gamoxatavs uaryofiT kavSirs am or cvlads Soris: X cvladis mniSvnelobis matebasTan erTad Y cvladis mniSvneloba mcirdeba.
uaryofiTi kavSiri (negative relationship): or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad meore cvladis mniSvneloba mcirdeba.
grafikze #13.3 xedavT, rom martoobis dabali qulebi dakavSirebulia depresiis maRal qulebTan da martoobis maRali qulebi dakavSirebulia depresiis dabal qulebTan. uaryofiT kavSirs zogjer Sebrunebul (inversiul) kavSirsac uwodeben, radgan cvladebi Sebrunebulad aris erTmaneTTan dakavSirebuli: X -is dabali mniSvnelobebi dakavSirebulia Y -is maRal mniSvnelobebTan da X -is maRali mniSvnelobebi dakavSirebulia Y -is dabal mniSvnelobebTan.
suraTze @#13.3 gamosaxuli uaryofiTi kavSiri ar aris sruli kavSiri martoobisa da depresiis qulebs Soris. magaliTad, ori cdis piris martoobis qulaa 20, magram maTi Sesabamisi depresiis qulebia 17 da 24. aseve, ori cdis piris depresiis qulaa 32, magram erTis martoobis qulaa 5 da meores martoobis qulaa 12.
depresiisa da martoobis qulebs Soris sruli uaryofiTi kavSiri naCvenebia cxrilze #13.4. am SemTxvevaSi wertilTa gabnevis diagrama qmnis wrfes da erTi monacemis (qulis) codna meore monacemis (qulis) zusti winaswarmetyvelebis saSualebas iZleva. magaliTad, Tu CemTvis cnobilia, rom cdis piris martoobis qulaa 8, aseve cnobilia, rom misi depresiis qula aris 32. aseve, Tu viciT, rom cdis pirma miiRo 22 qula depresiis skalaze, maSin misi martoobis qula aris 18. sruli dadebiTi kavSiris msgavsad, realuri monacemebis SemTxvevaSi sruli uaryofiTi kavSiric
Tavi XIII. korelacia _______________________________________________________________________
6
iSviaTad gvxvdeba. yuradReba miaqcieT imas, rom `dadebiTi~ da `uaryofiTi~ wertilTa gabnevis diagramasTan mimarTebaSi gamoxatavs am kavSiris mxolod mimarTulebas da ara siZlieres. erTi qulis mixedviT meores winaswarmetyvelebisaTvis dadebiTi da uaryofiTi kavSirebi Tanabrad gamosadegia. suraTi #13.3 10 cdis piris martoobisa da depresiis qulebs Soris uaryofiTi kavSiri.
suraTi #13.4 10 cdis piris martoobisa da depresiis qulebs Soris sruli uaryofiTi kavSiri.
SeamowmeT Tqveni codna XIII-1
1. ganmarteT: monacemebis organzomilebiani ganawileba, korelaciuri kvleva, korelacia, pirdapiri kavSiri, Sebrunebuli kavSiri, wrfivi kavSiri, uaryofiTi kavSiri, dadebiTi kavSiri, wertilTa gabnevis diagrama.
2. cxrilisaTvis #13.1 SeabruneT da cvladebis aRniSvna (e.i. depresiis qula aRniSneT X -iT da martoobis qula Y -iT) da Semdeg am wertilebisaTvis aageT suraTze #13.1 gamosaxulis msgavsi wertilTa gabnevis diagrama. cvladis X -iT an Y -iT aRniSvna cvlis Tu ara qulebs Soris kavSirs?
3. 1-li da me-2 cxrilebis monacemebisaTvis ujrian furcelze aageT wertilTa gabnevis diagrama da Semdeg TiToeul cxrilTan dakavSirebiT upasuxeT Semdeg kiTxvebs: a. qulebs Soris kavSiri dadebiTia, uaryofiTi Tu kavSiri ar
arsebobs? b. qulebs Soris kavSiri srulia Tu ara? g. martoobis qulebis codna gazrdida Tu ara depresiis
qulebis winaswarmetyvelebis sizustes?
cxrili #1
cdis piri
martooba
depresia
1 4 19 2 12 20 3 28 33 4 18 32 5 14 26 6 6 19 7 22 28 8 23 36 9 15 30 10 18 27
cxrili #2
cdis piri
martooba
depresia
Tavi XIII. korelacia _______________________________________________________________________
7
1 5 29 2 11 31 3 23 23 4 17 17 5 28 16 6 13 21 7 12 26 8 4 28 9 21 14 10 16 25
4. qvemoT mocemulia 10 cdis piris SerCevis sxeulis wonisa (funtebSi) da TviTpativiscemis monacemebi
cdis piri
sxeulis wona
KTviTpativiscema
1 100 39 2 111 47 3 117 54 4 124 23 5 136 35 6 139 30 7 143 48 8 151 20 9 155 28 10 164 46
a. ujrian furcelze aageT wertilTa gabnevis diagrama.
monacemebs Soris kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar arsebobs?
b. monacemebs Soris kavSiri srulia Tu ara? g. cdis piris sxeulis wonis codna gaaumjobesebda Tu ara
am cdis piris TviTpativiscemis qulis winaswarmetyvelebis sizustes?
pirsonis korelaciis koeficienti wertilTa gabnevis diagramis Seswavla warmodgenas gviqmnis imis Sesaxeb, arsebobs Tu ara mimarTeba or cvlads Soris da rogoria am mimarTebis mimarTuleba. magram mxolod wertilTa gabnevis diagrama ar aris sakmarisi imis dasadgenad, arsebobs Tu ara kavSiri or cvlads Soris. diagramaze gamosaxuli mimarTeba raodenobrivad unda aRiweros. aRweriT statistikebs, romlebic or cvlads Soris kavSiris xarisxs gamoxatavs, korelaciis koeficientebi ewodeba. intervalebis an Sefardebis skalebisaTvis gavrcelebuli koeficientia pirsonis korelaciis koeficienti, romelic aRiniSneba r -iT. am statistikas mieniWa karl pirsonis (1857-1936), im britaneli statistikosis saxeli, romelmac am
Tavi XIII. korelacia _______________________________________________________________________
8
koeficientis gamosaTvleli formula Seqmna. pirsonis korelaciis koeficienti ganisazRvreba, rogorc:
pirsonos korelaciis koeficienti = ( )( )
( )[ ] ( )[ ]∑∑∑
−−
−−=
22YY XX
YYXXr
sadac X aris cdis piris X cvladis Sesabamisi qula
X aris X cvladis Sesabamisi yvela qulis saSualo Y aris cdis piris Y cvladis Sesabamisi qula
Y aris Y cvladis Sesabamisi yvela qulis saSualo pirsonis korelaciis koeficienti (Pearson correlation coefficient): statistika, romelic aRiniSneba r -iT da asaxavs intervalebis an Sefardebis skalaze or cvlads Soris wrfivi kavSiris xarisxs. es formula SeiZleba Caiweros ufro martivi aRniSvnebis gamoyenebiT, rogorc:
( )( )YX
XY
SSSSCPr =
sadac XYCP aris ( )( )∑ −− YYXX , anu aris X –sa da Y –is jvaredina namravli
( X da Y indeqsebi niSnavs, rom jvaredina namravlebi gamoTvlilia X da Y cvladebisTvis)
XSS aris X monacemebis kvadratebis jami (anu ( )∑ −2
XX )
YSS aris Y monacemebis kvadratebis jami (anu ( )∑ −2
YY ) ganmartebiTi formula Cveulebriv r –is gamosaTvlelad ar gamoiyeneba. mis nacvlad gamoiyeneba an standartuli qulis formula, an gamTvliTi formula, romlebic am TavSi qvemoT aris warmodgenili. magram ganmartebiTi formula asaxavs, Tu ras zomavs korelaciis koeficienti. ganvixilavT am formulas da Semdeg warmovadgenT standartuli qulisa da gamoTvliT formulebs. korelaciis sazomis miReba: jvaredina namravlebi imisaTvis, rom davinaxoT, Tu rogor aris r imis sazomi, ramdenad Zlieria korelacia or cvlads Soris, davubrundeT cxrilis #13.1 monacemebs. am monacemebis wertilTa gabnevis diagrama (ix. suraTi 13.1) gamoxatavda, rom monacemebi dadebiT kavSirSia: martoobis dabali qulebi dakavSirebulia depresiis dabal qulebTan da martoobis maRali qulebi dakavSirebulia depresiis maRal qulebTan. me ganvixilav statistikas, romelic or cvlads Soris korelaciis siZlieres gamoxatavs sxvaobis qulis gamoyenebiT, romelic aCvenebs, Tu ramdenad gansxvavdeba qula Tavisi
saSualosagan, XX − an YY − . magaliTad, cxrilze #13.1 martoobis
Tavi XIII. korelacia _______________________________________________________________________
9
qulebis saSualoa 17. me-2 cdis pirs aqvs martoobis 27 qula. Tu
gamovTvliT XX −2 , anu 27-17 = +10-s, davinaxavT, rom cdis pirma miiRo martoobis qulebis saSualoze 10-iT maRali qula. me-4 cdis pirma miiRo 7 qula martoobis skalaze. am cdis piris martoobis qula SeiZleba aseve
gamoisaxos, rogorc XX −4 ; am SemTxvevaSi 7 – 17 = - 10, anu 10 qula martoobis qulis saSualos qvemoT. qulis gardaqmna am qulis saSualosagan gansxvavebad imasac gamoxatavs, qula saSualoze metia Tu naklebi da imasac, Tu ramdenad metia an naklebia. Y qulebi SeiZleba gamoisaxos msgavsadve am qulebis saSualosTan,
Y -Tan, mimarTebiT. magaliTad, depresiis qulebis saSualoa 28. me-2 cdis pirma miiRo depresiis 37 qula. am qulis saSualosgan gansxvavebaSi
gadayvaniT viRebT: YY −2 = 37-28 = +9. meores mxriv, me-4 cdis pirma miiRo depresiis 23 qula. saSualosgan gansxvavebis saxiT gamosaxuli, es qula
udris: YY −4 = 23 – 28 = -5, anu depresiis saSualo qulaze 5 quliT dabalia. sxvaobis am monacemebidan Cven vxedavT, rom me-2 cdis pirs aqvs saSualoze maRali qulebi rogorc martoobis, aseve depresiis skalebze, maSin, roca 44-e cdis piris orive qula saSualoze dabalia. davuSvaT, rom me yvela cdis pirisaTvis viTvli martoobisa da depresiis sxvaobis qulebs, rogorc es cxrilis #13.2 b da d svetebSia naCvenebi. yuradReba miaqcieT imas, rom im 6 cdis piridan, romelTa martoobis qula saSualoze maRalia (sveti b, cdis pirebi 2, 3, 5 da 8), 4 cdis piris depresiis qulac saSualoze maRalia (sveti d). msgavsadve, 3 cdis pirs im 4-dan, romelTa martoobis qula saSualoze dabalia (sveti b, cdis pirebi 1, 4 da 10), depresiis qulac saSualoze dabali aqvs (sveti d). cxrili #13.2 X -sa da Y -is jvaredina namravlebis gamoTvla. a da g svetebSi mocemuli qulebi
cxrilis #13.1 martoobisa da depresiis qulebia.
(a) (b) (g) (d) (e)
martooba depresia
cdis piri X XX − Y YY − ( )( )YYXX −−
1 4 -13 16 -12 +156 2 27 +10 37 +9 +90 3 18 +1 33 +5 +5 4 7 -10 23 -5 +50 5 30 +13 34 +6 +78 6 12 -5 32 +4 -20 7 18 +1 24 -4 -4 8 23 +6 29 +1 +6 9 19 +2 26 -2 -4 10 12 -5 26 -2 +10
X 17 Y 28 jami +367
Tavi XIII. korelacia _______________________________________________________________________
10
qulebis korelaciis siZlieris sazomi miiReba TiToeuli cdis piris
qulebis X cvladis saSualosgan sxvaobisa da Y cvladis saSualosgan
sxvaobis erTmaneTze gamravlebiT da Semdeg yvela cdis piris namravlis SekrebiT. es procesi naCvenebia cxrilis #13.2 e svetSi. magaliTad, me-4
cdis pirisaTvis, ( XX − )( YY − ) = (-10) (-5) = +50. am svetSi namravlebis jami aris +367.
( )( )∑ −− YYXX sidides ewodeba X -isa da Y -is jvaredina namravli,
rac aRiniSneba, rogorc XYCP da is warmoadgens r –is formulis
mricxvels. XYCP asaxavs monacemebis or simravles Soris kavSiris siZlieresa da mimarTulebas. Tu monacemebi dadebiT kavSirSia, rogorc 13.2 cxrilze, maSin jvaredina namravli dadebiT mniSvnelobas iRebs; rac ufro
Zlieria korelacia X –sa da Y –s Soris, miT ufro izrdeba XYCP –is
mniSvneloba. uaryofiTad dakavSirebuli monacemebisaTvis XYCP uaryofiTia da misi mniSvneloba miT ufro didia, rac ufro Zlieria kavSiri.
X -isa da Y -is jvaredina namravlebi (Cross products of X and Y ): ( )( )∑ −− YYXX mniSvneloba X da Y cvladebisTvis
cxrilze #13.3a naCvenebia XYCP –is mniSvneloba suraTis #13.3 Sesabamisi
X –isa da Y –is uaryofiTi kavSirisaTvis. am monacemebisaTvis XYCP aris -399. cxrili #13.3 a) X -isa da Y –is jvaredina namravlebi martoobisa da depresiis qulebisaTvis.
b) X -isa da Y –is jvaredina namravlebi im SemTxvevisaTvis, roca martoobisa da
depresiis qulebs Soris kavSiris ar aris.
Mmartooba depresia
cdis piri X XX − Y YY − ( )( )YYXX −−
1 5 -10 32 +7 -70 2 24 +9 23 -2 -18 3 20 +5 17 -8 -40 4 12 -3 32 +7 -21 5 14 -1 21 -4 +4 6 20 +5 24 -1 -5 7 28 +13 14 -11 -143 8 13 -2 25 0 0 9 6 -9 36 +11 -99 10 8 -7 26 +1 -7
X 15 Y 25 jami -399
Tavi XIII. korelacia _______________________________________________________________________
11
rodesac X da Y qulebi erTmaneTTan kavSirSi ar aris, XYCP nuls
uaxlovdeba. XYCP im SemTxvevisaTvis, roca X da Y ar aris kavSirSi, naCvenebia cxrilze #13.3b. suraTze #13.5 am qulebis wertilTa gabnevis diagramaa naCvenebi. yuradReba miaqcieT imas, rom X da Y qulebi ar korelirebs: X -is maRali maCveneblebi SeTanxmebulad ar aris dakavSirebuli Y -is maRal an dabal qulebTan. msgavsadve, X -is dabali qulebi ar aris SeTanxmebulad dakavSirebuli Y -is maRal an dabal
qulebTan. cxrilze #13.3b gamoTvlili XYCP = -12 Seesabameba kavSiris ararsebobas. suraTi #13.5 10 cdis piris martoobisa da depresiis qulebs Soris kavSiri ar arsebobs. qulebi aRebulia 13.3b cxrilidan.
r : XYCP -is Sedareba monacemebis saerTo cvalebadobis sazomTan
XYCP gamoxatavs or cvlads Soris kavSiris mimarTulebas, magram XYCP –is
ricxviTi mniSvneloba damokidebulia X –isa da Y –is gazomvis erTeulze da Cvens xelT arsebuli qulebis raodenobaze. qulebis or gansxvavebul
simravles SeiZleba hqondes erTnairi siZlieris kavSiri, magram XYCP -is gansxvavebuli mniSvnelobebi, radgan am qulebiT gazomilia gansxvavebuli
martooba depresia
cdis piri X XX − Y YY − ( )( )YYXX −−
1 24 +7 14 -12 -84 2 16 -1 36 +10 -10 3 4 -13 24 -2 +26 4 12 -5 16 -10 +50
5 22 +5 28 +2 +10
6 30 +13 26 0 0 7 10 -7 34 +8 -56 8 28 +11 30 +4 +44 9 14 -3 28 +2 -6 10 10 -7 24 -2 +14
X 17 Y 26 jami _12
Tavi XIII. korelacia _______________________________________________________________________
12
erTeulebi an qulebis erTi simravle ufro didi SerCevidanaa miRebuli,
vidre meore. imisaTvis, rom XYCP , rogorc X -isa da Y –is korelaciis sazomi gamosadegi iyos, is warmodgenili unda iyos iseTi formiT, romelic standartul mniSvnelobebs miiRebs gazomvis erTeulis da monacemebis raodenobis miuxedavad. jvaredina namravlebi standartul mniSvnelobebs maSin iRebs, rodesac maT qulebis TiToeuli simravlis saerTo cvalebadobis sazomebs vadarebT. cvalebadobis erT-erTi sazomi, romelsac Cveulebriv viyenebT, aris kvadratebis jami (SS). X cvladiT aRniSnuli monacemebisaTvis
XSS = ( )∑ −2
XX da Y cvladiT aRniSnuli monacemebisaTvis YSS = ( )∑ −2
YY . imisaTvis, rom SS gadaviyvanoT jvaredina namravlis erTeulebSi,
ToToeuli SS -dan viRebT kvadratul fesvs. Semdeg XSS -is kvadratuli
fesvis gamravlebiT YSS –is kvadratul fesvze viRebT monacemTa ori
simravlis saerTo cvalebadobis sazoms, ( )( )YX SSSS . pirsonis korelaciis
koeficienti, r , aris XYCP –is Sefardeba saerTo cvalebadobis am sazomTan. Sesabamisad, Cven vxedavT, rom Semdegi ori formula r -is misaRebad erTsa da imave ricxviT mniSvnelobebs iyenebs:
( )( )( )[ ] ( )[ ]∑∑∑
−−
−−=
22YYXX
YYXXr
da
( )( )YX
XY
SSSSCPr = .
yuradReba miaqcieT imas, rom r –is formulis mniSvneli yovelTvis
dadebiT mniSvnelobas iRebs, radgan sxvaobis kvadrats Seicavs. maSasadame, korelaciis koeficinetis dadebiT (+) Tu uaryofiT (_) niSans mricxvelis niSani gansazRvravs. cxrilSi #13.4 naCvenebia cxrilis #13.1 10 qulisaTvis ganmartebiTi formulis gamoyenebiT r –is gamoTvla. TiToeuli cdis pirisaTvis:
• sveti a aris martoobis qula
• sveti b aris cdis piris martoobis qulisa da martoobis qulebis
saSualos Soris sxvaoba (anu ( XX − )) • sveti g aris b svetis mniSvnelobis kvadrati
• sveti d aris depresiis qula
• sveti e aris cdis piris depresiis qulisa da depresiis qulebis
saSualos Soris sxvaoba (anu ( YY − )) • sveti v aris e svetis mniSvnelobis kvadrati
• sveti z warmoadgens b da e svetebis namravls (( XX − )( YY − )), anu
XYCP -s.
XSS miiReba g svetis monacemebis dajamebiT, YSS _ v svetis monacemebis
dajamebiT da XYCP _ z svetis monacemebis dajamebiT. Semdeg xdeba am jamebis formulaSi Casma r –is ricxviTi mniSvnelobis misaRebad, romelsac
Tavi XIII. korelacia _______________________________________________________________________
13
realurir ewodeba (ix. cxrilis #13.4 me-2 safexuri). cxrilis #13.1
monacemebisaTvis, realurir = +0,78. sanam realurisr interpretacias ganvixilavde,
warmogidgenT or alternatiul formulas, romlebic xSirad gamoiyeneba korelaciis koeficientis gamosaTvlelad. cxrili #13.4 r –is gamoTvla cxrilis #13.1 martoobisa da depresiis qulebisaTvis ganmartebiTi formulis gamoyenebiT.
safexuri 1. viTvliT ( )∑ −=2
XXSS X -s, ( )∑ −=2
YYSSY -s da
( )( )∑ −−= YYXXCPXY -s.
(a) (b) (g) (d) (e) (v) (z)
martooba
depresia
cdis piri
X XX − ( )2XX − Y YY − ( )2YY − ( )( )YYXX −−
1 4 -13 169 16 -12 144 +156 2 27 +10 100 37 +9 81 +90 3 18 +1 1 33 +5 25 +5 4 7 -10 100 23 -5 25 +50 5 30 +13 169 34 +6 36 +78 6 12 -5 25 32 +4 16 -20 7 18 +1 1 24 -4 16 -4 8 23 +6 36 29 +1 1 +6 9 19 +2 4 26 -2 4 -4 10 12 -5 25 26 -2 4 +10
X 17
XSS 630 Y 28
YSS 352 XYCP +367,0
safexuri 2. viTvliT r -s:
( )( )( )( ) ,367
,352
,6302
2
∑∑∑
+==−−
==−
==−
XY
Y
X
CPYYXX
SSYY
SSXX
Dda
( )( )( )[ ] ( )[ ]∑∑∑
−−
−−=
22YYXX
YYXXr
an
( )( ).
YX
XY
SSSSCPr =
ricxviTi mniSvnelobebis CasmiT:
Tavi XIII. korelacia _______________________________________________________________________
14
( )( ).78,0779,0
352630367
+=+=+
=realurir
pirsonis korelaciis koeficientis alternatiuli formulebi r -is standartuli qulebis formula r -is ganmartebiTi formula SeiZleba Caiweros mniSvnelis dayofiT mis Semadgenel kvadratebis or jamad:
( )( )( )[ ] ( )[ ]∑∑∑
−−
−−=
22YYXX
YYXXr
an
( )( )( )( )YX SSSS
YYXXr ∑ −−=
yuradReba miaqcieT, rom ( XX − ) / SSx Zalian hgavs standartul qulas, z = ( XX − )/S, romelic me-6 TavSia warmodgenili. imisaTvis, rom es
gamosaxuleba standartul monacemad gardavqmnaT da miviRoT XS , X
monacmebis SerCevis standartuli gadaxra, saWiroa XSS gavyoT N-ze, anu
XSS –is Semadgeneli monacemebis raodenobaze. igive msjeloba exeba ( YY − ) / YSS –sac. Sesabamisad, r -is standartuli qulis formulaa:
( )wyviliN
zzr YX∑= ,
sadac: Xz aris ( XX − )/ XS
Yz aris ( YY − ) / YS wyviliN aris monacemebis wyvilebis raodenoba
XS -s, X qulebis SerCevis standartul gadaxras, Semdegnairad vpoulobT:
( )N
XXS X
∑ −=
2
da YS -s, Y monacemebis SerCevis standartul gadaxras, Semdegnairad:
Tavi XIII. korelacia _______________________________________________________________________
15
( )N
YYSY
∑ −=
2
cxrili #13.5 gviCvenebs, rogor xdeba am formulis gamoyeneba realurisr
gamosaTvlelad cxrilis #13.1 monacemebisaTvis. pirveli safexuria z qulebis gamosaTvlelad saWiro XS –is da YS –is miReba. am safexurze
mocemuli ( )2XX −∑ = 630 da ( )2YY −∑ = 352 aRebulia cxrilidan #13.4. am
gamoTvlebiT Cven vigebT, rom X qulebis standartul qulebad
gardasaqmnelad Xz = ( X - 17,0)/7,937. msgavsadve, Y monacemebisaTvis Yz = (Y - 28,0)/5,933 (ixileT cxrilis me-2 safexuri). mesame safexurze xdeba TiToeuli nedli qulis standartul qulad gardaqmna z-is formulebis gamoyenebiT. me-3 safexuris a svetSi warmodgenilia TiToeuli cdis piris X qula. b svetSi warmodgenilia
XX − sxvaoba TiToeuli cdis pirisaTvis da g svetSi - TiToeuli cdis piris z qula. d svetSi warmodgenilia TiToeuli cdis piris Y qula, e
svetSi warmodgenilia YY − sxvaoba da v svetSi _ z qula ToToeuli Y –
Tvis. z svetSi gamravlebulia ToToeuli cdis piris Xz da Yz qulebi da
am svetis monacemebis jami, +7,794, gvaZlevs ( )∑ YX zz sidides, rac saWiroa r –is formulis mricxvelisaTvis. cxrilis me-4 safexuri gviCvenebs, Tu rogor xdeba ( )∑ YX zz –isa da
wyvilisN mniSvnelobebis Casma formulaSi. miRebulia realurir = +0,78, anu igive
mniSvneloba, rac ganmartebiTi formulis gamoyenebiT cxrilSi #13.4. rodesac Tqveni monacemebi standartuli qulebiT aris warmodgenili, girCevT, standartuli qulebis meTodi gamoiyenoT. Tu monacemebi standartuli qulebiT ar aris warmodgenili, maSin girCevT, gamoiyenoT qvemoT mocemuli gamoTvliTi formula. cxrili #13.5 r –is gamoTvla cxrilis #13.1 martoobisa da depresiis qulebisaTvis standartuli qulis formulis mixedviT
safexuri I. XS -isa da YS –is miReba:
( )wyviliN
XXS X
∑ −=
2
, ( )wyviliN
YYSY
2−
= ∑
cxrilis #13.4 g svetidan
cxrilis #13.4 v svetidan
( ) 6302=−∑ XX
wyviliN =10
.937,710
0,630==XS
( ) 3522=−∑ YY
wyviliN =10
933,510
0,352==YS
safexuri II. Xz -is, Yz -sa da YX zz∑ -is miReba:
X qulebis standartul qulebSi gadasayvanad
Y qulebis standartul qulebSi gadasayvanad
Tavi XIII. korelacia _______________________________________________________________________
16
0,17=X 937,7=XS
maSasadame,
937,70,17−
=Xz X
0,28=Y 933,5=YS
maSasadame,
.933,5
0,28−=
YzY
safexuri III. nedli qulebis z qulebSi gadayvana
(a) (b) (g) (d) (e) (v) (z) martooba depresia
cdis piri
X XX − Xz Y YY − Yz Xz Yz
1 4 -13,0 -1,638 16 -12,0 -2,023 +3,314 2 27 +10,0 +1,260 37 +9,0 +1,517 +1,911 3 18 +1,0 +0,126 33 +5,0 +0,843 +0,106 4 7 -10,0 -1,260 23 -5,0 -0,843 +1,062 5 30 +13,0 +1,638 34 +6,0 +1,011 +1,656 6 12 -5,0 -0,630 32 +4,0 +0,674 -0,425 7 18 +1,0 +0,126 24 -4,0 -0,674 -0,085 8 23 +6,0 +0,756 29 +1,0 +0,168 +0,128 9 19 +2,0 +0,252 26 -2,0 -0,337 -0,085 10 12 -5,0 -0,630 26 -2,0 -0,337 +0,212 X 17,0 Y 28,0 jami +7,794
safexuri IV. r -is gamoTvla:
( )∑ += 794,7YX zz (ixileT z sveti),
wyviliN =10,
( )wyviliN
zzr YX∑= .
ricxviTi mniSvnelobebis CasmiT,
.78,0779,0
10794,7
+=+=
+=realurir
cxrili #13.6 cxrilis #13.1 martoobisa da depresiis qulebisaTvis gamoTvliTi formulis mixedviT r –is gamoTvla
safexuri I. ∑ X -is, ∑Y -is, ∑ 2X -is, ∑ 2Y -sa da ∑ XY -is gamoTvla:
(a) (b) (g) (d) (e) cdis piri
X
Y
2X
2Y
XY 1 4 16 16 256 64 2 27 37 729 1369 999 3 18 33 324 1089 594 4 7 23 49 529 161 5 30 34 900 1156 1020 6 12 32 144 1024 384 7 18 24 324 576 432 8 23 29 529 841 667
Tavi XIII. korelacia _______________________________________________________________________
17
9 19 26 361 676 494 10 12 26 144 676 312
jamebi 170,0 280,0 3520,0 8192,0 5127,0
safexuri II. r -is gamoTvla:
( ) ( )
105127
3520
900.28170
170
2
22
=
=
=
==
=
∑∑∑∑
wyviliNXY
X
X
X
( ) ( )8192
400.78280
280
2
22
=
==
=
∑∑∑
Y
Y
Y
( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑
∑∑∑−
−=
YYNXXN
YXXYNr
wyviliwyvili
wyvili
ricxviTi mniSvnelobebis CasmiT miviRebT:
( )( ) ( )( )( )( )[ ] ( )( )[ ]
( )( ).78,0779,0
14,47093670
400.78920.81900.28200.35600.47270.51
400.78819210900.28352010280170512710
+=+=+
=−−
−=
−−−
=realurir
r-is gamoTvliTi formula r-is mniSvnelobis gamoTvliTi formulaa:
( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑
∑∑∑−
−=
YYNXXN
YXXYNr
wyviliwyvili
wyvili
es formula iyenebs uSualod nedl qulebs da ar saWiroebs qulasa da qulebis saSulos Soris sxvaobis gamoTvlas. cxrili #13.6 gviCvenebs, Tu rogor xdeba am formulis gamoyeneba cxrilis #13.1 monacemebisaTvis. cxrilis #13.6 a da b svetebi warmoadgens Sesabamisad, martoobis ( X ) da
depresiis (Y ) qulebs. am svetebis monacemebis jamebi, ∑ X = 170 da ∑Y = 280,
gvaZlevs formulisaTvis saWiro ∑ X –sa ∑Y –s. g sveti gviCvenebs
ToToeul X 2 –s da am svetis monacemebis jami, 3520, gvaZlevs 2∑ X -s. d
sveti warmoadgens TiToeul Y 2 -s da am svetis monacemebis jami, 8192,
∑ 2Y -ia. e svetSi miRebulia X Y -is mniSvneloba TiToeuli cdis
pirisaTvis da am svetis monacemebis jami, 5127, ∑ XY -is mniSvnelobaa. cxrilis #13.6 me-2 safexuri gviCvenebs r –is formulaSi ricxviTi
mniSvnelobebis Casmas. realurir = +0,78. yuradReba miaqcieT imas, rom
Tavi XIII. korelacia _______________________________________________________________________
18
miuxedavad imisa, Tu ra formula iqna gamoyenebuli (ganmartebiTi, z qulis,
Tu gamoTvliTi), cxrilis #13.1 monacemebisaTvis realurir = + 0,78.
aq saWiroa kidev erT mniSvnelovan sakiTxze gamaxvildes yuradReba. rogorc aRvniSneT, cvladebis X -iT an Y -iT aRniSvna absoluturad pirobiTia. Cvens magaliTSi martoobis qula aRiniSna X -iT da depresiis qula Y -iT. Tumca, aRniSvna rom Segvebrunebina da depresiis qula aRgveniSna X -iT da martoobis qula - Y -iT, TiToeuli formulis gamoyenebiT warmoebuli gamoTvlebis Sedegebi iseTive iqneboda, rogoric mocemulia cxrilebSi #13.4, 13.5 da 13.6. cxrilis #13.1 martoobisa da depresiis qulebs Soris korelacia aris 0,78 miuxedavad imisa, Tu gamoTvlis miznebisaTvis romeli cvladi aRiniSna X -iT da romeli - Y -iT. r -is formulebi - Sejameba
r -is 4 sxvadasxva formulis Sejameba mocemulia cxrilSi #13.7. yvela
formula realurisr erTsa da imave mniSvnelobas iZleva, roca maT qulebis
erTi simravlisTvis viyenebT. cxrili #13.7 r -is formulebis Sejameba ganaszRvrebiTi formula
( )( )( )[ ] ( )[ ]∑∑∑
−−
−−=
22YYXX
YYXXr
XYCP -iT, XSS -Ta da YSS -iT Cawerili ganmartebiTi formula
( )( )YX
XY
SSSSCPr =
standartuli qulis formula
( )wyviliN
zzr YX∑=
gamoTvliTi formula
( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑
∑∑∑−
−=
YYNXXN
YXXYNr
wyviliwyvili
wyvili
SeamowmeT Tqveni codna XIII-2
Tavi XIII. korelacia _______________________________________________________________________
19
1. gansazRvreT: jvaredina namravli, XYCP , r , XSS , YSS 2. axseniT, Tu ratom aris jvaredina namravlebi monacemTa or
simravles Soris korelaciis sazomi. 3. mocemulia Semdegi monacemebi:
cdis piri
martooba
depresia
1 4 19 2 12 20 3 28 33 4 18 32 5 14 26 6 6 19 7 22 28 8 23 36 9 15 30 10 18 27
a. am qulebisaTvis ganmartebiTi formulis gamoyenebiT
gamoTvaleT realurir .
b. am kiTxvis qulebi gadmotanilia davalebidan SeamowmeT
Tqveni codna XIII-1, kiTxva #3, cxrili #1. realurir SeadareT im kiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT, raTa TvalsaCinod warmoidginoT
mimarTeba, romelsac realurir aRwers.
4. mocemulia Semdegi monacemebi:
cdis piri
martooba
depresia
1 5 29 2 11 31 3 23 23 4 17 17 5 28 16 6 13 21 7 12 26 8 4 28 9 21 14 10 16 25
a. am monacemebisaTvis standartuli qulis formulis
gamoyenebiT gamoTvaleT realurir .
am kiTxvis monacemebi gadmotanilia davalebidan SeamowmeT
Tqveni codna XIII_1, kiTxva #3, cxrili #2. realurir
SeadareT im kiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT, raTa TvalsaCinod
warmoidginoT mimarTeba, romelsac realurir aRwers.
b. mocemulia Semdegi monacemebi:
Tavi XIII. korelacia _______________________________________________________________________
20
cdis piri
sxeulis wona
TviTpativiscema
1 100 39 2 111 47 3 117 54 4 124 23 5 136 35 6 139 30 7 143 48 8 151 20 9 155 28 10 164 46
a. am monacemebisaTvis ganmartebiTi formulis gamoyenebiT
gamoTvaleT realurir .
b. am kiTxvis monacemebi gadmotanilia davalebidan SeamowmeT
Tqveni codna XIII_1, kiTxva #4. realurir SeadareT im
SekiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT imisaTvis, raTa TvalsaConod
warmoidginoT mimarTeba, romelsac realurir aRwers.
r - is Tvisebebi korelaciis koeficienti gamoxatavs or cvlads Soris wrfivi kavSiris rogorc mimarTulebas, aseve xarisxs. kavSiris mimarTuleba gamoixateba korelaciis koeficientis dadebiTi an uaryofiTi niSniT. imis aRmniSvnel xarisxs, Tu ramdenad uaxlovdeba wertilTa gabnevis diagramis wertilebis ganlageba wrfes, gvaZlevs r -is absoluturi mniSvneloba, sidide, romelic SeiZleba iRebdes 0-dan 1-mde mniSvnelobebs. niSnisa (+ an _) da ricxviTi mniSvnelobis gaTvlasiwinebiT r iRebs mniSvnelobebs -1-dan 0-is gavliT +1-mde. kavSiris mimarTuleba zemoT ganvixileT wertilTa gabnevis diagramaze qulebs Soris dadebiTi da uaryofiTi kavSirebi. korelaciis koeficienti kavSiris mimarTulebas
plus an minus niSnis meSveobiT asaxavs. dadebiTi r (mag., realurir = +0,86),
asaxavs X –sa da Y –s Soris dadebiT kavSirs; X cvladis qulebis zrdasTan erTad izrdeba Y cvladis qulebic. qulebi 13.1 da 13.2 cxrilebze
asaxavs dadebiT kavSirebs; cxrilisTvis #13.1 realurir = +0,78 da
cxrilisTvis #13.2 realurir = +1,00.
uaryofiTi r (magaliTad, realurir = _0,86) aRniSnavs cvladebs Soris
Sebrunebul kavSirs: roca X cvladis qulebi izrdeba, Y cvladis qulebi klebulobs. uaryofiTi kavSiri naCvenebia cxrilebze #13.3 da 13.4.
Tavi XIII. korelacia _______________________________________________________________________
21
cxrilisTvis #13.3 realurir udris _0,83-s da cxrilisTvis #13.4 udris _1,00-
s. kavSiris xarisxi r -is ricxviTi mniSvneloba gamoxatavs, imas, Tu ramdenad kargad aRwers wrfe X da Y cvladebs Soris mimarTebas. 1,00-is toli r niSnavs, rom kavSiri idealurad wrfivia; wertilTa gabnevis diagramaze yvela wertili wrfeze Zevs. suraTi #13.2 gviCvenebs sruli dadebiTi kavSiris amsaxvel
wertilTa gabnevis diagramas, realurir = +1,00 martoobisa da depresiis
qulebisaTvis. sruli uaryofiTi kavSiri, realurir = _1,00 warmodgenilia
suraTze #13.4. rodesac r udris plus an minus 1,00-s, X -is yoveli mniSvnelobisTvis arsebobs misi Sesabamisi Y –is mxolod erTi mniSvneloba. aseT kavSirs xedavT suraTebze #13.2 da 13.4. martoobis ToToeuli qulisaTvis arsebobs depresiis mxolod erTi qula. aseTi kavSiri gvaZlevs saSualebas, X qulebis mixedviT zustad viwinaswarmetyveloT Y . suraTebze #13.2 da 13.4 Tu iciT martoobis qula, zustad SegiZliaT iwinaswarmetyveloT am qulasTan dakavSirebuli depresiis qula. meore ukidures SemTxveaSi, Tu cvladebs Soris kavSiri ar arsebobs, maSin korelaciis koeficientis mniSvneloba nuls uaxlovdeba. rodesac r daaxloebiT 0-is tolia, erTi qulis mniSvneloba ar aris kanonzomierad dakvSirebuli meore qulis mniSvnelobasTan. aseTi magaliTi naCvenebia
suraTze #13.5, sadac realurir = _0,2.
korelaciis koeficientis 0-dan +1,00-mde mniSvnelobebi gamoxatavs, rom X –sa da Y –s Soris arsebobs kavSiri, magram ara sruli kavSiri. TvalsaCinoebisTvis aRvniSnavT, rom suraTze #13.1, romlisTvisac r = +0,78, X da Y cvladebi ar aris srul kavSirSi. martoobis qulebis zrdasTan erTad depresiis qulebic izrdeba, magram martoobis qulis codna depresiis qulis zusti winaswarmetyvelebis saSualebas ar gvaZlevs. magaliTad, orma cdis pirma miiRo 12 qula martoobis skalaze, magram depresiis skalaze erT cdis pirs hqonda 32 qula da meores – 26 qula. cxrilze #13.3 warmodgenilia msgavsi uaryofiTi kavSiri,
romlisTvisac realurir = _0,83. am SemTxvevaSi martoobis qulebis zrdisas
depresiis qulebi mcirdeba. magram kavSiri aqac ar aris sruli; martoobis qulis codna ar gvaZlevs masTan dakavSirebuli depresiis qulis zusti winaswarmetyvelebis saSualebas. magaliTisaTvis, yuradReba miaqcieT, rom orma cdis pirma miiRo 20 qula martoobis skalaze, magram maTgan erTma depresiis skalaze miiRo 17, meorem ki _ 24 qula. xSirad Znelia, TvalsaCinod warmovidginoT, Tu rogor saxes miiRebda r –is mniSvneloba, monacemebi wertilTa gabnevis diagramaze rom yofiliyo warmodgenili. es SesaZlebeli xdeba mxolod r –is mniSvnelobebis im qulebis wertilTa gabnevis diagramasTan SedarebiT, romlebisTvisac korelacia gamovTvaleT. suraTi #13.6 warmoadgens r –is gansxvavebuli mniSvnelobebis Sesabamis wertilTa gabnevis diagramebs. a-dan e-mde grafikebi asaxavs 1,00-dan 0,00-mde korelaciebs. yuradReba miaqcieT imas, rom r –is klebasTan erTad X –sa da Y –s Soris wrfivi
Tavi XIII. korelacia _______________________________________________________________________
22
kavSiri naklebad Zlieri xdeba, sanam realurir = 0,00, roca kavSiri aRar
arsebobs. pirsonis r aris X da Y cvladebs Soris wrfivi kavSiris sazomi. grafiki #13.6v X da Y cvladebs Soris arawrfiv, U-s formis kavSirs warmoadgens. am kavSiris dros X –is mcire mniSvnelobebi dakavSirebulia Y –is did mniSvnelobebTan. garkveul wertilamde X -is matebasTan erTad Y mcirdeba; X –is Semdgomi mateba ki dakavSirebulia Y –is matebasTan. X –sa da Y –s Soris aseTi kavSiris wrfiT warmodgena SeuZlebelia da misi aRwerisaTvis r -is gamoyeneba ar SeiZleba. suraTi #13.6 wertilTa gabnevis diagramebi gamoxatavs X da Y cvladis monacemebs Soris sxvadasva korelaciebs 10 cdis piris qulebisaTvis. qulebis nakrebebs Soris
pirsonis korelaciis koeficientis mniSvneloba mocemulia realurir -iT.
(a) realurir =+1,0 (b) realurir =-1,0 (g) realurir =-0,49
(d) realurir =+0,25 (e) realurir =0,00 (v) arawrfivi
diapazonis SezRudva da r –is mniSvneloba r -is mniSvneloba mgrZnobiarea mniSvnelobaTa im diapazonisadmi, rac X –ma da Y –ma SeiZleba miiRos. erT-erTi an orive cvladis mniSvnelobaTa diapazonis SemcirebasTan erTad r –is mniSvnelobac mcirdeba. aseTi kavSiris sailustraciod ganvixiloT suraTis #13.1 monacemebi,
romelTaTvisac realurir = +0,78. davuSvaT, rom minda vikvlio kavSiri
martoobasa da depresias Soris mxolod im 6 cdis pirTan, romelTac martoobis yvelaze maRali qulebi aqvT. es aris martoobis qulebi 18-dan 30-mde (anu martoobis qulebi ## 2, 3, 5, 7, 8 da 9 cdis pirebisagan). depresiis qulebis diapazoni am cdis pirebisaTvis 24-dan 37-mdea. aTive cdis piris martoobisa da depresiis monacemebis diapazonTan SedarebiT, maRali qulebis mqone cdis pirebis martoobisa da depresiis monacemebis
diapazoni SeizRuda, anu damoklda. am eqvsi monacemisaTvis realurir aris
+0,69, rac ufro naklebia, vidre +0,78-is toli realurir , monacemebis sruli
simravlis SemTxvevaSi. am magaliTis mniSvneloba naTelia. SezRuduli diapazonis mqone monacemebisTvis gamoTvlilma korelaciis koeficientma SeiZleba gvafiqrebinos, rom or cvlads Soris kavSiri ar arsebobs, maSin, roca kavSiris aRmoCena SesaZlebeli iqneboda, monacemebis ufro didi diapazoni rom gamogveyenebina. amitom, korelaciis koeficientis gamoTvlamde kargad SeiswavleT mopovebuli monacemebi, raTa gansazRvroT, warmoadgenen Tu ara isini SezRudul diapazons. determinaciis koeficienti
determinaciis koeficienti (Coefficient of determination): 2r -is mniSvneloba, romelic X da Y cvladebis saerTo cvalebadobas asaxavs.
Tavi XIII. korelacia _______________________________________________________________________
23
me-9 TavSi warmovadgine 2η (eta kvadrati), rogorc eqsperimentSi damoukidebeli cvladis damokidebul cvladze gavlenis siZlieris sazomi.
korelaciur kvlevebSi determinaciis koeficienti, 2r ( r -is mniSvnelobis kvadrati), warmoadgens X –sa da Y –s Soris kavSiris siZlieris analogiur sazoms.
2r -is interpretacia 2η –is interpretaciis msgavsia. kerZod, 2r gamoxatavs, qulebis erTi simravlis cvalebadobis ra proporciaa dakavSrebuli qulebis meore simravlis cvalebadobasTan. magaliTad,
cxrilis #13.1 monacemebisaTvis, r = +0,78; maSasadame, realuri2r -s
mniSvnelobaa (+0,78)², anu 0,61. es sidide niSnavs, rom qulebis am wyvilebs saerTo aqvT cvalebadobis 61 procenti. sxva sityvebiT, martoobis qulebis cvalebadobis 61 procenti depresiis qulebis cvalebadobasTan aris dakavSirebuli. korelacia da mizezobrioba bevri cvladi Zlier kavSirSia erTmaneTTan, Tumca X da Y cvladebs Soris korelaciis arseboba ar niSnavs, rom erTi iwvevs meores. Tu X da Y cvladebi korelaciaSia, maSin am cvladebs Soris mizez-Sedegobrivi kavSiris sul mcire sami varianti arsebobs:
• X iwvevs Y –s.
• Y iwvevs X –s
• arc X da arc Y mizezobriv gavlenas ar axdens erTmaneTze, orive gamowveulia mesame, Z cvladiT.
ganvixiloT magaliTi martoobisa ( X cvladi) da depresiis (Y cvladi) korelaciis Sesaxeb. Cvens magaliTSi miviReT +0,78 korelacia, rac niSnavs, rom SedarebiT martosuli adamianebi ufro depresiulebic arian. romeli cvladi axdens meoreze mizez-Sedegobriv gavlenas, Tu aseTi gavlena arsebobs? martooba iwvevs depresias? Tu depresiuloba iwvevs martoobas? Tu am SemTxvevaSi arc erTi aseTi gavlena ar arsebobs, da mesame cvladi, Z, magaliTad megobruli urTierTobebis CamoyalibebisaTvis saWiro socialuri unarebis nakleboba, iwvevs adamianis rogorc martosulobas, aseve depresiulobas? es sami mizez-Sedegobrivi mimarTeba naCvenebia cxrilze #13.8. or cvlads Soris korelacia miuTiTebs mxolod imaze, rom isini wrfivad arian dakavSirebuli; is ar gvawvdis informacias imis Sesaxeb, Tu ra iwvevs maT kavSirs. sxva magaliTSi sigaretis moweva ( X cvladi) da stresis done (Y cvladi) dadebiT kavSirSia; sigaretis mwevelebs Cveulebriv stresis ufro maRali done aqvT, vidre aramwevelebs (Parrott, 1999). es kavSiri korelaciuria da miRebulia adamianebis gamokiTxviT sigaretis mowevis Cvevisa da stresis donis Sesaxeb. am SemTxvevaSi saeWvoa ara TavisTavad kavSiri, aramed am kavSiris mizezi (Kassel, 2000). am kavSiris asaxsnelad gamoTqmuli erT-erTi varaudis Tanaxmad, mwevelebs stresis ufro maRali done aqvT sxvadasxva mizezis gamo (Z cvladi), rogoricaa aramwevelebTan SedarebiT ufro Zlieri samsaxureobrivi an ojaxuri stresebi da moweva am stresebis Sesustebis meTodia. meore SemoTavazebuli axsnis mixedviT, stresis maRal dones realurad moweva iwvevs da is nikotinze
Tavi XIII. korelacia _______________________________________________________________________
24
damokidebulebis Sedegia. Tumca martiv korelaciur aRmoCenas ar SeuZlia upasuxos kiTxvas, Tu romeli axsnaa WeSmariti am oridan. sxva kvlevaSi miRebuli iyo uaryofiTi korelacia Ramis Zilis xangrZlivobasa ( X cvladi) da dRis ganmavlobaSi energiis dones (Y cvladi) Soris; is adamianebi, romelTac naklebad sZinavT, acxadeben, rom dRis ganmavlobaSi meti energia aqvT. SeiZleba Tu ara am mimarTebidan davaskvnaT, rom Tu dRis ganmavlobaSi energiis naklebobas ganicdi, RamiT naklebi unda iZino? pasuxia _ ara, radgan korelaciuri kavSiri ar gvaZlevs aseTi mizez-Sedegobrivi daskvnis gakeTebis saSualebas. albaT, miRebul mimarTebaze pasuxismgebelia mesame cvladi, rogoricaa fiziologiuri mdgomareoba, romelic gavlenas axdens dRis ganmavlobaSi energiis raodenobazec da Zilis saWiro xangrZlivobazec. am magaliTebidan gamomdinare, yovelTvis unda gvaxsovdes, rom korelaciuri kavSiri gamoxatavs mxolod or cvlads Soris kavSirs, magram arafers gveubneba am kavSiris mizezze. cxrili #13.8 ori cvladis, martoobisa ( X ) da depresiis (Y ), korelaciis Sesabamisi SesaZlo mimzez-Sedegobrivi kavSirebi.
a) miRebuli korelacia:
miRebulia X da Y cvladebs Soris dadebiTi korelacia
adamianebs, romelTac aqvT martoobis
maRali qula ( X ), aqvT depresiis maRali
qula (Y ).
adamianebs, romelTac aqvT martoobis
dabali qula ( X ), aqvT depresiis dabali
qula (Y ).
b) miRebuli korelaciis amxsneli SesaZlo mizez-Sedegobrivi kavSirebi:
pirveli SesaZlebloba X cvladi iwvevs Y cvlads
pirovnebis martosuloba ( X ) iwvevs mis depresiulobas (Y ).
meore SesaZlebloba Y cvladi iwvevs X cvlads
pirovnebis depresiuloba (Y ) iwvevs mis martosulobas ( X ). mesame SesaZlebloba
Z cvladi iwvevs X da Y cvladebs saTanado socialuri unarebis nakleboba ( Z cvladi) iwvevs pirovnebis martosulobasa ( X ) da depresiulobas (Y ).
Tavi XIII. korelacia _______________________________________________________________________
25
SeamowmeT Tqveni codna XIII-3
1. ras gveubneba r –is plus an minus miSani X da Y cvladebis kavSiris Sesaxeb?
2. ras gveubneba r –is ricxviTi mniSvneloba X da Y cvladebis kavSiris Sesaxeb?
3. ra gavlena SeiZleba hqondes X an Y cvladis monacemebis diapazonis SezRudvas r –is mniSvnelobaze?
4. gamoTvaleT determinaciis koeficienti realurisr Semdegi
mniSvnelobebisaTvis:
a. +0,12 g. _0,74 e. _0,59 b. _0,36 d. +0,23 v. +0,19
5. Tqven akvirdebiT kavSirs X da Y cvladebs Soris.
gansazRvreT is sami SesaZlo mizez-Sedegobrivi kavSiri, riTac SeiZleba korelaciis axsna.
6. mravalma mkvlevarma daadgina qvemoT CamoTvlili korelaciebi. gansazRvreT mesame cvladi, romelic, SesaZloa, miRebul korelaciaze pasuxismgebeli iyos a. uaryofiTi korelacia Semosavalsa da sisxlis wnevas
Soris b. dadebiTi korelacia yavis smasa da sigaretis mowevas
Soris g. dadebiTi korelacia bavSvebis mier televiziiT Zaladobis
yurebasa da maT agresiul qcevas Soris.
pirsonis r koeficientis statistikuri mniSvnelovnobis Semowmeba
populaciis korelacia, ρ
pirsonis r koeficienti aRwers SerCevis X da Y qulebs Soris wrfiv kavSirs. rogorc qulebis nebismieri simravlis SemTxvevaSi, SerCeva miRebulia qulebis ufro didi populaciidan da SeiZleba gvsurdes, SevawmoT, arsebobs Tu ara X da Y monacemebs Soris korelacia populaciaSi. populaciaSi X da Y monacemebs Soris wrfiv kavSirs warmoadgens parametri ρ (berZnuli anbanis me-17 aso, ro ). martoobisa da depresiis qulebis korelaciis Sesaxeb magaliTSi miRebuli realurir = +0,78,
gvaZlevs ρ–s savaraudo mniSvnelobas. Tumca, SerCevis Secdomis gamo,
realurir ρ–s toli ar iqneba. radgan SerCevis safuZvelzea miRebuli,
sxvadasxva SerCevisaTvis gamoTvlili realurir , iseve rogorc nebismieri sxva
statistika, sxvadasxva mniSvnelobas mogvcems, miuxedavad imisa, rom es SerCevebi aRebulia EerTi da imave populaciidan. aqedan gamomdinare,
SesaZlebelia, SevamowmoT hopoTeza imis Sesaxeb, rom realurir axasiaTebs
garkveuli ρ–s mqone populaciidan aRebul SerCevas. Tumca, SesaZlebelia SevamowmoT hopoTeza ρ–s nebismier mniSvnelobasTan dakavSirebiT, yvelaze
ρ : ro, populaciis korelaciis koeficientis aRmniSvneli simbolo.
Tavi XIII. korelacia _______________________________________________________________________
26
gavrcelebulia Semowmeba hopoTezisa imis Sesaxeb, rom im populaciaSi, saidanac aRebulia SerCeva, ρ nulis tolia. aranulovani korelaciis Semowmeba populaciaSi statistikuri Semowmeba imisa, SemTxveviTad gansxvavdeba Tu ara r populaciaSi nulovani korelaciisagan martivia. saWiroa hopoTezebis statistikuri Semowmebis nacnobi safexurebis gavla:
• vaxdenT nulovani hopoTezis, 0H -isa, da alternatiuli hipoTezis,
1H -is, formulirebas;
• vadgenT r -is SerCeviT ganawilebas im varaudiT, rom 0H WeSmaritia.
es ganawileba mocemulia cxrilSi #A.5; • virCevT mniSvnelovnobis dones;
• cxrilSi #A.5 mocemuli r –is SerCeviTi ganawilebis mixedviT vpoulobT r -is kritikul mniSvnelobas, romelic aRiniSneba,
rogorc kritikulir ;
• r -is SerCeviT ganawilebaSi ganvsazRvravT uaryofis ares.
• SerCevis monacemebis mixedviT viTvliT korelaciis koeficients,
romelic aRiniSneba realuriTr ;
• imis mixedviT, xvdeba Tu ara realurir uaryofis areSi, viRebT
gadawyvetilebas 0H –is miRebis an uaryofis Sesaxeb.
statistikuri hipoTeza nulovani hopoTeza. nulovani hopoTeza aseTia:
0H : ρ = 0 es hipoTeza ambobs, rom populaciaSi X da Y cvladebs Soris korelacia nulis tolia. alternatiuli hipoTeza. alternatiuli hipoTeza Semdegnairia:
1H : ρ ≠ 0 es hipoTeza ambobs, rom populaciaSi X da Y cvladebs Soris korelacia ar aris nulis toli. r -is SerCeviTi ganawileba da kritikuli mniSvnelobebi radgan r –s SeuZlia 0-dan 1-mde mniSvnelobebi miiRos, misi SerCeviTi
ganawileba 0H : ρ = 0–sTvis SeiZleba advilad ganisazRvros. Tumca, rogorc
sxva statistikuri kriteriumebis SemTxvevaSi, r -is SerCeviTi ganawileba
damokidebulia mis Tavisuflebis xarisxze. r –saTvis df udris wyviliN – 2,
sadac wyviliN aris im qulebis wyvilebis raodenoba, saidanac gamoTvlilia
r . r –is kritikuli mniSvnelobebi, anu kritikulir , r –is SerCeviTi
Tavi XIII. korelacia _______________________________________________________________________
27
ganawilebebidan im SemTxvevebisTvis, roca df meryeobs 1-dan 100-mde, da roca α = 0,05 da 0,01-s, warmoadgenilia danarTis cxrilSi #A.5. uaryofis zonis dadgena
Tu 1H WeSmaritia, mosalodnelia, rom r –is iseT mniSvnelobebs miviRebT,
romlebic aRemateba nuls. Sesabamisad, realurisr uaryofis zona aris r –is is
mniSvneloba, romelic udris an metia kritikulzer . roca realurir absoluturi
mniSvnelobiT (anu –is + an _ niSans yuradRebas Tu ar mivaqcevT,) skritikuliir
tolia an masze metia, is xvdeba uaryofis areSi. Tu 0H WeSmaritia, aseTi
realuriisr albaToba alfas mniSvnelobis tolia an masze naklebia.
gadawyvetilebis miRebisaTvis saWiro wesebi statistikuri hipoTezis Sesaxeb gadawyvetilebis miRebis wesebi nacnobi nimuSis mixedviT xdeba:
• Tu realurir uaryofis areSia, maSin:
uaryaviT 0H : ρ = 0 miiReT 1H : ρ ≠ 0
• Tu realurir ar xvdeba uaryofis areSi, maSin
ar uaryoT 0H
ar miiRoT 1H
am process giCvenebT qvemoT mocemuli magaliTi. cxrili #13.9 ajamebs im gadawyvetilebebsa da daskvnebs, rasac r –is mixedviT viRebT. cxrili #13.9 Sejameba: statistikuri hipoTezebis r –is mixedviT Semowmebisas gadawyvetilebebis miReba da daskvnebis gakeTeba. gamoyenebulia mniSvnelovnobis 0,05 done.
Tu realurir xvdeba uaryofis areSi, roca
05,0=α , maSin:
Tu realurir ar xvdeba uaryofis areSi,
roca 05,0=α , maSin:
• realurisr albaToba 0,05-ze naklebia an
misi tolia, anu 05,0≤p
• realurisr albaToba 0,05-ze metia, anu
p > 0,05.
• uarvyofT 0H : ρ = 0-s
• uarvyofT 0H -s
• viRebT 1H : ρ ≠ 0-s. realurir gviCvenebs,
rom populaciis ρ nulisagan
gansxvavebulia.
• uarvyofT 1H -s. realurir gviCvenebs,
rom populaciis ρ nulisagan
gansxvavebulia.
• realurir statistikurad
mniSvnelovania 0,05 doneze.
• realurir ar aris statistikurad
mniSvnelovani 0,05 doneze.
• viRebT gadawyvetilebas, rom im populaciaSi, saidanac SerCeva iyo
aRebuli, X da Y cvladebi korelaciaSia.
• viRebT gadawyvetilebas, rom im populaciaSi, saidanac SerCeva iyo
aRebuli, X da Y cvladebi ar aris korelaciaSi.
Tavi XIII. korelacia _______________________________________________________________________
28
daSvebebi r -is statistikuri mniSvnelovnobis SemowmebisaTvis r -is statistikuri mniSvnelovnobis Semowmebisas igulisxmeba, rom populaciaSi X da Y organzomilebian normalur ganawilebas Yqmnis. organzomilebiani normaluri ganawilebis maxasiaTeblebia:
• X qulebi, martoobis qulebi Cvens magaliTSi, normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva;
• Y qulebi, depresiis qulebi Cvens magaliTSi, normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva;
• ToToeuli X qulis Sesabamisi Y qulebis ganawileba populaciaSi normaluria. Cveni magaliTis mixedviT, es piroba niSnavs, rom martoobis TiToeuli qulisTvis populaciaSi arsebobs depresiis qulebis normaluri ganawileba;
• ToToeuli Y qulisaTvis X qulebis ganawileba populaciaSi normaluria. es piroba wina pirobis msgavsia, magram axla depresiis TiToeuli qulisTvis populaciaSi arsebobs martoobis qulebis normaluri ganawileba.
ori qcevis gazomvisas SeiZleba davuSvaT, rom isini populaciaSi oragnzomilebian normalur ganawilebas qmnian da Sesabamisad, SegviZlia SevamowmoT hipoTeza populaciis qulebis aranulovani korelaciis Sesaxeb. organzomilebiani normaluri ganawileba (Bivariate normal distribuition): X da Y cvladebis iseTi ganawileba, romlisTvisac (1) X qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (2) Y qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (3) TiToeuli X qulisaTvis Y qulebis ganawileba populaciaSi normaluria da (4) TiToeuli Y qulisaTvis X qulebis ganawileba populaciaSi normaluria.
magaliTi #13.1
martoobasa da depresias Soris realurisr statistikuri mniSvnelovnobis
Semowmeba
amocana cxrilis #13.1 martoobisa da depresiis qulebisTvis pirsonis korelaciis
koeficientia realurir = +0,78. gamoxatavs Tu ara SerCevisaTvis miRebuli es
mniSvneloba aranulovani korelaciis arsebobas depresiisa da martoobis qulebs Soris im populaciaSi, saidanac moxda cdis pirebis SerCeva? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna: amocana moiTxovs, rom SevamowmoT statistikuri hipoTeza imis Sesaxeb, rom martoobisa da depresiis qulebisTvis populaciis korelacia, ρ, nulis tolia.
Tavi XIII. korelacia _______________________________________________________________________
29
saWiro statistika: realurir = +0,78
daSveba gamoyenebisaTvis: martoobisa da depresiis monacemebs organzomilebiani normaluri ganawileba aqvs im populaciaSi, saidanac aRebulia SerCeva.
statistikuri hipoTezebi: 0H : ρ = 0 1H : ρ ≠ 0 mniSvnelovnobis done: 05,0=α
Tavisuflebis xarisxi (df): df = wyviliN _ 2 = 10 _ 2 = 8.
r -is kritikuli mniSvneloba: cxrilis #A.5 mixedviT, kritikulir (8) = 0,632.
uaryofis zonebi: realurisr is mniSvnelobebi, romelTa absoluturi
mniSvneloba udris an metia 0,632-ze.
gadawyvetileba: realurir = +0,78-is absoluturi mniSvneloba metia, vidre
kritikulir = 0,632. aqedan gamomdinare, is xvdeba uaryofis areSi. maSasadame,
uarvyofT 0H : ρ = 0-s viRebT 1H : ρ ≠ 0-s daskvna: populaciaSi martoobisa da depresiis qulebs Soris
mniSvnelovani korelacia arsebobs. realurir = +0,78 mxolod SemTxveviT ar
gansxvavdeba populaciaSi WeSmariti nulovani korelaciisgan. populaciis
korelaciis arsebuli saukeTeso Sefaseba aris realurir = +0,78. martosuli
adamianebi depresiulebic arian. determinaciis koeficienti, realuri2r =
(+0,78)² = 0,61, gamoxatavs, rom martoobisa da depresiis qulebs cvalebadobis 61 procenti saerTo aqvT.
magaliTi #13.2
dakavSirebulia Tu ara iumoris grZnoba fizikur janmrTelobasTan? amocana erT-erTi Tviseba, romelsac sxvebis daxasiaTebisas viyenebT, aris iumoris grZnoba. qerolis da Smidtis (Caroll & Shmidt, 1992) hopoTezis mixedviT, iumoris kargi grnoba SeiZleba dakavSirebuli iyos karg fizikur janmrTelobasTan, radgan is adamianebi, romlebic SfoTvis gamomwvev situaciebs iumoriT umklavdebian, SesaZlebelia, ufro advilad axerxebdnen stresuli situaciebis kontrols. davuSvaT, rom am hopoTezas amowmebT da amisaTvis 16 cdis pirs iumoris grZobasa da fizikur
Tavi XIII. korelacia _______________________________________________________________________
30
janmrTelobas uzomavT. iumoris grZnoba izomeba situaciaze iumoriT pasuxis kiTxvariT (Situation Humor Response Questionnaire, Martin & Lefcourt, 1984). am kiTxvarSi warmodgenilia 18 situaciis aRwera da cdis pirs moeTxoveba, TiToeuli situacia 1-dan 5-mde skalaze Seafasos imis mixedviT, Tu ramdenad saxalisod eCveneba is. Sesabamisad, am kiTxvarze miRebuli qulebi SeiZleba meryeobdes 18-dan 90-mde, sadac ufro maRali qulebi miuTiTebs adamianis mier situaciebSi meti iumoris gamoyenebaze. fizikuri janmrTelobis gasazomad cdis pirebs avsebineben 20-kiTxvian TviTSefasebis kiTxvars janmrTelobis problemebis Sesaxeb. am kiTxvaris qulebi SeiZleba meryeobdes nulidan (janmrTelobis problemebis ararseboba) 20-mde (janmrTelobis 20 problema). am or sazomze Tqven Semdeg monacemebs iRebT:
cdis piri iumoris kiTxvari:
X fizikuri
janmrTeloba: Y
1 36 9 2 68 3 3 61 12 4 47 12 5 35 15 6 48 17 7 42 10 8 30 17 9 56 11 10 39 17 11 51 14 12 54 8 13 60 3 14 29 13 15 65 7 16 47 8
aris Tu ara iumoris kiTxvaris qula dakavSirebuli janmrTelobis problemebis raodenobasTan? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna: amocana moiTxovs, SevamowmoT statistikuri hipoTeza imis Sesaxeb, rom populaciaSi iumoris kiTxvarisa da janmrTelobis problemebis monacemebs Soris korelacia, ρ nulis tolia. saWiro statistika: r. daSveba gamoyenebisaTvis: iumoris kiTxvarisa da janmrTelobis monacemebs organzomilebiani normaluri ganawileba aqvs im populaciaSi, saidanac aRebulia SerCeva.
statistikuri hipoTezebi: 0H : ρ = 0 1H : ρ ≠ 0 mniSvnelovnobis done: 05,0=α
Tavi XIII. korelacia _______________________________________________________________________
31
Tavisuflebis xarisxi (df): df = wyviliN _ 2 = 16 _ 2 = 14.
r -is kritikuli mniSvneloba: cxrilis #A.5 mixedviT, kritikulir (14) = 0,497.
uaryofis zonebi: realurisr is mniSvnelobebi, romelTa absoluturi
mniSvneloba udris an metia 0,497–ze.
gamoTvla: realurisr gamosaTvlelad gamoviyenebT gamoTvliT formulas:
( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑
∑∑∑−
−=
YYNXXN
YXXYNr
wyviliwyvili
wyvili
am formulisaTvis saWiro ricxviTi mniSvnelobebia:
wyviliN = 16
∑ XY = 7908
∑ X = 768
∑Y = 176
∑ 2X = 39,092
∑ 2Y = 2242
( )2∑ X = 589.824
( )2∑Y = 30.976.
am formulaSi ricxviTi mniSvnelobebis CasmiT miviRebT:
( ) ( )( )( )[ ] ( )[ ]
65,0079,211.13
8640976.30224216824.589092.3916
176768790816−=
−=
−−−
=realurir
realuri2r = (_0,65) 2 =0,42
gadawyvetileba: realurir = _0,65-is absoluturi mniSvneloba metia, vidre
kritikulir = +0,497 da maSasadame, xvdeba uaryofis areSi. gadawyvetilebebi 0H –
sa da 1H -Tan dakavSirebiT aseTia:
uarvyofT 0H : ρ = 0-s viRebT 1H : ρ ≠ 0-s daskvna: populaciaSi, saidanac aRebulia SerCeva, arsebobs mniSvnelovani uaryofiTi korelacia iumoris koTxvaris monacemebsa da janmrTelobis problemebs Soris. im cdis pirebs, romelTac iumoris kiTxvarSi maRali qulebi miiRes, naklebi janmrTelobis problemebi hqondaT, vidre im cdis pirebs, romelTac ufro dabali qulebi miiRes iumoris skalaze.
Tavi XIII. korelacia _______________________________________________________________________
32
determinaciis koeficienti aRniSnavs, rom iumoris kiTxvarisa da janmrTelobis monacemebs saerTo aqvT cvalebadobis 42 procenti. korelaciuri kvlevis Sedegebis warmodgena
korelaciuri kvlevis angariSi unda Seicavdes realurisr mniSvnelobas,
Tavisuflebis xarisxs (df) visrealurisaTr , gamoyenebul mniSvnelovnobis dones
da informacias imis Sesaxeb, iyo Tu ara realurir statistikurad
mniSvnelovani. xSirad angariSSi mocemulia monacemebis saSualo da standartuli gadaxrac. ilustraciisaTvis gamoviyeneb martoobasa da depresias Soris korelaciis magaliTs. alfas 0,05 doneze, martoobisa (M = 17,0; SD = 8,4) da depresiis (M = 28,0; SD
= 6,3) monacemebs Soris arsebobs mniSvnelovani korelacia, r (8) = +0,78, p < 0,01, martoobisa da depresiis qulebi mniSvnelovan dadebiT kavSirSia.
___________________________________________________________________ spirmenis ranguli korelaciis koeficienti
spirmenis ranguli korelaciis koeficienti, r s (Spearman rank-order correlation coefficient): korelaciis koeficienti, romelic rigis skalaze gamoiyeneba. pirsonis r koeficienti Sesaferisia maSin, roca im cvladebis Sesabamisi qulebi, romelTa korelaciis dadgenasac vapirebT intervalebis an Sefardebis skalazea. Tumca, zogjer qulebi X da Y cvladebis rangebs warmoadgens. rodesac rigis mixedviT dalagebuli qulebi gvaqvs, Cveulebriv monacemTa or simravles Soris mimarTebis ricxviTi mniSvnelobis dasadgenad gamoiyeneba spirmenis ranguli korelaciis
koeficienti, rac aRiniSneba Sr –iT. spirmenis ranguli korlaciis
koeficienti gamoiTvleba formuliT:
( )( )
−−= ∑
16
1 2
2
wyviliwyvili NND
rs
sadac D aris cdis piris monacemebis rangebs Soris gansxvaveba wyviliN aris kvlevaSi rangebis wyvilebis raodenoba
rogorc pirsonis r koeficientis SemTxveaSi, Sr -is mniSvnelobebic
SeiZleba iyos dadebiTi an uaryofiTi da iRebdes mniSvnelobas -1,00-dan nulis gavliT +1,00-mde.
Sr -is gamoyenebis magaliTi
yvelas SegvimCnevia, rom erTi da imave saxeobis cxovelebi gansxvavebulad iqcevian. cxovelebis qcevis gansxvavebebis Sesaswavlad fiverma, mendlma da beiTsonma (Feaver, Mendl, & Bateson, 1986) Seqmnes iseTi qcevebis gazomvis
Tavi XIII. korelacia _______________________________________________________________________
33
meTodi, rogoricaa TamaSi, devna, gaqceva, miwaze garTxma da ase Semdeg. davuSvaT, rom mkvlevars, romelic am meTods iyenebs, aqvs hopoTeza, rom katebis TamaSisa ( X ) da devnis (Y ) qcevebi korelaciaSia. garkveuli drois ganmavlobaSi akvirdebian 10 katas erTsa da imave garemoSi da maTi TamaSisa da devnis qcevebis rangebs aniWeben. is kata, romelic yvelaze mets TamaSobs, miiRebs rangs 1, xolo is, romelic yvelaze naklebs TamaSobs, _ rangs 10. igive wesis mixedviT, kata, romelic yvelaze xSirad avlens devnis qcevas, miiRebs rangs 1 da kata, romelic yvelze iSviaTad avlens devnis qcevas, rangs 10. 10 katis TamaSisa da devnis qcevebis qulebi warmodgenilia cxrilis #13.10 a da b svetebSi. 1-li da me-10 katebi devnis qcevis mixedviT erTnairad Seafases. maSasadame, orive katas mianiWes rangi 6,5, rac aris 6-isa da 7-is, im ori rangis saSualo, rac maT im SemTxvevaSi mieniWeboda, qulebi bmuli rom ar yofiliyo.
Sr -is gamoTvlisaTvis saWiro procedurebi naCvenebia cxrilis #13.10
momdevno nawilSi. pirveli safexuria sxvaobis, D–s, povna rangebis ToToeuli wyvilis erTi rangis meorisaTvis gamoklebiT. es safexuri naCvnebia g svetSi, romelSic TiToeuli katis TamaSis rangs ( X ) gamoklebulia devnis rangi (Y). Semdeg TiToeuli D ayvanilia kvadratSi,
rac naCvenebia d svetSi. am svetis monacemebis jami, 80,50, gvaZlevs ∑ 2D -s
mniSvnelobas. formulaSi vsvamT ∑ 2D -isa da wyvilisN mniSvnelobebs da
viTvliT Sr -s: realuriSr = +0,51.
spirmenis ranguli korelaciis koeficientis statistikuri mniSvnelovnoba
Sr -is statistikuri mniSvnelovnobis Semowmeba iseve xdeba, rogorc
pirsonis r –sa. nulovani hopoTezaa, rom populaciaSi ranguli korelacia,
ρs, nulis tolia, anu 0H : ρs = 0. alternatiuli hipoTezis mixedviT,
populaciaSi korelacia ar aris nuli toli, anu 1H : ρs ≠ 0. Sr –is
kritikuli mniSvnelobebi 0,05 da 0,01 doneebze warmodgenilia danarTis
cxrilSi #A-9. kritikuli mniSvnelobebi damokidebulia wyvilzeN da ara
Tavisuflebis xarisxze. Tu realuriSr metia an tolia kritikuli
mniSvnelobisa, maSin unda miviRoT 0H –is uaryofisa da 1H –is miRebis
gadawyvetileba.
magaliTisaTvis, Sr -is kritikuli mniSvneloba maSin, roca wyviliN = 10
da 05,0=α , aris 0,648. radgan realuriSr = +0,51 kritikul mniSvnelobaze
naklebia, 0,05 doneze nulovan hopoTezas ar uarvyofT. TamaSisa da devnis qcevebis rangebs Soris mimarTeba ar aris statistikurad mniSvnelovani; SerCevaSi, romelzec xdeboda dakvirveba, TamaSisa da devnis qcevebi erTmaneTTan dakavSirebuli ar iyo.
Sr -is Tvisebebi
spirmenis korelaciis koeficienti gviCvenebs ori cvladis rangebs Soris SeTanxmebas. Tu X da Y cvladebs erTnairi rangebi aqvs miniWebuli, maSin
Tavi XIII. korelacia _______________________________________________________________________
34
Sr iqneba +1,00. Tu cvladebi sapirispirod aris Sefasebuli ise, rom X
cvladis rangi 1 Seesabameba Y cvladis yvelaze dabal rangs da X
cvladis yvelaze dabali rangi Seesabameba Y cvladis rangs 1, maSin Sr
iqneba _1,00. Tu rangebs Soris arsebobs garkveuli, magram ara sruli SeTanxmeba, maSin 0-sa da 1,00-s Soris mniSvnelobas miiRebs. kavSiris mimarTulebas gamoxatavs korelaciis plus an minus niSani. cxrili #13.10
spirmenis ranguli korelaciis koeficientis ( sr ) gamoTvla 10 katis TamaSisa da
devnis qcevebisaTvis. a da b svetebSi warmodgenilia TiToeuli katis TamaSisa da devnis qcevebis rangebi.
(a) (b) (g) (d) kata TamaSi: X devna: Y ( )YXD −= ( )22 YXD −=
1 2 3 4 5 6 7 8 9 10
2 6 8 4 7 5 10 3 1 9
6,5 8 4 1 5 9 10 2 3 6,5
-4,5 -2 4 3 2 -4 0 1 -2 2,5
20,25 4,00 16,00 9,00 4,00 16,00 0,00 1,00 4,00 6,25
jami 80,50
Sr -is formulaa
( )( )
−−= ∑
16
1 2
2
swyviliwyvili
realuri NND
r
Tu CavsvamT ∑ = 50,802D da wyviliN = 10 mniSvnelobebs,
( )( )( )( )
ebTdavamrgval demeasedebam Tu
realuri
1,5,0512,0448,01
1101050,8061 2 s
+=+=−=
−
−=r
SeamowmeT Tqveni codna XIII-4
1. gansazRvreT: organzomilebiani normaluri ganawileba, ρ. 2. CamoayalibeT statistikuri hopoTeza, romelic saWiroa
imisTvis, rom SevamowmoT, mniSvnelovnad gansxvavdeba Tu ara
realurir nulisagan.
3. wyvilisN mocemuli mniSvnelobebidan TiToeulisaTvis
gansazRvreT Tavisuflebis xarisxi (df) da Semdeg cxrilis #A.5 mixedviT ipoveT kritikulir , roca α = 0,05 da α = 0,01. wyviliN =
Tavi XIII. korelacia _______________________________________________________________________
35
a. 8 g. 15 e. 20 b. 32 d. 43 v. 75
4. qvemoT mocemuli ToToeuli realurisar da iswyvilisaTvN CamoayalibeT
nulovani da alternatiuli hipoTezebi; Semdeg ipoveT kritikulir
mniSvnelovnobis 0,05 doneze. gansazRvreT, xvdeba Tu ara realurir
uaryofis areSi da Semdeg miuTiTeT, Tu ra gadawyvetilebas iRebT statistikur hipoTezebTan dakavSirebiT.
rrealuri
wyviliN
a. b. g. d. e. v.
+0,580 -0,346 -0,682 +0,204 +0,249 -0,731
12 12 20 24 80 9
5. kolejis profesorebi xSirad miiCneven, rom leqciebze daswreba da saboloo Sefaseba (niSani) korelaciaSia. davuSvaT, rom profesori aRricxavda 12 studentis daswrebas da miiRo gacdenebisa da gamocdis Sefasebebis Semdegi saSualoebi:
studenti
gacdenebi
Ggamocdis Sefasebebis saSualo
1 22 72 2 6 88 3 1 99 4 8 68 5 6 66 6 11 77 7 10 52 8 6 78 9 10 76 10 0 96 11 5 91 12 10 86
a. am monacemebisaTvis gamoTvliTi formulis meSveobiT ipoveT
realurisr mniSvneloba
b. SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara realurir
populaciaSi nulovani korelaciisagan. g. ra daskvnas akeTebT gacdenebisa da gamocdebis saSualo
maCveneblebis mimarTebis Sesaxeb?
Tavi XIII. korelacia _______________________________________________________________________
36
6. Sr -isa da wyvilisN TiToeuli mniSnelobisaTvis gansazRvreT
statistikuri nulovani hipoTezebi; Semdeg ipoveT kritikulir
mniSvnelovnobis 0,05 doneze. miuTiTeT, xvdeba Tu ara miRebuli
Sr uaryofis areSi da Semdeg miuTiTeT, ra gadawyvetilebas
iRebT statistikur hipoTezebTan dakavSirebiT.
Sr wyviliN
a. b. g. d. e. v.
+0,401 -0,620 +0,541 -0,366 +0,544 -0,377
17 12 24 10 21 30
Sejameba
• korelaciuri kvlevis mizania or an met cvlads Soris kavSiris xarisxis dadgena.
• wertilTa gabnevis diagramaze, ori monacemidan erT-erTi ( X cvladi) warmodgenilia horizontalur RerZze (abscisaze) da meore (Y cvladi) gadazomilia vertikalur RerZze (ordinataze). cdis piris qulas oridan TiToeul cvladze wertilTa gabnevis diagramaze erTi wertili warmoadgens.
• dadebiTi anu pirdapiri kavSiris SemTxvevaSi X cvladis mniSvnelobis zrdasTan erTad izrdeba Y cvladis mniSvnelobac.
• uaryofiTi, anu Sebrunebuli kavSiris SemTxvevaSi X cvladis mniSvnelobis zrdasTan erTad Y cvladis mniSvneloba mcirdeba.
• pirsonis korelaciis koeficienti ganisazRvreba Semdegnairad:
( )( )( )[ ] ( )[ ]∑∑∑
−−
−−=
22YYXX
YYXXr
• ( )( )∑ −− YYXX aris X da Y cvladebis jvaredina namravli da
aRiniSneba XYCP –iT. Sesabamisad, korelaciis koeficienti SeiZleba aseve gamoisaxos, rogorc:
( )( )YX
XY
SSSSCPr =
• standartuli qulebis gamoyenebiT r –is mniSvneloba mocemulia formuliT:
( )wyviliN
zzr YXΣ=
• r -is mniSvneloba gamoTvliTi formulis gamoyenebiT Semdegia:
Tavi XIII. korelacia _______________________________________________________________________
37
( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑
∑∑∑−
−=
YYNXXN
YXXYNr
wyviliwyvili
wyvili
• kavSiris mimarTulebas gamoxatavs r –is dadebiTi an uaryofiTi niSani
• xarisxi imisa, Tu ramdenad qmnis wertilTa gabnevis diagramis wertilebi wrfes, warmodgenilia r –is absoluturi mniSvnelobiT, romelic SeiZleba icvlebodes 0,00-dan 1,00-mde.
• populaciaSi X da Y cvladebs Soris wrfivi kavSiri warmodgenilia ρ –Ti.
• r -is statistikuri mniSvnelovnobis SemowmebisaTvis gamoiyeneba cxrili #A.5.
• determinaciis koeficienti 2r gamoxatavs X cvladis cvalebadobis im proporcias, romelic dakavSirebulia Y cvladis cvalebadobasTan.
• Tu X da Y cvladebs Soris korelacia arsebobs, maSin cvladebs Soris mizez-Sedegobrivi mimarTebis, sul mcire, sami SesaZlebloba arsebobs: X aris Y –is mizezi Y aris X –is mizezi arc X da arc Y ar axdens erTmaneTze mizez-Sedegobriv
gavlenas da orives mizezia mesame, Z cvladi.
• spirmenis ranguli korelaciis koeficienti gamoiTvleba formuliT:
( )( )
−−= ∑
16
1 2
2
wyviliwyvili NND
rs
• Sr -is statistikuri mniSvnelovnobis Semowmeba xdeba cxrilis #A.6 gamoyenebiT.
ZiriTadi terminebi da simboloebi
• dadebiTi kavSiri
• determinaciis koeficienti ( 2r )
• korelacia
• korelaciis koeficienti
• korelaciuri kvleva
• organzomilebiani ganawileba
• organzomilebiani normaluri ganawileba
• pirsonis korelaciis koeficienti ( r )
• ρ
• Sρ
• spirmenis ranguli korelaciis
koeficienti ( Sr )
• XSS
• YSS
• uaryofiTi kavSiri • wertilTa gabnevis diagrama
• wrfivi kavSiri • X -sa da Y -is jvaredina namravlebi
( XYCP )
Tavi XIII. korelacia _______________________________________________________________________
38
kiTxvebi
1. pirveli Tavis kiTxvaSi #3 mocemuli iyo kvleva mamakacebis
gabrazebis donesa da sisxlis wnevas Soris kavSiris Sesaxeb. davuSvaT, rom am sakiTxiT dainteresebuli eqsperimentatori 16 mamakacisagan moipovebs gabrazebis donis qulebs da sisxlis sistolur wnevas (merkuris milimetrebSi). gabrazebis donis qulebi miRebuli iyo Sefasebis skalis gamoyenebiT, romelzec SesaZlo qulebi meryeobda 10-dan 100-mde da maRali qulebi niSnavda brazis ufro maRal dones. miRebuli iyo Semdegi qulebi:
cdis piri Bbrazis done sisxlis wneva
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
64 30 37 39 17 29 36 34 44 32 37 47 50 41 46 61
170 132 129 144 145 122 124 145 137 115 127 148 148 144 133 158
a. aageT wertilTa gabnevis diagrama gabrazebis
qulisa (abscisaze) da sisxlis wnevis monacemebisaTvis (ordinataze).
b. werilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi Tu kavSiri ar vlindeba?
g. gamoTvliTi formulis gamoyenebiT gamoTvaleT
realurir da SeamowmeT, mniSvnelovnad gansxvavdeba Tu
ara is populaciaSi nulovani korelaciisagan 0,05 doneze.
d. SerCevis safuZvelze ra daskvnas akeTebT gabrazebis donesa da sisxlis sistolur wnevas Soris kavSiris Sesaxeb?
e. gamoTvaleT 2r . am ori sazomisaTvis gabrazebis donisa da sisxlis wnevis cvalebadobis ra proporciaa saerTo?
v. Tqveni kvlevis monacemebi warmoadgineT am Tavis Sesabamis ganyofilebaSi (korelaciuri kvlevis
Tavi XIII. korelacia _______________________________________________________________________
39
Sedegebis warmodgena) mocemuli magaliTis mixedviT.
2. adamianebisadmi Cvens reaqcias xSirad maTi garegnoba gansazRvravs. rogorc Cans, mamakacebis SemTxvevaSi mainc, simaRle xSirad sxvaTa mxridan dadebiT reaqciebs iwvevs. rogorc Zveli gamonaTqvami ambobs, `maRali, Savgvremani da simpaTiuri~. aseTi tipis msjelobidan gamomdinare, vilimezs, aisenbergsa da kerols (Villimez, Eisenberg, & Carroll, 1986) ainteresebdaT daedginaT, arsebobda Tu ara raime kavSiri dawyebiTi skolis bavSvebis fizikur sididesa da akademiur moswrebas Soris. maT miiRes bavSvebis sididis ramdenime maCvenebeli da Seadares isini testis qulebsa da niSnebis furclis niSnebs. imisaTvis, rom gaekontrolebinaT simaRleSi asakTan dakavSirebuli gansxvavebebi, yvela gamokvleuli
bavSvis asaki erTi da igive iyo. maT miiRes –0,12-is toli realurir
simaRlesa da niSnebis furclis niSnebs Soris 53 mesameklasel
gogonasTan da +0,32-is toli realurir 42 mesameklasel biWTan.
a. daadgineT, mniSvnelovnad gansxvavdeba Tu ara TiToeuli korelaciis koeficienti populaciaSi nulovani korelaciisagan 0,05 doneze.
b. realurisr TiToeuli mniSvnelobisaTvis gamoTvaleT
determinaciis koeficienti. g. SerCevis safuZvelze ra daskvnas akeTebT dawyebiTi
skolis bavSvebis simaRlesa da niSnebis furclis niSnebs Soris mimarTebis Sesaxeb?
3. studentebs xSirad ainteresebT, iwinaswarmetyvelon Tavisi saboloo Sefaseba Sualeduri gamocdis Sefasebis mixedviT. aseTi winaswarmetyvelebis gasakeTeblad saWiroa Sualeduri da saboloo gamocdebis Sefasebebis korelacia. davuSvaT, rom elementaruli statistikis klasSi Sualedur da saboloo gamocdebze 10-ma studentma miiRo Semdegi Sefasebebi:
gamocda
studenti Sualeduri: X saboloo: Y
1 2 3 4 5 6 7 8 9 10
87 89 97 80 73 85 81 74 80 89
84 91 96 87 66 90 79 80 89 86
a. daxazeT wertilTa gabnevis diagrama Sualeduri
(gadazomeT abscisaze) da saboloo (gadazomeT ordinataze) gamocdebisaTvis.
Tavi XIII. korelacia _______________________________________________________________________
40
b. wertilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar vlindeba?
g. gamoTvleT realurir gamoTvliTi formulis
gamoyenebiT da SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.
d. SerCevis safuZvelze ra daskvnas akeTebT Sualeduri gamocdis niSnebsa da saboloo gamocdis niSnebs Soris kavSiris Sesaxeb?
4. krancma (Krantz, 1987) daadgina, rom fizikuri momxibvleloba da popularoba dadebiT korelaciaSia sabavSvo baRis aRsazrdel gogonebSi, magram ara biWebSi. davuSvaT, rom Tqven CaatareT msgavsi kvleva, romlisTvisac damkvirvebeli sabavSvo baRis moswavleebs afasebda fizikuri momxibvlelobisa da popularobis mixedviT. 12 gogonasa da 14 biWis Sefasebis Sedegad miiReT Semdegi rangebi:
gogonebi
fizikuri mimzidveloba:
X
popularoba: Y
3 6 2 9 12 1 7 8 4 10 5 11
1 7 3 10 11 3 8 7 5 9 6 12
biWebi fizikuri
mimzidveloba:
X
popularoba: Y
5 11 3 9 4 8 12 1
9 1 10 4 6 12 5 13
Tavi XIII. korelacia _______________________________________________________________________
41
14 10 6 13 2 7
8 14 2 11 7 3
a. gamoTvaleT realuriSr gogonebis monacemebisaTvis da
SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.
b. SerCevis safuZvelze ra daskvnas akeTebT gogonebSi fizikuri momxibvlelobisa da popularobis rangebs Soris kavSiris Sesaxeb?
g. gamoTvaleT realuriSr biWebis monacemebisaTvis da
SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.
d. SerCevis safuZvelze ra daskvnas akeTebT biWebSi fizikuri momxibvlelobisa da popularobis rangebs Soris kavSiris Sesaxeb?
5. uongis (Wong, 1993) kvleva afasebda kolejis 76 students veqsleris inteleqtis skaliT (Wechsler Adult Intelligence Scale - Revised) da iTvlida miRebuli IQ-s korelacias studentebis niSnebis saSualosTan. IQ-s qulebis diapazoni iyo 83-dan 133-mde.
IQ qulebsa da niSnebis saSualos Soris realurir iyo +0,18, rac
aramniSvnelovania 0,05 doneze. gansazRvreT erTi SesaZlo mizezi imisa, Tu ratom ar aris korelacia or cvlads Soris am kvlevaSi.
6. samsaxurSi da saxlSi kompiuterebis sul ufro da ufro xSirad gamoyenebasTan erTad izrdeba interesi imis mimarT, Tu ra gavlenas axdens kompiuteris ekranis xangrZlivad yureba aRqmis unarze. vaisi, qimeli da Staini (Weiss, Kimmel, & Stein, 1993) adgendnen korelacias siRrmis aRqmis maCvenebelsa da uSualod aRqmis testis Catarebamde kompiuterTan gatarebuli saaTebis raodenobas Soris. isini zomavdnen cdis piris mier kompiuterTan gatarebuli saaTebis raodenobas da im Secdomis sidides (santimetrebSi), rasac cdis piri uSvebda siRrmis aRqmis testSi ori vertikaluri xazis erTmaneTis paralelurad dalagebisas. Tu cdis piris siRrmis aRqma idealuri iyo, Secdomis qula nuli santimetri iqneboda. Secdomad CaiTvleboda SemTxveva, roca cdis piri ver SeZlebda ori xazis paralelurad ganlagebas; rac ufro didia Secdoma, miT ufro cudia siRrmis aRqma. davuSvaT, rom Tqven SeafaseT 13 cdis piri maT mier kompiuterTan gatarebuli saaTebisa da siRrmis aRqmis testis mixedviT, uSualod kompiuterTan muSaobis damTavrebis Semdeg da miiReT Semdegi monacemebi:
cdis piri kompiuterTan
muSaobis xangrZlivoba
siRrmis aRqmis Secdoma (sm.):
Y
Tavi XIII. korelacia _______________________________________________________________________
42
(sT.): X
1 2 3 4 5 6 7 8 9 10 11 12 13
0,5 1,7 4,2 6,0 2,8 5,3 0,9 3,4 1,4 5,1 1,9 4,8 2,6
4,3 3,1 2,1 0,4 6,3 3,7 1,6 1,2 2,4 5,2 0,6 4,1 5,4
a. aageT wertilTa gabnevis diagrama kompiuterTan
muSaobis xangrZlivobisa (gadazomeT abscisaze) da siRrmis aRqmis Secdomis (gadazomeT ordinataze) monacemebisaTvis.
b. werilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar vlindeba?
g. gamoTvliTi formulis gamoyenebiT gamoTvaleT
realuriSr da SeamowmeT, mniSvnelovnad gansxvavdeba Tu
ara is populaciaSi nulovani korelaciisagan 0,05 doneze.
d. SerCevis safuZvelze ra daskvnas akeTebT kompiuterTan muSaobis saaTebsa da siRrmis aRqmis Secdomas Soris kavSiris Sesaxeb?
7. hubneri da makkalafi (Huebner & McCullough, 2000) ikvlevdnen mozardebis skoliT kmayofilebasTan dakavSirebul faqtorebs. erT-erTi nakvlevi cvladi iyo skolasTan dakavSirebuli yovedlRiuri uaryofiTi movlenebis raodenoba. yoveldRiuri uaryofiTi movlenebi moicavs iseT aspeqtebs rogoricaa Relva akademiuri moswrebis gamo, cudi maswavleblis yola, an aramosawon adamianebTan urTierToba. davuSvaT, rom am sakiTxiT dainteresebuli mkvlevari 15 mozards sTxovs, rom awarmoon skolasTan dakavSirebuli yoveldRiuri uaryofiTi movlenebis Cawera da skoliT kmayofileba Seafason skalaze, romelzec qulebi meryeobs 6-dan (Zalian dabali kmayofileba) 48-mde (Zalian maRali kmayofileba). mkvlevarma Semdegi monacemebi miiRo:
studenti uaryofiTi
movlenebi skoliT
kmayofileba 1 2 3 4 5
7 14 8 19 6
34 14 24 15 29
Tavi XIII. korelacia _______________________________________________________________________
43
6 7 8 9 10 11 12 13 14 15
10 17 7 12 17 6 9 13 22 5
27 14 38 19 21 29 27 18 18 30
a. daxazeT wertilTa gabnevis diagrama uaryofiTi movlenebis raodenobisa (gadazomeT abscisaze) da skoliT kmayofilebisaTvis (gadazomeT ordinataze).
b. wertilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar vlindeba?
g. gamoTvaleT realurir gamoTvliTi formulis
gamoyenebiT da SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.
d. SerCevis safuZvelze ra daskvnas akeTebT uaryofiTi movlenebis raodenobasa da skoliT kmayofilebas Soris kavSiris Sesaxeb?
8. pirvel TavSi warmodgenili iyo voerisa da qeivelis (Woehr & Cavell, 1993) da qeivelisa da voeris (Cavell & Woehr, 1994) naSromi fsiqologiis Sesavlis kursis pirveli gamocdaze warmatebis korelatebis Sesaxeb. erT-erTi gazomili cvladi iyo im saaTebis raodenoba, romelTa ganmavlobaSic studenti kviris manZilze televizors uyurebda. davuSvaT, rom Tqvens 17 megobars sTxoveT, CaeweraT kviraSi ramdeni saaTis ganmavlobaSi uyurebdnen televizors, raTa am monacemebis maTi pirveli gamocdebis SedegebTan korelacia dagedginaT. monacemebis mixedviT miiReT Semdegi mniSvnelobebi:
televizoris yurebisaTvis XSS =652; pirveli gamocdis
niSnisaTvis YSS =2094 da XYCP = _859.
a. am monacemebis gamoyenebiT ipoveT realurir
televizoris yurebasa da gamocdis niSans Soris.
b. realurisr mniSvneloba statistikurad mniSvnelovania
0,05 doneze? g. davuSvaT, rom erTi megobari televozors uyurebs
kviraSi 4 saaTis ganmavlobaSi, xolo meore – 15 saaTis ganmavlobaSi. Tqvens mier gamoTvlili korelacia ra winaswarmetyvevlebis gakeTebisaken gibiZgebT?
9. maswavlebelma, romelic dainteresebuli iyo bavSvis klaselebis mier mowonebasa da bavSvis klasSi CarTulobas Soris kavSiriT, am ori ganzomilebis mixedviT mexuTeklaseli
Tavi XIII. korelacia _______________________________________________________________________
44
bavSvebis ranJireba moaxdina. davuSvaT, rom 16 baSvisaTvis miRebulia Semdegi rangebi:
bavSvi mowonebadobis
done CarTulobis
done 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6 9 1 2 16 3 8 4 10 5 15 12 7 13 14 11
8 12 2 4 14 1 11 3 5 6 13 9 7 10 16 15
a. am qulebisaTvis ipoveT Sr -is mniSvneloba.
b. SeamowmeT da gansazRvreT, mniSvnelovnad
gansxvavdeba Tu ara miRebuli Sr populaciaSi
nulovani korelaciisagan. g. ra daskvnas akeTebT mowonebadobasa da klasSi
CarTulobas Soris kaSiris Sesaxeb?
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
1
Tavi XIV
regresia da winaswarmetyveleba __________________________________________________________
wrfivi kavSiri wrfivi kavSiris gansazRvreba wrfis daxriloba wrfis Y -TanakveTa wrfivi kavSiris magaliTebi da maTi gantolebebi
regresiis wrfis dadgena magaliTi ufro realuri magaliTi umcires kvadratTa meTodi
X -is mixedviT Y -is winaswarmetyvelebisaTvis umcires kvadratTa
regresiis wrfis daxriloba da Y -TanakveTa umcires kvadratTa regresiis wrfis morgeba monacemebze
winaswarmetyvelebis Secdoma naSTi: winaswarmetyvelebis Secdomis sazomi Sefasebis standartuli Secdoma
XYs ⋅ -is gamoTvliTi formula
XYs ⋅ -is gamoTvla r -is mixedviT informacia, romelsac gvaZlevs Sefasebis standartuli Secdoma
Y -is mixedviT X -is winaswarmetyveleba regresiis wrfis gamoyeneba mravlobiTi regresia
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
2
me-13 TavSi korelaciis ganxilvisas aRvniSneT, rom Tu ori cvladi srul kavSirSia, ise, rogorc monacemebi suraTebze #13.2 da 13.4, erTi cvladis mniSvnelobis codna meore cvladis mniSvnelobis zusti winaswarmetyvelebis saSualebas gvaZlevs. magram erTi cvladis qulebi meore cvladis qulebis winaswarmetyvelebisaTvis maSinac SegviZlia
gamoviyenoT, roca X da Y cvladebi srul kavSirSi ar arian, magram maT
Soris aris statistikurad mniSvnelovani kavSiri. bevri mizezi arsebobs imisa, Tu ratom SeiZleba gvindodes erTi cvladis mixedviT meores winaswarmetyveleba. magaliTad, davuSvaT, rom gindaT icodeT, ras SeiZleba niSnavdes pirvel gamocdaze miRebuli qula Tqveni saboloo niSnisaTvis. am SemTxvevaSi Tqven gindaT, pirveli gamocdis niSnis mixedviT iwinaswarmetyveloT saboloo Sefaseba. kolejis misaRebi komisiis wevrebs surT, standartizebuli miRwevis testebis mixedviT iwinaswarmetyvelon niSnebis saSualo. personalis menejerebs SeuZliaT TanamSromlebis samsaxurSi warmateba samsaxureobrivi testis qulebis mixedviT iwinaswarmetyvelon. Tqvens eqims ki SeuZlia Tqveni samedicino gamokvlevis Sedegebis mixedviT iwinaswarmetyvelos garkveuli daavadebis riski. sxva magaliTebi qvemoTaa moyvanili. es Tavi exeba erTi cvladis mixedviT meores winaswarmetyvelebas im SemTxvevaSi, roca es ori cvladi wrfiv kavSirSia. Cvens mier gamoyenebuli midgoma gulisxmobs erTi qulis mixedviT meores winaswarmetyvelebis mizniT ori cvladis wertilTa gabnevis diagramisTvis wrfis morgebas, romelsac regresiis wrfe ewodeba. or cvladi SeiZleba arawrfivadac iyos dakavSirebuli, magram Cveni msjeloba mxolod wrfiv kavSirs exeba. Semdeg gagacnobT martoobis kiTxvaris pasuxebis mixedviT depresiis qulebis winaswarmetyvelebis magaliTs.
wrfivi kavSiri wrfivi kavSiris gansazRvra
wrfivi kavSiri: kavSiri or cvlads Soris, romlis drosac X cvladis
yoveli erTi erTeuliT cvlilebisas Y cvladi mudmivi sididiT icvleba. or cvlads Soris wrfivi kavSiri aris cvladebs Soris kavSiri, romelic
wrfiT aRiwereba. suraTze #14.1 warmodgenilia ori cvladi, X da Y , romlebic srul wrfiv kavSirSia. es suraTi saSualebas gvaZlevs wrfivi kavSiri ufro kargad ganvsazRvroT. wrfivi kavSiris dros, yovelTvis, roca erTi cvladi erTi erTeuliT icvleba, meore cvladic ganicdis
mudmiv cvlilebas. suraTze #14.1 X -is yoveli erTi erTeuliT zrdisas Y
0,5 erTeuliT izrdeba. magaliTad, roca X izrdeba 2-dan 3-mde (erTi
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
3
erTeuliT X RerZze), Y izrdeba 1,0-dan 1,5-mde (0,5 erTeuli Y RerZze). msgavsadve, roca X izrdeba 4-dan 5-mde (erTi erTeuliT X RerZze),
Y izrdeba 2,0-dan 2,5-mde (0,5 erTeuli Y RerZze). Sesabamisad, am magaliTSi,
X -is erTi erTeuliT cvlilebas Sesabamis sazomze mosdevs Y sazomze 0,5 erTeuliT mudmivi cvlileba. suraTi #14.1
X da Y cvladebs Soris sruli dadebiTi wrfivi kavSiri +0,5 daxrilobiT
wrfivi kavSiris moTxovna, rom X cvladis erTi erTeuliT cvlilebas
Tan unda axldes Y cvladis mudmivi sididiT cvlileba, ZalaSia rogorc
dadebiTi, aseve uaryofiTi kavSiris SemTxvevaSi. suraTi #14.2 gviCvenebs X -
sa da Y -s Soris srul uaryofiT kavSirs. yovelTvis, roca X 1 erTeuliT
icvleba, Y mudmivi sididiT icvleba. magaliTad, X -is erTi erTeuliT, 2-
dan 3-mde cvlilebisas Y 9,4-dan 8,6 mde icvleba, anu Y cvladis _0,8-iT
cvlileba xdeba. msgavsadve, X -is cvlilebas 8-dan 9-mde Tan axlavs Y –is
4,6-dan 3,8-mde cvlileba, anu Y cvladi isev _0,8-iT icvleba. suraTis #14.2
wrfivi kavSirisaTvis, X -is yoveli 1 erTeuliT cvlilebas Tan axlavs Y–is Sesabamisi da mudmivi cvlileba _0,8 erTeuliT. wrfis gantoleba: abXY += wrfis gantolebas Semdegi saxe aqvs:
abXY += ,
sadac X aris X cvladis Sesabamisi qula
Y aris Y cvladis Sesabamisi qula b aris wrfis daxriloba a aris konstanta, romelsac Y–is TanakveTa ewodeba es gantoleba gamoiyeneba X cvladis mniSvnelobis mixedviT Y cvladis mniSvnelobis winaswarmetyvelebisaTvis. wrfis daxriloba wrfis daxriloba: wrfis daxriloba mocemulia tolobiT
cvlileba ismniSvnelob is
cvlileba ismniSvnelobis
−−
=XY
b
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
4
suraTebze #14.1 da 14.2, X -is yovel 1 erTeuliT cvlilebas Tan axlavs
Y cvladis Sesabamisi da mudmivi cvlileba. X cvladis cvlilebasa da mis
Sesabamisad Y cvladis cvlilebis kavSirs warmoadgens im wrfis
daxriloba, romelic am ori cvladis mimarTebas asaxavs. kerZod, X da Y
cvladebis damakavSirebeli wrfis daxriloba aRiniSneba b–Ti da ganisazRvreba, rogorc
daxriloba = cvlileba ismniSvnelob is
cvlileba ismniSvnelobis
−−
=XY
b
suraTi #14.2
X da Y cvladebs Soris sruli uaryofiTi wrfivi kavSiri _0,8 daxrilobiTa da +11
Y -TanakveTiT. es kavSiri gamoisaxeba, rogorc:
12
12
XXYYb
−−
=
sadac 1X da 2X aris X -is ori mniSvneloba
1Y da 2Y aris 1X -is da 2X –is Sesabamisi Y-is mniSvnelobebi suraTze #14.1 warmodgenili ricxviTi mniSvnelobebi saSualebas
gvaZlevs, 1X = 2 da 2X = 3 mniSvnelobebis gamoyenebiT X -isa da Y-is mniSvnelobebze gamavali wrfis daxriloba ganvsazRvroT. X –is am
mniSvnelobebis Sesabamisi Y –is mniSvnelobebia 1Y = 1 da 2Y = 1,5. maSasadame,
.5,015,0
2315,1
12
12 +=++
=−−
=−−
=XXYYb
am wrfis daxriloba aris +0,5; X -is yoveli erTi erTeuliT gazrdisas Y izrdeba 0,5 erTeuliT. wrfis daxriloba mudmivia mis mTel sigrZeze.
maSasadame, miuxedavad imisa, Tu X -is ra mniSvnelobebs gamoviyenebT, am
magaliTSi b mainc +0,5 iqneba. magaliTad, davuSvaT, rom gamoviyeneT
mniSvnelobebi 1X = 3 da 2X = 5. X -is am mniSvnelobebisTvis, Y -is Sesabamisi
mniSvnelobebia 1Y = 1,5 da 2Y = 2,5. aqedan gamomdinare,
5,021
355,15,2
+=++
=−−
=b .
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
5
roca viciT suraTze #14.1 gamosaxuli wrfis daxriloba, SegviZlia
davweroT am wrfis X da Y mniSvnelobebis damakavSirebeli gantoleba. es
gantolebaa:
XY 5,0+=
es gantoleba gvaZlevs X -is TiToeuli mniSvnelobis Sesabamisi Y-is mniSvnelobas da SegviZlia XY 5,0+= X -is sxvadasxva mniSvnelobebisTvis
Semdegnairad amovxsnaT:
Tu X = ____, maSin Y = ____. 0 1 2 3 4 5 6 7 8
0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
SeadareT Y –is es mniSvnelobebi grafikze #14.1 datanil Y-is mniSvnelobebs
X –is TiToeuli mniSvnelobis SemTxvevaSi. mxedvelobaSi miiReT, rom
XY 5,0+= aris gantoleba, romlic aRwers suraTze #14.1 warmodgenil wrfiv kavSirs. maSasadame, rom gvcodnoda mxolod es gantoleba da cdis
piris X qula, zustad SevZlebdiT Y qulis winaswarmetyvelebas.
bXY = gantoleba warmoadgens wrfis zogadi gantolebis nawils da
aRwers X -sa da Y-s Soris nebismier kavSirs im pirobiT, rom roca X = 0,
Y = 0. magram Tu Y ar udris nuls, roca X = 0? am problemas qvemoT davubrundebiT. wrfis Y -TanakveTa suraTi #14.3
X da Y cvladebs Soris wrfivi kavSiri +0,5 daxrilobiT da +3 Y -TanakveTiT
daakvirdiT suraTze #14.3 gamosaxul kavSirs X da Y cvladebs Soris. am
wrfis daxrilobac +0,5-ia, X -is yoveli erTi erTeuliT cvlilebisas Y
+0,5 erTeuliT icvleba. magram ar SeiZleba, es wrfe XY 5,0+= gantolebiT
aRiweros, radgan, am gantolebis mixedviT, Y unda iyos nulis toli, roca
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
6
X = 0. magram, suraTze #14.3 vxedavT, rom Y = 3, roca X = 0. modiT,
XY 5,0+= gantolebaSi CavsvaT X –is sxva mniSvneloba, magaliTad, X = 10.
amis Sedegad aRmovaCenT, rom Y unda iyos 5-is toli, roca X = 10. Tumca,
suraTze #14.3 vxedavT, rom Y = 8, roca X = 10. XY 5,0+= gantolebidan
miRebuli Y -isa da grafikze #14.3 datanili Y -is SedarebiT davadgenT, rom XY 5,0+= gantoleba gvaZlevs Y -is iseT mniSvnelobas, romelic 3 erTeuliT
naklebia grafikze datanil Y -Tan SedarebiT. maSin, davuSvaT, rom viyenebT XY 5,0+= gantolebas da misgan miRebuli Y -is TiToeul mniSvnelobas
vumatebT 3-s. maSin gantoleba aseTia:
35,0 ++= XY
es gantoleba ukve aRwers suraTze #14.3 X -sa da Y -s Soris kavSirs. 35,0 ++= XY gantolebis amoxsniT X -is sxvadasxva mniSvnelobebisaTvis
viRebT Semdegs:
Tu X = ____, maSin Y = ____. 0 2 4 10 13 19 24 26 30
3,0 4,0 5,0 8,0 9,5 12,5 15,0 16,0 18,0
Y -is es mniSvnelobebi zustad Seesabameba suraTze #14.3 aRniSnul Y-is mniSvnelobebs. maSasadame, 35,0 ++= XY aris gantoleba, romelic aRwers suraTze #14.3 gamosaxul wrfes.
es magaliTi gvaZlevs X -sa da Y -s Soris wrfivi kavSiris
gantolebis kerZo SemTxvevas. wrfis zogadi gantolebis gamosaxvisaTvis, am 35,0 ++= XY gantolebaSi +0,5 daxrilobas CavanacvlebT b asoTi da
konstanta 3-s _ a asoTi. Sedegad miviReT abXY += , wrfis zogadi gantoleba b daxrilobiTa da a-s toli Y –TanakveTiT. Y-TanakveTa aris Y-is mniSvneloba, roca X = 0. roca X = 0, wrfe kveTs Y RerZs da am wertilSi Y-is mniSvneloba udris Y-TanakveTas. suraTze #14.3 Y-TanakveTa aris +3, maSin roca suraTze #14.1 Y-TanakveTa aris 0.
Y TanakveTa: abXY += gantolebaSi Y-is mniSvneloba, roca X = 0.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
7
wrfivi kavSirebis magaliTebi da maTi gantolebebi wrfivi kavSirebis da maTi aRmweri gantolebebis gasacnobad suraTi #14.4
warmoadgens X –sa da Y –s Soris rogorc dadebiTi, aseve uaryofiTi
wrfivi kavSirebis magaliTebs. TiToeuli naCvenebi kavSirisaTvis warmodgenilia wrfis gantoleba daxrilobiTa da Y TanakveTiT. magaliTad,
(b) grafiki X -sa da Y -s Soris uaryofiTi kavSiris saCveneblad
warmoadgens suraTze #14.2 gamoyenebul wrfiv kavSirs. am kavSiris Sesabamisi gantolebaa 118,0 +−= XY . aq Cven vxedavT, rom daxriloba aris
_0,8, minus niSani ki miuTiTebs, rom kavSiri uaryofiTia. erTi erTeuliT
X -is yoveli matebisas Y 0,8 erTeuliT mcirdeba. roca X nulis tolia,
wrfe Y RerZs kveTs +11 mniSvnelobaze; maSasadame, am gantolebisaTvis Y-TanakveTa aris +11. SeiswavleT TiToeuli magaliTi da TiToeuli warmodgenili kavSirisaTvis gansazRvreT b da a. suraTi #14.4
X da Y cvladebs Soris sxvadasxvagvari wrfivi kavSirebi: a) Y =+1,2 X +2;
b) Y = - 0,8 X +11; g) Y = -0,1 X + 30; d) Y = +0,2 X + 25; e) Y = +1,5 X _ 2; v) Y = -1,4 X +100
SeamowmeT Tqveni codna XIV-1
1. gansazRvreT: a, b, wrfivi kavSiri, wrfis daxriloba, Y-TanakveTa. 2. dawereT wrfis gantoleba. 3. dawereT wrfis daxrilobis gantoleba. 4. qvemoT mocemuli TiToeuli grafikisaTvis ipoveT daxriloba da Y-
TanakveTa. Semeg dawereT TiToeuli wrfis gantoleba. Aa. b.
g. d.
regresiis wrfis dadgena magaliTi
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
8
wina magaliTebi exeboda cvladebs, romlebic aRniSnuli iyo, rogorc
ubralod X da Y da abXY += gantoleba gamoiyeneboda X -is mixedviT
Y -is winaswarmetyvelebisaTvis. imisaTvis, rom gagacnoT, Tu rogor vadgenT gantolebas, romelic monacemTa or simravles Soris wrfiv kavSirs aRwers, davubrundebi me-13 TavSi warmodgenil amocanas martoobisa da depresiis qulebis kavSiris Sesaxeb. Cveni mizania, SevadginoT iseTi gantoleba, romelic saSualebas mogvcems, martoobis qulebis mixedviT depresiis qulebi viwinaswarmetyveloT.
davuSvaT, rom cxrilis #14.1 monacemebi miRebuli iyo 10
adamianisagan. es qulebi srul dadebiT kavSirSia, realurir = +1,00.
Sesabamisad, moveliT, rom gabnevis diagrama gamoavlens, rom qulebi wrfezea ganlagebuli, rogorc amas suraTi #14.5 gviCvenebs. martoobisa da depresiis qulebs Soris wrfivi kavSiris gantolebas viRebT wrfis daxrilobisa da Y-TanakveTis dadgeniT. orive es mniSvneloba vipoveT grafikis mixedviT: b = +1 da a = +10. aqedan gamomdinare, gantoleba,
romelic cxrilis #14.1 depresiis qulebs (Y ) martoobis qulebTan ( X )
akavSirebs, aris:
101 += XY anu
depresiis qula = (+1)(martoobis qula) + 10.
cxrili #14.1 martoobisa da depresiis hopoTeturi qulebi cdis 10 monawilisaTvis
monawile martoobis qula: X depresiis qula: Y
1 2 3 4 5 6 7 8 9 10
6 15 28 2 24 8 25 30 14 18
16 25 38 12 34 18 35 40 24 28
suraTi #14.5 cxrilSi #14.1 mocemuli martoobisa da depresiis qulebis wertilTa gabnevis diagrama
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
9
radgan qulebs Soris korelacia aris +1,00, Tu viciT ToToeuli cdis piris mxolod martoobis qula, am gantolebis gamoyenebiT SegviZlia zustad viwinaswarmetyveloT misi depresiis qula. cxrili #14.2 martoobis qulebis mixedviT nawinawarmetyvelebi depresiis qulebi Semdegi gantolebis gamoyenebiT:
depresiis qula = (+1,00) (martoobis qula) +10 a svetSi mocemulia martoobis realuri qula, b svetSi _ gantolebis mixedviT nawinaswarmetyvelebi depresiis qula da g svetSi _ depresiis realuri qula cxrilidan #14.1.
monawile
(a) martoobis qula: X
(b) nawinaswarmetyvelebi depresiis qula: Y′
Y (g) depresiis
realuri qula: Y 1 2 3 4 5 6 7 8 9 10
6 15 28 2 24 8 25 30 14 18
16 25 38 12 34 18 35 40 24 28
16 25 38 12 34 18 35 40 24 28
nawinaswarmetyvelebi depresiis qulebi mocemulia cxrilis #14.2 b svetSi. magaliTad, Ppirvel monawiles aqvs depresiis 6 qula. am gantolebis gamoyenebiT,
nawinaswarmetyvelebi depresiis qula = (+1) (martoobis qula) +10
da gamovTvliT, rom
nawinaswarmetyvelebi depresiis qula = (+1)(6) + 10 = 6 + 10 = 16.
depresiis qula, romelic martoobis 6-is toli mniSvnelobis mixedviT viwinaswarmetyveleT, 16-ia. martoobis sxva monacemebis mixedviT nawinaswarmetyvelebi depresiis qulebi warmodgenilia cxrilis #14.2 (b) svetSi. es nawinaswarmetyvelebi qulebi aRniSnulia Y′–iT, (ikiTxeba, rogorc `Y primi~ an `Y nawinaswarmetyvelebi~), imis gamosaxatad, rom isini nawinaswarmetyvelebia martoobis qulebis mixedviT Semdegi gantolebis meSveobiT:
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
10
101 +=′ XY . TiToeuli monawilis mier miRebuli realuri depresiis qula (rac aRiniSneba Y–iT), warmodgenilia (g) svetSi. yuradReba miaqcieT imas, rom nawinaswarmetyvelebi da realuri depresiis qulebi identuria. radgan
realurir = +1,00, Y ′ da Y erTi da igivea da martoobis qulebis mixedviT
depresiis qulebis winaswarmetyveleba SecdomasTan ar aris dakavSirebuli. ufro realuri magaliTi
cxrili #14.3 10 monawilis martoobisa da depresiis hipoTeturi qulebi. qulebis TiToeuli nakrebisaTvis mocemulia saSualo da standartuli gadaxra.
monawile
Mmartoobis qula:
X
Kdepresiis qula:
Y
1 4 16 2 27 37 3 18 33 4 7 23 5 30 34 6 12 32 7 18 24 8 23 29 9 19 26 10 12 26
X 17,0 Y 28,0
Xs 8,37 Ys 6,25
gamoyenebiTi TvalsazrisiT, cxrilis #14.1 monacemebi (qulebi) uCveuloa. iSviaTad an arasdros aris qulebi iseT srul kavSirSi, rogorc cxrilis #14.1 monacemebia. ufro realistur SemTxvevebSi ori cvladis qulebi gamoavlens srulze nakleb korelacias, rogorc es cxrilis #14.3 monacemebis SemTxvevaSia. es monacemebi gamoviyeneT me-13 TavSi pirsonis korelaciis koeficientis warmosadgenad (ix. cxrili #13.1).
realurir am monacemebisTvis aris +0,78, rac statistikurad mniSvnelovania
0,05 doneze.
suraTi #14.6 cxrilSi #14.3 mocemuli martoobisa da depresiis qulebis wertilTa gabnevis
diagrama. wertilta gabnevis diagramaze A da B wrfeebi pirobiTad aris gavlebuli
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
11
cxrilis #14.3 monacemebi warmodgenilia suraTze #14.6 gamosaxul
wertilTa gabnevis diagramaze. rogorc realurir = +0,78 gviCvenebs, es
monacemebi dadebiT kavSirSia; depresiis maRali qulebi dakavSirebulia martoobis maRal qulebTan da depresiis dabali qulebi dakavSirebulia martoobis dabal qulebTan. miuxedavad imisa, rom kavSiri martoobisa da depresiis qulebs Soris suraTze #15.6 ar aris sruli, is wrfivia. magram romeli iqneboda am monacemebis aRsaweri saukeTeso wrfe? cxadia, SeiZleba, rom gabnevis diagramaze ramdenime wrfe gaivlos. magaliTad, Cven am grafikze ori, A da B wrfe gavavleT. wrfeebi pirobiTad aris
gavlebuli garda im pirobisa, rom TiToeuli wrfe gadis X = 17 da Y = 28 wertilze. A wrfe ukeTesad Seesabameba da aRwers monacemebs Tu B wrfe? am kiTxvaze pasuxis gasacemad, raTa gadavwyvitoT ramdenad kargad Seesabameba TiToeuli wrfe korelaciaSi myofi monacemebis nakrebs, saWiroa garkveuli meTodi. umcires kvadratTa meTodi suraTze #14.6 TiToeuli, A da B wrfe, X –is TiToeuli mniSvnelobisTvis Y –is mniSvnelobis (anu Y ′ -is) winaswarmetyvelebis saSualebas gvaZlevs. magaliTad, martoobis 7 qulas A wrfis mixedviT Seesabameba 24,0-is toli Y ′ da 27-is tol martoobis qulas Seesaameba 32,0-is toli Y ′ . meores mxriv, B wrfe prognizirebs 20-is tol Y ′–s 36,0-is toli martoobis 7 qulisaTvis da 36-is tol Y ′ -s martoobis 27 qulisaTvis. es nawinaswarmetyvelebi Y qulebi warmodgenilia cxrilis #14.4 g svetSi A wrfisaTvis da v svetSi B wrfisaTvis. cxrili #14.4
A an B wrfeebis gamoyenebiT suraTis #14.6 X qulebis mixedviT Y qulebis
winaswarmetyvelebis Secdoma. Secdoma warmodgenilia realur Y qulasa da X -is
mixedviT nawinawarmetyveleb Y ′qulas Soris sxvaobiT, anu YY ′− -iT. a da b svetebSi warmodgenilia realuri X da Y qulebi. g svetSi warmodgenilia A
wrfis mixedviT nawinaswarmetyvelebi Y , svetebSi d da e - A wrfis mixedviT nawinaswarmetyvelebi Y qulis Secdoma da Secdomis kvadrati, v svetSi – B wrfis
mixedviT nawinaswarmetyvelebi Y , z da T svetebSi – B wrfis mixedviT nawinaswarmetyvelebi qulebis Secdoma da Secdomis kvadrati.
(a) (b) (g) (d) (e) (v) (z) (T)
A wrfis mixedviT
nawinaswarmetyvelebi
B wrfis mixedviT
nawinaswarmetyvelebi
realuri qulebi
qulebi Secdoma qulebi Secdoma
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
12
monawile X Y Y ′ YY ′− ( )2YY ′−
Y ′ YY ′− ( )2YY ′−
1 4 16 22,8 _6,8 46,24 17,6 _1,6 2,56 2 27 37 32,0 +5,0 25,00 36,0 +1,0 1,00 3 18 33 28,4 +4,6 21,16 28,8 +4,2 17,64 4 7 23 24,0 _1,0 1,00 20,0 +3,0 9,00 5 30 34 33,2 +0,8 0,64 38,4 _4,4 19,36 6 12 32 26,0 +6,0 36,00 24,0 +8,0 64,00 7 18 24 28,4 _4,4 19,36 28,8 _4,8 23,04 8 23 29 30,4 _1,4 1,96 32,8 _3,8 14,44 9 19 26 28,8 _2,8 7,84 29,6 _3,6 12,96 10 12 26 26,0 0,0 0,00 24,0 +2,0 4,00
jami 0,0 159,20 0,0 168,00 TiToeuli Y ′qula cdomilia realurad miRebul Y qulasTan SedarebiT da
YY ′− sxvaoba aris X qulidan Y qulis winaswarmtyvelebisas daSvebuli Secdomis sazomi. cxrili #14.4 gviCvenebs suraTis #14.6 A (d sveti) da B (z sveti) wrfeebis gamoyenebiT X qulebis mixedviT TiToeuli Y qulis winaswarmetyvelebasTan dakavSirebul Secdomebs. TiToeuli am wrfis mixedviT winaswarmetyveleba SecdomasTan aris dakavSirebuli, magram romelime wrfes mivyavarT Tu ara ufro mcire winaswarmetyvelebis SecdomasTan, vidre meores? SegviZlia, vcadoT, am kiTxvas vupasuxoT cxrilis #14.4 e da T svetebSi mocemuli Secdomebis SekrebiT. magram Tu svetebs SevajamebT, rogorc es cxrilSi #14.4,
aRmovaCenT, rom Secdomebis (cdomilebebis) jami ( )∑ ′−YY TiToeuli
wrfisaTvis nulis tolia. amitom Secdomebis jams ver gamoviyenebT, rogorc kriteriums wertilTa gabnevis diagramisa da wrfis Sesabamisobis dasadgenad; is yovelTvis nulis toli iqneba. igive problema gvqonda me-5 TavSi, rodesac vcdilobdiT saSualos garSemo qulebis cvalebadobis
sazomis dadgenas. im SemTxvevaSi davadgineT, rom ( )∑ − XX yovelTvis
nulis toli iyo. problemis gadasaWrelad XX − sxvaobebis dajamebamde
moxda TiToeulis kvadratSi ayvana. Semdeg ( )∑ −2
XX gamoviyeneT qulebSi
cdomilebis cvalebadobis sazomis misaRebad. amJamindeli problemac SeiZleba msgavsadve gadaiWras TiTeuli
YY ′− sxvaobis kavadratSi ayvaniT da YY ′− sxvaobebis kvadratebis SejamebiT. Semdeg ( )∑ ′− 2YY –s gamoviyenebT, rom ganvsazRvroT, romeli
wrfe Seesabameba saukeTesod wertilTa gabnevis diagramas. cxrili #14.4 e
da z svetebSi gviCvenebs TiToeuli, A da B , wrfisaTvis ( )∑ ′− 2YY –is
miRebas. ( )∑ ′− 2YY A wrfisaTvis aris 159,20 (ix. sveti e), maSin, roca B
wrfisaTvis es aris 168,0 (ix. sveti z). ( )∑ ′− 2YY -ze, rogorc
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
13
winaswarmetyvelebis Secdomis sazomze dayrdnobiT, A wrfe nakleb cdomilebas iwvevs, vidre B wrfe. umcires kavadratTa meTodiT miRebuli regresiis wrfe: wrfe,
romelic minimumamde amcirebs ( )∑ ′− 2YY -is mniSvnelobas.
wrfes, romelic minimumamde amcirebs ( )∑ ′− 2YY –is mniSvnelobas,
umcires kavadratTa meTodiT miRebuli regresiis wrfe ewodeba, radgan is minimalur mniSvnelobas aniWebs TiToeuli prognizirebuli qulis realuri qulisagan sxvaobis kvadratebis jamis mniSvnelobas. anu, is minimumamde
amcirebs ( )∑ ′− 2YY –s. suraTze #14.7 naCvenebia umcires kvadratTa regresiis
wrfe, romelic Seesabameba cxrilis #14.3 monacemebis gabnevis diagramas. am wrfis gantolebaa 1,18583,0 ++=′ XY . am wrfis maxasiaTebelia is, rom,
( )∑ ′− 2YY minimaluria; cxrilis #14.3 monacemebis Sesabamis sxva arc erT
wrfes ar eqneba ( )∑ ′− 2YY –is ufro mcire mniSvneloba. axla ki avxsni, Tu
rogor unda vipovoT umcires kvadratTa regresiis wrfis daxriloba da Y –TanakveTa. umcires kvadratTa regresiis wrfis daxriloba da Y –TanakveTa Y -is X –is mixedviT winaswarmetyvelebisaTvis
imisaTvis, rom X cvladis mixedviT Y cvladis winaswarmetyvelebisaTvis
vipovoT umcires kvadratTa regresiis wrfe, unda ganvsazRvroT im wrfis
daxriloba (b ) da Y –TanakveTa ( a ), romelic ( )∑ ′− 2YY –is mniSvnelobas
minimumamde amcirebs. suraTi #14.7 umcires kvadratTa meTodiT miRebuli regresiis wrfe 14.3 cxrilSi mocemuli martoobisa da depresiis qulebisaTvis
umcires kvadratTa meTodiT miRebuli wrfis daxriloba arsebobs umcires kvadratTa meTodiT miRebuli wrfis daxrilobis povnis ramdenime alternatiuli formula; TiToeuli formula gvaZlevs b –s erTsa da imave mniSvnelobas. gadaxraTa da jvaredina namravlebis formula. ori cvladis umcires kvadratTa regresiis wrfis daxriloba am qulebis maTi saSualodan gadaxrebis gamoyenebiT mocemulia formuliT:
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
14
daxriloba = ( )( )
( )∑∑
−
−−= 2
XX
YYXXb
gaixseneT, rom am formulis mricxveli ( )( )∑ −− YYXX , aris X –sa da Y –is
jvaredina namravli da aRiniSneba XYCP –iT. am formulis mniSvneli,
( )∑ −2
XX ,
aris X –is kvadratebis jami, anu XSS . Sesabamisad, es gantoleba SemiZlia
aseve Cavwero b –s jvaredina namravlebisa da kvadratebis jamis saxiT, rogorc:
daxriloba = X
XY
SSCPb =
gamoTvliTi formula. formula, romelic gamoTvlis ufro advil xerxs gvTavazobs, Semdegia:
daxriloba = ( )( ) ( )( )( )( ) ( )22 ∑∑
∑∑∑−
−=
XXN
YXXYNb
wyvili
wyvili
sadac wyviliN aris qulebis wyvilebis raodenoba.
korelaciuri formula. b –s mniSvneloba SeiZleba agreTve miviRoT r -is meSveobiT Semdegi gantolebis gamoyenebiT:
daxriloba =
=
X
Y
ssrb
sadac r aris realurir monacemTa or wyebas Soris
Ys aris populaciisaTvis gamoTvlili Y qulebis standartuli gadaxra
Xs aris populaciisaTvis gamoTvlili X qulebis standartuli gadaxra. am gantolebis gamoyenebas giCvenebT magaliTi #14.1. romeli formula unda gamoiyenoT b –s gamosaTvlelad? cxrilSi #14.5 warmodgenilia daxrilobis gamosaTvleli TiToeuli zemoT mocemuli
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
15
formula. monacemTa simravleze gamoyenebisas TiToeuli formula gvaZlevs b –s erTsa da imave mniSvnelobas. romeli formula unda amovirCioT monacemTa simravlis mixedviT b –s gamosTvlelad? zogadad, gamoTvliTi formulisaTvis saWiro gamoTvlebi ufro martivi Sesasrulebelia da naklebad mosalodnelia, rom Secdomamde migviyvans, vidre is gamoTvlebi, romlebic saWiroa gadaxraTa da jvaredina namravlebis formulebisaTvis. aqedan gamomdinare, gamoTvliTi formula umjobesia gamoiyenoT monacemTa mixedviT b –s gamosaTvlelad da misi gamoyeneba qvemoT iqneba naCvenebi.
Tumca, Tu Tqven ukve ipoveT realurisr mniSvneloba, yvelaze martivi iqneba
korelaciuri formula, radgan am SemTxvevaSi realurisr mniSvnelobis garda
Tqven mxolod Xs da Ys unda ipovoT. cxrili #14.5 umcires kvadratTa meTodiT miRebuli wrfis daxrilobis formulebi
gadaxraTa formula
( )( )( )∑
∑−
−−= 2
XX
YYXXb
jvaredina namravlebisa da kvadratebis jamis formula
X
XY
SSCPb =
gamoTvliTi formula
( )( ) ( )( )( )( ) ( )22 ∑∑
∑∑∑−
−=
XXN
YXXYNb
wyvili
wyvili
wyviliN aris qulebis wyvilebis raodenoba.
korelaciis formula
=
X
Y
ssrb
b -s povna gamoTvliTi formulis meSveobiT cxrili #14.6 gviCvenebs, Tu rogor xdeba gamoTvliTi formulis gamoyeneba imisaTvis, rom vipovoT suraTis #14.7 umcires kvadratTa regresiis wrfis daxriloba, romelic cxrilis #14.3 depresiisa da martoobis qulebs Seesabameba. cxrilis #14.7 a da b svetebi warmoadgens X da Y qulebs. am
svetebis jamebi gvaZlevs saWiro ∑ X –sa da ∑Y –s: ∑ X =170 da ∑Y = 280. g
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
16
svetSi miRebulia TiToeuli 2X da am svetis monacemebis jami, 3520,
gvaZlevs ∑ 2X -s. d sveti gvaZlevs TiToeuli XY -is mniSvnelobas (anu
TiToeuli X qulis mis Sesabamis Y qulaze namravls) da am svetis
monacemebis jami, 5127, gvaZlevs ∑ XY -s.
cxrilis #14.6 meore safexuri gviCvenebs am gantolebaSi ∑ X –is,
∑Y -is, ∑ XY -isa da ∑ 2X –is mniSvnelobebis Casmas. cxrilis #14.3
qulebisaTvis miRebuli b –s mniSvneloba aris +0,583. plus niSani miuTiTebs, rom daxriloba dadebiTia. Y -TanakveTa da umcires kvadratTa regresiis wrfe umcires kvadratTa regresiis wrfis Y -TanakveTa gamoiTvleba b –s mniSvnelobis mixedviT Semdegi formuliT:
XbYa −=
sadac Y aris Y qulebis saSualo
X aris X qulebis saSualo b aris umcires kvadratTa regresiis wrfis daxriloba, miRebuli
wina qveTavSi mocemuli erT-erTi formulis meSveobiT. vaCvenoT, Tu rogor xdeba am formulis gamoyeneba 14.3 cxrilis
qulebisaTvis, sadac Y = 28, X = 17 da b = +0,583. Sesabamisad,
( )( ) 1,18089,18911,92817583,028 ==−=+−=a meaTedebamde damrgvalebis Semdeg. am qulebisaTvis umcires kvadratTa meTodiT dadgenili regresiis wrfe Y RerZs kveTs Y =18,1 mniSvnelobaze. cxrili #14.6
b -s gamoangariSeba gamoTvliTi formulis gamoyenebiT. martoobis qulebi ( X ) a svetSi da depresiis qulebi (Y ) b svetSi aRebulia 14.3 cxrilidan.
safexuri I. ∑ X -is, ∑Y -is, ∑ 2X -isa da
∑ XY -is gamoTvla:
(a) (b) (g) (d) cdis piri
X
Y
2X
XY 1 4 16 16 64 2 27 37 729 999 3 18 33 324 594
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
17
4 7 23 49 161 5 30 34 900 1020 6 12 32 144 384 7 18 24 324 432 8 23 29 529 667 9 19 26 361 494 10 12 26 144 312
jamebi 170 280 3520 5127 safexuri II. b -s gamoTvla:
( )
,10,5127
,3520
,280
,900.28170
,170
2
22
=
=
=
=
==
=
∑∑∑∑∑
wyviliNXY
X
Y
X
X
da
( )( ) ( )( )( )( ) ( )22 ∑∑
∑∑∑−
−=
XXN
YXXYNb
wyvili
wyvili.
Tu CavsvamT ricxviT mniSvnelobebs,
( )( ) ( )( )( )( )
.583,063003670
900,28352010280170512710
+=+
=
−−
=b
umcires kvadratTa meTodiT dadgenili regresiis wrfis gantoleba roca a -sa da b -s mniSvlelobebs CavsvamT wrfis gantolebaSi
abXY +=′ ,
miviRebT, rom 1,18583,0 ++=′ XY .
maSasadame, umcires kvadratTa regresiis wrfes cxrilis #14.3 qulebisaTvis aRwers gantoleba 1,18583,0 ++=′ XY . imisaTvis, rom miviRoT Y –is
nawinaswarmetyvelebi mniSvnelobebi, am gantolebaSi unda CavsvaT X –is mniSvneloba da gantoleba amovxsnaT Y ′–is mimarT. Y –is nawinaswarmetyvelebi mniSvnelobebi cxrilis #14.3 qulebisaTvis Semdegia:
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
18
cdis piri
Tu X = ____, maSin Y ′= ____
1 4 20,4 2 27 33,8 3 18 28,6 4 7 22,2 5 30 35,6 6 12 25,1 7 18 28,6 8 23 31,5 9 19 29,2 10 12 25,1
Y ′ -is es mniSvnelobebi Zevs suraTze #14.7 gamosaxul wrfeze. umcires kvadratTa regresiis wrfis morgeba monacemebisaTvis axla giCvenebT, Tu rogor xdeba 1,18583,0 ++=′ XY -iT warmodgenili umcires kvadratTa regresiis wrfis morgeba suraTze #14.7 datanili qulebisaTvis. Tu cnobilia nebismieri ori wertili wrfeze, maSin Cven sakmarisi informacia gvaqvs grafikze wrfis asagebad. regresiis wrfisaTvis, wrfis erT-erTi wertilia Y -TanakveTa, _ Y ′ -is mniSvneloba, roca X =0. Cveni magaliTisaTvis Y –TanakveTa aris 18,1. maSasadame, wrfis erTi wertilis koordinatebia X =0, Y = 18,1. meore wertilis gageba SegviZlia gantolebaSi X –is sxva mniSvnelobis CasmiT da Y ′–is mimarT amoxsniT. miuxedavad imisa, rom SegviZlia CavsvaT X –is nebismieri mniSvneloba, viyenebT X –is
mniSvnelobas, roca, magaliTad, X =17,0. gantolebaSi X =17,0-is CasmiT viRebT:
( ) 0,281,1817583,0 =++=′Y
meaTedebamde damrgvalebiT.
Sesabamisad, roca X =17, 0,28=′Y . Y –is nawinaswarmetyvelebi mniSvneloba,
0,28=′Y , udris Y –is mniSvnelobas. umcires kvadratTa meTodiT miRebuli
nebismieri wrfisaTvis, roca X –is mniSvneloba X qulebis saSualos tolia, Y ′ toli iqneba realuri Y qulebis saSualosi. maSasadame, meore
wertili regresiis wrfeze yovelTvis iqneba X ,Y koordinatebis mqone
wertili. suraTis #14.7 qulebisaTvis, X =17, Y =28 da suraTze #14.7 regresiis wrfe am koordinatebze gadis.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
19
SeamowmeT Tqveni codna XIV-2
1. gansazRvreT: umcires kvadratTa regresiis wrfe, Y ′ . 2. CawereT wrfis meSveobiT X –is mixedviT Y –is mniSvnelobis
gansasazRvri zogadi gantoleba. 3. testi #2-is niSnebis (Y ) testi #1-is niSnebze ( X ) regresiis wrfis
daxriloba da Y-TanakveTa aris b = +0,08 da a =9. a. dawereT testi #2-is niSnebis testi #1-is niSnebze regresiis
gantoleba b. ipoveT testi #2-is nawinaswarmetyvelebi niSnebis mniSvnelobebi
(anu Y ′ ) testi #1-is Semdegi qulebisaTvis (anu X -sTvis): 60, 65, 80, 85, 90.
g. ujrian furcelze aageT regresiis wrfe. 76=X . 4. davaleba #2-is qulebis (Y ) davaleba #1-is qulebze ( X ) regresiis
wrfis daxriloba da Y-TanakveTa aris b = _1,4 da a =23. a. dawereT davaleba #2-is qulebis davaleba #1-is qulebze
regresiis gantoleba b. ipoveT davaleba #2-is nawinaswarmetyvelebi qulebis
mniSvnelobebi (anu Y ′ ) davaleba #1-is Semdegi qulebisaTvis: (anu X -Tvis): 5, 8, 11, 14, 17.
g. ujrian furcelze aageT regresiis wrfe. .11=X winaswarmetyvelebis Secdoma
naSTi: winaswarmetyvelebis Secdomis sazomi naSTi: YY ′− -is mniSvneloba. rogorc suraTidan #14.7 davinaxeT, Y cvladis winaswarmetyvelebisaTvis X cvladis gamoyeneba prognozirebis garkveul SecdomasTan aris dakavSirebuli, Tu, ra Tqma unda, X -sa da Y –s Soris sruli korelacia ar aris. sxvaoba qulasa da mis nawinaswarmetyveleb mniSvnelobas Soris, YY ′− , aris am Secdomis sazomi da mas naSTi ewodeba. naSTi gamoiyeneba, rogorc safuZveli winaswarmetyvelebis sizustis sazomisa, romelsac Sefasebis standartuli Secdoma ewodeba. Sefasebis standartuli Secdomis gansaxilvelad davubrundeT cxrils #14.7. cxrilis #14.7 a da b svetebSi warmodgenilia cxrilis #14.3 martoobisa da depresiis qulebi. regresiis gantolebidan 1,18583,0 ++=′ XY
miRebuli depresiis nawinaswarmetyvelebi qulebi (anu Y ′ ) naCvenebia g svetSi. magaliTad, Y ′ me-6 cdis pirisTvis, romlis martoobis qulaa 12, udris
( )( ) 10,251,1812583,06 =++=′Y
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
20
imisaTvis, rom minimumamde Segvemcirebina damrgvalebis Secdoma Semdegi gamoTvlebisas, g svetis nawinaswarmetyvelebi mniSvnelobebi measedebamde davamrgvaleT. kargi iqneba, Tu gamoTvliT danarCen Y ′ mniSvnelobebs g svetSi, raTa darwmundeT, rom SegiZliaT am qulebisTvis regresiis gantolebaze muSaoba. TiToeuli cdis pirisaTvis naSTi, YY ′− , naCvenebia d svetSi. am naSTebis jami nulis tolia (damrgvalebis Secdomis farglebSi), amitom am jams winaswarmetyvelebis Secdomis sazomad ver gamoviyenebT. maSasadame, winaswarmetyvelebis Secdomis sazomis misaRebad naSTebi kvadratSi agvyavs, rogorc es e svetSia naCvenebi. e svetSi YY ′− mniSvnelobebis kvadratebis
SejamebiT viRebT kvadratebis jams, rasac naSTiSS , naSTebis kvadratebis
jami, ewodeba. am magaliTisaTvis naSTiSS udris ( )∑ ′− 2YY -s, anu 138,1597-s.
naSTiSS : ( )∑ ′− 2YY -is mniSvneloba.
cxrili #14.7 regresiis 1,18583,0 ++=′ XY gantolebis gamoyenebiT cxrilis #14.3 martoobis
qulebis ( X , a sveti) mixedviT depresiis nawinaswarmetyvelebi qulebis (Y ′ , g sveti) naSTis gamoTvla. naSTi, anu winaswarmetyvelebis Secdoma, mocemulia d svetSi da naSTis kvadratebi warmodgenilia e svetSi.
Sefasebis standartuli Secdoma
(a) (b) (g) (d) (e)
cdis piri X Y Y ′ YY ′− ( )2YY ′−
1 4 16 20,43 -4,43 19,6249 2 27 37 33,84 3,16 9,9856 3 18 33 28,59 4,41 19,4481 4 7 23 22,18 0,82 0,6724 5 30 34 35,59 -1,59 2,5281 6 12 32 25,10 6,90 47,6100 7 18 24 28,59 -4,59 21,0681 8 23 29 31,51 -2,51 6,3001 9 19 26 29,18 -3,18 10,1124 10 12 26 25,10 0,90 0,8100
jami: 138,1597
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
21
Sefasebis standartuli Secdoma ( XYs ⋅ ): ( )
2
2
−
′−∑wyviliN
YYgamosaxulebis
mniSvneloba me-5 TavSi qulebis simravlisaTvis cdomilebis cvalebadobis sazomi, standartuli gadaxra, s , miviReT qulebis am nakrebis SS -is gayofiT mis Tavisuflebis xarisxze da Semdeg miRebuli mniSvnelobidan kvadratuli fesvis amoRebiT. Y –is X –is mixedviT winaswarmetyvelebis Secdomis cvalebadobis msgavs sazoms, Sefasebis standartuli Secdomas, viRebT
naSTsSS mis Tavisuflebis xarisze ganayofidan kvadratuli fesvis amoRebiT.
naSTisTvisSS Tavisuflebis xarisxi (df ) udris wyviliN _ 2-s, sadac wyviliN aris
qulaTa wyvilebis raodenoba. aqedan gamomdinare, gazomvis standartuli
Secdoma, romelic aRiniSneba rogorc XYs ⋅ da gamoxatavs, rom
Y nawinaswarmetyvelebia X –is mixedviT, ganisazRvreba, rogorc:
gazomvis standartuli Secdoma = 2−
=⋅wyvili
naSTi
NSS
s XY
es igivea, rac
gazomvis standartuli Secdoma = ( )
2
2
−
′−= ∑
⋅wyviliN
YYs XY .
am magaliTSi, naSTiSS =138,1597 da wyviliN = 10; Sesabamisad, martoobis qulebis
mixedviT, depresiis qulebis winaswarmetyvelebis Sefasebis standartuli Secdoma aris
demeasadebam ulidamrgvaleb ,16,4210
1597,138=
−=⋅XYs
XYs ⋅ -is gansazRvreba
( )2
2
−
′−= ∑
⋅wyviliN
YYs XY
gviCvenebs, rom is warmoadgens nawinaswarmetyvelebi qulebis realur (miRebul) qulebTan Sedarebis Secdomis cvalebadobis sazoms. Tumca, radgan am formulisaTvis saWiroa YY ′− sxvaobis gamoTvla, is Zalian bevr dros moiTxovs da gamoTvlis SecdomasTan aris dakavSirebuli. amitom Seiqmna ori alternatiuli formula: erTia gamoTvliTi formula, romelic SeiZleba gamoviyenoT YY ′− sxvaobis gamoTvlis gareSe da meorea formula r –is gamoyenebiT.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
22
XYs ⋅ -is gamoTvliTi formula
gamoTvliTi formula XYs ⋅ -is uSualod qulebis mixedviT gamoTvlisaTvis, naSTis povnis gareSe Semdegia:
( ) ( ) ( ) ( )( )[ ]( )
−
−−−
−=
∑ ∑∑∑∑∑ ∑⋅ 22
222
21
XXN
YXXYNYYN
NNs XY
wyvili
wyvili
wyvili
wyviliwyvili
sadac wyviliN aris qulaTa wyvilebis raodenoba.
es gantoleba ar aris iseTi rTuli, rogoric erTi SexedviT SeiZleba Candes. is gamoTvlebi, rac am formulis gamoyenebiT martoobisa da
depresiis qulebisTvis XYs ⋅ -is misaRebad aris saWiro, naCvenebia cxrilSi #14.8. am cxrilis pirveli safexuri moicavs am formulis amosaxsnelad saWiro ricxviT mniSvnelobebs. ToToeuli cdis pirisaTvis:
• a svetSi mocemulia martoobis qula, X
• b svetSi mocemulia depresiis qula, Y
• g svetSi mocemulia martoobis qulis kvadrati, ( 2X )
• d svetSi mocemulia depresiis qulis kvadrati ( 2Y )
• e svetSi mocemulia martoobisa da depresiis qulis namravli ( XY ) cxrili #14.8
XYs ⋅ -is gamoangariSeba gamoTvliTi formulis gamoyenebiT. martoobis qulebi ( X , a
sveti) da depresiis qulebi (Y , b sveti) aRebulia cxrilidan #14.3.
safexuri I. ∑ X -is, ∑Y -is, ∑ 2X -is, ∑ 2Y -sa da ∑ XY -is gamoTvla:
(a) (b) (g) (d) (e) cdis piri
X
Y
2X
2Y
XY 1 4 16 16 256 64 2 27 37 729 1369 999 3 18 33 324 1089 594 4 7 23 49 529 161 5 30 34 900 1156 1020 6 12 32 144 1024 384 7 18 24 324 576 432 8 23 29 529 841 667 9 19 26 361 676 494 10 12 26 144 676 312
jamebi 170 280 3520 8192 5127
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
23
safexuri IIi. XYs ⋅ -is gamoTvla:
( )
,10,5127
,3520
,280
,900.28170
,170
2
22
=
=
=
=
==
=
∑∑∑∑∑
wyviliNXY
X
Y
X
X
( )8192
400.78280
280
2
22
=
==
=
∑∑∑
Y
Y
Y
da
( ) ( ) ( ) ( )( )[ ]( )
( ) ( ) ( ) ( )( )[ ]( )
( )
Sedegad isdamrgvaleb demeasedebam
wyvili
wyvili
wyvili
wyviliwyvili
4,16
079,1382801
630036703520
801
900.28352010280170512710400.78819210
8101
21
2
2
22
222
=
=
−
=
−−
−−
=
−
−−−
−=
∑ ∑∑∑∑∑ ∑⋅
XXN
YXXYNYYN
NNs XY
∑ X aris a svetis jami, ∑Y aris b svetis jami, 2∑ X aris g svetis
jami, ∑ 2Y aris d svetis jami da 2∑ X aris e svetis jami. me-2 safexuri
gviCvenebs am formulaSi mniSvnelobebis Casmas. XYs ⋅ –is miRebuli mniSvneloba aris 4,16, igive, rac ganmartebiTi formulis gamoyenebiT miviReT.
XYs ⋅ -is gamoTvla r –is mixedviT gazomvis standartuli Secdoma SeiZleba agreTve gamoiTvalos X -sa da Y -s
Soris realurisar da Y cvladis standartuli gadaxris gamoyenebiT Semdegi
formuliT:
( )2realuri
wyvili
wyvili rNN
ss YXY −
−
−=⋅ 1
21
cxrilis #14.3 monacemebisaTvis,
25,6=Ys ,
78,0+=realurir ,
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
24
10=wyviliN .
maSasadame,
( ) ( )[ ] ( )( ) ,15,46638,025,678,0121011025,6 2 ==+−
−−
=⋅XYs
damrgvalebis Secdomis sazRvrebis gaTvaliswinebiT es igivea, rac ganmartebiTi an gamoTvliTi formulis meSveobiT miRebuli mniSvneloba.
rodesac wyviliN -s mniSvneloba didia, wyviliN _ 1-is wyviliN _ 2-Tan
Sefardeba uaxlovdeba 1,00-s da formula xSirad Caiwereba, rogorc
21 realurirss YXY −=⋅
XYs ⋅ -is gamosaTvleli formulebi Sejamebulia cxrilSi #14.9. cxrili #14.9
Sefasebis standartuli Secdomis ( XYs ⋅ ) formulebi
naSTebis formula
2−=⋅
wyvili
naSTi
NSS
s XY
an
( )2
2
−
′−= ∑
⋅wyviliN
YYs XY
gamoTvliTi formula
( ) ( ) ( ) ( )( )[ ]( )
−
−−−
−=
∑ ∑∑∑∑∑ ∑⋅ 22
222
21
XXN
YXXYNYYN
NNs XY
wyvili
wyvili
wyvili
wyviliwyvili
formula realurisr gamoyenebiT
( )2realuri
wyvili
wyvili rNN
ss YXY −
−
−=⋅ 1
21
informacia, romelsac gvaZlevs Sefasebis standartuli Secdoma
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
25
imisaTvis, rom gavigoT Tu ra informacias gvaZlevs Sefasebis standartuli Secdoma, cota xniT mivubrundebiT standartuli gadaxris, s -is, ganxilvas. qulebis simravlis standartuli gadaxra gvaZlevs am qulebis Secdomis (cdomilebis) cvalebadobas. Tu qulebis Sesaxeb araferi viciT garda maTi saSualosi, saukeTeso winaswarmetyveleba am qulebisaTvis aris maTi saSualo. ra Tqma unda, savaraudoa, rom es winaswarmetyveleba mcdari iqneba da saSualod, es Secdoma (cdomileba) standartuli gadaxris mniSvnelobis toli iqneba. magaliTad, davuSvaT, Tqven giTxres, rom cxrilis #14.3 monacemebis depresiis saSualo qula aris 28. Tu gTxoven, rom iwinaswarmetyveloT calkeuli qula, maSin, roca araferi iciT imis
garda, rom 0,28=Y , Tqveni saukeTeso winaswarmetyveleba iqneba 28,0. magram cxrilis #14.3 nebismieri qulisaTvis es winaswarmetyveleba mcdaria; arc erT cdis pirs miuRia depresiis 28 qula. qulaTa am simravlisaTvis aseTi winaswarmetyvelebis saSualo Secdoma aris standartuli gadaxris
mniSvneloba, 25,6=Ys . axla davuSvaT, TqvenTvis cnobili gaxda, rom martoobisa da depresiis qulebs Soris arsebobs wrfivi kavSiri, gamoxatuli, rogorc
1,18583,0 ++=′ XY . garda amisa, davuSvaT, TqvenTvis aseve cnobili gaxda, rom pirvel cdis pirs aqvs depresiis 4 qula. am qulis mixedviT da regresiis gantolebis gamoyenebiT Tqven iwinaswarmetyvelebT, rom pirovnebis Y qula iqneba 20,43. miuxedavad imisa, rom es winaswarmetyvelebac mcdaria (cdis piris realuri qula iyo 16), nawinaswarmetyvelebi 20,43 ufro axlosaa realur qulasTan,
vidre Y –is mixedviT nawinaswarmetyvelebi 28,0. Sefasebis standartuli Secdoma aris nawinaswarmetyvelebi qulebis Secdomis standartuli
gadaxra. maSasadame, XYs ⋅ gamoxatavs regresiis wrfis mixedviT prognozirebisas daSvebul saSualo Secdomas. yuradReba miaqceT imis, rom
Cvens magaliTSi XYs ⋅ = 4,16 naklebia 25,6=Ys -ze. aqedan gamomdinare, Tu Y –s
viwinaswarmetyvelebT X –is mixedviT, winaswarmetyvelebis Secdoma,
Cveulebriv, naklebi iqneba imasTan SedarebiT, ubralod Y rom gamogveyenebina yvela Y mniSvnelobis winaswarmetyvelebisaTvis. cxrilSi
#14.10 warmodgenilia Ys -sa da XYs ⋅ –s Soris kavSiri, rogorc X da
Y cvladebs Soris korelaciis funqcia. cxrili Sedgenilia realurisr
mniSvnelobebis CasmiT Semdeg formulaSi:
2 XY1 realurirss YXY −=⋅
yuradReba miaqcieT imas, rom roca realurir nulis tolia, maSin XYs ⋅ = Ys da
cdis piris martoobis qulis codna misi depresiis qulis prognozirebis sizustes ar zrdis. r –is mcire mniSvnelobebisTvis (magaliTad, r -is mniSvnelobaa 0,1, 0,2 an 0,3), X cvladis codna Y cvladis qulis winaswarmetyvelebis sizustis gazridisaTvis bevrs arafers akeTebs. meores mxriv, r -is didi mniSvnelobebi (magaliTad, r =0,7; 0,8 an 0,9) mniSvnelovnad
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
26
aumjobesebs winaswarmetyvelebis sizustes. magaliTad, Tu X –sa da Y –s
Soris korelacia aris 0,8, maSin XYs ⋅ 0,6-jer naklebia Ys –ze. roca r udris 1,00-s, maSin XYs ⋅ = 0; Y qulis winaswarmetyveleba X qulis mixedviT zustia da winaswarmetyvelebisas Secdomas ar vuSvebT. cxrili #14.10
XYs ⋅ -sa da Ys –s Soris kavSiri, rogorc r –is funqcia.
Tu realurir = _____ , maSin XYs ⋅ =_____ Ys . 0 1,000 0,1 0,995 0,2 0,980 0,3 0,954 0,4 0,917 0,5 0,866 0,6 0,800 0,7 0,714 0,8 0,600 0,9 0,436 1,0 0,000
X -is winaswarmetyveleba Y -is mixedviT
aqamde moyvanil magaliTebSi martoobis qulebi ( X ) gamoiyeneboda depresiis qulebis (Y ) winaswarmetyvelebisaTvis. magram SegveZlo, Segvebrunebina procesi da depresiis qulebi gamogveyenebina martoobis winaswarmetyvelebisaTvis. es mogvcemda imisgan gansxvavebuol regresiis gantolebas, rac depresiis martoobis mixedviT winaswarmetyvelebisaTvis gvqonda. cxrilSi #14.11 warmodgenilia martoobisa da depresiis qulebi cxrilis #14.3 Sebrunebuli X da Y aRniSvnebiT; depresiis qulebi axla aRniSnulia X -iT (a sveti) da martoobis qulebi Y -iT (b sveti). g da d
svetebSi mocemulia 2∑ X -sa da ∑ XY -is mniSvnelobebi, rac saWiroa
gamoTvliTi formulis gamoyenebiT wrfis daxrilobis gamoTvlisaTvis, rogorc amas cxrilis me-2 safexuri gviCvenebs. me-3 safexurze miRebulia Y -TanakveTa, a . depresiis mixedviT martoobis winaswarmetyvelebis regresiis wrfe warmodgenilia me-4 safexurze. es regresiis wrfe gamoiyeneboda e svetSi naCvenebi Y ′ mniSvnelobebis gamosaTvlelad. naSTebi warmodgenilia v svetSi da naSTebis kvadratSi ayvanili mniSvnelobebi – z svetSi. me-5
safexurze naCvenebia naSTebis mixedviT XYs ⋅ -is gamoTvla.
yuradReba miaqcieT imas, rom regresiis wrfis daxriloba da Y -TanakveTa damokidebulia imaze, Tu romeli cvladia aRniSnuli X -iT da
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
27
romeli - Y -iT. Tu martooba aris aRniSnuli X -iT, maSin regresiis wrfea 1,18583,0 ++=′ XY . magram, Tu depresiaa aRniSnuli X -iT, maSin regresiis
wrfea 2,12043,1 −+=′ XY . Tumca orive wrfe Y -iT aRniSnuli cvladis erTnairad zusti winaswarmetyvelebis saSualebas gvaZlevs. es Tviseba
SeiZleba davinaxoT TiToeuli wrfis XYs ⋅ -is Ys -Tan SedarebiT. Tu martooba
aris X cvladi, maSin XYs ⋅ aris 4,16 da Ys aris 6,25. am SemTxvevaSi
XYs ⋅ / Ys udris 4,16/6,25, rac udris 0,67-s. meores mxriv, Tu depresiaa X
cvladi, XYs ⋅ udris 5,56 da Ys aris 8,37. aq XYs ⋅ / Ys udris 5,56/8,37, rac udris 0,66-s. damrgvalebis Secdomis farglebSi, TiToeuli wrfe qulebis winaswarmetyvelebisas erTnair SecdomasTan aris dakavSirebuli. SemTxvevaTa umetesobaSi mkvlevars garkveuli mizezi aqvs imisaTvis, rom romelime cvladi aRniSnos X -iT da is Y -is winaswarmetyvelebisaTvis gamoiyenos. maSasadame, monacemebis umetesobisaTvis, xdeba mxolod erTi regresiis wrfis dadgena. cxrili #14.11 umcires kvadratTa regresiis wrfis, nawinaswarmetyvelebi qulebis da naSTebis gamoTvla martoobis qulebis depresiis qulebis mixedviT winaswarmetyvelebisaTvis.
safexuri I. ∑ X -is, ∑Y -is, ∑ 2X -isa da ∑ XY -is gamoTvla:
cdis piri
(a) (b) (g) (d) (e) (v) (z) depresia
( X )
martooba
(Y ) 2X XY Y ′ YY ′− ( )2YY ′−
1 16 4 256 64 4,488 -0,488 0,238 2 37 27 1369 999 26,391 0,609 0,371 3 33 18 1089 594 22,219 -4,219 17,800 4 23 7 529 161 11,789 -4,789 22,935 5 34 30 1156 1020 23,262 6,738 45,401 6 32 12 1024 384 21,176 -9,176 84,199 7 24 18 576 432 12,832 5,168 26,708 8 29 23 841 667 18,047 4,953 24,532 9 26 19 676 494 14,918 4,082 16,663 10 26 12 676 312 14,918 -2,918 8,515
jamebi 280 170 8192 5127 247,362
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
28
safexuri II. viTvliT b -s gamoTvliTi formulis gamoyenebiT:
( )
.105127
8192
170
400.78280
280
2
22
=
=
=
=
==
=
∑∑∑∑∑
wyviliNXY
X
Y
X
X
( )( ) ( )( )( )( ) ( )22 ∑∑
∑∑∑−
−=
XXN
YXXYNb
wyvili
wyvili
ricxviTi mniSvnelobebis CasmiT:
( )( ) ( )( )( )( ) 043,1
35203670
78400819210170280512710
+=+=−
−=b
safexuri 3. a -s gamoTvla:
.2,12)28(043,117
17
,28
−=−=−=
=
=
XbYa
Y
X
safexuri 4. regresiis wrfea:
2,12043,1 −+=′ XY
safexuri 5. naSTebis mixedviT XYs ⋅ -is gamoangariSeba:
( )
( ) ( )56,58362,247
2
82362,247
2
2
=÷=
−÷′−=
=−
=′−
∑
∑
⋅ wyvili
wyvili
NYYs
NYY
XY
regresiis wrfis gamoyeneba wrfivi regresiis saTanadod gamoyenebisaTvis saWiroa ori piroba. pirveli,
am qulebis realurir statistikurad mniSvnelovani unda iyos. Tu qulebs Soris
korelacia ar aris, maSin Y qulis saukeTeso winaswarmetyveleba iqneba Y qulebis saSualos gamoyeneba. Tu X da Y cvladebs Soris korelacia ar aris da Tqven gaZleven X qulas da gTxoven Y qulis winaswarmetyvelebas,
Y -is waukeTeso winaswarmetyveleba iqneba ubralod Y . X -is codna araviTar informacias ar gaZlevT Y -is mniSvnelobis Sesaxeb. meore piroba aris is, rom kavSiri X da Y cvladebs Soris wrfivi unda iyos. umcires kvadratTa meTodiT miRebuli wiri aris wrfe. Tu X da Y cvladebi ar aris wrfivad dakavSirebuli, maSin maTTvis umcires kvadratTa regresiis wrfis misadageba marTebuli ar aris. magaliTad, Tu X da Y cvladebs Soris kavSiri aris U –s an Sebrunebuli U –s (anu ) formis, maSin am qulebisaTvis umcires kvadratTa regresiis wrfis ageba ar migviyvanda zust winaswarmetyvelebamde, radgan es qulebi wrfivad dakavSirebuli ar aris. aseve frTxilad unda viyoT da ar gavavrceloT winaswarmetyveleba im SerCevis qulebis miRma, romlidanac winaswarmetyvelebis gantoleba iqna miRebuli. Cveni SerCevis sazRvrebs miRma mniSvnelobebisaTvis es kavSiri SeiZleba arawrfivi iyos.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
29
SeamowmeT Tqveni codna XIV-3
1. ganmarteT: umcires kvadratTa meTodiT miRebuli regresiis wrfe,
naSTi, naSTiSS , Sefasebis standartuli Secdoma, XYs ⋅ , YY ′− . 2. ratom ar gamoiyeneba ( )∑ ′−YY , rogorc X –is mixedviT Y –is
mniSvnelobis winaswarmetyvelebis Secdomis sazomi?
3. ratom aris SesaZlebeli gamoviyenoT ( )∑ ′− 2YY , rogorc X –is
mixedviT Y –is mniSvnelobis winaswarmetyvelebis Secdomis sazomi? 4. dawereT Sefasebis standartuli Secdomis ganmartebiTi formula. 5. kvlevaSi, romlis mizani iyo gamocdis niSnebis winaswarmetyveleba
mecadineobisaTvis daTmobili saaTebis mixedviT, qulaTa 16
wyvilisaTvis miiRes Semdegi mniSvnelobebi: 72=∑ X , ∑ = 4082X ,
( ) 51842=∑ X , 1264=∑Y , ∑ = 636,1022Y , ( ) 696.597.12
=∑Y da ∑ XY =6140.
ras udris XYs ⋅ -is mniSvneloba am kvlevis monacemebisaTvis?
6. fiziologma 45 adamianze 3 Tvis dakvirvebis Sedegad daadgina realurir =
+0,41-is toli korelacia fizikuri varjiSis odenobasa da wonis
klebas Soris. Ys -is mniSvneloba iyo 5,2 funti. am klvevis
monacemebisaTvis ras udris XYs ⋅ ? 7. maswavlebelma, romelsac surs, rom Sualeduri gamocdis Sedegebis
mixedviT iwinaswarmetyvelos saboloo gamocdis niSnani, 27
studentisagan Sedgenil klasSi daadgina, rom naSTiSS =462,25. ras udris
am klevis monacemebisaTvis XYs ⋅ ?
magaliTi #14.1
regresiis wrfis dadgena korelaciis koeficientis mixedviT amocana kvebiTi darRvevebis - anoreqsiisa da bulimiis - kvlevis erTi mimarTuleba Seiswavlis pirovnebis sxeulis xatsa da mis sxva maxasiaTeblebs (sqesobrivi roli, aRqmuli momxibvlvloba, an TviTSefaseba) Soris mimarTebas. magaliTad, minci da betci (Mintz & Betz, 1986) kolejis studendebSi TviTSefasebis skalebis gamoyenebiT zomavdnen sakuTari sxeuliT kmayofilebasa da depresiisadmi midrekilebas. sxeuliT kmayofilebis qulebi meryeobda 1-dan (rac gamoxatavda sakuTari sxeulis mimarT Zlier dadebiT damokidebulebas) 7-mde (rac gamoxatavda sakuTari sxeulis mimarT Zlier uaryofiT damokidebulebas). depresiisadmi midrekilebis qulebic aseve meryeobda 1-dan (rac gamoxatavda depresiisadmi Zlier mdgradobas) 7-mde (gamoxatavda depresiisadmi midrekilebas). 23
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
30
saSualoze odnav naklebi wonis mdedrobiTi sqesis studentis jgufSi mincma da betcma aRmoaCines statistikurad mniSvnelovani korelacia sxeuliT kmayofilebis Sefasebasa ( X cvladi) da depresiisadmi
midrekilebis maCvenebels (Y cvladi) Soris, realurir = +0,44. davuSvaT, rom
gavimeoreT es kvleva saSualoze odnav naklebi wonis mqone 15 qalze da miviReT sxeuliT kayofilebasa da depresiisadmi midrekilebas Soris +0,58-
is toli korelacia realurir . roca Tavisuflebis xarisxi (df ) aris 13,
kritikulir = 0,514, maSasadame, realurir statistikurad mniSvnelovania. sxeuliT
kmayofilebis qulebis saSualo iyo X = 3,7 da standartuli gadaxra iyo 1,8.
depresiisadmi midrekilebisaTvis Y = 3,8 da Ys =1,9. ipoveT: (a) depresiisadmi midrekilebis sxeuliT kmayofilebaze umcires kvadratTa regresiis wrfe, (b) depresiisadmi midrekilebis nawinaswarmetyvelebi mniSvneloba, rogorc sxeuliT kmayofilebis qulebis funqcia da (g) Sefasebis standartuli Secdoma. amoxsna: amocana moiTxovs regresiis wrfis daxrilobisa da Y-TanakveTis mniSvnelobebis povnas. radgan realurir mocemulia, daxrilobas vpoulobT
korelaciis formulis gamoyenebiT, romelic warmodgenilia cxrilze #14.5. daxrilobis dadgena: daxrilobas, b -s, vpoulobT r –is mixedviT Semdegnairad:
=
X
Y
ssrb
realurisr , Xs -isa da Ys -is mniSvnelobebis CasmiT viRebT:
61,08,19,158,0 +=
+=b .
Y-TanakveTis povna: Y-TanakveTas vpoulobT Semdegnairad
( )( ) 54,17,361,08,3 =+−=−= XbYa
regresiis gantolebis miReba: depresiisadmi midrekilebis (Y ) sxeuliT kmayofilebaze ( X ) regresiis gantoleba aseTia:
54,161,0 ++=′ XY .
depresiisadmi midrekilebis nawinaswarmetyvelebi qulebis (Y ′ ) miReba: sxeuliT kmayofilebis qulebis ( X ) mixedviT depresiisadmi midrekilebis nawinaswarmetyvelebi qulebi (Y ′ ) miiReba regresiis gantolebis X –is
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
31
mimarT amoxsniT. magaliTad, sxeuliT kmayofilebis 1-is toli SefasebisaTvis regresiis ganotoleba Semdeg saxes iRebs:
15,254,1)1(61,0 =++=′Y .
1-is toli sxeulis kmayofilebis Sefasebis mixedviT nawinaswarmetyvelebi depresiisadmi midrekilebis qula aris 2,15, anu, Tu meaTedebamde davamrgvalebT, 2,2. regresiis wrfis ageba: regresiis wrfe, romlis gantolebaa
54,161,0 ++=′ XY , gavlebulia suraTze #14.8. sxeuliT kmayofilebis mniSvneloba, _ is cvladi, romlis mixedviTac xdeba winaswarmetyveleba ( X ), gadazomilia abscisaze da nawinaswarmetyvelebi cvladis, depresesiisadmi midrekilebis (Y ′ ), mniSvneloba gadazomilia ordinataze. regresiis wrfis gavleba xdeba
grafikze Y-TanakveTisa da X ,Y koordinatebis (Sesabamisad 3,7 da 3,8) aRniSvniT da am ori wertilis SemaerTebeli wrfis gavlebiT. depresiisadmi midrekilebis nawinaswarmetyvelebi qulebi X –is ToToeuli mniSvnelobisaTvis wrfeze wertilebiT aris aRniSnuli.
Sefasebis standartuli Secdomis povna: XYs ⋅ gamoiTvleba Semdegnairad:
( )2121
realuri
wyvili
wyvili rNN
ss YXY −
−
−=⋅
Tu CavsvamT: Ys =1,9, 15=wyviliN , realurir = +0,58-s, miviRebT
( ) ( )( ) 6,158,012151159,1 2 =+−
−−
=⋅XYs
meaTedebamde davamrgvalebis Sedegad. suraTi #14.8 umcires kvadratTa meTodiT miRebuli regresiis wrfe sxeulis xatisa da depresiisadmi midrekilebis qulebisaTvis. depresiisadmi midrekilebis nawinaswarmetyvelebi mniSvnelobebi regresiis wrfeze wertilebiT aris aRniSnuli.
es XYs ⋅ naklebia, vidre Ys =1,9. pirovnebis sxeuliT kmayofilebis qulis codna aumjobesebs depresiisadmi midrekilebis qulis winaswarmetyvelebis
sizustes Y =3,8–s, yvela cdis pirisaTvis nawinaswarmetyvelebi mniSvnelobis gamoyenebasTan SedarebiT.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
32
magaliTi #14.2
regresiis wrfisa da Sefasebis standartuli Scdomis dadgena gamoTvliTi formulis mixedviT amocana amocanaSi #14.1 moyvanili iyo mincisa da betcis (Minz & Betz, 1986) kvleva sxeuliT kmayofilebasa da depresiisadmi midrekilebis qulebis mimarTebis Sesaxeb. cdis pirebisagan maT aseve miiRes socialuri TviTpativiscemis monacemebi. TviTpativiscemis skalebi iZleoda 20-dan (rac aRiniSnavda dabal TviTpativiscemas) 100-mde (aRniSnavda maRal TviTpativiscemas) qulebs. sxeuliT kmayofilebis qulebi icvleboda 1-dan (rac aRniSnavda Zlier dadebiT damokidebulebas sakuTari sxeulis mimarT) 7-mde (rac aRniSnavs Zlier uaryofiT damokidebulebas sakuTari sxeulis mimarT). saSualoze odnav naklebi wonis mqone qalebis kvlevisas mincma da betcma miiRes sxeuliT kmayofilebis qulebsa ( X ) da socialur TviTpativiscemas
(Y ) Soris statistikurad mniSvnelovani korelacia realurir = _ 0,60. davuSvaT,
rom am kvlevis 15 studentze gameorebisas miviReT sxeuliT kmayofilebisa da socialuri TviTSefasebis Semdegi qulebi:
studenti
sxeuliT kmayofileba: X
TviTpativiscema:
Y 1 2 76 2 4 48 3 6 27 4 1 70 5 3 41 6 5 38 7 6 37 8 3 61 9 7 33 10 1 91 11 4 36 12 2 55 13 3 49 14 5 51 15 4 52
gvinda, rom vipovoT: (a) socialuri TviTpativiscemis qulebis sxeuliT kmayofilebis qulebze umcires kvadratTa regresiis wrfe, (b) TviTpativiscemis nawinaswarmetyvelebi qulebi, rogorc sxeuliT
kmayofilebis funqcia, (g) XYs ⋅ –is mniSvneloba YY ′− naSTebis mixedviT da (d)
XYs ⋅ –is mniSvneloba gamoTvliTi formulis gamoyenebiT.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
33
amoxsna: cxrili #14.12 gvaZlevs moTxovnili informaciis mosaZieblad saWiro mniSvnelobebs. am cxrilSi mocemulia Semdegi mniSvnelobebi:
sveti Aa sxeuliT kmayofilebis qulebi ( X ) da ∑ X
Bb TviTkmayofilebis qulebi (Y ) da ∑Y
Gg sxeuliT kmayofilebis qulebis kvadrati ( 2X ) da
∑ 2X
Dd TviTpativiscemis qulebis kvadrati ( 2Y ) da ∑ 2Y
Ee sxeuliT kmayofilebisa da TviTpativiscemis
qulebis jvaredina namravlebi da ∑ XY
v dadgenili regresiis wrfis mixedviT nawinaswarmetyvelebi Y ′ -is mniSvnelobebi
z naSTi TviTpativiscemis TiToeuli qulisaTvis cxrili #14.12 umcires kvadratTa regresiis wrfis, nawinaswarmetyvelebi qulebis da naSTebis dadgena TviTpativiscemisa da sxeuliT kmayofilebis qulebisaTvis
(a) (b) (g) (d) (e) (v) (z) studenti X Y 2X 2Y X Y Y ′ YY ′− 1 2 76 4 5776 152 64,859 +11,141 2 4 48 16 2304 192 48,837 -0,837 3 6 27 36 729 162 32,815 -5,815 4 1 70 1 4900 70 72,870 -2,870 5 3 41 9 1681 123 56,848 -15,848 6 5 38 25 1444 190 40,826 -2,826 7 6 37 36 1369 222 32,815 +4,185 8 3 61 9 3721 183 56,848 +4,152 9 7 33 49 1089 231 24,804 +8,196 10 1 91 1 8281 91 72,870 +18,130 11 4 36 16 1296 144 48,837 -12,837 12 2 55 4 3025 110 64,859 -9,859 13 3 49 9 2401 147 56,848 -7,848 14 5 51 25 2601 255 40,826 +10,174 15 4 52 16 2704 208 48,837 +3,163 jamebi 56 765 256 43,321 2480 saSualo 3,73 51,00 s 1,83 17,54
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
34
regresiis wrfis povna b -s gamoangariSeba gamoTvliTi formulis saSualebiT:
( )
,2480
,765
,256
,313656
,56
2
22
=
=
=
==
=
∑∑∑∑∑
XY
Y
X
X
X
15=wyviliN ,
da
( )( ) ( )( )( )( ) ( )22 ∑∑
∑∑∑−
−=
XXN
YXXYNb
wyvili
wyvili
ricxviTi mniSvnelobebis CasmiT,
( )( ) ( )( )( )( ) ( ) .011,8
7045640
31362561576556248015
−=−
=−−
=b
gamovTvaloT a : X = 3,73 da Y = 51,00. am mniSvnelobebis CasmiT
XbYa −= –Si viRebT
( )( ) .881,8073,3011,800,51 =−−=a L regresiis wrfe: regresiis wrfe warmodgenilia gantolebiT:
881,80011,8 +−=′ XY .
TviTpativiscemis nawinaswarmetyvelebi qulebi: regresiis gantolebidan nawinaswarmetyvelebi TviTpativiscemis qulebi mocemulia cxrilis v svetSi. imisaTvis, rom minimumamde SevamciroT damrgvalebis Secdoma naSTebis
mixedviT XYs ⋅ -is gamoTvlisas, Y ′ -is mniSvnelobebs vamrgvalebT meaTasedebamde.
XYs ⋅ naSTebis mixedviT: naSTebis mniSvnelobebi mocemulia cxrilis z
svetSi. XYs ⋅ -is mniSvneloba am naSTebidan Semdegnairad miiReba:
( )2
2
−
′−= ∑
⋅wyviliN
YYs XY
( )∑ ′− 2YY -is misaRebad z svetSi mocemuli TiToeuli naSTi agvyavs
kvadratSi da Semdeg vkrebT kvadratSi ayvanil 15 mniSvnelobas.
( )∑ =′− 7380,12932YY . wyviliN _ 2 = 15 _ 2 = 13.
ricxviTi mniSvnelobebis CasmiT,
.98,913
7380,1293==⋅XYs
XYs ⋅ gamoTvliTi formulis mixedviT: gamoTvliTi formulaa:
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
35
( ) ( ) ( ) ( )( )[ ]( )
−
−−−
−=
∑ ∑∑∑∑∑ ∑⋅ 22
222
21
XXN
YXXYNYYN
NNs XY
wyvili
wyvili
wyvili
wyviliwyvili
ricxviTi mniSvnelobebis CasmiT,
( )( ) ( ) ( ) ( )( ) ( )( )[ ]( )( ) ( )
( )
.98,9909,405.19195
1
7045640590.64
1951
562561576556248015765321.4315
13151
2
2
22
=
=
−−
=
−−
−−
=⋅XYs
am formuliT miRebuli XYs ⋅ -is mniSvneloba igivea, rac ganmartebiTi formulisa da naSTebis gamoyenebiT miRebuli mniSvneloba. daskvna: regresiis gantoleba
881,80011,8 +−=′ XY saSualebas gvaZlevs, winaswarmetyvelebis mniSvnelovnad Semcirebuli SecdomiT viwinaswarmetyveloT TviTpativiscemis qulebi sxeuliT
kmayofilebis qulebis mixedviT. XYs ⋅ = 9,98 sidide, regresiis gantolebis mixedviT gamoTvlili TviTpativiscemis qulis winaswarmetyvelebasTan
dakavSirebuli saSualo Secdoma ufro mcirea, vidre 54,17=Ys - Secdomis
is saSualo sidide, rac Y -is (anu 51,0), rogorc yvela cdis pirisaTvis TviTSefasebis nawinaswarmetyvelebi qulis gamoyenebasTan aris dakavSirebuli.
magaliTi #14.3
samsaxureobrivi warmatebis winaswarmetyveleba regresiis gantolebis mixedviT amocana Sromis fsiqologiis ZiriTadi aspeqtia biznes da samTavrobo organizaciebSi samsaxuris dawyebis msurvelebis (aplikantebis) samsaxureobrivi warmatebis winaswarmetyveleba. am winaswarmetyvelebebis gasakeTeblad xSirad regresiuli analizi gamoiyeneba. aplikantebs utardebaT samuSaosTan dakavSirebuli testi da am testis qulebi gamoiyeneba samuSaoze potenciuri warmatebis winaswarmetyvelebisaTvis. ganvixiloT martivi magaliTi. davuSvaT, rom kompania samuSaos msurvels
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
36
(aplikants) utarebs tests `samuSaos nimuSi~. testi `samuSaos nimuSi~ aris kompaniaSi Sesabamisi samsaxuris miniaturuli versia. testis `samuSaos nimuSi~ Sedegebi ( X cvladi) gamoiyeneba samsaxurSi realuri warmatebis (Y ′ ) prognozirebisaTvis regresiis gantolebaze dayrdnobiT:
557,0 ++=′ XY
davuSvaT, rom 12 aplikants Cautarda testi `samuSaos nimuSi~ da miRebulia Semdegi qulebi:
aplikanti testis `samuSaos nimuSi~ qula: X 1 12 2 37 3 27 4 34 5 17 6 39 7 24 8 40 9 47 10 15 11 31 12 21
kompania ar daiqiravebs iseT adamians, romlis samuSao kvalifikacia 75 procentze naklebia. romel aplikantebs SeuZliaT samsaxurSi moxvedra maTi `samuSaos nimuSze~ qulebis mixedviT? amoxsna: amoxsnisaTvis saWiroa TiToeuli aplikantis `samuSaos nimuSis~ testis qulis mixedviT regresiis gantolebis gamoyenebiT miviRoT samuSao kvalifikaciis nawinaswarmetyvelebi qula. qvemoT naCvenebia TiToeuli aplikantis samuSao kvalifikaciis nawinaswarmetyvelebi qula:
aplikanti NsamuSaoze efeqturobis
nawinaswarmetyvelebi qula: Y ′ 1 64,1 2 83,1 3 75,5 4 80,8 5 67,9 6 84,6 7 73.2 8 85,4 9 90,7
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
37
10 66,4 11 78,6 12 71,0
# 1, 5, 7, 10 da 12 aplikantebis nawinaswarmetyvebi qulebi saWiro 75 qulaze dabalia; Sesabamisad, es aplikantebi samuSaoze misaRebad Sesaferisebi ar arian. yvela sxva aplikantis qula 75 an ufro maRalia, maSasadame, es aplikantebi samuSaoze misaRebad Sesaferisebi arian.
557,0 ++=′ XY regresiis wrfe naCvenebia suraTze #14.9. suraTi #14.9 regresiis wrfe 557,0 ++=′ XY , romelic gamoiyeneba samuSaoSi efeqturobis
dasadgenad samsaxuris nimuSis testis miedviT. samuSaos nimuSisaTvis nawinaswarmetyvelebi samuSaoSi efeqturobis qulebis regresiis wrfeze wertilebiT aris aRniSnuli. gaiTvaliswineT, rom samuSao kvalifikaciis nawinaswarmetyvelebi 75 qula Seesabameba `samuSaos nimuSis~ testis 26,3 qulas. nebismieri, 26,3-ze dabali samuSaos nimuSis testis qula niSnavs samuSao kvalifikaciis efeqturobis nawinaswarmetyveleb 75 qulaze naklebs. aqedan gamomdinare, im aplikantebs, romelTac eqnebaT 26,3 qulaze naklebi samuSaos nimuSis testSi, samsaxurSi ar miiReben (ar daiqiraveben). mravlobiTi regresia mravlobiTi regresia: Y qulis mniSvnelobebis winaswarmetyveleba X cvladis ramdenime mniSvnelobis gamoyenebiT regresiuli analizi farTod gamoiyeneba bihevioristul mecnierebebsa da mTel rig praqtikul SemTxvevebSi. maSin, roca Cvens yvela amocanaSi warmodgenili iyo X da Y cvladebi, praqtikaSi X cvladis mixedviT Y cvladis winaswarmetyvelebisaTvis SesaZlebelia cnobili iyos mxolod X cvladi da regresiis gantoleba, sadac regresiis gantoleba miRebulia im SemTxvevebidan, roca cnobili iyo X –ca da Y –c. magaliTad, saTbobi sawvaviT movaWres wina kviris saSualo temperaturis ( X cvladi) mixedviT SeuZlia iwinaswarmetyvelos, ramdeni sawvavis SekveTa dasWirdeba mas Semdegi kvirisaTvis (Y cvladi). anda gamyidvels SeuZlia, iwinaswarmetyvelos vaWrobis moculoba (Y cvladi) rekalmaze daxarjuli Tanxis odenobis ( X cvladi) mixedviT. Cvens TiToeul magaliTSi Y cvladis winaswarmetyvelebisTvis gamoiyeneboda mxolod erTi X cvladi. Tumca praqtikaSi regresiis gantolebebi ufro rTulia imasTan SedarebiT, rac warmogidgineT da Y -is winaswarmetyvelebisaTvis iyenebs ramdenime X cvlads. am midgomas mravlobiTi regresia ewodeba, radgan gamoiyeneba
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
38
mravali prediqtori. magaliTad, damqiravebels SeiZleba ainteresebdes potenciuri TanamSromlis samuSaoze warmateba, rogorc miRwevis testis qulis, standartizebuli intervius qulisa da samuSaos nimuSis testis qulis funqcia. am midgomis gamomxatveli zogadi regresiis gantoleba aseTia:
aXbXbXbY +++=′ 332211
sadac Y ′aris samsaxurSi warmatebis nawinaswarmetyvelebi qula, 1X , 2X da
3X aris prediqtori cvladebi (am SemTxvevaSi 1X aris miRwevis testis
qula, 2X aris standartizebuli intervius qula da 3X aris samuSaos
nimuSis testis qula); 1b , 2b da 3b aris Sesabamisad 1X -is, 2X -sa 3X -is
regresiis koeficientebi, xolo a aris Y -TanakveTa. regresiis koeficientebi warmoadgens TiToeuli X cvladis wonas Y
cvladis winaswarmetyvelebisas. magaliTad, Tu 1b = +0,18, 2b = +0,10, 3b = +0,34
da a = 30,3, maSin gantoleba aseT saxes iRebs:
3,3034,010,018,0 321 ++++=′ XXXY
Tu aplikantis miRwevis testze aqvs 75 qula, interviuze 20 qula daagrova da samuSaos nimuSis testze - 90 qula, maSin
( )( ) ( )( ) ( )( ).4,763,306,300,25,13
3,309034,02010,07518,0=+++=
+++=′Y
mravlobiTi regresiis gantolebebi farTod gamoiyeneba qceviT mecnierebebSi winaswarmetyvelebisaTvis. magaliTad, fridmeni da kolegebi (Friedman et al., 1993) sicocxlis xangrZlovobis winaswarmetyvelebisaTvis iyenebdnen bavSvis pirovnebis Svid ganzomilebas: komunikabelurobas, fizikur energias, mxiarulobas, keTilsindisierebas, motivacias, guneba-ganwyobilebis mdgradobas da inteleqts. sxva SemTxvevaSi mozardebis mier sigaretis mowevis winaswarmetyvelebisaTvis uiliamsma da kovingtonma (Williams & Covington, 1997) mravlobiTi regresiis gantolebisaTvis gamoiyenes Semdegi cvladebi: rasa, asaki, TanatolebTan urTierToba, ojaxuri siaxlove, mweveli megobrebis raodenoba, mweveli ojaxis wevrebis raodenoba da mozardis mier janmrTelobisaTvis riskis aRqma. Tumca am rTuli gantolebebisaTvis regresiis koeficientebis gamoTvla iseT codnas moiTxovs, rac am Sesavali Tavis farglebs scildeba.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
39
SeamowmeT Tqveni codna XIV-4
1. pirovnuli Tvisebebi da garkveuli davalebebis warmatebiT Sesruleba
xSirad dakavSirebulia. magaliTad, gormlim da gormlim (Gormly & Gormly, 1986) ikvlies korelacia socialuri introversiis Sefasebebsa da kubikebis awyobis tests Soris, romelic sivrciT unarebs zomavda. maT miiRes statistikurad mniSvnelovani dadebiTi korelacia: socialurad ufro introvertma adamianebma ufro male aawyves kubikebiT saWiro konstruqcia. davuSvaT, rom Tqven gaimeoreT es kvleva da cdis pirebs afasebdiT socialuri introversiis 9-qulian skalaze, sadac 1 niSnavs Zlier socialurad intoverts da 9 aRniSnavs sruliad araintroverts. aRricxavdiT dros (wamebSi), rac TiToeul cdis pirs 9 feradi kubikis gamoyenebiT garkveuli konstruqciis asawyobad dasWirda. 12 cdis pirisagan miRebuli qulebi Semdegnairia:
cdis piri
introversia: ( X )
Kkubikebis awyobis dro (Y )
1 6 217 2 3 230 3 9 315 4 1 224 5 4 271 6 2 198 7 3 256 8 5 263 9 8 321 10 7 289 11 8 265 12 6 291
kubikebis awyobis saSualo dro iyo 261,67 wami da Ys = 38, 82 wami. am
qulebis realurir statistikurad mniSvnelovania 0,05 doneze. b –s gamoTvliTi formulis gamoyenebiT gaavleT kubikebis awyobis itroversiis qulebze regresiis wrfe.
a. rogoria regresiis wrfis daxriloba? b. ras udris regresiis wrfis Y –TanakveTa? g. CawereT regresiis wrfis gantoleba. d. aageT wertilTa gabnevis diagrama introversiisa da kubikebis
awyobis qulebisaTvis. diagramaze gaavleT regresiis wrfe. e. ras udris introversiis TiToeuli qulisaTvis kubikebis agebis
nawinaswarmetyvelebi qula (Y ′ )? v. TiToeuli cdis pirisaTvis mocemuli Y ′–is gamoyenebiT ipoveT
naSTi.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
40
z. ipoveT naSTiSS .
T. naSTiisSS gamoyenebiT ipoveT XYs ⋅ .
i. SeadareT am am qulebis XYs ⋅ da Ys = 38,82. introversiis qulebis mixedviT kubikebis awyobis qulebis winaswarmetyvelebisaTvis regresiis gantolebis gamoyeneba amcirebs Tu ara winaswarmetyvelebis Secdomas imasTan SedarebiT, roca kubikebis awyobis yvela qulis nawinaswarmetyveleb mniSvnelobad kubikebis awyobis saSualo qula (anu 261,7 wami) gamoiyeneboda?
2. frinvelebze damkvirveblebs SemCneuli aqvT, rom frinvelTa bevri saxeoba garemos daTvalierebas sakmao dros uTmobs. elgari, makkei da vuni (Elgar, McKey, & Woon, 1986) akvirdebodnen beRurebs, raTa daedginaT kavSiri garemos daTvalierebis sixSiresa da kvebis siswrafes Soris. maT am or cvlads Soris statistikurad mniSvnelovani kavSiri aRmoaCines. davuSvaT, rom Tqven gaimeoreT es kvleva 15 beRuraze dakvirvebiT. Tqven iwerdiT TiToeuli beRuris mier wuTSi ganxorcielebuli garemos daTvalierebis raodenobas ( X cvladi) da TiToeuli frinvelis mier wuTSi ganxorcielebuli sakvebis akenkvis
raodenobas (Y cvladi). daTvalierebis saSualo raodenoba ( X ) iyo
26,3 daTvaliereba wuTSi, Xs = 8,7. akenkvebis saSualo raodenoba iyo
13,3 wuTSi da Ys =4,5. am or cvlads Soris realurir aris _0,84. b –s gamosTvleli korelaciuri formulis gamoyenebiT aageT akenkvis sixSiris (Y ) daTvalierebebis sixSireze ( X ) regresiis wrfe.
a. rogoria regresiis wrfis daxriloba? b. ras udris regresiis wrfis Y -TanakveTa? g. CawereT regresiis wrfis gantoleba. d. rogoria akenkvis nawinaswarmetyvelebi sixSire im
SemTxvevisaTvis, roca garemos daTvalierebis raodenoba wuTSi 41-iaA? roca wuTSi 18-ia?
Sejameba
• wrfivi regresiis SemTxvevaSi X cvladis yoveli erTi erTeuliT cvlilebisas Y cvladi ganicdis mudmiv cvlilebas
• wrfis zogadi gantoleba gadmocemulia formuliT: abXY +=
• daxriloba b , ganisazRvreba, rogorc:
cvlileba ismniSvnelob is
cvlileba ismniSvnelobis
−−
=XY
b
• Y -TanakveTa, a , aris Y –is mniSvneloba, roca X =0.
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
41
• wrfis zofadi gantoleba SeiZleba gamoviyenoT X is cnobili qulebis mixedviT Y qulebis winaswarmetyvelebisaTvis: abXY +=′ . Y ′aris Y -is nawinaswarmetyvelebi mniSvneloba.
• naSTi, YY ′− , aris sxvaoba Y realur qulasa da X –is mixedviT nawinaswarmetyveleb qulas Y ′Soris.
• umcires kvadratTa regresiis wrfe minimumamde amcirebs naSTebis kvadratebis jamis mniSvnelobas.
• Sefasebis standartuli Secdoma Semdegnairad gamoiTvleba:
2−=⋅
wyvili
naSTi
NSS
s XY
an
( )2
2
−
′−= ∑
⋅wyviliN
YYs XY .
• Sefasebis standartuli Secdoma gamoxatavs X –is mixedviT Y qulebis winaswarmetyvelebis Secdomis sidides.
• mravlobiTi regresia gulisxmobs Y cvladis winaswarmetyvelebisaTvis ramdenime X cvladis gamoyenebas.
ZiriTadi terminebi da simboloebi
• mravlobiTi regresia
• Sefasebis standartuli
Secdoma ( XYs ⋅ )
• wrfis daxriloba (b )
• naSTi ( YY ′− ) • wrfivi kavSiri • naSTiSS
• umcires kvadratTa regresiis wrfe
• wrfis gantoleba
• Y -TanakveTa ( a ) • Y ′
kiTxvebi
1. jeimim gaigo, Tu ra Sualeduri Sefaseba aqvs statistikis kursSi.
misma maswavlebelma aRniSna, rom wina semestrSi jgufis 10 studentis mier miRebuli Sualeduri da saboloo gamocdebis niSnebi Semdegia:
studenti gamocda
Sualeduri: X saboloo: Y
1 87 84 2 89 91 3 97 96 4 80 87
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
42
5 73 66 6 85 90 7 81 79 8 74 80 9 80 89 10 89 86
a. gamoTvliTi formulis gamoyenebiT ipoveT im regresiis wrfis
daxriloba, romlis mixedviTac xdeba Sualeduri Sefasebis mixedviT saboloo gamocdis Sefasebis winaswarmetyveleba.
b. am qulebisaTvis realurir aris +0,781, Xs = 7,37 da Ys = 8,34.
korelaciuri formulis gamoyenebiT gamoTvaleT b . g. am qulebisaTvis daadgineT umcires kvadratTa regresiis wrfis
gantoleba X -is mixedviT Y qulis winaswarmetyvelebisaTvis.
d. jeimim Sualedur gamocdaze miiRo 77 qula. g punqtSi dadgenili regresiis wrfis mixedviT, rogoria misi nawinaswarmetyvelebi saboloo gamocdis qula?
e. rogoria XYs ⋅ -is mniSvneloba regresiis am wrfisaTvis? 2. mkvlevari akvirdeboda kviris ganmavlobaSi studentebis mier video-
TamaSebiT garTobis xangrZlovobas ( X cvladi) da iTvlida
korelacias maTi semestruli niSnebis saSualo ariTmetikulTan (Y cvladi). videoTamaSebiT garTobis 0-dan 15 saaTamde
mniSvnelobebisaTvis realurir iyo _0,632 ( p < 0,05). Xs = 4,10 saaTi da
Ys =0,75 saaTi. regresiis wrfis Y –TanakveTa aris 2,23. a. ipoveT videoTamaSebiT garTobis xangrZlivobis mixedviT
niSnebis saSualo ariTmetikulis gamosaTvleli umcires kvadratTa regresiis wrfe.
b. braiani videoTamaSebiT kviraSi daaxloebiT 3 saaTis ganmavlobaSi erToba. ra aris misi niSnebis saSualo ariTmetikulis nawinaswarmetyvelebi mniSvneloba?
g. Cvens SerCevaSi videoTamaSebiT garTobis saSualo xangrZlivoba iyo 5,1 saaTi kviraSi. ras udris niSnebis saSualo ariTmetikuli am SerCevisaTvis?
3. kvlevis Sedegad dadginda wrfivi kavSiri orsuli qalis mier sigaretis mowevasa da axalSobilis mcirewonianobas Soris (`Birth Weight~, 1986). dedis mier orsulobis dros dReSi erTi Reri sigaretis mowevasTan erTad bavSvis dabadebis wona mcirdeba 4,3 unciiT. maSasadame, Tu deda orsulobis dros eweoda saSualod 2 Rer sigarets dReSi, bavSvis dabadebis wona Semcirdeboda 8,6 unciiT. CaTvaleT, rom Tu orsuli deda sigarets ar eweva, axalSobilis saSualo wona aris 128 uncia.
a. dawereT wrfivi kavSiris gantoleba am monacemebisTvis (miniSneba: mocemulia daxrilobaca da Y -TanakveTac)
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
43
b. daxazeT regresiis wrfe ujrebian furcelze. abscisaze
gadazomeT moweuli sigaretis Rerebis raodenoba ( X ).
g. rogoria im axalSobilis nawinaswarmetyvelebi dabadebis wona, romlis dedac orsulobis dros eweoda Semdegi raodenobis sigarets dReSi:
0 1 5 8 10 13 20
4. me-13 TavSi, kiTxvaSi gameorebisTvis #1, 16 mamakacis gabrazebis
qulebisa da sisxlis sitoluri wnevis Semdegi mniSvnelobebi warmogidgineT:
cdis piri Bbrazis done sisxlis wneva
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
64 30 37 39 17 29 36 34 44 32 37 47 50 41 46 61
170 132 129 144 145 122 124 145 137 115 127 148 148 144 133 158
a. am monacemebis mixedviT daadgineT sisxlis wnevis (Y ) gabrazebis
doneze ( X ) regresiis gantoleba. gamoiyeneT gamoTvliTi
formula. b. ras udris TiToeuli adamianisaTvis nawinaswarmetyvelebi
sisxlis wneva, rogorc gabrazebis donis funqcia? g. daxazeT wertilTa gabnevis diagrama gabrazebis donisa
(abscisaze) da sisxlis wnevisaTvis (ordinataze). amave diagramaze gaavleT regresiis wrfec.
d. am monacemebisaTvis Ys -is ra proporcias warmoadgens XYs ⋅ ? 5. SeamowmeT Tqveni codna XIV-4-is savarjiSoSi #2 warmodgenili iyo
kvleva wuTSi garemos daTvalierebis raodenobasa ( X ) da sakvebis
akenkvis raodenobas (Y ) Soris mimarTebis Sesaxeb 15 beRurisaTvis. davuSvaT, rom msgavsi dakvirveba awarmoeT 13 WinWraqaze da miiReT Semdegi monacemebi:
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
44
frinveli
garemos
daTvaliereba: X
sakvebis akenkva: Y
1 29 12 2 34 8 3 24 14 4 14 20 5 37 11 6 22 13 7 11 23 8 33 9 9 24 11 10 30 14 11 36 9 12 20 23 13 32 7
a. rogoria am regresiis wrfis daxriloba? b. rogoria am regresiis wrfis Y –TanakveTa? g. CawereT regresiis wrfis gantoleba. d. aageT wertilTa gabnevis diagrama garemos daTvalierebisa da
sakvebis akenkvis qulebisaTvis. am diagramazeve gaavleT regresiis wrfe.
e. am monacemebisaTvis gamoTvliTi formulis gamoyenebiT
gamoTvaleT XYs ⋅ .
v. SeadareT am monacemebis mixedviT gamoTvlili XYs ⋅ da Ys = 5,41. garemos daTvalierebis qulebis mixedviT akenkvis qulebis winaswarmetyvelebisaTvis regresiis gantolebis gamoyeneba amcirebs Tu ara winaswarmetyvelebis Secdomas imasTan SedarebiT, roca garemos daTvalierebis yvela qulis nawinaswarmetyveleb mniSvnelobad am qulebis saSualo (anu 13,4) gamoiyeneboda?
6. kompania samsaxuris msurvelTa samuSaoSi warmatebis winaswarmetyvelebisaTvis iyenebs standartizebul interviuSi miRebul qulas. gamoyenebuli regresiis gantoleba aris 182,1 ++=′ XY ,
sadac Y ′ aris nawinaswarmetyvelebi warmateba samsaxurSi da X aris
aplikantis mier standartizebul interviuSi miRebuli qula. kompania samsaxurSi miiRebs mxolod im aplikantebs, visi samsaxurSi warmatebis nawinaswarmetyvelebi qula 60 an metia. qvemoT moyvanilia rva aplikantis standartizebuli intervius qulebi. romeli aplikanti daakmayofilebs miRebis pirobebs?
aplikanti standartizebuli intervius qula: X
Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________
45
daiana 31 rodrigo 36 kolini 30 maria 41 rubeni 39 luka 29 holi 27 blaiTi 44
7. me-13 Tavis me-8 kiTxva warmoadgenda kvlevas kviraSi televizoris yurebis xangrZlivobasa da pirveli gamocdis niSans Soris
korelaciis Sesaxeb 17 studentisaTvis. am qulebisaTvis realurir iyo
_0,735, televizoris yurebis saSualo xangrZlivoba ( X ) iyo 10,0 da Xs
iyo 6,38-s. gamocdaze miRebuli saSualo niSani (Y ) iyo 76,0 da Ys iyo
11,44. XSS iyo 652,0 da XYCP iyo _859,0. a. ipoveT regresiis wrfis gantoleba televizoris yurebis
xangrZlivobis mixedviT gamocdis niSnebis winaswarmetyvelebisaTvis.
b. ipoveT nawinaswarmetyvelebi gamocdis niSnebi im studentebisaTvis, romlebic televizors kviraSi 4 da 15 saaTis ganmavlobaSi uyureben.
g. am monacemebisaTvis ras udris XYs ⋅ ? Ys -is ra nawils warmoadgens
XYs ⋅ ?
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
1
Tavi XV araparametruli statistikuri kriteriumebi __________________________________________________________ sixSireTa analizi: xi-kvardat kriteriumi
damoukideblobis xi-kvadrat kriteriumi Tanxmobis χ² kriteriumi xi-kvardat kriteriumis Sedegebis warmodgena man-uitnis U kriteriumi
statistikuri hipoTezebis Semowmeba man-uitnis U kriteriumis gamoyenebiT man-uitnis U kriteriumis daSvebebi U-s gamosaTvleli formulebi
man-uitnis U kriteriumis Sedegebis warmodgena uilkoqsonis niSniani rangebis kriteriumi
statistikuri hipoTezebis Semowmeba uilkoqsonis niSniani rangebis kriteriumis gamoyenebiT
uilkoqsonis niSniani rangebis kriteriumis Sedegebis warmodgena araparametruli kriteriumebis gamoyeneba
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
2
araparametruli statistikuri kriteriumi (Nonparametric statistical test): statistikur kriteriums ewodeba araparametruli, Tu hipoTezaSi ar aris gamoTqmuli varaudi populaciis parametris Sesaxeb. me-8-dan me-12 Tavamde ganixileboda statistikuri hipoTezis Semowmeba, romelic yuradRebas amaxvilebs t kriteriumsa da dispersiul analizze. es orive kritieriumi parametruli kriteriumia, radgan populaciis parametrebis (μ) Sesaxeb daskvnebis gasakeTeblad SerCevis statistikebs
(magaliTad, X -s) iyenebs. araparametruli statistikuri kriteriumebi amowmeben hipoTezebs, romlebic ar exeba populaciis parametrebs. maT ufro naklebi daSvebebi aqvT populaciis formis Sesaxeb, vidre parametrul kriteriumebs. am mizezis gamo araparametrul kriteriumebs zogjer ganawilebisgan Tavisufal kriteriumebs uwodeben. gaixseneT, rom parametruli kriteriumebis daSvebebia:
• populaciebSi, romlebidanac aRebulia SerCeva, qulebi normalurad aris ganawilebuli;
• sxvadasxva populaciaSi qulebis dispersiebi erTmaneTis tolia. praqtikaSi miRebuli monacemebi, SesaZloa, ar Seesabamebodes am daSvebebs. miuxedavad amisa, radgan parametruli kriteriumebi sandod iTvleba, mkvlevrebs SeuZliaT gamoiyenon t kriteriumi an dispersiuli analizi iseTi monacemebisTvisac ki, romlebic daSvebebs srulyofilad ar akmayofilebs. Tumca, zogierT SemTxvevaSi am daSvebebTan Sesabamisobis xarisxi sakmarisi ar aris imisaTvis, rom SeiZlebodes parametruli kriteriumis gamoyeneba. magaliTad, adamianis reaqciis drois gazomvisas reaqciis droebis umravlesoba Zalian xanmoklea, (magaliTad, daaxloebiT 500 milisekundi an naklebi). reaqciis dro 0 wamze naklebi ver iqneba; magram zeda zRvari SeiZleba nebismieri iyos. Zalian cota adamians SeiZleba hqondes sxva adamianebTan SedarebiT reaqciis Zalian didi dro. maSasadame, reaqciis drois monacemebis sixSireebis ganawileba, savaraudod, dadebiTad gadaxrili iqneba; reaqciis droebis umravlesoba Zalian mcirea da cota Tu iqneba bevrad xangrZlivi. cxadia, rom aseTi ganawileba ar aris miaxloebuli normalur ganawilebasTan. parametruli kriteriumebi aseve akeTebs daSvebas, rom azri aqvs
miRebuli qulebis saSualos ( X ) gamoTvlas. es moTxovna gulisxmobs, rom monacemebi intervalebis an Sefardebis skalaze unda iyos. magram xSirad qcevis gazomva am doneebze ver xdeba. eqsperimentatori, SesaZlebelia, ubralod iTvlides im adamianebis raodenobas, romlebic TiToeul qcevas avlenen da amdagvarad iRebdes qcevis gamovlenis sixSires, romelic saxeldebis skalazea. an miRebuli qulebi SeiZleba warmoadgendes rigis skalas, magaliTad, garkveuli pirovnuli niSniT individebis rigis
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
3
mixedviT ganlagebisas. radgan parametruli kriteriumebisTvis saWiroa X -is gamoTvla, isini Sesaferisi ar aris im SemTxvevebisTvis, roca qulebi saxeldebis an rigis skalazea. amgvarad, araparametruli kriteriumebi gamoiyeneba maSin, roca:
• parametruli kriteriumebis daSvebebi ar aris dakmayofilebuli an
• qulebi saxeldebis an rigis skalazea warmodgenili
cxrili #15.1 warmoadgens parametrul kriteriumebs da Sesabamis araparametrul kriteriumebs erTfaqtoriani sqemisaTvis. am TavSi ganvixilavT xi-kvadrats, man-uitnis U da uilkoqsonis niSniani rangebis kriteriumebs. korelaciis araparametruli kriteriumi, spirmenis ranguli korelaciis koeficienti warmodgenilia me-13 TavSi. cxrili #15.1 parametruli da araparametruli kriteriumebi erT-faqtoriani sqemisaTvis, rogorc sqemis tipisa da gazomvis skalis funqcia
sqemis tipi cdis pirebs Soris cdis pirTa Sida monacemebis
gazomvis skala
ori done
sami an meti done
ori done sami an meti done
parametruli kriteriumebi
Sefardebis an intervalebis
lobisdamoukidebt
an cdis pirebs Soris ANOVA
cdis pirebs Soris ANOVA
lidamokidebut an cdis pirTa Sida ANOVA
cdis pirTa Sida ANOVA
araparametruli kriteriumebi
rigis
man-uitnis U
kruskal-uolisis ANOVA rangebis mixedviT
uilkoqsonis niSniani rangebi
fridmenis ANOVA rangebis mixedviT
saxeldebis
xi-kvadrati
xi-kvadrati
____________________________________________________________________
sixSireTa analizi: xi-kvadrat kriteriumi
xi-kvadrat kriteriumi (Chi-square test): statistikuri kriteriumi, romelic gamoiyeneba saxeldebis skalis monacemebis gasaanalizeblad, roca mocemulia sxvadasxva kategoriis Sesabamisi sixSireebi.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
4
bevri kvlevis Sedegad viRebT iseT monacemebs, romlebic mxolod imas gviCvenebs, qceva ganxorcielda Tu ara. magaliTad, bunebriv pirobebSi cxovelebze dakvirvebisas mecnierma SeiZleba Caiweros, axorcielebs Tu ara cxoveli garkveul qcevas. aseTi monacemebi saxeldebis skalazea; isini ubralod aRniSnaven qcevis urTierTgamomricxavi kategoriebis xdomilebas. SesaZlebelia, aRiricxos sxvadasxva kategoriis qcevis gamovlenis sixSire, magram TiToeuli qula gamoxatavs mxolod imas, qceva gamovlinda Tu ara; is ar Seicavs pasuxis Sesaxeb ricxvobriv informacias. araparametruli kriteriumi, xi-kvadrat kriteriumi, am monacemebisaTvis Sesaferisi sazomia, rodesac cdis pirebs Soris sqemaa gamoyenebuli, monacemebi saxeldebis skalazea da mxolod qcevis gamovlenis sixSires aRniSnavs. ganvixilavT xi-kvadratis or formas: damoukideblobis xi-kvadrat kriteriums da Tanxmobis xi-kvadrat kriteriums. damoukideblobis xi-kvadrat kriteriumi
damoukideblobis xi-kvadrat kriteriumi (Chi-square test of independence): xi-kvadrati, romelic gamoiyeneba maSin, roca monacemebis kategorizacia ori damoukidebeli niSnis mixedviT xdeba. damoukideblobis xi-kvadrat kriteriumi (romelsac aseve ewodeba ormimarTulebiani xi-kvadrati), gamoiyeneba, roca qulis kategorizacia xdeba ori gansxvavebuli ganzomilebis mixedviT. am kriteriumis gamoyenebas qvemoT moyvanili magaliTi giCvenebT. samyaros Sesaxeb codnisa da mis mimarT damokidebulebebis mniSvnelovan wyaros televiziaSi nanaxi warmoadgens. bihevioristi mecnierebi SeSfoTebulebi arian imiT, rom televiziam, misi informaciuli SesaZleblobebis gamo, SeiZleba warmoadginos sazogadoebis arazusti saxe. ganvixiloT iseTi martivi ram, rogoricaaa satelevizio gmiris Tmis feri. populaciaSi garkveuli Tmis feris mqone mamakacebisa da qalebis Tanafardoba daaxloebiT Tanabaria (magaliTad, WaRara Tmis mqone mamakacebisa da qalebis Tanafardoba daaxloebiT Tanabaria). Tumca, deivisma (Davis, 1990) aRmoaCina, rom satelevizio gmirebis SemTxvevaSi xuTjer meti iyo albaToba imisa, rom qalebs qera Tma eqnebodaT da eqvsjer meti iyo albaToba imisa, rom mamakacebs WaRara Tma eqnebodaT. msgavsadve, elasmarma, hasegavam da breinma (Elasmar, Hasegawa, & Brain, 1999) daadgines, rom siuJetis mixedviT, ufro meti qeraTmiani qali personaJi aRwevda mizans, vidre muqTmiani. davuSvaT, rom mkvlevars ainteresebs daadginos, arsebobs Tu ara kavSiri satelevizio gadacemebis personaJi qalebis Tmis fersa da karieras Soris. misi hipoTezis Tanaxmad, satelevizio gadacemebis qeraTmian personaJ qalebs ufro xSirad eqnebodaT profesiuli donis samuSao, vidre muqTmian personaJebs. am hipoTezis Sesamowmeblad man SemTxveiTad SearCia satelevizio gadacemebis garkveuli raodenoba, romlebSic monawileobdnen zrdasruli personaJi qalebi da Semdeg es personaJebi daajgufa Tmis feris mixedviT (qera an muqTmiani) da karieris mixedviT (profesiuli an
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
5
araprofesiuli). TiToeuli persoinaJi SerCevaSi mxolod erTxel aRiricxeboda. am kvlevis hipoTeturi Sedegebi naCvenebia cxrilSi #15.2. am cxrilSi satelevizio personaJis kategorizacia ori ganzimilebis mixedviT xdeba: (1) Tmis feri: qera an muqi da (2) samuSaos done: profesiuli an araprofesiuli. am cxrilSi O–Ti aRniSnuli mniSvnelobebi (observed frequency – realuri sixSire) warmoadgens TiToeul kategoriaSi miRebuli Sedegebis sixSires. am kvlevaSi daakvirdnen sul 180 qals da TiTeuli maTganisagan am cxrilisaTvis mxolod erTi pasuxi miiRes: personaJi an qera iyo, an muqTmiani da misi samuSao an profesiuli iyo, an – araprofesiuli.
niSanTa SeuRlebis cxrilebi cxrili #15.2 satelevizio gadacemebis qali personaJebis Tmis ferisa da karieris mixedviT
kategorizaciis hipoTeturi monacemebi. O -s saxiT mocemuli mniSvnelobebi
warmoadgens realur sixSireebs, xolo E -s saxiT warmodgenilia marginaluri
sixSireebis mixedviT miRebuli mosalodneli sixSireebi.
Tmis feri
Qqera muqi M(marginaluri sixSire)
karieris done
profesiuli O=36 E =28
O=48 E =56
84
araprofesiuli
O=24 E =32
O=72 E =64
96
(marginaluri sixSire)
60
120
(sul) 180
niSanTa SeuRlebis cxrili (Contingency table): striqonebiani da svetebiani cxrili, romelSic warmodgenilia yoveli kategoriis sixSireebi mocemuli striqonisa da svetis cvladebis gadakveTis Sedegad. cxrils #15.2 ewodeba striqonebiani ( r ) da svetebiani ( c ) niSanTa SeuRlebis cxrili an sixSireebis ormimarTulebiani cxrili. svetebi (aRiniSneba c -Ti) warmoadgens erT kategorias, romelSic SesaZloa cdis pirebi moxvdnen. striqonebi (aRiniSneba r -iT) warmoadgens pasuxis meore katagorias. maSasadame, cxrili #15.2 aris 2 striqoni ( r = samsaxuris donis Sesabamisi ori kategoria: profesiuli da araprofesiuli) 2 svetze ( c=
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
6
Tmis feris Sesabamisi ori kategoria: qera da muqi) niSanTa SeuRlebis cxrili. cxrils ewodeba niSanTa SeuRlebis cxrili, radgan Cven vcdilobT davadginoT, cdis piris samsaxuris done, profesiuli an araprofesiuli, SesabamisobaSia an SeuRlebulia Tu ara misi Tmis ferTan. marginaluri (Sejamebuli) sixSireebi, romlebic naCvenebia cxrilSi, aris striqonebis jamebi (anu striqonis marginaluri sixSireebi) da svetebis jamebi (anu svetebis marginaluri sixSireebi) miRebuli sixSireebisaTvis.
marginaluri (Sejamebuli) sixSireebi (Marginal Frequencies): niSanTa SeuRlebis cxrilSi sixSireebis jami striqonis an svetis mixedviT. mosalodneli siSireebi
mosalodneli siSireebi (Expected Frequencies): niSanTa SeuRlebis cxrilidan miRebuli sixSireebi im daSvebis safuZvelze, rom cxrilSi warmodgenili ori ganzomileba damoukidebelia. cxrilSi #15.2 warmodgenilia 60 qeraTmiani da 120 muqTmiani personaJi. sul 180 personaJidan 84-s, 46,7%-s, hqonda profesiuli donis samsaxuri da 96-s, anu 53,3%-s, hqonda araprofesiuli samsaxuri. Tu Tmis feri ar aris dakavSirebuli samsaxuris donesTan, maSin mosalodnelia, rom 60 qeraTmiani personaJis daaxloebiT 46,7%-s (anu 84/180) anu 28 personaJs, unda hqondes profesiuli donis samsaxuri. ZalaSia igive molodini, Tu Tmis fersa da samsaxuris dones Soris kavSiri ar arsebobs. am SemTxvevaSi moveliT, rom profesiuli donis samsaxuri eqneba 120 muqTmiani personaJis daaxloebiT 46,7 procents, anu 56 qals, da araprofesiuli donis samsaxuri eqneba 120 muqTmiani personaJebis 53,3 procents, anu 64 qals. am mniSvnelobebs mosalodneli sixSireebi ewodeba, _ pasuxis mosalodneli sixSireebi im SemTxvevaSi, Tu Tmis feri ar aris dakavSirebuli samsaxuris donesTan. mosalodneli sixSireebi gamoxatulia E aRniSvniT (Expected frequency, mosalodneli sixSiris aRsaniSnavad) cxrilis TiToeul ujraSi. amgvarad, mosalodneli sixSireebi miiReba im varaudis safuZvelze, rom ar arsebobs kavSiri im or ganzomilebas Soris, romlis mixedviTac xdeba cdis monawilis kategorizacia. am magaliTSi mosalodneli sixSireebi uSvebs, rom Tmis feri da samsaxuris done erTmaneTisgan damoukidebelia. xi-kvadrat kriteriumis misaRebad realuri da mosalodneli sixSireebi gamoiyeneba. Tu SerCevis or ganzomilebas Soris kavSiri ar aris, miRebuli sixSireebi da mosalodneli sixSireebi daaxloebiT toli unda iyos. Tu svetisa da striqonis cvladebi ar aris dakavSirebuli erTmaneTTan, miRebulsa da mosalodnel sixSireebs Soris didi gansxvavebebi mxolod iSviaTad da SemTxveviT miiReba. magram Tu SerCevaSi svetisa da striqonis cvladebi dakavSirebulia, maSin miRebuli sixSireebi ufro metad unda gansxvavdebodes mosalodneli sixSireebisagan. magaliTad, Tu Tmis feri dakavSirebulia karieris donesTan, rogorc hipoTeza
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
7
varaudobda, maSin mosalodnelia, rom qeraTmian personaJebs ufro meti profesiuli donis samsaxuri eqnebaT, vidre SemTxveviT mosalodneli 28 da ufro naklebi araprofesiuli samsaxuri, vidre SemTxveviT mosalodneli 32. msgavsadve, Tu hipoTezaSi gamoTqmuli varaudi Tmis ferisa da samsaxuris donis kavSiris Sesaxeb gamarTlda, aseve mosalodnelia, rom muqTmian personaJebs SemTxveiT mosalodnel 56-ze naklebi profesiuli donis da 64-ze meti araprofesiuli donis samsaxuri eqnebaT.
2χ statistika xi-kvadrat kriteriumi (aRiniSneba 2χ -iT) zomavs miRebuli sixSireebis mosalodneli sixSireebisagan gansxvavebas:
( )∑∑= =
−=
r
r
c
c rcErcErcO
1 1
2
2χ
sadac χ aris `xi~, berZnuli anbanis ocdameerTe aso. am formulaSi
rcO aris rc ujris realuri sixSire, sadac r striqonis nomeria da c - svetis nomeri.
rcE aris rcujris mosalodneli sixSire
r aris striqonis cvladis kategoriebis raodenoba (Cvens magaliTSi samsaxuris done, r = 2)
c aris svetis cvladis doneebis raodenoba (Cvens magaliTSi Tmis feri, c = 2).
ormagi jamis niSnebi da zRvrebi (∑=
r
r 1∑=
c
c 1
) miuTiTebs, rom niSanTa
SeuRlebis cxrilis yvela svetisa da striqonis ( ) rcrcrc EEO 2−
mniSvnelobebi jamdeba.
statistikuri hipoTezebis Semowmeba χ²-iT Cvens magaliTSi 2χ -is gamoyeneba imis dasadgenad, dakavSirebulia Tu ara Tmis feri samsaxuris donesTan, moiTxovs stastikuri hipoTezis Semowmebas Semdegi nacnobi safexurebis mixedviT:
• nulovani, 0H , hipoTezisa da alternatiuli, 1H , hipoTezis
Camoyalibeba. nulovani hipoTeza saSualebas gvaZlevs, avagoT 2χ statistikis SerCeviTi ganawileba
• mniSvnelovnobis donis SerCeva
• cxrilSi #A.7 mocemuli 2χ -is SerCeviTi ganawilebidan 2χ -is kritikuli mniSvnelobis moZebna, rac aRiniSneba, rogorc
kritikuli2χ .
• kritikulis2χ gamoyenebiT 2χ is SerCeviTi ganawilebaSi uaryofis
zonis gansazRvra
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
8
• realuris2χ gamoTvla niSnanTa SeuRlebis cxrilis sixSireebis
mixedviT
• gadawyvetilebis miReba 0H -is uaryofis an miRebis Sesaxeb imis
mixedviT, xvdeba Tu ara realuri2χ uaryofis areSi.
statistikuri hipoTezebi
nulovani hipoTeza. nulovani hipoTeza, romlis farglebSic
yalibdeba 2χ -is SerCeviTi ganawileba da romlis farglebSic dgindeba TiToeuli ujrisaTvis mosalodneli sixSireebi, aris:
0H : populaciaSi striqonis cvladi da svetis cvladi
damoukidebelia.
nimuSad moyvanili kvlevis SemTxvevaSi 0H SeiZleba konkretulad
Camoyalibdes, rogorc:
0H : satelevizio personaJis samsaxuris done ar aris damokidebuli
misi Tmis ferze. alternatiuli hipoTeza. alternatiuli hipoTezaa:
1H : striqonisa da svetis cvladebi populaciaSi dakavSirebulia.
nimuSad moyvanili kvlevisaTvis 1H SeiZleba konkretulad Camoyalibdes, rogorc
1H : satelevizio personaJis samsaxuris done dakavSirebulia misi Tmis ferTan. mniSvnelovnobis donis SerCeva mniSvnelovnobis donis SerCevis procesi identuria imisa, rac wina TavebSia ganxiluli. miRebuli praqtikis Sesabamisad, virCevT α = 0, 05-s. χ²kritikulis povna da uaryofis zonis dadgena
2χ -is Tavisuflebis xarisxi. 2χ -is, da Sesabamisad kritikulis2χ , konkretuli
SerCeviTi ganawileba damokidebulia statistikuri sididis Tavisuflebis
xarisxze. Tavisuflebis xarisxi (df ) 2χ -Tvis aris
Tavisuflebis xarisxi = ( )( )11 −− cr
sadac r SeuRlebis cxrilSi striqonebis raodenobaa da c - svetebis raodenoba. magaliTisaTvis, r = 2 da c= 2, maSasadame Tavisuflebis xarisxi
df = (2 - 1)(2 - 1), rac 1-is tolia. kritikulis2χ mniSvnelobebi α = 0, 05 da 0,01
doneebze sxvadasxva Tavisuflebis xarisxis mqone 2χ -is SerCeviTi
ganawilebisTvis mocemulia damatebis cxrilSi #A-5. cxrilis #A.7 mixedviT, roca Tavisuflebis xarisxi 1-ia da α = 0, 05, 2χ -is kritikuli mniSvneloba aris 3,84.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
9
uaryofis zonis dadgena. Tu nulovani hipoTeza WeSmaritia da pasuxis
siswore studentis tipze ar aris damokidebuli, maSin realuri2χ SedarebiT
mcire unda iyos; mosalodneli da miRebuli sixSireebi niSanTa SeuRlebis cxrilis TiToeuli ujrisaTvis bevrad msgavsi unda iyos. meores mxriv, Tu
1H WeSmaritia da striqonis cvladi dakavSirebulia svetis cvladTan,
maSin realuris2χ mniSvneloba gaizrdeba da realuri da mosalodneli
sixSireebi erTmaneTisagan gansxvavebuli iqneba. Sesabamisad, 2χ -is SedarebiT didi mniSvnelobebi iSviaTi unda iyos, Tu nulovani hipoTeza WeSmaritia, magram xSiri unda iyos, Tu alternatiuli hipoTezaa WeSmariti.
maSasadame, realuris2χ mniSvneloba, romelic kritikulze
2χ metia an tolia, statistikurad mniSvnelovania α-s SerCeul mniSvnelobaze. Sesabamisad, Cvens
magaliTSi Tu realuri2χ 3,84-ze metia an tolia, is xvdeba uaryofis areSi da
moxdeba 0H –is uaryofa da 1H –is miReba.
realuris2χ gamoTvla
realuris2χ gamoTvlisaTvis saWiroa pasuxebis realuri da mosalodneli
sixSireebis codna. mosalodneli sixSireebi niSanTa SeuRlebis cxrilis TiToeuli ujrisaTvis marginaluri sixSireebisagan Semdegi formuliT miiReba:
( )( )raodenoba saerTo pasuxebis
ujrisaTvis sixSire imarginalur svetisujrisaTvis sixSire imarginalur striqonis
sixSire
imosalodnel
ujris
=
magaliTad, mosalodneli sixSire ujrisaTvis `qera Tma/profesiuli donis samsaxuri~ (striqoni 1, sveti 1) gamoiTvleba Semdegnairad:
( )( )jami
sixSire imarginalur svetis pirvelisixSire imarginalur striqonis pirveli=11E
anu
( )( ) 281805040
1806084
11 ===E
es aris mosalodneli sixSire , E , cxrilis #15.2 `qera Tma/profesiuli donis samsaxuri~ ujrisaTvis.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
10
mas Semdeg, rac gavigebT mosalodnel sixSireebs TiToeuli
ujrisaTvis, realuri2χ gamoiTvleba Semdegnairad:
( )∑∑= =
−=
r
r
c
c rcErcErcO
1 1
2
2χ
realuri2χ -is gamosaTvlelad,
• TiToeuli ujrisaTvis miRebul sixSires (O ) gamoakeliT mosalodneli sixSire ( E )
• TiToeuli O - E sxvaoba aiyvaneT kvadratSi • O - E sxvaobis kvadrati gayaviT am ujris mosalodnel sixSireze • realuris
2χ misaRebad SeajameT niSanTa SeuRlebis cxrilSi TiToeuli
ujrisaTvis miRebuli (O - E )² ∕ E mniSvnelobebi
realuris2χ ricxviTi mniSvneloba cxrilSi #15.2 mocemuli sixSireebisaTvis
Semdegnairad gamoiTvleba:
( ) ( ) ( ) ( )
( ) ( )
Sedegad. isdamrgvaleb demeasedebam
realuri
,43.6,000,1143,1000,2286,2
,648
568
328
288
,64
647256
564832
322428
2836
2222
22222
=+++=
+−
+−
+=
−+
−+
−+
−=χ
gadawyvetilebis miReba statistikuri hipoTezis Sesaxeb
realuri2χ , romlis mniSvnelobaa 6,43, metia kritikulze
2χ 3,84-is toli mniSvnelobiT da Sesabamisad, uaryofis areSi xvdeba. aqedan gamomdinare,
xdeba 0H -is uaryofa da 1H -is miReba. striqonisa da svetis cvladebs
Soris arsebobs statistikurad mniSvnelovani kavSiri. kriteriumis mixedviT gakeTebuli daskvnebi
0H -is uaryofa miuTiTebs, rom Tmis feri da samsaxuris done
urTierTdakavSirebulia. kavSiris buneba Cans cxrilidan #15.2: profesiuli donis samsaxuris SemTxveviTze meti sixSire dakavSirebulia qera TmasTan da araprofesiuli samsaxuris SemTxveviTze meti sixSire dakavSirebulia muq TmasTan. daSvebebi xi-kvadrat kriteriumis gamoyenebisaTvis damoukideblobis xi-kvadrat kriteriumis gamoyeneba SesaZlebelia nebismieri zomis niSanTa SeuRlebis cxrilis SemTxvevaSi (magaliTad, 2×3, 3×4, an 4×5). Tumca misi gamoyenebisaTvis arsebobs garkveuli daSvebebi:
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
11
• TiToeuli cdis piri gvaZlevs mxolod erT pasuxs niSanTa SeuRlebis cxrilisaTvis
• miRebuli pasuxebis raodenoba imdenad didi unda iyos, rom arcerTi mosalodneli sixSire ar iyos 10-ze naklebi 2×2 Sesabamisobis cxrilSi an 5-ze naklebi im cxrilSi, romelic 2×2-ze didia. Tu es piroba dakmayofilebuli ar aris, an meti pasuxi unda Segrovdes an gamoyenebul unda iqnas raime alternatiuli kriteriumi, magaliTad fiSeris zusti kriteriumi.
xi-kvadratisa da sixSiris monacemebis sxva kriteriumebis gamoyenebis Sesaxeb dawvrilebiTi informaciis moZieba SeiZleba statistikis ufro rTul saxelmZRvaneloebSi.
magaliTi #15.1
xi-kvadrat kriteriumis gamoyeneba 3×2 niSanTa SeuRlebis cxrilisaTvis amocana ojaxSi pirveli Sviloba bavSvs xSirad garkveul upiratesobebs aniWebs. sxvadasxva kvelvebma aCvena, rom pirvel Svilebs, rogorc wesi, ukeTesi Sedegebi aqvT akademiur testebSi, vidre rigiT momdevno Svilebs, aisenmanma (Eisneman, 1987) ki aRmoaCina kavSiri kreatulobasa da dabadebis rigs Soris kolejis student mamakacebSi. mis SerCevaSi pirveli Svili mamkacebis qulebi xSirad kreatulobis maCveneblebis medianaze zeviT iyo da arapirveli Svili mamakacebis qulebi - medianaze qveviT. davuSvaT, rom Tqven ganavrcobT am kvlevas da ikvlevT zrdasrul mamakacebs: 100 pirvel Svils da 100 momdevno Svils, da sTxovT, moifoqron Cveuli sagnebis kreatuli gamoyenebis gzebi. maTi pasuxebis kategorizacias axdenT imis mixedviT, maTi qulebi kreatulobis qulebis ganawilebis zeda, Sua Tu qveda mesamedSia. dabadebis rigisa da kreatulobis testis mixedviT kategorizaciis Sedegad miRebulia Semdegi realuri sixSireebi:
dabadebis rigi pirveli Svili momdevno Svili
kreatulobis testis qula
zeda mesamedi Sua mesamedi qveda mesamedi
47 29 29 35 24 36
aris Tu ara kavSiri dabadebis rigsa da kreatulobas Soris mamakacebSi? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna am kvlevaSi cdis pirebis dajgufebis erTi kategoria aris dabadebis rigi ori mniSvnelobiT: pirveli Svili da momdevni Svili. meore kategoria
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
12
dafuZnebulia imaze, kreatulobis testis qula kolejis populaciis qulebis zeda, Sua Tu qveda mesamedSia. TiToeuli cdis piri gvaZlevs sixSirisaTvis mxolod erT qulas, anu ricxvs, maSasadame, am monacemebisaTvis xi-kvadratis gamoyeneba mizanSewonilia.
saWiro kriteriumi: 2χ daSvebebi χ²–is gamoyenebisaTvis:
1. TiToeuli cdis pirisagan viRebT mxolod erT pasuxs niSanTa SeuRlebis cxrilisaTvis.
2. arc erTi mosalodneli sixSire ar unda iyos 5-ze naklebi. es minimaluri mosalodneli sixSire aviReT imitom, rom niSanTa SeuRlebis 3×2 cxrili ufro didia, vidre 2×2.
statistikuri hipoTeza:
0H : SerCevaSi dabadebis rigi da kreatuloba urTierTdamoukidebelia;
1H : SerCevaSi dabadebis rigsa da kreatulobas Soris aris kavSiri. mniSvnelovnobis done: α = 0,05 Tavisuflebis xarisxi, df : df = ( r -1)( c -1) = (3-1)(2-1) = 2 kritikuli mniSvneloba: cxrilis #A.7 mixedviT, χ²kritikuli(2) = 5,99146, anu Tu measedebamde davamrgvalebT, 5,99.
uaryofis are: realuris2χ 5,99-is toli an meti mniSvneloba.
gamoTvla: realuris2χ formulisaTvis saWiroa mosalodneli sixSireebis
povna; amisaTvis ki saWiroa marginaluri sixSireebis gamoTvla. miRebuli (realuri) sixSireebi (aRniSnuli O-Ti), mosalodneli sixSireebi (aRniSnuli E-Ti) da marginaluri sixSireebi mocemulia Semdeg cxrilSi: Ddabadebis rigi pirveli
Svili momdevno Svili
(striqonis marginaluri sixSire)
kreatulobis testis qula
zeda mesamedi O=47
E =38
O=29
E =38
76
Sua mesamedi O=29
E =32 O=35
E =32
64
qveda mesamedi O=24
E =30 O=36
E =30
60
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
13
(svetis marginaluri sixSire)
100 100 (jami) 200
svetis marginaluri sixSireebi miRebulia TiToeul svetSi mocemuli realuri sixSireebis SejamebiT. striqonis marginaluri sixSireebic miRebulia msgavsadve TiToeuli striqonis realuri sixSireebis SejamebiT. marginaluri sixSireebis codnis SemTxvevaSi mosalodneli sixSireebis gamoTvla Semdegnairad xdeba:
( )( )raodenoba saerTo pasuxebis
ujrisaTvis sixSire imarginalur svetisujrisaTvis sixSire imarginalur striqonis
sixSire
imosalodnel
ujris
=
mosalodneli sixSireebi cxrilis TiToeuli ujrisaTvis aseTia:L
ujra: pirveli Svili - zeda ⅓: ( )( ) 38200
7600200
1007611 ===E
ujra: momdevno Svili - zeda ⅓: ( )( ) 38200
7600200
1007612 ===E
ujra: pirveli Svili - Sua ⅓: ( )( ) 32200
6400200
1006421 ===E
ujra: momdevno Svili - Sua ⅓: ( )( ) 32200
6400200
1006422 ===E
ujra: pirveli Svili - qveda ⅓: ( )( ) 30200
6000200
1006031 ===E
ujra: momdevno Svili - qveda ⅓: ( )( ) 30200
6000200
1006032 ===E
realuris2χ mniSvneloba gamoiTvleba formuliT:
( )∑∑
==
−=
2
1
23
1
2
c rc
rcrc
r EEO
realuriχ
ricxviTi mniSvnelobebis Casmis Semdeg miviRebT:
( ) ( ) ( ) ( ) ( ) ( )
22,7,20,120,128,028,013,213,2
,30
303630
302432
323532
322938
382938
3847 2222222
=+++++=
−+
−+
−+
−+
−+
−=realuriχ
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
14
gadawyvetileba: realuri2χ = 7,22 aRemateba kritikuli
2χ = 5,99-s; maSasadame, is
xvdeba uaryofis areSi. uarvyofT 0H -s da viRebT 1H -s. daskvna: Cvens SerCevaSi dabadebis rigi da kreatuloba erTmaneTTan dakavSirebulia. realuri da mosalodneli siSireebis cxrilze dakvirvebis Sedegad vxedavT, rom pirveli Svili mamakacebis qulebi ufro xSirad da momdevno Svili mamkacebis qulebi ufro iSviaTad xvdeba kreatulobis qulebis zeda mesamedSi, vidre SemTxveviT iqneboda mosalodneli. aseve vamCnevT, rom pirveli Svili mamakacebis qulebi ufro iSviaTad xvdeba qveda mesamedSi momdevno Svili mamakacebis qulebTan SedarebiT, vidre amas SemTxveviTobis hipoTeza winaswarmetyvelebs. Tanxmobis χ²²kriteriumi
Tanxmobis χ² kriteriumi (Chi-square test of goodness of fit): xi-kvadrati, roca monacemebis kategorizacia xdeba mxolod erTi ganzomilebis mixedviT. damoukideblobis xi-kvadrat kriteriumis SemTxvevaSi qulebis dajgufeba (kategorizacia) xdeboda ori damoukidebeli ganzomilebis mixedviT: satelevizio personaJi qera iyo Tu muqTmiani da am personaJis kariera profesiuli iyo Tu ara. Tanxmobis χ² kriteriumis SemTxvevaSi qulebis kategorizacia xdeba cvladis mxolod erTi ganzomilebis mixedviT. ganvixiloT magaliTi. bevri adamiani miiCnevs, rom paTologiuri qceva savse mTvaris dros ufroa mosalodneli. am mosazrebis empiriulad Semowmebis mizniT, davuSvaT, rom moaxdineT erTi wlis ganmavlobaSi fsiqiatriul ganyofilebaSi Semosuli axali klientebis miRebis kategorizacia mTvaris fazebis mixedviT da miiReT Semdegi ganawileba:
mTvaris faza savse mTvare axali mTvare pirveli meoTxedi mesame meoTxedi
62 50 60 56 gansxvavebulia Tu ara miRebuli klientebis raodenoba mTvaris fazis
mixedviT? am kiTxvaze pasuxis gasacemad SeiZleba gamoviyenoT Tanxmobis 2χ kriteriumi.
mocemuli sixSireebi axdenen qcevis kategorizacias mxolod mTvaris fazis mixedviT. xi-kvadrat kriteriumis gamosayeneblad saWiroa, davadginoT mTvaris TiToeuli fazis dros klientebis miRebis mosalodneli sixSireebi. radgan striqonisa da svetebis marginaluri sixSireebi ar gvaqvs, raime sxva safuZveli unda gamaCndes mosalodneli sixSireebis winaswarmetyvelebisaTvis. Tu fsiqiatriul ganyofilebaSi miReba ar aris kavSirSi mTvaris fazasTan, maSin axali pacientebis miRebis sixSire TiToeul fazaSi erTnairi unda iyos da raime gamovlenili gansxvaveba
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
15
mxolod SerCevis Secdomas unda asaxavdes. es msjeloba SeiZleba gamoixatos nulovani hipoTeziT:
0H : SerCevaSi fsiqiatriul ganyofilebaSi axali klientebis miReba
da mTvaris faza erTmaneTisgan damoukidebelia.
maSin alternatiuli hipoTeza iqneba:
1H : SerCevaSi fsiqiatriul ganyofilebaSi axali klientis miReba da mTvaris faza urTierTdamokidebulia.
Tu es alternatiuli hipoTeza WeSmaritia, maSin axali miRebebis raodenoba mTvaris sxvadasxva fazebis SemTxvevaSi erTnairi ar unda iyos. magram Tu
WeSmaritia 0H , maSin TiToeuli fazis dros pacientebis meoTxedis miReba
unda xdebodes. maSasadame, miRebebis mosalodneli sixSireebi SeiZleba miviRoT, Tu miRebebis saerTo raodenobas gavyofT 4-ze, anu mTvaris fazebis raodenobaze. Tu aseve vimsjelebT Cvens magaliTzec, radgan 228 axali pacientia miRebuli, mosalodneli sixSire TiToeuli fazisTvis aris
4228 , anu 57. mTvaris TiToeuli fazisaTvis realuri da mosalodneli
sixSireebi aseTia:
mTvaris faza savse mTvare Aaxali mTvare Ppirveli meoTxedi mesame meoTxedi
O=62 O=50 O=60 O=56 E=57 E=57 E=57 E=57
am monacemebis mixedviT xi-kvadrati miiReba Semdegnairad:
( )∑=
−=
c
c c
cc
EEO
1
22χ
Tavisuflebis xarisxi am isrealurisTv
2χ aris c – 1. aqedan gamomdinare,
( ) ( ) ( ) ( )
47,15784
,571
579
5749
5725
,571
573
577
575
,57
575657
576057
575057
5762
2222
22222
==
+++=
−++
−+=
−+
−+
−+
−=realuriχ
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
16
χ²²-is kritikuli mniSvneloba 3-is toli Tavisuflebis xarisxisaTvis da
roca α = 0,05, mocemulia A.7 cxrilSi, kritikuli2χ (3) = 7,81. realuri
2χ = 1,47 da
naklebia kritikuli2χ -ze, amitom ar xvdeba uaryofis zonaSi. anu ver vaxerxebT
nulovani hipoTezis uaryofas. ar arsebobs damamtkicebeli sabuTi imisa, rom gamokvleul SerCevaSi fsiqiatriul ganyofilebaSi axali miRebebis raodenoba mTvaris fazebzea damokidebulia. daSvebebi Tanxmobis xi-kvadrat kriteriumisaTvis iseTivea, rogoric damoukideblobis xi-kvadrat kriteriumisaTvis: TiToeulma cdis pirma unda mogvces erTi pasuxi da mosalodneli sixSireebi ar unda iyos 10-ze naklebi, - Tu gvaqvs mxolod ori kategoria da ar unda iyos 5-ze naklebi, Tu gvaqvs sami an meti kategoria.
xi-kvardat kriteriumis Sedegebis warmodgena amerikis fsiqologiuri asociaciis saxlmZRvanelo publikaciebis Sesaxeb (The Publication Manual of the American Psychological Association, 1994) moiTxovs, rom xi-kvadrat kriteriumis Sesaxeb warmodgenili Sedegebi Seicavdes Tavisuflebis xarisxsa (df ) da SerCevis zomas, rasac χ²realuri eyrdnoba. sailustraciod gamoviyeneb satelevizio personaJebis Sesaxeb amocanisaTvis gamoTvlili xi-kvadratis magaliTs. 1-li cxrili warmoadgens Tmis ferisa da karieris donis Sesabamis
sixSireebs. alfas 0,05 mniSvnelovnobis donis SemTxvevaSi xi-kvadrati am
sixSireebisaTvis statistikurad mniSvnelovani aRmoCnda. 2χ (1, N =180)²=6,43,
p < 0,05. ufro meti iyo albaToba imisa, rom satelevizio gadacemis
qeraTmian personaJebs eqnebodaT profesiuli donis kariera, vidre muqTmianebs.
cxrili 1 satelevizio gadacemebis qali personaJebis kategorizaciis hipoTeturi monacemebi.
Tmis feri
qera muqi
kariera profesiuli 36 48 araprofesiuli
24
72
Sedegebis warmodgenaSi:
cxrili 1 gvaZlevs miRebul sixSireebs α = 0,05 aRniSnavs SemowmebisaTvis arCeul
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
17
mniSvnelobis dones χ² (1, N = 180)² aRniSnavs gazomvis statistikas, xi-kvadrats.
frCxilebSi mocemuli cifri 1 miuTiTebs Tavisuflebis xarisxs df= 1 da N= 180 gviCvenebs SerCevis zomas.
= 16,57 gviCvenebs realuri2χ -s mniSvnelobas (da ara
kritikuli2χ -s mniSvnelobas A.7 cxrilidan)
p < 0,01. gamoxatavs, rom
a. Tu 0H WeSmaritia, realuri2χ -s albaToba
naklebia an tolia 0,01-sa. es mniSvneloba
aris realuri2χ -s an ufro ukiduresi
mniSvnelobis albaToba, Tu 0H WeSmaritia;
es albaToba SeiZleba α-s arCeuli mniSvnelobisgan gansxvavebuli iyos.
b. uaryofil iqna 0H :,,satelevizio gadacemis
personaJis samsaxuris done ar aris damokidebuli personaJis Tmis ferze”
dadasturda 1H : ,,satelevizio personaJis samsaxuris done dakavSirebulia personaJis Tmis ferTan”.
SeamowmeT Tqveni codna 15.1
1. ganmarteT c, xi-kvadrat kriteriumi, mosalodneli sixSireebi,
marginaluri sixSireebi, araparametrulikriteriumi, parametruli kriteriumi, r , niSanTa SeuRlebis cxrili, ormimarTulebiani sixSireebis cxrili.
2. ra SemTxvevebSi ixmareba araparametruli kriteriumi eqsperimentis monacemebis analizisaTvis?
3. gansazRvreT saTanado statistikuri kriteriumi CamoTvlilTagan TiToeuli erT-faqtoriani sqemisTvis, damoukidebeli cvladis ori donisaTvis da miTiTebuli gazomvis skalisaTvis: a. cdis pirebs SigniT, Sefardebis skala b. cdis pirebs Soris, saxeldebis skala g. cdis pirebs Soris, Sefardebis skala d. cdis pirebs Soris, rigis skala e. cdis pirebs SigniT, rigis skala
4. ra saxis monacemebis analizi aris xi-kvadrat kriteriumisTvis
Sesaferisi? 5. ra aris warmodgenili niSanTa SeuRlebis cxrilis svetebSi? 6. ra aris warmodgenili niSanTa SeuRlebis cxrilis striqonebSi?
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
18
7. qvemoT mocemulia realuri2χ -s mniSvnelobebi Tavisuflebis
sxvadasxva xarisxisaTvis (romelic miTiTebulia frCxilebSi).
realuri
2χ -s da Tavisuflebis xarisxis (df) ToToeuli
mniSvnelobisaTvis ipoveT kritikuli2χ -s Sesabamisi mniSvneloba 0,05
doneze. Semdeg gansazRvreT, xvdeba Tu ara realuri2χ uaryofis areSi
da ra gadawyvetilebas iRebT 0H -sa da 1H -Tan dakavSirebiT.
8. gamoTvaleT mosalodneli sixSireebi qvemoT mocemuli TiToeuli
niSanTa SeuRlebis cxrilisaTvis, Semdeg ipoveT realuri2χ da
mouTiTeT, aris Tu ara statistikurad mniSvnelovani 0,05 doneze. ToToeul ujraSi mocemuli mniSvnelobebi warmoadgens miRebul sixSireebs.
a.
sveti
1 2
striqoni
1 12 32
2 16 38
b.
sveti
1 2
striqoni
1 27 22
2 17 46
g.
sveti
realuri2χ (2) = 7,26 realuri
2χ (4) = 8,33
realuri2χ (7) = 14,19 realuri
2χ (20) = 26,94
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
19
1 2
striqoni
1 33 66
2 27 14
9. nimuSis saxiT mocemuli amocanis magaliTze xi-kvadratis
gamoyenebiT amoxseniT Semdegi amocana. gamoiyeneT 0,05 mniSvnelovnobis done. yofila Tu ara iseTi SemTxveva, roca SuqniSanis wiTel Suqze ar gaCerebulxarT? SesaZlebelia - ara, magram albaT sxvebis amgvari qcevis momswre mainc yofilxarT. ramdenad xSirad xdeba, rom mZRolebi ar Cerdebian wiTel Suqze da aris Tu ara aseTi qcevis sixSire damokidebuli dRis droze da kviris dReze? davuSvaT, rom mkvlevari ikvlevda am sakiTxs da akvirdeboda im avtomobilebis raodenobas, romlebic gzajvaredinze `sdeq~ niSanze ar gaCerdnen. is akvirdeboda transports dilis 10-dan 11 saaTamde da SuadRis 2-dan 3 saaTamde erTi kviris samSabaTs, oTxSabaTs da xuTSabaTs. mkvlevari iniSnavda manqanis nomrebs da TiToeul manqanas aRricxavda mocemuli kviris ganmalobaSi mocemul gzajvaredinze mxolod pirvelad gavlisas. davuSvaT, rom mkvlevarma im manqanebis sixSireebi mogvawoda, romlebic ar Cerdebodnen `sdeq~ niSanze mocemuli kviris dReebis or monakveTSi.
Kkviris dRe
samSabaTi
oTxSabaTi MxuTSabaTi
dro
dilis 10:00-dan 11:00-mde
38 46 58
SuadRis 2:00-dan 3:00-mde
46
54
63
aris Tu ara kavSiri kviris dRes an dRis monakveTsa da mZRolebis mier `sdeq~ niSnis ignorirebas Soris?
10. nimuSis saxiT mocemuli amocanis formatis mixedviT xi-kvadratis gamoyenebiT amoxseniT Semdegi amocana. gamoiyeneT 0,05 mniSvnelovnobis done. fsiqologebi varaudoben, rom qalebSi kvebiTi darRvevebis ganviTarebis erT-erTi mizezi isaa, rom arsebobs bevri iseTi signali, romelic xazs usvams sigamxdres, rogorc qalisaTvis sasurvel Tvisebas. smiTma, ualdorfma da trembaTma (Smith, Waldorf, & Trembath, 1990) ikvlies aseTi signalis erT-erTi forma, piradi gancxadeba (romelsac partnioris mosaZebnad
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
20
iyeneben), raTa SeemowmebinaT, am gancxadebebSi naxsenebi iyo Tu ara sigamxdre, rogorc sasurveli partnioris mniSvnelovani Tviseba. isini varaudobdnen, rom mamakacebis mier qalebis mosaZebnad ganTavsebul gancxadebebSi ufro meti albaTobiT iqneboda aRniSnuli wonis moTxovna, vidre qalebis mier mamkacebis mosaZebnad ganTavsebul gancxadebebSi. am hipoTezis Sesamowmeblad maT gamoikvlies piradi gancadebebis garkveuli raodenoba da moaxdines maTi kategorizacia gancxadebis avtoris sqesisa da gancxadebaSi sasurveli wonis aRniSvna-araRniSvnis mixedviT. maTi kvlevis Sedegebs gamoxatavs qvemoT mocemuli hipoTeturi sixSireebi. arsebobs Tu ara am monacemebs Soris is kavSiri, rasac hopoTeza varaudobs?
aRniSnuli iyo Tu ara
sasurveli wona diax
ara
mamakacis mier ganTavsebuli gancxadeba
89 186
qalis mier ganTavsebuli gancxadeba
6
219
man-uitnis U kriteriumi
man-uitnis U kriteriumi (Mann-Whitney U test): araparametruli kriteriumi cdis pirebs Soris sqemisaTvis, roca gamoyenebulia damoukidebeli cvladis ori done da monacemebi sul mcire, rigis skalazea.
man-uitnis U kriteriumi aris araparametruli kriteriumi cdis pirebs Soris sqemisaTvis damoukidebeli cvladis ori donis gamoyenebiT, roca qulebi, sul mcire, rigis skalazea. man-uitnis U kriteriums ganvixilavT orazrovani figurebis aRqmis kvlevis magaliTze (Reisberg, 1983, Strueber and Stadler, 1999). suraTi 15.1 Sroderis kibe, orazrovani figura orazrovani figura aris ori an meti SesaZlo perceptuli organizaciis mqone stimuli. stimulis magaliTad 15.1 suraTze gamosaxulia Sroderis kibe. am figuras gaaCnia ori SesaZlo perceptuli organizacia: pirvel SemTxvevaSi Cans Cveulebrivi kibe, meore SemTxvevaSi ki kibe amobrunebuli gveCveneba. safexurebi amobrundeba, Tu Tqven maT garkveuli
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
21
xnis ganmavlobaSi uyurebT. Tu erTi perceptuli organizaciis meoreTi Sesacvlelad yuradReba da mexsiereba aris saWiro, maSin yuradRebisa da mexsierebis momcveli damatebiTi amocana gazrdis Sebrunebis aRqmisaTvis saWiro dros. am hipoTezis Sesamowmeblad Seiqmna cdis pirebis ori ekvivalenturi jgufi. Srederis kibe orive jgufs aCvenes da cdis pirebma aRniSnes, Tu rodis dainaxes stimulis orive organizacia. damoukidebeli cvladi iyo xelis SemSleli amocanis tipi. sakontrolo jgufis cdis pirebi ubralod uyurebdnen naxats da acxadebdnen gamosaxulebis Sebrunebis Sesaxeb. eqsperimentuli jgufis cdis pirebi stimulis danaxvis paralelurad iTvlidnen samniSna ricxvidan klebis mimarTulebiT da 3-is bijiT, anu asrulebdnen amocanas, romelic moiTxovs yuradRebisa da mexsierebis CarTvas. Tu am figuris Sebrunebuli variantis danaxva moiTxovs yuradRebasac da mexsierebasac, maSin klebis mimarTulebiT Tvlam xeli unda SeuSalos figuris aRqmas da misi SebrunebiT danaxvis dro unda gazardos. eqsperimentatorebi iniSnavdnen, Tu ramdeni dro dasWirdaT cdis pirebs figuris Sebrunebulad dasanaxad. Tu cdis piri ver dainaxavda am figuris Sebrunebas meore wuTis dasasrulisaTvis, eqsperimenti Cerdeboda da cdis pirs mianiWebdnen ,,usasrulo” (∞) latentobas. 15.3 cxrili warmoadgens am eqsperimentis hipoTetur monacemebs im SemTxvevisaTvis, roca TiToeul jgufSi cxra cdis piria. qulebi naCvenebia TiToeuli jgufisaTvis mzardi latentobis mixedviT, Tumca isini am TanmimdevrobiT ar Cawerila, radgan sinamdvileSi cdis pirebisgan iyo miRebuli. yuradReba miaqcieT, rom eqsperimentul jgufSi erT-erTma cdis pirma 2 wuTis ganmavlobaSi ver dainaxa figuris Sebruneba. am cdis pirs mianiWes usasrulo latentoba ricxviTi mniSvnelobis miniWebis gareSe. radgan t kriteriumica da dispersiuli analizic moiTxovs TiToeuli cdis piris ricxviT mniSvnelobas, maT ver gamoviyenebT am monacemebis analizisaTvis. magram man-uitnis U kriteriumi mxolod imas moiTxovs, rom qulebi rigis mixedviT iyos dalagebuli, usasrulo latentobis rigis mixedviT dalageba ki SesaZlebelia. maSasadame, am monacemebis analizi xdeba man-uitnis U kriteriumis meSveobiT. statistikuri hipoTezebis Semowmeba man-uitnis U kriteriumiT Tavdapirvelad warmovadgenT U kriteriumis cnebis mimoxilvas da Semdeg misi gamoTvlis sqemas. orive midgoma moiTxovs monacemebis rigis moxedviT dalagebas gazomvis statistikis misaRebad da am statistikis SerCeviTi ganawilebis misaRebad. es dalageba SesaZlebelia maSinac, roca monacemebi intervalebis an Sefardebis skalazea; qulebis rigis mixedviT dalagebiT xdeba maTi rigis skalaze gadayvana. cxrili 15.3 perceptuli Sebrunebis hipoTeturi dro (wamebSi) Srederis kibisaTvis. sakontrolo jgufis cdis pirebi ubralod uyurebdnen am figuras. eqsperimentuli jgufis cdis pirebi figuris yurebisas iTvlidnen klebis mimarTulebiT samis toli bijiT.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
22
jgufi
sakontrolo
( 1A ) Eeqsperimentuli
( 2A ) 2 4 5 10 6 11 8 12 9 14 13 17 15 85 21 98 42 a∞
msjelobis gaadvilebis mizniT, TiToeuli pirobisaTvis qulebi warmodgenilia latentobis zrdadi TanmimdevrobiT. aes qula warmoadgens Sebrunebis `usasrulo~
latentobas; am cdis pirma ver dainaxa figuris Sebruneba 2-wuTiani Sualedis ganmavlobaSi
qulebis rigis mixedviT dalageba (ranJireba) qulebis rigis mixedviT dalagebis pirveli safexuria orive jgufis qulebis gaerTianeba (N raodenobis qulebis jamis misaRebad) da Semdeg am qulebis zrdadi mniSvnelobis mixedviT, umciresidan udidesisaken dalageba. es procedura 15.3 cxrilis monacemebisaTvis naCvenebia 15.4 cxrilis pirvel safexurSi. rigis mixedviT dalagebisas SenarCunebulia qulebis jgufisadmi mikuTvnebuloba (ix. me-2 safexuri 15.4 cxrilSi). Semdeg xdeba am qulebisaTvis 1-dan N-mde rangis miniWeba, umciresi maCveneblisaTvis 1-is da udidesisaTvis N rangis miniWebiT. rangebis miniWebis procesi naCvenebia 15.4 cxrilis me-3 safexurSi. xSirad SesaZlebelia bmuli qulebis miReba, magaliTad, roca ori adamianis latentobaa 12 wami. 15.5 cxrilSi aRwerilia procedurebi bmuli qulebis SemTxvevisaTvis, romelic gamoyenebulia 15.2 amocanaSi. U statistika U statistika aris is raodenoba, ramdenjerac erT jgufis qulis rangi win uswrebs meore jgufis qulis rangs. es gamoTvlebi naCvenebia 15.4 cxrilis me-4 da me-5 safexurebiT. magaliTad, 2 qula (pirvel safexurSi naCvenebi) A1 jgufidan (naCvenebia me-2 safexurSi) rangis mixedviT win uswrebs yvela A2 maCvenebels (gansazRvrulia me-3 safexurSi). maSasadame, es qula rangis mixedviT win uswrebs cxrave A2 qulas. A1 jgufis qulebi: 5, 6, 8 da 9 rangis mixedviT win uswrebs A2 jgufis rva qulas. 15.4 cxrilis me-4 safexuri gviCvenebs, Tu rangis mixedviT ramdenjer uswrebs win A1 qula A2 qulas.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
23
daakvirdiT ricxviT maCveneblebs am safexurisTvis da gaiazreT, Tu rogor iqna isini miRebuli. saerTo jamSi 56 SemTxveva iyo imisa, rom A1 qula A2
qulas rangis mixedviT win uswrebda, rac aris me-4 safexurze naCvenebi ricxvebis jami (anu 9 + 8 + 8 + 8 + 8 + 5 + 4 + 3 + 3 = 56). Sesabamisad, UA1realuri (A1 jgufisaTvis miRebuli U) aris 56. gvaqvs U -s mniSvneloba A2 jgufisTvisac. es ricxvi warmoadgens im raodenobas, Tu ramdenia iseTi SemTxveva, roca A2 qulis rangi win uswrebs A1 jgufis qulis rangs. 15.4 cxrilis me-5 safexuri gviCvenebs, Tu ramdenjer uswrebs win rangis mixedviT A1 qula A2 qulas. am safexurze ricxvebis jami gvaZlevs UA2realuri= 25 (anu 8 + 4 + 4 + 4+ + 3 + 2 = 25). cxrili 15.4 15.3 cxrilis qulebis rigis mixedviT dalageba 1. umciresidan udidesisaken dalagebuli latentobis qulebi 2 4 5 6 8 9 10 11 12 13 14 15 17 21 41 85 98 ∞ 2. jgufis
wevrobaa 1 2 1 1 1 1 2 2 2 1 2 1 2 1 1
2 2 2
3. rangi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 4. ramdenjer
uswrebs win 1A
qula 2A qulas 9 8 8 8 8 5 4 3 3
5. ramdenjer uswrebs win
2A qula
1A qulas 8 4 4 4 3 2
a1 aRniSnavs, rom qula 1A , sakontrolo jgufidanaa; 2 aRniSnavs, rom qula 2A ,
eqsperimentuli jgufidanaa.
U statistikis Tvisebebi Tu eqsperimentul Carevas Sedegi ar mohyveba, maSin mosalodnelia, rom qulebis gaerTianebisa da rigis mixedviT dalagebis Semdeg, rogorc es 15.4 cxrilzea mocemuli, TiToeuli jgufis qulebi rangebis mixedviT daaxloebiT erTnairad iqneba ganawilebuli. ar aris mosalodneli, rom erTi jgufis qulebi warmodgenili iqneba mxolod qveda mcire rangebiT da meore jgufis qulebi mxolod zeda rangebiT. aqedan gamomdinare, UA1 realuri
daaxloebiT toli iqneba UA2 realuri-sa. magaliTad, eqsperimentul Carevas Sedegi rom ar hqonoda Cvens magaliTSi, UA1 realuri-ca da UA2realuri-c daaxloebiT 40-is toli iqneboda. Carevas gansakuTrebiT Zlieri efeqti rom hqonoda da yvela A1 qulas yvela A2 qulaze dabali rangi hqonoda, UA1realuri
iqneboda 81 da UA2 realuri iqneboda 0. UA1 realuri-sa da UA2realuri-s mniSvnelobebi
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
24
idealur Sebrunebul kavSirSia. roca UA1 realuri izrdeba, UA2realuri Tanabari sididiT mcirdeba. maSasadame, U-s mxolod erTi, umciresi mniSvneloba gamoiyeneba statistikuri gazomvisas. Semdegi mimarTebis saSualebiT advilad SegviZlia imis gansazRvra, moxda Tu ara am umciresi mniSvnelobis gamoTvla:
realurirealuri 1212 AAAA UnnU −= sadac nA1 aris monacemebis raodeonoba A1 jgufSi da nA2 aris monacemebis raodenoba A2 jgufSi. Cvens magaliTSi UA2realuri gamoiTvleba UA1 realuri-s mixedviT Semdegnairad:
UA2 realuri = (9)(9) – 56 = 81 – 56 = 25, da 25 aris 15.3 cxrilis monacemebisTvis gamoTvlili U -s ori mniSvnelobidan umciresi. maSasadame, UA2 realuri = 25. U statistika gamoiyeneba hopoTezebis statistikuri Semowmebisas hopoTezebis statistikuri Semowmebis Tanmimdevrobis mixedviT. statistikuri hipoTeza
nulovani hipoTeza. nulovani hipoTeza, romlis farglebSic U -s SerCeviTi ganawileba miiReba, aris Semdegi:
0H : A1 monacemebis populaciaSi ganawileba da A2 monacemebis
populaciaSi ganawileba erTnairia.
Tu 0H WeSmaritia, maSin UA1realuri-sa da UA2realuri-s mniSvnelobebi daaxloebiT
toli unda iyos. alternatiuli hopoTeza. alternatiuli hipoTeza U kriteriumisaTvis aris:
1H : A1 monacemebis populaciaSi ganawileba da A2 monacemebis populaciaSi ganawileba erTnairi ar aris.
Tu 1H WeSmaritia, maSin UA1realuri da UA2realuri erTmaneTisgan gansxvavebuli unda iyos. U -s SerCeviTi ganawileba da kritikuli mniSvnelobebi U -s SerCeviTi ganawileba gansxvavebulia nA1–sa da nA2–sTvis. U statistikis gamoyenebis gasaadvileblad Seiqmna U –s SerCeviTi ganawilebisa da kritikuli mniSvnelobebis cxrilebi sandoobis 0,01 da 0,05 doneebze nA1–sa da nA2–is sxvadasxva zomisaTvis. radgan U –s maqsimaluri mniSvneloba ganisazRvreba nA1–sa da nA2–is meSveobiT, U -s SerCeviTi ganawilebis
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
25
cxrilebSi warmodegenilia U –s is umciresi mniSvnelobebi, romelTac kvlevebis Sedegad viRebT. 1-dan 20-mde cdis pirisagan Sedgenili jgufebisaTvis U –s mniSvnelobebi mniSvnelvnebis α= 0,01 da α= 0,05 doneebze warmodgenilia danarT cxrilSi A.8. am cxrilSi mocemuli mniSvnelobebi warmoadgens umciresi U realuris Sesabamis U –s kritikul mniSvnelobebs, rac aRniSnulia U kritikuliT. α – s arCeuli mniSvnelobisaTvis statistikurad mniSvnelovania umciresi U realuris is mniSvnelobebi, romlebic U kritikul –ze naklebia an tolia. yuradReba miaqcieT imas, rom t , F da
2χ ²statistikebisagan gansxvavebiT, U realuris uaryofis zona warmodgenilia
im mniSvnelobebiT, romlebic naklebia an tolia U kritikulis. Cveni magaliTis monacemebis SemTxvevaSi nA1= 9 da nA2 = 9, Sesabamisad U kritikuli = 17 mniSvnelvnebis 0,05 doneze. gadawyvetilebis miReba statistikuri hipoTezis Sesaxeb statistikuri hipoTezis Sesaxeb gadawyvetilebis miReba xdeba Urealuris ori mniSvnelobidan umciresis gamoyenebiT, Semdegnairad:
umciresi Urealuris mniSvneloba
statistikuri gadawyvetileba
Urealuri-ze naklebi an Urealuris toli
uarvyofT H0-s viRebT H1- s
Urealuri-ze meti ar uarvyofT H0-s ar viRebT H1- s
Cveni magaliTis SemTxvevaSi, 15.3 cxrilSi warmodgenili Sebrunebuli gamosaxulebis danaxvis latentobis maCveneblebisaTvis, UA1realuri = 56 da UA2realuri = 25. UA2realuri or miRebul mniSvnelobas Soris umciresia da swored es mniSvneloba gamoiyeneba hopoTezis SemowmebisaTvis. Urealuri = 25 metia, vidre Ukritikuli = 17. aqedan gamomdinare, Urealuris mniSvneloba ar
xvdeba uaryofis zonaSi; Cven ver uarvyofT 0H –s d ar viRebT 1H –s.
kriteriumis mixedviT gakeTebuli daskvnebi
radgan ver vaxerxebT 0H –is uaryofas, mivdivarT im daskvnamde, rom Cveni
magaliTis SemTxvevaSi ar arsebobs imis damadasturebeli monacemebi, rom sakontrolo jgufis qulebis populaciaSi ganawileba gansxvavdeba eqsperimentuli jgufis qulebis populaciaSi ganawilebisagan. sakontrolo pirobasTan SedarebiT klebis mimarTulebiT Tvlam ar Secvala Sebrunebuli gamosaxulebis danaxvisaTvis saWiro dro (latentoba).
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
26
man-uitnis U kriteriumis daSvebebi
U kriteriumis daSvebebia: • TiToeuli cdis piri iZleva mxolod erT qulas
monacemebisaTvis
• monacemebi, sul mcire, rigis skalazea da warmoadgens uwyvet cvlads. bevr SemTxvevaSi, rogorc es Cvens magaliTSi iyo, monacemebi iqneba an intervalebis an Sefardebis skalaze da U
kriteriumis gamoyenebisaTvis moxdeba am monacemebis rangebSi gadayvana.
U-s gamosaTvleli formulebi 15.4 cxrilSi UA1–sa da UA2 - is misaRebad gamoyenebuli midgoma rTulia, magram TvalsaCinod asaxavs, Tu ras warmoadgens UA1 da UA2. gamoTvlis miznebisTvis Seqmnilia gamartivebuli formulebi. UA1 realuri
UA1 realuri-is mniSvneloba gamoiTvleba formuliT:
( ) ∑−++= 1
1121 A1 2
1A
AAAA RnnnnU realuri
sadac nA1 aris A1 jgufSi monacemebis raodenoba
nA 2 aris A2 jgufSi monacemebis raodenoba
∑ 1AR aris A1 jgufis monacemebisaTvis miniWebuli rangebis jami
∑ 1AR miiReba 15.4 cxrilis me-3 safexurze A1 monacemebisaTvis miniWebuli
rangebis SekrebiT. magaliTad, A1 jgufSi monacemebis rangebia 1, 3, 4, 5, 6, 10,
12, 14 da 15. am rangebis jamia 70; Sesabamisad ∑ 1AR = 70. UA1 –is
gantolebaSi ricxvebis CasmiT viRebT:
( )( ) ( ) 56702
19999 A1 =−+
+=realuriU
UA2 realuri
UA2 realuri-is mniSvneloba gamoiTvleba formuliT:
( ) ∑−++= 2
2221 A2 2
1A
AAAA RnnnnU realuri
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
27
sadac ∑ 2AR aris A2 jgufis monacemebisaTvis miniWebuli rangebis jami. A2 jgufSi monacemebis rangebia 2, 7, 8, 9, 11, 13, 16, 17 da 18 (ix. 13.4 cxrilis me-3
safexuri). am rangebis jamia 101; anu ∑ 2AR = 101. formulaSi Casmis Sedegad,
( )( ) ( ) 251012
19999 A2 =−+
+=realuriU
13.5 cxrili warmoadgens U kriteriumis Sejamebas am gamoTvliTi formulebis gamoyenebiT. es cxrili agreTve Seicavs procedurebs, romlebic unda Sesruldes bmuli rangebis SemTxvevaSi. 13.2 magaliTSi naCvenebia am cxrilSi mocemuli safexurebis gamoyeneba. cxrili 15.5
man-uitnis U kriteriumis gamoTvliTi formulis gamoyeneba
safexuri 1. orive jgufis qulebi daalageT rigis mixedviT umciresi
mniSvnelobidan (rangi = 1) udides mniSvnelobamde (rangi = N ).
SeinarCuneT TiToeuli qulis jgufis wevroba ( 1A da 2A ). bmul qulebs mianiWeT im rangebis saSualo, rasac mianiWebdiT im SemTxvevaSi, bmuli rom ar yofiliyo.
safexuri 2. ipoveT ∑ 1AR da ∑ 2AR .
safexuri 3. gamoTvaleT UA1 realuri
( ) ∑−++= 1
1121 A1 2
1A
AAAA RnnnnU realuri
safexuri 4. gamoTvaleT UA2 realuri
( ) ∑−++= 2
2221 A2 2
1A
AAAA RnnnnU realuri
an
realurirealuri 1212 AAAA UnnU −=
safexuri 5. amoirCieT U realuri-s mniSvnelobebidan umciresi. SeadareT es U realuri Ukritikuli-s.
magaliTi 15.2
man-uitnis U kriteriumis gamoyeneba amocana bevr kolejsa da universitetSi janmrTelobis problemebs Soris yvelaze mniSvnelovania alkoholis borotad gamoyeneba. davuSvaT, rom fsiqologma
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
28
studentebisaTvis SeimuSava alkoholis Sesaxeb ganaTlebis programa. saswavlo wlis dasawyisSi fsiqologma SemTxveviTad SearCia 8 pirvelkurseli mamakaci da maT gaacno ganaTlebis programa. am programis gacnobidan 6 kviris Semdeg man studentebs Seavsebina kiTxvari alkoholis moxmarebis Sesaxeb. amavdroulad man SemTxveviTad SearCia 10 sxva pirvelkurseli mamakaci, romlebic ar gascnobian alkoholis ganaTlebis programas da maTac Seavsebina kiTxvari. yoveldRiurad moxmarebuli alkoholis raodenoba sufTa alkoholis unciebSi Semdegnairi aRmoCnda:
ganaTlebis programis piroba
gaiara programa (A1)
ar gauvlia programa (A2)
0,31 0,53 0,58 0,14 0,16 0,52 0,53 0,02
0,41 0,63 1,14 0,21 0,89 0,55 0,89 0,91 0,08 0,59
radgan es maCveneblebi asaxavs qcevis gazomvas da ara TviTon qcevas, fsiqologma CaTvala, rom isini intervalebis skalis moxovnebs ar akmayofilebda. Sesabamisad, fsiqologi am monacemebs mxolod rigis skalaze ganixilavda. moaxdina Tu ara alkoholis Sesaxeb ganaTlebis programam gavlena im adamianebis alkoholis moxmarebaze, vinc am programas gaecno? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna :kvlevis sqema aris cdis pirebs Soris sqema damoukidebeli cvladis, ganaTlebis programis pirobis, ori doniT. miCneulia, rom damokidebuli cvladi, moxmarebuli alkoholis raodenoba, rigis skalazea. Sesabamisad, man-uitnis U kriteriumi am monacemebis Sesabamisi statistikuri kriteriumia. saWiro kriteriumi: umciresi U realuri.
daSvebebi man-uitnis U kriteriumis gamoyenebisTvis: 1. TiToeuli cdis piris Semowmeba xdeba mxolod erTi
eqsperimentuli Carevis pirobaSi 2. damokidebuli cvladi, sul mcire, rigis skalazea da uwyvetia.
statistikuri hipoTeza
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
29
0H : populaciaSi im adamianebis qulebis ganawileba, romelTac
gaiares saganmanaTleblo programa (A1), identuria populaciaSi im adamianebis qulebis ganawilebisa, romelTac ar gauvliaT saganmanaTleblo programa (A1).
1H : populaciaSi im adamianebis qulebis ganawileba, romelTac
gaiares saganmanaTleblo programa (A1), ar aris identuri populaciaSi im adamianebis qulebis ganawilebisa, romelTac ar gauvliaT saganmanaTleblo programa (A2).
mniSvnelovnobis done: α= 0,05 U-s kritikuli mniSvneloba: roca nA1 = 8, nA 2 = 10 da α= 0,05, A.8 cxrilis mixedviT Ukritikuli = 17. uaryofis are: umciresi U realuri-s mniSvnelobebi, romlebic udris an naklebia 17-ze. gamoTvla: U realuri-s gamosaTvlelad unda mivyveT 15.5 cxrilSi mocemul safexurebs
safexuri 1: orive jgufis gaerTianebuli monacemebi daalageT rigis mixedviT umciresidan udidesisaken (zrdadobis mixedviT).
rigis mixedviT dalagebuli monacemebi: 0,02 0,08 0,14 0,16 0,21 0,31 0,41 0,52 0,53 0,53 0,55 0.58 0,59 0,63 0,89 0,89 0,91 1,14 jgufTan mikuTvnebuloba: 1 2 1 1 2 1 2 1 1 1 2 1 2 2 2 2 2 2
rangi: 1 2 3 4 5 6 7 8 9,5 9,5 11 12 13 14 15,5 15,5 17 18
safexuri 2: gamoTvaleT TiToeuli jgufis rangebis jami
∑ 1AR = 1+3+4+6+8+9,5+9,5+12 = 53 ∑ 2AR = 2+5+7+11+13+14+15,5+15,5+17+18= 118
safexuri 3: gamoTvaleT U A1realuri
( )( ) ( )
( ).63533680
532/7280
532
188108 A1
=−+=−+=
−+
+=realuriU
safexuri 4: gamoTvaleT U A2realuri
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
30
( )( ) ( )
( ).171185580
1182/11080
1182
11010108 A2
=−+=−+=
−+
+=realuriU
safexuri 5: airCieT umciresi U realuri
U A1realuri = 17 aris umciresi U A realuri
gadawyvetilebis miReba: miRebuli U realuri = 17 udris Ukritikuli = 17-s.
maSasadame, is xvdeba uaryofis areSi, uarvyofT 0H -s da viRebT 1H -s.
daskvna: 0H -is uaryofa niSnavs, rom saganmanaTleblo programis jgufis
monacemebis ganawileba gansxvavebulia arasaganmanaTleblo jgufis monacemebis ganawilebisgan. alkoholis Sesaxeb ganaTlebis programam Seamcira moxmarebuli alkoholis raodenoba, rac izomeboda studentebis mier mowodebuli maCveneblebiT maT mier dReSi moxmarebuli alkoholis raodenobis Sesaxeb. man-uitnis U kriteriumis Sedegebis warmodgena
U kriteriumis Sedegebis warmodgena Sedegebis statistikuri warmodgenis zogadi principebis Sesabamisad xdeba. sailustraciod gamoviyenebT amocanas imis Sesaxeb, Tu rogoria klebis mimarTulebiT Tvlis gavlena Sroderis kibis Sebrunebebis danaxvaze. sakontrolo (n = 9) da eqsperimenul (n = 9) jgufebSi figuris SebrunebisaTvis saWiro drois ganawilebebis Sesadareblad gamoiynes man-uitnis U krteriumi. alfas 0,05
mniSvnelobisas ganawilebebi mniSvnelovnad ar gansxvavdeboda erTmaneTisgan, U = 25, p > 0,05
Sedegebis aseTi warmodegena gvawvdis Cveul informacias gamoyenebuli kriteriumis Sesaxeb. U = 25 umciresi U realuri–ia da p > 0,05 gamoxatavs, rom U aramniSvnelovania 0,05 doneze. mowodebuli informacia jgufis zomis Sesaxeb (nA1 = nA 2 = 9) saWiroa imisaTvis, rom U kritikuli movZebnoT A.8 cxrilSi.
SeamowmeT Tqveni codna 15.2
1. gaaerTianeT A1 da A2 jgufebis qvemoT mocemuli monacemebi da daalageT rigis mixedviT 1-dan N-mde. SeinarCuneT TiToeuli qulis jgufTan mikuTvnebuloba. Semdeg gamoTvaleT: UA1realuri,
ramdenjer uswrebs win A1 monacemis rangi A2 monacemis rangs,
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
31
UA2realuri, ramdenjer uswrebs win A2 monacemis rangi A1 monacemis rangs.
jgufi
1A 2A
15 23 10 19
7 26 14 16
a. U realuri–s mniSvnelobebidan romelia umciresi? b. aris Tu ara U realuri statistikurad mniSvnelovani 0,05
doneze? 2. pirveli kiTxvis qulebisaTvis ipoveT UA1realuri da UA2realuri U–s
gamosaTvleli formulis gamoyenebiT. a. formulebiT miRebuli UA1realuri -sa da UA2realuri-s mniSvnelobebi eTanxmeba Tu ara pirvel kiTxvaSi miRebul mniSvnelobebs?
3. CamoayalibeT statistikuri hipoTeza man-uitnis U kriteriumisaTvis.
4. Tu 0H WeSmaritia, rogori unda iyos UA1realuri–isa da UA2realuri–is
mimarTeba?
5. Tu 1H WeSmaritia, rogori unda iyos UA1realuri–isa da UA2realuri –is mimarTeba?
6. Semdeg cxrilSi mocemulia Urealuri-is umciresi mniSvnelobebi sxvadasxva zomis jgufebisaTvis. U realuri–s da jgufis zomis TiToeuli nakrebisaTvis ipoveT Ukritikuli–s mniSvneloba 0,01 doneze. Semdeg miuTiTeT, xvdeba Tu ara U realuri uaryofis
zonaSi da ra gadawyvetilebs miiRebT 0H –sa da 1H -Tan
dakavSirebiT.
Urealuri 1An 2An
a. b. g. d. e. v.
3 12 28 21 4 126
6 10 16 7 13 18
4 10 9 19 19 19
7. amocanis nimuSis formatis mixedviT man-uitnis kriteriumis gamoyenebiT amoxseniT Semdegi amocana. gamoiyeneT mniSvnelovnobis 0,05 done. musikas sul ufro da ufro xSirad iyeneben stomatologiuri pacientebis relaqsaciisaTvis.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
32
davuSvaT, rom somatologma Seamowma pacientebis relaqsaciisaTvis musikis efeqturoba, risTvisac pacientebi SemTxveviT miakuTvna ori jgufidan erT-erTs. cameti pacienti musikis pirobaSi stomatologiuri procedurebis dros usmenda musikas, 12 pacienti musikis arqonis pirobaSi igive stomatologiuri procedurebis dros ar usmenda musikas. stomatologTan vizitis naxevari drois gasvlis Semdeg TiToeul pacients sTxovdnen, ganesazRvraT sakuTari SfoTva 1-dan (sruliad ar vSfoTav) 100-mde (Zalian vSfoTav) skalaze. SfoTvis miRebuli monacemebi Semdegia:
Emusikis piroba
moismina musika
( 1A ) ar mousmenia
musika
( 2A ) 52 75 66 33 17 78 94 89 27 49 53 78
44 21 66 54 37 84 30 29 54 49 33 10 78
gansxvavdeba Tu ara jgufebi SfoTvis maCveneblebis ganawilebis mixedviT?
uilkoqsonis niSniani rangebis kriteriumi
uilkoqsonis niSniani rangebis kriteriumi (Wilcoxon signed-ranks test): araparametruli kriteriumi cdis pirebs Soris sqemisaTvis damoukidebeli cvladis ori doniT.
uilkoqsonis niSniani rangebis kriteriumi araparametruli kriteriumia cdis pirebs SigniT sqemisaTvis damoukidebeli cvladis ori doniT.
maSasadame, is aris lidamokidebut kriteriumis araparametruli analogi.
uilkoqsonis kriteriumis daSvebis Tanaxmad, qcevis ganzomileba uwyvetia da monacemebi rigis mixedvis SeiZleba dalagdes. uilkoqsonis niSniani rangebis kriteriumis sailustraciod warmovadgenT eqsperiments xmaSi emociis gamoxatvis Sesaxeb (Cosmides, 1983).
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
33
davuSvaT, rom fsiqologis hipoTezaa, rom xmis sixSireebi icvleba sxvadasxva fsiqologiuri mdgomareobis dros. am hopoTezis Sesamowmeblad is iyenebs erTfaqtorian, cdis pirebs SigniT sqemas damoukidebeli cvladis ori doniT. damoukidebeli cvladia scenaris tipi, romelic cdis pirma unda waikiTxos: mxiaruli situaciis amsaxveli scenari (A1) an sevdiani situaciis amsaxveli scenari (A2). scenaris mizania cdis pirSi mxiaruli an sevdiani emociis gamowveva. ToToeul pirobaSi cdis piris amocanaa, Cumad waikiTxos scenari da eqsperimentatoris miniSnebaze xmamaRla warmoTqvas: ,,me jer kidev vkiTxulob~. TiToeuli cdis pirisaTvis damokidebuli cvladia gamoTqmis ,,mer jer kidev vkiTxulob~ xmovani sixSire da izomeba wamSi rxevebis raodenobiT (rxeva wamSi anu herci). maSasadame, monacems warmoadgens diapazoni cdis piris mier am sityvebis warmoTqmis umaRles xmovan sixSiresa da yvelaze dabal xmovan sixSires Soris. davuSvaT, 15.6 cxrilSi warmodgenilia scenaris orive pirobaSi monawile 10 cdis piris sixSireebi. aq naCvenebia am monacemebis saSualoebi, standartuli gadaxrebi da dispersiebi. yuradReba miaqcieT imas, rom gansxvavdeba ara mxolod sixSireebis diapazonebis saSualoebi, aramed didi
gansxvavebaa ori pirobis cvalebadobas Soris. gaixseneT, rom lidamokidebut moiTxovs, rom ori populaciis dispersia Tanabari iyos, am magaliTSi ki ori dispersia erTmaneTisgan Zlier gansxvavdeba (mxiaruli scenarisaTvis 704,37 da sevdiani scenarisaTvis 10281,96). maSasadame, es monacemebi arRvevs
lidamokidebut is erT-erT daSvebas. uilkoqsonis niSniani rangebis kriteriums ar
aqvs moTxovna dispersiebis Sesaxeb da aqedan gamomdinare, is am monacemebisaTvis Sesaferisi kriteriumia. cxrili 15.6 xmovani sixSiris hipoiTeturi qulebi (hercebSi), rogorc wakiTxuli scenaris tipis funqcia
scenaris tipi cdis piri mxiaruli ( 1A ) sevdiani ( 2A )
1 2 3 4 5 6 7 8 9 10
305 275 360 299 317 326 284 347 331 314
214 385 519 307 483 454 501 463 382 297
AX
As
315,80 26,54
400,50 101,40
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
34
2s 704,37 10.281,96
statistikuri hipoTezebis Semowmeba uilkoqsonis niSniani rangebis kriteriumiT T statistika uilkoqsonis kriteriumi rigis mixedviT dalagebul monacemebs iyenebs statistikis misaRebad, romelsac ewodeba T da romelic ori eqsperimentuli pirobidan aRebuli rangebis umciresi jamia. statistikuri hipoTeza nulovani hipoTeza. nulovani hipoTeza, romelic T–s gamoyenebiT mowmdeba, aris:
0H : erTmaneTTan dakavSirebuli A1 da A2 qulebis populaciaSi
ganawilebebi identuria.
alternatiuli hopoTeza. alternatiuli hipoTezaa:
1H : erTmaneTTan dakavSirebuli A1 da A2 qulebis populaciaSi ganawilebebi ar aris identuri.
SerCeviTi ganawileba da T-s kritikuli mniSvnelobebi
kritikuliT -s mniSvnelobebi, roca α= 0,05 da α= 0,01 da SerCevis zoma 6-dan 50-
mdea, warmodgenilia danarT A.9 cxrilSi. am cxrilSi Segvaqvs wyviliN -s
mniSvnelobebi, rac aris qulebis wyvilebis raodenoba, anu cdis pirebis
raodenoba. realuriT –s is mniSvnelobebi, romlebic metia an udris kritikuliU –s,
xvdeba uaryofis areSi da statistikurad mniSvnelovania α–s arCeul doneze.
magaliTisaTvis, roca wyviliN = 10 da α= 0,05, kritikuliT = 8. maSasadame, uaryofis
areSi xvdeba kritikuliT -s is mniSvnelobebi, romlebic metia an tolia 8-is.
T -s gamoTvla T -s mniSvnelobis gamoTvla moicavs qvemoT CamoTvlil safexurebs. TiToeuli safexuris Sedegi xmovani sixSiris diapazonis Sesaxeb eqsperimentis moncemebisaTvis naCvenebia 15.7 cxrilSi.
safexuri 1
TiToeuli cdis pirisaTvis ipoveT
d = 21 AA XX − , anu monacemebs Soris gansxvaveba, rogorc es 15.7 cxrilSia mocemuli. sxvaobas
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
35
miuwereT + an – niSani. safexuri 2
ipoveT TiToeuli sxvaobis absoluturi
mniSvneloba ( d , mniSvneloba niSnis
gaTvaliswinebis gareSe), rogorc es naCvenebia b svetSi
safexuri 3
zrdadobis mixedviT (umciresi mniSvnelobis rangia 1 da udidesis _ N) daalageT absoluturi mniSvnelobebi, rogorc es g svetSia naCvenebi. 0-is toli sxvaobebis
ranJireba ar xdeba da kritikuliT –s
gamosaTvlelad gamoyenebuli N - is mniSvneloba mcirdeba 0-s toli sxvaobebis raodenobiT. d –s bmul qulebs eniWeba im rangebis saSualo, rasac mivaniWebdiT maT, bma rom ar momxdariyo.
safexuri 4
gGsvetis ToToeul rangs miuwereT a svetis sxvaobis niSani. niSniani rangebi naCvenebia d svetSi.
safexuri 5
SeajameT msgavsi niSnis mqone rangebi d svetidan. magaliTad, plus niSnis mqone rangebis jami d svetSi aris 6 (anu, (+4) + (+2)) da minus niSnis mqone rangebis jami aris – 49 (anu, (-5) + (-8) + (-1) + (-9) + (-7) + (-10) + (-6) + (-3)).
safexuri 6 ipoveT rangebis TiToeuli jamis absoluturi mniSvneloba. Semdeg amorCieT rangebis ori jamidan umciresi. am SemTxvevaSi es aris 6, dadebiTi rangebis jami. swored es jami aris
realuriT ; maSasadame, realuriT = 6.
cxrili 15.7 15.6 cxrilis qulebisTvis uilkoqsonis T -s gamoTvla
scenaris tipi (a) (b) (g) (d) cdis piri
mxiaruli
( 1A )
sevdiani
( 2A ) d d ranJirebuli
d
niSniani rangi
1 2 3 4 5 6 7 8 9
305 275 360 299 317 326 384 347 331
214 285 519 307 483 454 501 463 382
+91 _110 _159 _8 _166 _128 _217 _116 _51
91 110 159 8 166 128 217 116 51
4 5 8 1 9 7 10 6 3
+4 _5 _8 _1 _9 _7 _10 _6 _3
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
36
10 314 297 +17 17 2 +2 statistikuri hipoTezis Sesaxeb gadawyvetilebis miReba T realuri= 6 ufro mcirea, vidre Tkritikuli= 8; aqedan gamomdinare, T realuri
xvdeba uaryofis areSi. uarvyofT 0H -s da viRebT 1H -s.
daskvna kriteriumis mixedviT
0H -is uaryofas da 1H –is miRebas mivyavarT daskvnamde, rom monacemebis ori
ganawileba erTmaneTisagan gansxvavdeba; scenaris tipma ganapiroba sxvaobebi xmis sixSiris dipazonSi. xmis sixSiris diapazoni ufro didia sevdiani scenaris SemTxvevaSi, vidre mxiaruli scenaris SemTxvevaSi.
magaliTi 15.3
uilkoqsonis niSniani rangebis kriteriumis gamoyeneba amocana hipoTezis Tanaxmad, musikis tempi gavlenas axdens qcevis siswrafeze; kerZod, swrafi tempis musika zrdis qcevis ganxorcielebis SesaZlo siswrafes (McElrea & Standing, 1992; Roballey et al., 1985). warmoidgineT, rom am hopoTezis Sesamowmeblad mkvlevarma Caatara Semdegi kvleva: ori dRis ganmavlobaSi akvirdebodnen kafeteriis erTi da igive 11 klientis sadilobas musikis fonis ori gansxvavebuli pirobis SemTxvevaSi: pirvel dRes neli musikis pirobaSi (A1) da meore dRes swrafi musikis pirobaSi (A2). Caiweres TiToeuli cdis piris mier musikis TiToeul pirobaSi wuTSi miRebuli lukmebis raodenoba. davuSvaT, rom TiToeuli cdis piris mier musikis tempis TiToeuli pirobis SemTxvevaSi wuTSi miRebuli lukmebis raodenobebi Semdegia:
musikis tempi cdis piri neli ( 1A ) swrafi ( 2A )
1 2 3 4 5 6 7 8 9 10 11
3 4 2 3 3 4 1 5 3 5 2
4 6 3 3 8 2 4 4 6 9 8
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
37
eqsperimentatori miiCnevda, rom es monacemebi rigis skalazea. axdens Tu ara gavlenas musikis tempi miRebuli lukmebis raodenobaze? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna: kvlevis sqema warmoadgens erTfaqtorian, cdis pirTa Sida sqemas damoukidebeli cvladis, musikis tempis, ori doniT. damokidebuli cvladi izomeba, sul mcire, rigis skalaze. maSasadame, uilkoqsonis niSniani rangebis kriteriumi am monacemebisaTvis Sesabamisi kriteriumia. saWiro kriteriumi: T, rangebis jamebs Soris umciresi jami. daSvebebi kriteriumis gamoyenebisaTvis:
1. TiToeul cdis pirs vafasebT damoukidebeli cvladis orive doneze.
2. damokidebuli cvladi izomeba, sul mcire, rigis skalaze statistikuri hipoTeza:
0H : erTmaneTTan dakavSirebuli A1 da A2 monacemebis populaciaSi
ganawilebebi identuria.
1H : erTmaneTTan dakavSirebuli A1 da A2 monacemebis populaciaSi ganawilebebi ar aris identuri.
mniSvnelovnobis done: α = 0,05 T-s kritikuli mniSvneloba: im SemTxvevaSi, roca wyviliN = 11 da α = 0,05,
kritikuliT = 11. T-s gamoTvlisas cdis piri #4 gviCvenebs nulovan cvlilebas
musikis or pirobas Soris. es cdis piri gamoeTiSeba analizs. Sesabamisad,
wyviliN xdeba 10 da N = 10-is SemTxvevisaTvis, A.9 cxrilis mixedviT,
kritikuliT gaxdeba 8.
uaryofis are: realuriT -s is mniSvnelobebi, romlebic naklebia an tolia 8-is.
gamoTvla:
safexuri 1
TiToeuli cdis pirisaTvis viTvliT d = XA 1 – XA (15.8 cxrilis a sveti)
safexuri 2
vpoulobT d–s absolutur mniSvnelobebs (15.8 cxrilis b sveti)
safexuri 3
zrdadobis mixedviT, 1-dan 10-mde valagebT d–s absolutur mniSvnelobebs; d = 0 ar
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
38
monawiloeobs ranJirebaSi (g sveti). yuradReba miaqcieT bmuli d monacemebis ramdenime SemTxvevis ranJirebas. bmul qulebs aniWeben maT mier dakavebuli rangebis saSualos. magaliTad,
magaliTad, b svetSi aris sami sxvaoba d =1. es
mniSvnelobebi ikavebs rangebs 1, 2 da 3. am
rangebis saSualo aris 2; maSasadame, d =1-is
TiToeul mniSvnelobas eniWeba rangi 2. safexuri 4
d-s rangs (d sveti) vuwerT TiToeuli d –s niSans (a svetidan).
safexuri 5
vajamebT msgavsi niSnis mqone rangebs. plus niSnis mqone rangebis jamis absoluturi mniSvneloba aris 6,5 (anu, (+4,5) + (+2) = 6,5). minus niSnis mqone rangebis jamis absoluturi mniSvneloba aris 48,5 (anu, (-2) + (-4,5) + (-2) + (-9) + (-6,5) + (-6,5) + (-8) + (-10) = 48,5).
safexuri 6 plus niSnis mqone rangebis jamis absoluturi mniSvneloba rangebis jamebs Soris umciresia,
amitom realuriT = 6,5.
cxrili 15.8 uilkoqsonis T -s gamoTvla lukmebis raodenobis qulebis, rogorc musikis tempis funqciisaTvis
musikis tempi (a) (b) (g) (d) cdis piri
swrafi
( 1A )
neli
( 2A ) d d ranJirebuli
d
niSniani rangi
1 2 3 4 5 6 7 8 9 10 11
3 4 2 3 3 4 1 5 3 5 2
4 6 3 3 8 2 4 4 6 9 8
_1 _2 _1 0 _5 +2 _3 +1 _3 _4 _6
1 2 1 0 5 2 3 1 3 4 6
2 4,5 2 _ 9 4,5 6,5 2 6,5 8 10
_2 _4,5 _2 _ _9 +4,5 _6,5 +2 _6,5 _8 _10
gadawyvetileba: realuriT = 6,5 ufro naklebia, vidre kritikuliT = 8 da is xvdeba
uaryofis areSi, uarvyofT 0H -s da viRebT 1H –s.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
39
daskvna: musikis ori tempis pirobebSi wuTSi miRebuli lukmebis raodenobis ganawilebebi mniSvnelovnad gansxvavdeba; nel tempTan SedarebiT swrafi tempis dros met lukmas iReben. uilkoqsonis niSniani rangebis kriteriumis Sedegebis warmodgena uilkoqsonis niSniani rangebis kriteriumis Sedegebis warmodgenas giCvenebT im magaliTis gamoyenebiT, romelSic xdeboda scenaris tipiT manipulireba da cdis pirebis xmis sixSiris diapazonis gazomva.
mxiaruli da sevdiani scenarebis pirobebSi xmis sixSireebis diapazonebis Sesadareblad gamoiyeneboda uilkoqsonis niSniani rangebis kriteriumi. alfas 0,05 doneze scenaris pirobebi mniSvnelovnad gansxvavdeboda erTmaneTisagan, T = 6 (N = 10), p < 0,05. mxiaruli scenaris pirobasTan SedarebiT sevdiani scenaris pirobaSi sixSireTa diapazoni ufro didi iyo.
ase warmodgenili Sedegebi gvaZlevs realuriT –s mniSvnelobis
interpretaciisaTvis saWiro informacias. radgan wyviliN -s mniSvneloba
viciT, SegviZlia saWiroebis SemTxvevaSi movZebnoT kritikuliT .
araparametruli kriteriumebis gamoyeneba rogorc vnaxeT, araparametruli kriteriumebi ar ar akeTeben daSvebebs SerCevis qulebis populaciaSi ganawilebis Sesaxeb. ufro metic, isini Cveulebriv moiTxoven mxolod imas, rom miRebuli monacemebis rigis mixedviT dalageba SegveZlos. parametruli kriteriumebi, rogoricaa t an dispersiuli analizi, ufro mkacr daSvebebs akeTeben eqsperimentis monacemebis bunebis Sesaxeb, _ iseT daSvebebs, romelTa dakmayofileba kvlevaSi miRebuli monacemebisaTvis zogjer saeWvoa. miuxedavad amisa, araparametruli statistikuri kriteriumebi ufro naklebad gamoiyeneba qceviTi mecnierebebis kvlevebSi, vidre parametruli kriteriumebi. ratom?
am kiTxvaze ramdenime pasuxi arsebobs. zogadad, Tu gasaanalizebeli monacemebi parametruli kriteriumebis daSvebebs akmayofilebs, maSin parametruli kriteriumi ufro aRmoaCens damoukidebeli cvladis efeqts, vidre misi Sesabamisi araparametruli kriteriumi. anu araparametruli kriteriumebi parametrul kriteriumebTan SedarebiT naklebad mZlavria da amitom, roca monacemebi akmayofilebs parametruli kriteriumis daSvebebs, ufro mosalodnelia, rom isini meore gvaris Secdomamde migviyvans. radgan iTvleba, rom parametruli kriteriumebis umravlesoba daSvebebTan Sesabamisobis TvalsazrisiT ufro mdgradia, bevri mkvlevari monacemebs parametruli kriteriumebis gamoyenebiT aanalizebs.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
40
araparametruli kriteriumebi ufro sxva saxis informacias gvaZlevs,
vidre parametruli kriteriumebi. magaliTad, lobisdamoukidebt –as mivyavarT
daskvnamde populaciebis saSualoebis toloba-aratolobis Sesaxeb, xolo U kriteriumiT gakeTebuli daskvnebi exeba populaciis ganawilebis toloba-aratolobas. garda amisa, ufro rTuli sqemebis SemTxvevaSi, rogoricaa me-11 TavSi ganxiluli faqtoruli sqema, ar arsebobs iseTi araparametruli kriteriumebi, romlebic imdenive informacias gvaZlevs, ramdensac dispersiis parametruli faqtoruli analizi. rogorc aRvniSneT, parametruli kriteriumis gamoyenebis mkacrad dadgenili wesebi ar arsebobs. Sroderis kibis magaliTi, romelic moyvanili iyo U kriteriumis gasacnobad, warmoadgens araCveulebrivad naTel magaliTs im SemTxvevebisa, roca saWiroa araparametruli kriteriumis gamoyeneba. am eqsperimentSi miRebuli erT-erTi monacemi iyo `usasruloba~. roca aseTi monacemi gvaqvs, monacemebis saSualos an standartuli gadaxras ver gamovTvliT. amitom, cxadia, rom am monacemebisaTvis parametreuli kriteriumis gamoyeneba ar SeiZleba. sxva magaliTebi xSirad aseTi erTmniSvnelovani ar aris. magaliTad, ramdenad unda gansxvavdebodes populaciis qulebis ganawileba normaluri ganawilebisagan, rom parametruli kriteriumis gamoyeneba Seusabamod CaiTvalos? rodis iTvleba ori an meti populaciis dispersiebi tolad? qceviTi mecnierebi ver Tanxmdebian am kiTxvebis pasuxebze. Sesabamisad, isini SeiZleba imazec ver SeTanxmdnen, garkveuli monacemebis analizisaTvis parametruli kriteriumi unda gamoviyenoT, Tu araparametruli.
SeamowmeT Tqveni codna 15.3
1. rogori tipis kvlevis sqemisaTvis aris Sesaferisi uilkoqsonis
kriteriumi? 2. romel gazomvis statistikas iyenebs uilkoqsonis niSniani
rangebis kriteriumi? 3. CamoayalibeT statistikuri hipoTeza uilkoqsonis niSniani
rangebis kriteriumisaTvis. 4. qvemoT mocemulia niSniani rangebis jamebis umciresi
absoluturi mniSvnelobebi sxvadasxva zomis jgufebze Catarebuli eqsperimentebidan. TiToeuli umciresi rangisa da
wyviliN –s mniSvnelobisaTvis gansazRvreT realuriT is mniSvneloba.
Semdeg ipoveT kritikuliT mniSvnelovnobis 0,05 doneze da miuTiTeT,
xvdeba Tu ara uaryofis areSi. aseve aRniSneT, ra
gadawyvetilebas iRebT 0H –sa da 1H -Tan dakavSirebiT.
Uumciresi rangi
wyviliN
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
41
a. b. g. d. e.
2 9 17 40 98
8 10 13 19 25
5. amocanis nimuSis formatis mixedviT uilkoqsonis kriteriumis
gamoyenebiT amoxseniT Semdegi amocana. gamoiyeneT mniSvnelovnobis 0,05 done.
sportis fsiqologebi dainteresebulebi arian sportuli Sejibrebebis dros SfoTvasa da sportul Sedegebs Soris kavSiriT. magaliTad, meinardma da houim (Maynard & Howe, 1987) gazomes ragbis moTamaSeebis SfoTva finalis pirveli da meore TamaSebis win. davuSvaT, rom Tqven CaatareT msgavsi kvleva da miiReT SfoTvis qvemoT moyvanili monacemebi. SesaZlo monacemebis diapazoni meryeobs 12-dan (Zalian dabali SfoTva) 84-mde (Zalian maRali SfoTva).
TamaSi moTamaSe 1 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
68 56 78 49 66 43 77 68 28 69 75 80 75 65 71
72 65 77 54 64 53 70 78 29 75 78 72 75 76 70
mosalodnelia, rom SfoTvis qulebis ganawileba populaciaSi uaryofiTad gadaxrili iqneba (moTamaSeebis umravlesobas eqneba Zalian maRali SfoTva da mxolod ramdenime maTgans eqneba Zalian dabali SfoTva). amitom Tqven apirebT araparametruli statistikuri kriteriumis gamoyenebas. gansxvavdeba Tu ara pirveli da meore TamaSis win miRebuli SfoTvis maCveneblebi?
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
42
Sejameba
• xi-kvadrat kriteriumi gamoiyeneba im monacemebisaTvis, romlebic warmoadgenen qcevis (pasuxis) gamovlenis sixSires.
• damoukideblobis xi-kvadrat kriteriumi gamoiyeneba ori damoukidebeli ganzomilebis kategorias Soris mimarTebis dasadgenad.
• damoukideblobis xi-kvadrat kriteriumi gamoiTvleba formuliT:
( )∑∑= =
−=
r
r
c
c rcErcErcO
1 1
2
2realuriχ
• Tanxmobis xi-kvadrat kriteriumi gamoiyeneba imisaTvis, rom dadgindes, Tu ramdenad kargad Seesabameba erT kategoriaSi moTavsebuli monacemebi am kategoriisaTvis Teoriulad miRebul mosalodnel sixSireebs.
• Tanxmobis xi-kvadrat kriteriumi gamoiTvleba formuliT:
( )∑=
−=
c
c c
cc
EEO
1
22realuriχ
• man-uitnis U kriteriumi araparametruli kriteriumia cdis pirebs Soris sqemisaTvis damoukidebeli cvladis ori donis gamoyenebiT, roca qulebi, sul mcire, rigis skalazea.
• U statistika is raodenobaa, ramdenjerac erTi jgufis qulis rangi win uswrebs meore jgufis qulis rangs.
• U A1realuri -s mniSvneloba gamoiTvleba formuliT:
( ) ∑−++= 1
1121 A1 2
1A
AAAA RnnnnU realuri
xolo U A2realuri-s _ formuliT:
( ) ∑−++= 2
2221 A2 2
1A
AAAA RnnnnU realuri
realuriU -s umciresi mniSvnelobebi gamoiyeneba, rogorc gazomvis
statistika.
• uilkoqsonis niSniani rangebis kriteriumi aris araparametruli kriteriumi cdis pirTa Sida sqemisaTvis, damoukidebeli cvladis ori doniT.
• uilkoqsonis niSniani rangebis kriteriumisTvis gamoiyeneba gazomvis statistika, T , ori eqsperimentuli pirobis monacemebis rangebis jamebs Soris umciresi jami.
• roca gasaanalizebeli monacemebi parametruli kriteriumis daSvebebs akmayofilebs, parametruli kriteriumi ufro mdgradia, vidre misi Sesabamisi araparametruli kriteriumi.
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
43
ZiriTadi terminebi da simboloebi
kiTxvebi
1. bevri adamiani fsiqoTerapiis kurss winaswar dagegmil vadamde
wyvets. aseTi adamianebi fsiqoTerapiiT mosargebleebis umravlesobas warmoadgenen (Stahler & Eisenman, 1987). davuSvaT, rom am problemis ukeT Sesaswavlad Tqven moipovebT monacemebs fsiqikuri janmrTelobis centrebidan da maT, vinc Terapias vadamde wyvets da maT, vinc ar wyvets, ajgufebT sqesisa da fsiqologiuri problemis simZimis mixedviT. a. Tqven miiReT im mamakacebisa da qalebis Semdegi sixSireebi,
romlebic vadamde wyveten an ar wyveten Terapiis kurss: sqesi
mamrobiTi mdedrobiTi ar Sewyvita Sewyvita
99 75
130 78
arsebobs Tu ara mimarTeba sqessa da Terapiis Sewyvetis sixSires Soris mniSvnelovnobis 0,05 doneze?
b. Semdeg moxda cdis pirebis dajgufeba maTi fsiqologiuri problemis simZimis mixedviT: msubuqi, zomieri, mZime. miRebulia Semdegi sixSireebi:
Ffsiqologiuri problemis
• araparametruli statistikuri kriteriumi
• damoukideblobis xi-kvadrat kriteriumi
• ganawilebisagan Tavisufali kriteriumebi
• Tanxmobis xi-kvadrat kriteriumi
• man-uitnis U kriteriumi
• marginaluri sixSireebi
• mosalodneli siSireebi
• niSanTa SeuRlebis cxrili
• parametruli kriteriumi
• niSanTa SeuRlebis cxrili
• T • kritikuliT
• realuriT
• uilkoqsonis niSniani rangebis kriteriumi
• U • kritikuliU
• realuriA1U
• realuriA2U
• xi-kvardat kriteriumi
• 2χ
• kritikuli2χ
• realuri2χ
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
44
simZime msubuqi zomieri mZime
ar Sewyvita Sewyvita
51 64
66 54
112 35
arsebobs Tu ara mimarTeba TerapiisTvis vadamde Sewyvetis maCvenebelsa da fsiqologiuri problemis simZimes Soris mniSvnelovnobis 0,05 doneze? Tu aseTi mimarTeba arsebobs, rogoria is?
2. 12.2 magaliTSi ganxiluli iyo kvleva imis Sesaxeb, aris Tu ara mTqnareba mowyenilobis niSani. erTfaqtoriani cdis pirTa Sida sqemis farglebSi, cdis pirebs aCvenebdnen rokmusikis 30-wuTian videorgols (damainteresebeli stimuli) da feradzolian sacdel satelevizio gamosaxulebas xmis gareSe (aradamainteresebeli stimuli) 30 wuTis ganmavlobaSi. akvirdebodnen da aRricxavdnen, Tu ramdenjer daamTqnarebda cdis piri TiToeuli sesiis ganmavlobaSi. davuSvaT, rom es eqsperimenti 11 cdis pirze gavimeoreT da ToToeuli sesiisaTvis miviReT damTqnarebis Semdegi raodenoba:
3. videorgolis tipi ( A ) cdis piri saintereso uintereso
1 2 3 4 5 6 7 8 9 10 11
5 2 4 3 0 4 7 6 3 1 8
7 1 7 8 2 5 6 23 3 4 9
a. uilkoqsonis niSniani rangebis kriteriumis gamoyenebiT mniSvnelovnobis 0,05 doneze daadgineT, axdens Tu ara davalebis tipi (esqperimentuli piroba) gavlenas damTqnarebis raodenobaze.
b. ratom SeiZleba iyos uilkoqsonis niSniani rangebis kriteriumi ufro Sesaferisi am monacemebisaTvis, vidre
lidamokidebut ? (miniSneba: daadgineT monacemebis ganawileba.)
g. ra daskvnebs akeTebT uilkoqsonis kriteriumis mixedviT? 3. me-9 Tavis #1 kiTxva exeboda klvevas imis Sesaxeb, Tu ra
gavlenas axdenda xmauri sisxlis wnevaze. Seqmnes cdis pirebis ori jgufi. siwynaris jgufis cdis pirebi 30 wuTis ganmavlobaSi isvenebdnen mosaxerxebel savarZelSi da 30 wuTis
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
45
gasvlis Semdeg maT uzomavdnen sisxlis sistolur wnevas. xmauris jgufis cdis pirebi aseve 30 wuTis ganmavlobaSi isxdnen savarZelSi, Tumca am 30 wuTis ganmavlobaSi isini usmendnen transportis xmaurs, romelic Cawerili iyo did qalaqSi, pikis saaTze. 30 wuTis ganmavlobaSi am xmauris mosmenis Semdeg maTac uzomavdnen sisxlis sistolur wnevas. davuSvaT, rom miRebulia sisxlis wnevis Semdegi maCveneblebi (merkuris milimetrebSi):
jgufi
siwynare xmauri 106 117 124 129 115 147 121 115 128 136 127
141 136 124 139 121 119 147 128 115 134 140
gamoiyeneT U kriteriumi da mniSvnelovnobis 0,05 doneze gamoTvaleT, gansxvavdeba Tu ara ganawilebebi. axseniT Tqveni daskvnebi.
4. imis dasadgenad, arsebobda Tu ara sqesebs Soris gansxvavebebi SfoTvaSi, rogorc pirovnul TvisebaSi, mkvlevarma 9 qalsa da 13 mamakacs Seavsebina SfoTvis kiTxvari. kiTxvarSi miRebuli qulebi SesaZloa meryeobdes 20-dan (dabali SfoTva) 80-mde (maRali SfoTva). davuSvaT, rom miRebulia Semdegi qulebi:
sqesi
mdedrobiT mamrobiTi 36 25 47 39 55 41 29 59 38
54 70 34 53 41 56 32 44 40 61 50 30
Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________
46
37 gamoiyeneT man-uitnis kriteriumi da mniSvnelovnobis 0,05 doneze daadgineT, gansxvavdeba Tu ara qalebis SfoTvis qulebis ganawileba mamakacebis SfoTvis qulebis ganawilebisagan. axseniT Tqveni daskvnebi.
5. me-9 TavSi mocemuli iyo amocana, romelic exeboda wonis matebas sigaretis mowevis Sewyvetis Semdeg. im kvlevaSi gazomes 8 adamianis wina maSin, roca isini mowevis Sewyvetis programaSi gawevriandnen da erTi wlis Semdeg. miiRes Semdegi wonebi:
wonis gazomvis dro cdis piri programis
dawyebisas Mmowevis
Sewyvetidan erTi wlis Semdeg
1 2 3 4 5 6 7 8
122,4 117,1 127,8 123,2 112,5 118,7 121,4 131,7
127,9 119,4 127,1 126,5 114,6 125,3 119,8 134,2
gaaanalizeT es monacemebi uilkoqsonis niSniani rangebis kriteriumis meSveobiT, raTa daadginoT, gansxvavdeba Tu ara ganawilebebi 0,05 doneze. aRwereT Tqveni daskvnebi.
danarTebi
A statistikuri cxrilebi B maTematikis mimoxilva C terminTa leqsikoni D statistikuri simboloebi E amocanebis pasuxebi F xSirad gamoyenebuli formulebi
danarTi A statistikuri cxrilebi cxrili # A.1 standartuli normaluri ganawilebis qveS mdebare
farTobis wili cxrili # A.2 t ganawilebis kritikuli mniSvnelobebi, rodesac
α = .05 da α = .01. cxrili # A.3a da b F ganawilebis kritikuli mniSvnelobebi,
rodesac (a) α = .05 da (b) α = .01. cxrili # A.4a da b studentis rangobrivi statistikis - q-s
mniSvnelobebi, rodesac (a) α = .05 da (b) α = .01.
cxrili # A.5 r-s kritikuli mniSvnelobebi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi).
cxrili # A.6 rS-s kritikuli mniSvnelobebi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi).
cxrili # A.7 xi kvadrat ganawilebis kritikuli mniSvnelobebi,
rodesac α = .05 da α = .01. cxrili # A.8a da b U-s kritikuli mniSvnelobebi man-uitnis U
kriteriumSi, rodesac (a) α = .05 da (b) α = .01. cxrili # A.9 T-s kritikuli mniSvnelobebi uilkoqsonis
kriteriumSi, rodesac α = .05 da α = .01.
cxrili # A.1 standartuli normaluri ganawilebis qveS mdebare farTobis wili cxrili # A.1 (gagrZeleba) wyaro: amoRebulia cxrilidan II.1, normaluri albaTobis funqcia da masTan dakavSirebuli funqciebi, CRC Handbook of Tables for Probability and Statistics (2nd ed.). Copyright 1968, CRC Press, Inc., Boca Raton, Florida. gamoiyeneba nebarTviT. cxrili # A.2 t ganawilebis kritikuli mniSvnelobebi, rodesac α = .05 da α = .01. trealuri-s (df) mniSvneloba, romelic tolia an metia cxrilSi moyvanili mniSvnelobisa, statistikurad mniSvnelovania SerCeul doneze. wyaro: amoRebulia cxrilidan 12, t-ganawilebis procentuli wertilebi, Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.3a F ganawilebis kritikuli mniSvnelobebi, rodesac α = .05. Frealuris mniSvneloba (dfmricxveli, dfmniSvneli), romelic tolia an metia cxrilSi moyvanili mniSvnelobisa, statistikurad mniSvnelovania .05 mniSvnelovnobis doneze. wyaro: amoRebulia cxrilidan 18, F ganawilebis procentuli wertilebi, (dispersiis koeficienti), zeda 5% wertilebi. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.3b F ganawilebis kritikuli mniSvnelobebi, rodesac α = .01. Frealuris mniSvneloba (dfmricxveli, dfmniSvneli), romelic tolia an metia cxrilSi moyvanili mniSvnelobisa, statistikurad mniSvnelovania .01 mniSvnelovnobis doneze. wyaro: amoRebulia cxrilidan # 18, F ganawilebis procentuli wertilebi, (dispersiis koeficienti), zeda 1% wertilebi. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.4a studentis rangobrivi statistikis - q-s mniSvnelobebi, rodesac α = .05. wyaro: amoRebulia cxrilidan # 29, studentis rangobrivi statistikis procentuli wertilebi, q = (xn - x1)/sv. zeda 5% wertilebi. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. Tu danarTi A.4 gamoiyeneba faqtorul sqemaSi ubralo efeqtebis kriteriumis CD-saTvis q-s mosaZebnad, saWiro svetis sapovnelad gamoiyeneT Semdegi gardaqmna.
sqemis tipi cxrilidan # A.4 gamoiyeneT q, romelic moTavsebulia damoukidebeli cvladis doneebis svetSi
2x2 2x3 3x2 3x3 3x4 4x3 4x4
3 5 5 7 8 8 10
cxrili # A.4b studentis rangobrivi statistikis - q-s mniSvnelobebi, rodesac α = .01. wyaro: amoRebulia cxrilidan # 29, studentis rangobrivi statistikis procentuli wertilebi, q = (xn - x1)/sv. zeda 1% wertilebi. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. Tu danarTi A.4 gamoiyeneba faqtorul sqemaSi ubralo efeqtebis kriteriumis CD-saTvis q-s mosaZebnad, saWiro svetis sapovnelad gamoiyeneT Semdegi gardaqmna.
sqemis tipi cxrilidan # A.4 gamoiyeneT q, romelic moTavsebulia damoukidebeli cvladis doneebis svetSi
2x2 2x3 3x2 3x3 3x4 4x3 4x4
3 5 5 7 8 8 10
cxrili # A.5 r-s kritikuli mniSvnelobebi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi). rrealuris mniSvneloba, romelic tolia an metia cxrilSi
moyvanili mniSvnelobisa, statistikurad mniSvnelovania α-s SerCeul doneze. adf udris Nwyvilebi - 2, sadac Nwyvilebi aris dawyvilebuli dakvirvebebis raodenoba. wyaro: amoRebulia cxrilidan # 13, korelaciis koeficientis, r, ganawilebis
procentuli wertilebi, rodesac ρ=0. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.6
rS-s kritikuli mniSvnelobebi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi). rsrealuris mniSvneloba, romelic cxrilSi moyvanili
mniSvnelobis tolia an metia, statistikurad mniSvnelovania α-s SerCeul doneze. wyaro: E.G. Olds, ranguli gasxvavebebis kvadratebis jamebis ganawileba individebis mcire raodenobebisTvis. Annals of Mathematical Statistics, Volume 9, 1938, 133-148 da E.G. Olds, ranguli gasxvavebebis kvadratebis jamebis 5%-iani mniSvnelovnobis doneebi da koreqcia. Annals of Mathematical Statistics, Volume 20, 1949, 117-118. nebarTviT amoRebulia cxrilidan # 12.12,
spirmenis ranguli korelaciis koeficientis kritikuli mniSvnelobebi, rodesac ρ=0, CRC Handbook of Tables for Probability and Statistics (2nd ed.). Copyright 1968, CRC Press, Inc., Boca Raton, Florida. cxrili # A.7 xi kvadrat ganawilebis kritikuli mniSvnelobebi, rodesac α = .05 da α = .01. χ2realuris (df) mniSvneloba, romelic tolia an metia cxrilSI
moyvanili mniSvnelobisa, statistikurad mniSvnelovania α-s SerCeul doneze. wyaro: amoRebulia cxrilidan # 8, xi-kvadratis ganawilebis procentuli wertilebi, Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.8a U-s kritikuli mniSvnelobebi man-uitnis U kriteriumSi, rodesac (a) α = .05 (ormxrivi kriteriumi). Tu jgufis zomebi toli ar aris, n1–iT ufro mcire zomis jgufi aRiniSneba. Urealuris mniSvneloba, romelic cxrilSi moyvanili mniSvnelobis tolia an naklebia, statistikurad mniSvnelovania .05 mniSvnelovnobis doneze. wyaro: D. Auble, U-s kritikuli mniSvnelobebi uilkoqsonis (man-uitnis) statistikaSi ori SerCevisaTvis, Institute of Educational Research, Indiana University, Bloomington, Indiana, Volume 1, Number 2, 1953, romelic nebarTviT amoRebulia cxrilidan # 10.4, U-s kritikuli mniSvnelobebi uilkoqsonis (man-uitnis) statistika ori SerCevisaTvis, CRC Handbook of Tables for Probability and Statistics (2nd ed.). Copyright 1968, CRC Press, Inc., Boca Raton, Florida. cxrili # A.8b U-s kritikuli mniSvnelobebi man-uitnis U kriteriumSi, rodesac (a) α = .01 (ormxrivi kriteriumi). Tu jgufis zomebi toli ar aris, n1–iT ufro mcire zomis jgufi aRiniSneba. Urealuris mniSvneloba, romelic tolia an naklebia cxrilSi moyvanili mniSvnelobisa, statistikurad mniSvnelovania .01 mniSvnelovnobis doneze. wyaro: D. Auble, U-s kritikuli mniSvnelobebi uilkoqsonis (man-uitnis) statistikaSi ori SerCevisaTvis, Institute of Educational Research, Indiana University, Bloomington, Indiana, Volume 1, Number 2, 1953, romelic nebarTviT amoRebulia cxrilidan # 10.4, U-s kritikuli mniSvnelobebi uilkoqsonis (man-uitnis) statistika ori SerCevisaTvis, CRC Handbook of Tables for Probability and Statistics (2nd ed.). Copyright 1968, CRC Press, Inc., Boca Raton, Florida. cxrili # A.9
T-s kritikuli mniSvnelobebi uilkoqsonis kriteriumSi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi). Trealuris mniSvneloba, romelic tolia an naklebia cxrilSi moyvanili mniSvnelobisa, statistikurad
mniSvnelovania α-s SerCeul doneze. tireebi aRniSnavs, rom am doneze jgufis am zomisaTvis ver viRebT statistikur mniSvnelovnobas. wyaro: Robert L. McCornack, Extended Tables of the Wilcoxon Matched Pair Sighed Rank Statistic. Journal of the American Statistical Association, 1965, 60, 864-871. amoRebulia nebarTviT The Journal of the American Statitical Assocation, Copyright 1965 by the American Statistical Association. yvela ufleba daculia.
danarTi B maTematikis mimoxilva aq mokled aris mimoxiluli is maTematikuri simboloebi da operaciebi, romelic gvWirdeba am saxelmZRvaneloSi warmodgenili statistikis gasagebad. statistikuri simboloebi simbolo, romelsac xandaxan niSansac uwodeben, aris aso an aRniSvna, romelsac iyeneben raimes aRsaniSnad. Semdeg cxrilSi warmodgenilia statistikaSi gamoyenebuli zogadi simboloebi. simbolo da magaliTi
aRwera
X an Xi X gamoiyeneba cdis pirisagan miRebuli qulis an monacemis aRsaniSnad. indeqsi i, rodesac mas iyeneben, aRniSnavs konkretuli cdis piris qulas. magaliTad, X1 aRniSnavs 1-li cdis piris qulas.
= tolobis niSani.
X1 = 3 1-li cdis piris qula udris 3-s.
X5 = 27 me-5 cdis piris qula udris 27-s.
a = 2 a-s mniSvneloba 2-is tolia.
≠ utolobis niSani.
6 ≠ 3 6 ar udris 3-s.
+ plusi - mimatebis niSani. es niSani miuTiTebs, rom + niSniT dakavSirebuli ricxvebi unda daematos erTmaneTs. mimatebis Sedegs jami ewodeba. iseTi debuleba, rogoricaa "5 + 1-is jami" niSnavs, rom es ricxvebi unda mivumatoT erTmaneTs.
5 + 1 = 6 5-isa da 1-is jami udris 6-s. 5 + 1 + 6 = 12 5 plus 1 plius 6 udris 12-s.
a + b a plus b: b-s mniSvneloba unda mivumatoT a-s mniSvnelobas.
_ minusi - gamoklebis niSani. es niSani miuTiTebs, rom ricxvi, romelic mosdevs _-is niSans, unda gamoakldes ricxvs, romelic dgas _ niSnis win. gamoklebis Sedegs sxvaobas uwodeben. iseTi debuleba, rogoricaa "5-isa da 1-is sxvaoba" niSnavs, rom erTi ricxvi meores unda gamoakldes.
5 _ 1 = 4 5-s gamovakloT 1 udris 4-s.
a _ b a minus b: b-s mniSvneloba unda gamovakloT a-s mniSvnelobas.
()() an x gamravlebis niSani. frCxilebSi moTavsebuli an x-iT gamoyofili mniSvnelobebi unda gavamravloT erTmaneTze. gamravlebis Sedegs namravls uwodeben.
(2)(3) = 6 2-isa da 3-is namravli aris 6. orjer sami aris eqvsi.
2 x 3 = 6 2-isa da 3-is namravli aris 6. orjer sami aris eqvsi.
(a)( b) an a x b a-jer b: a-s mniSvneloba unda gamravldes b-s
mniSvnelobaze. arsebobs am wesis gamonaklisi: faqtorul sqemaSi A
da B faqtorebis urTierTqmedeba aRiniSneba AxB. am SemTxvevaSi, aRniSvna unda wavikiTxoT ase: "A-sa da B-s urTierTqmedeba". es ar niSnavs, rom A unda gamravldes B-ze.
÷ an / an __ gayofis niSani. ricxvi, romelic win uswrebs ÷ an / an ricxvi, romelic __ xazis zeviT weria (mricxveli), iyofa ricxvze, romelic mosdevs ÷ an / an ricxvze, romelic __ xazis qveviT weria (mniSvneli). gayofis Sedegs ganayofs uwodeben.
6 ÷ 2 = 3 eqvsi, gayofili orze, aris sami.
6/2 = 3 eqvsi, gayofili orze, aris sami.
26 = 3
eqvsi, gayofili orze, aris sami.
a ÷ b, an a/b, an ba
a, gayofili b-ze
< naklebobis simbolo. ricxvi, romelic weria <-is win, naklebia, vidre ricxvi, romelic weria <-is Semdeg.
2 < 3 ori naklebia samze
a < b a naklebia b-ze X3 < 10 mesame cdis piris qula naklebia aTze.
> metobis simbolo. ricxvi, romelic weria >-is win, metia, vidre ricxvi, romelic weria >-is Semdeg.
3 > 2 sami metia orze. a > b a metia b-ze. X3 > 10 mesame cdis piris qula metia aTze.
≤ "naklebia an tolia" simbolo. ricxvi, romelic weria ≤ -is win, naklebia an tolia, vidre ricxvi, romelic weria ≤ -is Semdeg.
2 ≤ 3 ori naklebia an tolia samis. 4 ≤ 4 oTxi naklebia an tolia oTxis. a ≤ b a naklebia an tolia b-si. X3 ≤ 10 mesame cdis piris qula naklebia an tolia 10-is.
≥ "metia an tolia" simbolo. ricxvi, romelic weria ≥-is win, metia an tolia, vidre ricxvi, romelic weria
≥-is Semdeg. 3 ≥ 2 sami metia an tolia oris.
4 ≥ 4 oTxi metia an tolia oTxis.
a ≥ b a metia an tolia b-s. X3 ≥ 10 mesame cdis piris qula metia an tolia 10-is.
5 < X1 < 10 rogorc xedavT, metobis an naklebobis siimboloebi SesaZlebelia erT aRniSvnaSi moTavsdes. es aRniSvna ase ikiTxeba: "pirveli cdis piris qula 5-ze metia da 10-ze naklebi".
5 ≤ X1 ≤ 10 "naklebia an tolia" da "metia an tolia" simboloebi aseve SeiZleva moTavsdes erT aRniSvnaSi. es aRniSvna ase ikiTxeba: "pirveli cdis piris qula metia an
tolia 5-ze da naklebia an tolia 10-is".
absoluturi mniSvnelobis simbolo. es simbolo niSnavs, rom Cven ignorirebas vukeTebT + an _ niSnebs, romlebic mibmulia ricxvebze.
_ 6 = 6 minus eqvsis absoluturi mniSvneloba aris eqvsi.
+ 6 = 6 plius eqvsis absoluturi mniSvneloba aris eqvsi.
( )2 frCxilebSi moTavsebuli ricxvi ayvanilia kvadratSi, anu gamravlebulia Tavis Tavze. zogjer frCxilebi ar gamoiyeneba da kvadratis indikatori, 2, ubralod mosdevs kvadratSi asayvan ricxvs.
(5)2 = 25 xuTis kvadrati, anu 5 x 5, aris 25. 52 = 25 xuTis kvadrati udris 25-s.
kvadratuli fesvis simbolo. niSnavs, vipovoT
ricxvi, romelic, Tu mas Tavis Tavze gavamravlebT
(anu kvadratSi aviyvanT) udris niSnis qveS
moTavsebul ricxvs. 25 = 5 kvadratuli fesvi 25-dan udris 5-s, radganac 5
gamravlebuli Tavis Tavze (anu 5 x 5) 25-is tolia. 36 = 6 kvadratuli fesvi 36-dan udris 6-s, radganac Tavis
Tavze gamravlebuli 6 (anu 6 x 6) 36-is tolia.
∑ dajamebis simbolo. ricxvebi, romlebic mosdeven ∑ simbolos, unda mivumatoT erTmaneTs.
∑ (3+2) = 5 3-isa da 2-is jami udris 5-s.
∑ (3+2+4) = 9 3 plius 2 plius 4 udris 9-s.
∑ X an ∑ Xi X-is jami. daajameT yvela qula, romelic warmodgenilia X-iT. magaliTad, Tu sami cdis piris
qulebia: X1 = 3, X2 = 6 da X13 = 4, maSin ∑ X = 3 + 6 + 4 = 13. xandaxan dajamebis simbolos Tavze zRvrebia
miTiTebuli, magaliTad, ∑=
3
1iiX . es zRvrebi miuTiTeben,
rom unda davajamoT pirvelidan mesame cdis piramde
qulebi. Tu zRvrebi ar gamoiyeneba, maSin ∑ X niSnavs, rom unda davajamoT X-iT aRniSnuli yvela qula.
maTematikuri operaciebi uaryofiTi ricxvebi uaryofiTi ricxvi SeiZleba miviRoT gamoklebis Sedegad, magaliTad 6 _ 8 = _2. 8, romelic 6-ze 2-iT metia, gvaZlevs _2 sxvaobas, rodesac mas eqvss vaklebT. uaryofiTi ricxvebi xSirad gvxvdeba statistikaSi da gamoiyeneba ZiriTad maTematikur operaciebSi. Tu ricxvs ar aqvs win minusis niSani, igulisxmeba, rom is dadebiTia. uaryofiTi ricxvebis mimateba 6 + (_4) = 2 uaryofiTi ricxvis dadebiTisaTvis mimateba igivea, rac
dadebiTi ricxvidan uaryofiTis gamokleba. ase rom, 6 + (_4) = 6 _ 4 = 2.
(_6) + (_4) = _10 am SemTxvevaSi, _4 akldeba _6-s, anu (_6) + (_4) = _6 _4 =
_10. uaryofiTi ricxvebis gamokleba 6 _ (_4) = 10 uaryofiTi ricxvis gamokleba igivea, rac uaryofiTi
ricxvis absoluturi mniSvnelobis mimateba. ase rom, 6 _ (_4) = 6 + 4 = 10.
uaryofiTi ricxvebis gamravleba (_6)(_4) = 24 Tu orive gasamravlebeli ricxvi uaryofiTia, maSin
namravli dadebiTia. (6)(_4) = _24 Tu ori ricxvidan mxolod erTia uaryofiTi, maSin (_6)(4) = _24 namravli uaryofiTia. uaryofiTi ricxvebis gayofa
_10 ÷ 5 = _2 Tu mricxveli an mniSvneli uaryofiTia, maSin ganayofi
10 ÷ _5 = _2 uaryofiTia.
_10 ÷ _5 = _2 Tu mricxvelic da mniSvnelic uaryofiTia, maSin ganayofi dadebiTia.
wiladebi wiladebTan muSaobis yvelaze martivi midgomaa mricxvelis mniSvnelze
gayofis gziT maTi aTwiladad gardaqmna, magaliTad, 41 = 0.25,
42 = 0.50,
da 43 = 0.75. Tu aTwiladi Semdgom gamoTvlebSi unda gamoviyenoT, maSin
is unda davamrgvaloT sul mcire meaTasedebamde (mag., 114 = 0.3636). ase
rom, 52 +
81 +
31 = 0.4000 + 0.1250 + 0.3333 = 0.8583.
wiladebi da procentebi wiladebi wiladi aris mTelis nawili. magaliTad, Tu 100 adamianma Seavso kiTxvari
da maTgan 45 mamakacia, maSin mamakaci respondentebis wiladi aris 10045
anu .45. qali respondentebis wiladi ki aris 10055
= .55.
procenti procenti miiReba, rodesac wiladi 100-ze mravldeba. amdenad, mamakaci respondentebis wiladi .45 udris 45 x 100 anu mamakaci respondentebis 45 procents. xolo qali respondentebis procenti ki aris 55 x 100 = 55 procenti.
maTematikuri operaciebis Tanmimdevroba momdevno magaliTebi gviCvenebs, Tu ra TanmimdevrobiT sruldeba maTematikuri operaciebi Cveulebriv statistikur aRniSvnebSi:
∑ − )( XX 1. saSualo akldeba yovel qulas qulaTa erTobliobaSi.
2. sxvaobebi jamdeba. 2
)(∑ − XX 1. saSualo akldeba yovel qulas qulaTa erTobliobaSi.
2. yoveli sxvaoba aiyvaneba kvedratSi. 3. kvadratSi ayvanili sxvaobebi jamdeba.
∑ 2X 1. yoveli X mniSvneloba aiyvaneba kvadratSi. 2. kvadratSi ayvanili X mniSvnelobebi jamdeba.
(∑ 2X ) 1. X mniSvnelobebi jamdeba.
2. X mniSvnelobebis jami (anu ∑ X ) aiyvaneba
kvadratSi.
∑ XY 1. qulaTa erTobliobaSi Sesatyvisi X da Y mniSvnelobebi mravldeba.
2. gadamravlebuli X Y mniSvnelobebi jamdeba qulaTa erTobliobisaTvis.
(∑ X )(∑Y ) 1. X mniSvnelobebi jamdeba, raTa miviRoT ∑ X .
2. Y mniSvnelobebi jamdeba, raTa miviRoT ∑Y .
3. X mniSvnelobebis jami (anu ∑ X ) mravldeba Y
mniSvnelobebis jamze (anu ∑Y ).
ტერმინები 1
თავი I • aRweriTi statistika (Descriptive statistics): _ statistikuri procedurebi, romlebic
gamoiyeneba SerCevidan miRebuli monacemebis Sesajameblad da aRsawerad.
• damokidebuli cvladi (Dependent variable): _ cvladi eqsperimentSi, romelic damokidebulia damoukidebel cvladze. SemTxvevaTa umravlesobaSi damokidebuli cvladi qcevis garkveuli sazomia.
• damoukidebeli cvladi (Independent variable): _ cvladi, romliTac manipulireben eqsperimentSi, damokidebul cvladze misi gavlenis gansazRvris mizniT.
• ekvivalenturi jgufebi (Equivalent groups): _ cdis pirTa jgufebi, romlebic, savaraudod, ar gansxvavdebian erTmaneTisgan raime mudmivi maxasiaTebliT eqsperimentis damoukidebeli cvladis miwodebamde.
• kvlevis hipoTeza (Research hypothesis): _ debuleba or an met cvlads Soris mosalodneli an nawinaswarmetyvelebi kavSiris Sesaxeb. eqsperimentSi kvlevis hipoTeza aris nawinaswarmetyvelebi kavSiri damoukidebel da damokidebul cvladebs Soris.
• korelacia (Covary): _ ori cvladi korelaciaSi imyofeba erTmaneTTan, rodesac erTi cvladis cvlileba ukavSirdeba meore cvladis SeTanxmebul sistematur cvlilebas.
• korelaciis koeficienti (Correlation coefficient): _ statistika, romelic ricxobrivad gviCvenebs maCvenebelTa ori jgufis dakavSirebulobis xarisxs da am kavSiris mimarTulebas.
• monacemebi (Data _ mravlobiTi, datum _ mxolobiTi): _ is qulebi an qcevis Tu maxasiaTeblebis gazomvis Sedegebi, romelic miiReba adamianebze an cxovelebze dakvirvebis Sedegad.
• nedli monacemebi (Raw data): _ yvela cdis pirisgan miRebuli qula, romelTac jer ar gauvliaT statistikuri analizi.
• parametri (Parameter): _ ricxvi, romelic aRwers populaciis maxasiaTebels.
• populacia (Population): _ aris saerTo maxasiaTeblis mqone adamianTa, cxovelTa an movlenaTa sruli erToblioba.
• regresiuli analizi (Regression analysis): _ statistikuri meTodebis gamoyeneba qulaTa erTi erTobliobis mixedviT qulaTa meore erTobliobis Sesaxeb winaswarmetyvelebisTvis.
• statistika (mecniereba) (Statistics): _ is meTodebi Tu procedurebi, romlebic gamoiyeneba monacemebis sistematizaciisTvis, gasaanalizeblad da maTgan daskvnebis gamosatanad.
• statistika (statistikuri sidide) (Statistic): _ monacemebze dayrdnobiT gamoTvlili calkeuli ricxvi, romelic SeiZleba gamoyenebul iqnes SerCevis monacemTa erTobliobis aRsawerad, an am monacemTa ufro sruli gaanalizebisTvis.
• statistikuri daskvna (Statistical inference): _ populaciis maxasiaTeblis Sefaseba SerCevidan miRebul statistikebze dayrdnobiT.
• statistikuri hipoTezis Semowmeba (Statistical hypothesis testing) _ procedurebi, romlebic jgufebs Soris SemTxveviTi gansxvavebis zomis winaswarmetyvelebisTvis gamoiyeneba.
• qula (Score): _ cdis piris mier davalebis Sesrulebis gazomviT miRebuli ricxviTi mniSvneloba.
ტერმინები 2
• SemTxveviTi gansxvaveba (Chance difference): _ gansxvaveba ekvivalentur jgufebs Soris, romelic miiReba SemTxveviT an gauTvaliswinebeli mizeziT.
• SemTxveveiTi SerCeva (Random sample): _ SerCeva, romelSic individebi SerCeulni arian ise, rom populaciis TiToeuls wevrs SerCevaSi moxvedris toli Sansi aqvs da erTi wevris arCeva ar aris damokidebuli populaciis sxva wevrebis arCevaze.
• SerCeva (Sample): _ populaciidan mTeli rigi wesebis dacviT (raTa SerCeva populaciis reprezentaciuli iyos) arCeuli qvesimravle anu qvejgufi.
• cdis piri (Subject): _ pirovneba, romelic monawileobs eqsperimentSi.
• cdis pirTa Soris sqema (Between-subject design): _ kvlevis sqema, romelSic monawileobs ori an meti jgufi
• cvladi (variable): _ garemos nebismieri mdgomareoba, nebismieri stimuli, pirovnuli maxasiaTebeli an qceva, romelmac SeiZleba miiRos sxvadasxva mniSvneloba sxvadasxva dros an sxvadasxva adamianTan.
თავი II • bunebrivi dakvirveba (Naturalistic observation): _ kvleva, romelic gulisxmobs bunebriv
pirobebSi ganxorcielebul qcevebze dakvirvebas.
• gazomva (Measurement): _ garkveuli wesebis mixedviT cvladebisTvis ricxviTi mniSvnelobebis miniWeba.
• gamokiTxva (Survey research): _ kvleva, romelSic monacemebi adamianebis zepiri an werilobiTi interviuebiT miiReba.
• diskretuli (wyvetili) cvladi (Discrete variable): _ cvladi, romelmac SeiZleba miiRos sasruli an Tvladi mniSvneloba.
• empiriuli monacemebi (Empirical data): _ dakvirvebasa da sensorul gamocdilebaze dafuZnebuli gazomvis Sedegad miRebuli qulebi.
• eqsperimenti (Experiment): _ zedmiwevniT gakontrolebuli situacia, sadac mecnieri axdens erTi an meti damoukidebeli cvladis manipulirebas da akvirdeba, Tu ra gavlenas axdens es damokidebul cvladze.
• Tvisebrivi monacemebi (Qualitative data): _ saxeldebis skalaze miRebuli monacemebi, romlebic miuTiTeben, rom cvladebi Tvisebrivad gansxvavdebian erTmaneTisgan.
• intervalebis skala (Interval measurement): _ cvladis mniSvnelobebi dalagebulia skalaze da miniWebul ricxvebs Soris gansxvaveba asaxavs cvladis mniSnelobebs Soris erTi da igive sididis sxvaobas. intervalTa skalaze nuli pirobiTad sawyisi wertilia.
• kvlevis meTodi (Research method): _ midgoma monacemebis mogrovebisadmi.
• operaciuli gansazRvreba (Operational definition): _ dakvirvebisas, damoukidebeli cvladiT manipulirebisas an damokidebuli cvladis gazomvisas gamoyenebuli operaciebis detaluri aRwera.
• rigis skala (Ordinal measurement): _ skalaze cvladis sidide dalagebulia mniSvnelobebis mixedviT.
• raodenobrivi monacemebi (Quantitative data): _ rigis, intervalebisa an Sefardebis skalaze miRebuli monacemebi, romlebic miuTiTeben, Tu ra sididis aris cvladi.
• ricxvis realuri zeda sazRvari(Upper real limit of a number) _ ricxvisa da misi momdevno ricxvis Suawertili.
ტერმინები 3
• ricxvis realuri sazRvrebi (Real limits of a number): _ ricxvisa da misi wina ricxvis Suawertili an ricxvisa da misi momdevno ricxvis Suawertili im skalaze, romelsac viyenebT mocemuli cvladis gasazomad.
• ricxvis realuri qveda sazRvari (Lower real limit of a number) _ ricxvisa da misi wina ricxvis Suawertili.
• saarqivo Canawerebi (Archival recodrs): _ kvleva, romelSic gamoiyeneba arsebuli Canawerebi.
• saxeldebis skala (Nominal measurement): _ gazomili cvladis sxvadasxva kategoriebSi gaerTianeba.
• uwyveti cvladi (Continuous variable): _ cvladi, romelmac SeiZleba miiRos usasrulo mniSvneloba cvladis nebismier or dones Soris.
• Sefardebis skala (Ratio measurement): _ cvladis sidide dalagebulia skalaze da miniWebul ricxvebs Soris gansxvaveba asaxavs cvladis mniSnelobebs Soris igive sididis gansxvavebas. Sefardebis skalaze arsebobs WeSmaritad nulovani wertili, romelic Seesabameba gasazomi maxasiaTeblis srul ararsebobas.
თავი III • asimetriuli ganawileba (Skewed distribution): _ sixSireTa ganawileba, sadac qulebi
Tavmoyrilia ganawilebis erT boloSi; ganawilebis meore boloSi (anu kudSi) ki ganlagebulia iSviaTad dafiqsirebuli qulebi.
• daujgufebel sixSireTa ganawileba (Ungroup frequency distribution): _ sixSireTa ganawileba, romelic agebulia miRebul minimalur da maqsimalur qulaTa Soris yvela SesaZlo mniSvnelobis dalagebiT da Semdeg qulis gaswvriv Tvlis Sedegis aRniSvnis CxiriT (/) imdenjer moTavsebiT, ramdenjerac iqna miRebuli es qula.
• dagrovili intervaluri sixSire (Cumulative frecuency of a class interval): _ intervalSi qulebis Sexvedris raodenobas damatebuli yvela ufro dabal intervalSi qulebis sixSireTa jami.
• dagrovili intervaluri sixSiris ganawileba procentebSi ( Cumulative grouped percentage frecuency of a class interval): _ intervalSi qulebis sixSires procentebSi damatebuli yvela ufro dabal intervalSi qulaTa sixSireTa jami procentebSi.
• dagrovili intervaluri fardobiTi sixSire (Cumulative grouped relative frecuency of a class interval): _ intervalSi qulebis fardobiT sixSires damatebuli yvela ufro dabal intervalSi qulaTa fardobiT sixSireTa jami.
• dajgufebul sixSireTa ganawileba (Grouped frequency distribution): _ ganawileba, sadac qulebi dajgufebulia intervalebSi da TiToeuli maTganSi moxvedrili qulebis sixSireebi gamoisaxeba cxrilis saxiT.
• dajgufebul sixSireTa ganawileba procentebSi (Grouped percentage frecuancy distribution): _ dajgufebul sixSireTa ganawileba, sadac intervalSi qulaTa sixSire Sefardebulia ganawilebaSi qulaTa saerTo raodenobasTan.
• dajgufebul fardobiT sixSireTa ganawileba (Grouped relative frequancy distribution): _ dajgufebul
• sixSireTa ganawileba miiReba qulaTa fardobiTi sixSiris 100-ze gamravlebiT.
• intervali (Class interval): _ intervalis sidide, romelic gamoiyeneba dajgufebul sixSireTa ganawilebaSi nedli qulebis dasajgufeblad. intervalis sidide aRiniSneba i-iT.
ტერმინები 4
• intervalis dadgenili sazRvari (Stated limits of a class interval): _ maqsimaluri da minimaluri qulebi, romelic SeiZleba moxvdes am intervalSi.
• intervalis realuri sazRvari (Real limits of a class interval): _ intervalis dadgenil sazRvrebsa da am intervalis wina an momdevno intervalebis dadgenil sazRvrebs Soris Suawertili.
• intervalis Suawertili (Midpoint of a class interval): _ intervalis realur sazRvrebs Soris Suawertili.
• moda (Mode): _ qulaTa ganawilebaSi yvelaze xSirad dafiqsirebuli qula.
• nedli monacemebi an nedli qulebi (raw data or raw scores) _ kvlevis monawileebisgan miRebuli qulebi, romelTac jer ar gauvliaT statistikuri analizi.
• procentili (Percentile): _ qula, romelzec qulaTa garkveuli procenti ganawilebaSi tolia an naklebia am qulaze.
• svetebiani diagrama (Bar graph) _ raodenobrivi monacemebis sixSireTa ganawilebis grafikulad gamoxatvis saSualeba. mocemul kategoriaSi qulaTa gamoCenis sixSires svetis simaRle Seesabameba.
• sixSireTa ganawileba (Frequency distribution): _ cxrili, romelSic mocemulia TiToeuli qula da is sixSire, ra sixSiriTac gvxvdeba es qula.
• sixSireTa simetriuli ganawileba (Symmetrical frecuency distribution): _ sixSireTa ganawileba, romlis Suawertilze orad gadakecvis SemTxvevaSi miiReba erTnairi formis ori naxevari.
• sixSireTa poligoni (Frequency polygon): _ grafiki, romlis abscisebia intervalTa Suawertilebi, ordinatebi ki _ am intervalTa Sesabamisi sixSireebi.
• sixSiris procenti (%ƒ) (Relative frequency): _ qulis fardobiTi sixSire gamravlebuli 100-ze.
• fardobiTi sixSire (rƒ) (Relative frequency): _ qulis sixSire gayofili miRebul qulaTa saerTo raodenobaze.
• foTlebiani Reroebis diagrama (Stem-and-leaf display): _ monacemTa gamosaxvis forma, sadac qulis pirveli (an pirveli da meore) cifri aris Rero, meore (an bolo) ki _ foToli.
• qulis dagrovili sixSire (Cumulative frecuency of a score): _ qulis Sexvedris sixSires damatebuli yvela ufro naklebi qulis sixSireTa jami.
• qulis procentuli rangi (Percentile rank of a score): _ im qulaTa procenti ganawilebaSi, romlebic tolia an naklebia am qulaze.
• histograma (Histogram): _ svetebiani grafikis nairsaxeoba, sadac intervalebSi qulaTa Sexvedris sixSire gamosaxulia svetis simaRliT, TiToeuli intervalis sigrZe ki _ abscisaze svetis siganiT.
• ƒ: _ sixSireTa ganawilebaSi TiToeuli qulis Sexvedris martivi sixSire.
• i _ intervalis zoma dajgufebul sixSireTa ganawilebaSi.
• N: _ jgufSi miRebuli qulebis saerTo raodenoba
• X _ simbolo, romelic calkeuli, individualuri qulis aRsaniSnad gamoiyeneba.
ტერმინები 5
თავი IV • bimodaluri (Bimodal): _ ganawileba, romelsac aqvs ori moda.
• kvadratebis jami (SS) (Sum of squares): _ ricxvi, romelic miiReba ganawilebaSi Semavali TiTouli qulidan ganawilebis saSulos gamoklebiT, TiToeuli sxvaobis kvadratSi ayvaniTa da miRebuli sidideebis SejamebiT.
• mediana (Median, Mdn): _ qula ganawilebaSi, romlis zemoT da qvemoT ganlagebulia qulaTa toli raodenoba, mediana aris 50-e procentili ganawilebaSi.
• moda (Mode): _ qula, romelic yvelaze xSirad gvxvdeba ganawilebaSi.
• multimodaluri (Multimodal): _ ganawileba, romelsac aqvs orze meti moda.
• populaciis saSualo (Population mean – µ) _ populaciaSi yvela SemTxvevis jamis fardoba qulebis saerTo raodenobasTan.
• unimodaluri (Unimodal): _ ganawileba, romelsac aqvs erTi moda.
• SerCevis saSualo ( X ) (Sample mean): _ miRebul qulaTa jami gayofili qulaTa mTlian raodenobaze.
• centraluri tendenciis sazomebi (Measures of central tendency): _ ricxvebi, romlebic warmoadgenen SerCevidan miRebul saSualo an tipur qulas.
• cvalebadobis sazomi (measures of central tendency) _ ricxvi, romelic miuTiTebs, Tu ramdenad gansxvavdebian qulebi erTmaneTisgan da ramdenad gansxvavdeba maTgan scentraluri tendenciis sazomi.
• M _ simbolo, romelic amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvaneloSi (1994) miTiTebuli normebiT Sesrulebul naSromSi gamoiyeneba SerCevis saSualos aRsaniSnad.
თავი V • diapazoni (Range): _ sxvaoba ganawilebis maqsimalur da minimalur qulebs Soris.
• kvartilTSorisi gabnevis diapazoni (Interquartile range, IQR): _ ganawilebaSi Semaval qulaTa SuaTana 50%.
• kvartilTSorisi gabnevis diapazonis naxevari (Semi-interquartile range, SIQR) _ kvartilTSorisi gablevis diapazonis naxevari.
• populaciis dispersia (Population variance, σ2) _ populaciis yvela qulis gazomviT miRebuli dispersia.
• populaciis dispersiis Sefaseba (Estimated population variance, s2): _ cvalebadobis sazomi, romelic miiReba ganawilebaSi Semavali ToToeuli qulisa da am ganawilebis saSualos sxvaobebis kvadratSi ayvaniT, miRebuli sidideebis dajamebiTa da miRebuli jamis gayofiT ganawilebaSi Semavali qulebis raodenobaze erTiT nakleb ricxvze.
• populaciis standartuli gadaxra (σ) _ kvadratuli fesvi populaciis dispersiidan.
• populaciis standartuli gadaxris Sefaseba (Estimated populaation standard deviation) _ kvadratuli fesvi populaciis dispersiis Sefasebidan.
• SerCevis dispersia (Sample variance, S2): _ cvalebadobis sazomi, romelic miiReba ganawilebaSi Semavali ToToeuli qulisa da am ganawilebis saSualos sxvaobebis
ტერმინები 6
kvadratSi ayvaniT, miRebuli sidideebis dajamebiTa da miRebuli jamis gayofiT ganawilebaSi Semavali qulebis raodenobaze.
• SerCevis standartuli gadaxra (Sample standard deviation, S): _ kvadratuli fesvi SerCevis dispersiidan.
• cvalebadoba (Variability): _ ramdenad gansxvavdebian qulebi erTmaneTisgan da ganawilebis centraluri tendenciis sazomisgan.
• X75 _ ganawilebis 75-e procentilis Sesabamisi qula.
• X25 _ ganawilebis 25-e procentilis Sesabamisi qula.
თავი VI • albaTuri Teoriuli ganawileba (Theoretical probability distribution) _ Teoriuli
albaTobis modelis gamoyenebiT miRebuli albaTuri ganawileba.
• albaTuri klasikuri ganawileba (Empirical probability distribution) _ xdomilobis realurad ganxorcielebis sixSiris daTvliT miRebuli albaTuri ganawileba.
• diskretuli Sedegebi (discrete outcomes) _ mniSvnelobebaTa sasruli raodenobis mqone Sedegebi ganawilebaSi.
• normaluri ganawileba (Normal distribution): _ Teoriuli maTematikuri ganawileba, romelic aRwers populaciis qulebis fardobiT sixSireebs.
• standartuli normaluri gadaxra (Standard normal deviate): _ normaluri ganawileba, romlis μ = 0, σ = 1.
• standartuli normaluri ganawileba (Standard normal distribution): _ normaluri ganawileba, romlis μ = 0, σ = 1.
• standartuli qula (Standard score): _ qula, romelic miiReba formuliT ( ) SXXz /−= .
• urTierTgamomricxavi Sedegebi (Mutually exclusive outcomes) _ Sedegebi, romelic ver ganxorcieldeba erTdroulad.
• Sedegi (outcome) _ albaTobis ganawilebaSi TiToeuli SesaZlo varianti.
• xdomiloba (event) _ albaTobis ganawilebaSi Sedegebis konkretuli simravlis ganxorcieleba.
• xdomilobis ganxorcielebis albaToba (probability of occurrence of en event – p) _
raodenoba saereTo
Sedegebis SesaZlo
raodeoba Sedegebis
iSemadgenel xdomilobis
xdomiloba =)(p
თავი VII • martivi SemTxveviTi SerCeva (Simple random sampling): _ populaciidan wevrebis SerCevis
iseTi procesi, rodesac populaciis TiToeul wevrs SerCevaSi moxvedris toli Sansebi aqvT da erTi wevris arCeva populaciis nebismieri sxva wevris arCevisgan damoukideblad xdeba.
• ndobis intervali (Confidence interval): _ qulis mniSvnelobaTa diapazoni, romelic, savaraudod, moicavs μ-is mniSvnelobas ndobis garkveul doneze.
ტერმინები 7
• ndobis sazRvrebi (Confidence limits): _ ndobis intervalis SemomsazRvreli qveda da zeda qulebi.
• saSualos standartuli Secdoma, Xσ (Standard error of the mean): _ saSualos SerCeviTi
ganawilebis standartuli gadaxra, romelic miiReba σ-as SerCevis moculobis kvadratul fesvTan SefardebiT.
• saSualos standartuli Secdomis Sefaseba, Xs (Estimated standard error of the mean) _ s-is gamoyenebiT σ-as, saSualos standatruli Secdomis Sefaseba.
• saSualos SerCeviTi ganawileba (Sampling distribution of the mean): _ X -is mniSvnelobebis ganawileba, rodesac populaciidan aRebulia N moculobis mqone yvela SesaZlo SerCeva.
• statistikuri daskvna (Statistical inference) _ SerCevisgan miRebuli statistikis mixedviT populaciis Sefaseba.
• stratificirebuli SemTxveviTi SerCeva (Stratified random sampling): _ SerCevis meTodi, rodesac populaciis wevrebi erTiandeba homogenur qvejgufebSi, romelTac stratebi ewodeba. populaciis wevrebi SemTxveviT airCevian stratidan im proporciiT, romliTac es strata populaciaSi gvxvdeba.
• SerCeviTi ganawileba (Sampling distribution): _ populaciidan aRebuli N moculobis mqone yvela SesaZlo SerCevidan miRebuli statistikis mniSvnelobebis albaTuri Teoriuli ganawileba.
• SerCevis Secdoma (Sampling error): _ sidide, romliTac konkretuli SerCevis saSualo gansxvavdeba populaciis saSualosgan.
• Caunacvlebeli Sefaseba (Unbiased estimation): _ statistika, romlis mniSvnelobaTa saSualo, gamoTvlili usasrulod bevri SerCevidan, Sesafasebeli parametris mniSvnelobis tolia.
• centraluri zRvris Teorema (Central limit theorem): _ maTematikuri Teorema, romlis Tanaxmad, SerCevis moculobis zrdasTan erTad saSualos SerCeviTi ganawileba normalur ganawilebas uaxlovdeba.
• Zalmosili Sefaseba (Consistent estimation): _ statistika, romlisTvisac albaToba imisa, rom es statistika ufro axlos iqneba Sesafasebel parametrTan, izrdeba SerCevis moculobis zrdasTan erTad.
• wertilovani Sefaseba (Point estimation): _ parametris mniSvnelobis Sefaseba statistikis mniSvnelobiT (anu erTi ricxviT).
• Xµ _ saSualos Teoriuli SerCeviTi ganawilebis saSualo.
• XX _ saSualos empiriuli SerCeviTi ganawilebis saSualo.
ტერმინები 8
VIII Tavi. statistikuri hipoTezebis Semowmebis Sesavali: z kriteriumi da erTi SerCevis t kriteriumi
parametruli kriteriumi (parametric test) - statistikur kriteriums ewodeba parametruli, Tu hipoTezaSi aris gamoTqmuli varaudi populaciis parametris Sesaxeb.
araparametruli kriteriumi (nonparametric test) - statistikur kriteriums
ewodeba araparametruli, Tu hipoTezaSi ar aris gamoTqmuli varaudi populaciis parametris Sesaxeb.
statistikuri hipoTeza (statistical hypothesis) _ debuleba populaciis
parametris Sesaxeb(parametruli kriteriumis Sesaxeb). nulovani hipoTeza (null hypothesis) - aris im pirobis Camoyalibeba, romelic
mecniers populaciis Sesaxeb apriorulad WeSmaritad miaCnia. es aris hipoTeza, romelic mowmdeba statistikuri kriteriumiT.
statistikuri alternatiuli hipoTeza (statistical alternative hypothesis) _ aris
debuleba imis Sesaxeb, Tu ra unda iyos WeSmariti, Tu nulovani hipoTeza mcdaria.
kriteriumis statistika (test statistic) aris ricxvi, romelic nulovani
hipoTezis Semowmebis saSualebas gvaZlevs da gamoiTvleba SerCevis monacemebze dayrdnobiT.
alfa(α): mniSvnelovnobis donis sidide (albaToba). uaryofis are (rejection region) - kriteriumis statistikis (SerCeviTi
ganawilebis) is mniSvnelobebi, romelTa albaToba tolia an naklebia α-ze nulovani hipoTezis WeSmaritebis pirobebSi. Tuki kriteriumis statistika moxvdeba uaryofis areSi, maSin H0 uaryofili iqneba.
ormxrivi kriteriumi (two-tailed test) - statistikuri kriteriumi, roca
uaryofis aris asagebad gamoyenebulia kriteriumis statistikis SerCevis ganawilebis orive kudi.
calmxrivi kriteriumi (one-tailed test)- statistikuri kriteriumi, roca
uaryofis aris asagebad gamoyenebulia kriteriumis statistikis SerCevis ganawilebis mxolod erTi kudi.
kritikuli mniSvnelobebi (critical value) - konkretuli ricxviTi
mniSvnelobebi, romlebic adgenen uaryofis aris sazRvrebs.
mniSvnelovnobis done (significance level) esaa albaTobis is mniSvneloba, romelic iZleva nulovani hipoTezis uaryofis pirobas statistikur kriteriumSi.
ტერმინები 9
statistikurad mniSvnelovani gansxvaveba (statistically significant difference) -
kriteriumis statistikis realuri mniSvneloba xvdeba uaryofis areSi da H0 uaryofilia.
aramniSvnelovani gansxvaveba: (nonsignificant difference) - kriteriumis
statistikis realuri mniSvneloba ar xvdeba uaryofis areSi da nulovani hipoTeza ar aris uaryofili.
erTi SerCevis t kriteriumi (one-sample t test): t kriteriumi, romelic
gamoiyeneba SerCevis saSualosa da populaciis hipoTetur saSualos Soris gansxvavebis statistikuri mniSvnelovnobis Sesamowmeblad, roca
σ X Sefasebulia s X -is saSualebiT. Tavisuflebis xarisxi (degrees of freedom): qulaTa raodenoba, romelic
Tavisufalia cvalebadobis dros statistikis gamoTvlisas. pirveli gvaris Secdoma (Type I error) - Secdoma statistikuri
gadawyvetilebis miRebis dros, romelic gamoixateba nulovani hipoTezis uaryofaSi, maSin roca is realurad WeSmaritia.
meore gvaris Secdoma (Type II error) - Secdoma statistikuri
gadawyvetilebis miRebis dros, romelic gamoixateba imiT, rom ar xdeba nulovani hipoTezis uaryofa, maSin roca is realurad mcdaria da alternatiuli H1 hipoTeza ki WeSmariti.
beta (β): meore gvaris Secdomis albaToba. simZlavre (Power) – H0 uaryofis albaToba, roca H0 ar aris samarTliani
da H1 ki aris WeSmariti. statistikuri kriteriumis simZlavre mocemulia
1-β-s saSualebiT.
IXFTavi. eqsperimentis safuZvlebi da saSualoTa Soris gansxvavebis Semowmeba
ტერმინები 10
cdis pirTa Soris sqema (Between–subject design) – eqsperimenti, romelSic
ori an meti jgufi.
SemTxveviTi SerCeva (Random sampling) - SerCevis meTodi, romelSic individebi iseTnairad arian arCeulni, rom populaciis TiToeul wevrs aqvs am SerCevaSi moxvedris erTnairi Sansi da erTi wevris SerCeva ar aris damokidebuli populaciis nebismieri sxva wevris SerCevaze.
xelmisawvdomi SerCeva (Convenience sampling): cdis pirebis iseTi SerCeva, romelic mkvlevrisaTvis ufro xelmisawvdomi an xelsayrelia.
ekvivalenturi jgufebi (Equivalent groups): cdis pirTa jgufebi, romlebic, savaraudod, ar gansxvavdebian erTmaneTisgan raime mudmivi an sistematuri maxasiaTebliT eqsperimentis damoukidebeli cvladis miwodebamde.
SemTxveviTi ganawileba (Random assignment): cdis pirTa jgufebSi
ganawilebis meTodi, romlis mixedviT, eqsperimentisaTvis SerCeul nebismier individs aqvs nebismier jgufSi ganawilebis toli albaToba da erTi cdis piris ganawileba jgufSi ar moqmedebs nebismieri sxva individis ganawilebaze imave jgufSi.
damoukidebeli cvladi ( Independent variable) _ cvladi, romliTac manipulireben eqsperimentSi, damokidebul cvladze misi gavlenis gansazRvris mizniT.
damoukidebeli cvladis done (Level of an independent variable) _
damoukidebeli cvladis erTi mniSvneloba. imisaTvis, rom cvladi gaxdes, damoukidebeli cvlads unda hqondes sul cota ori gansxvavebuli done.
faqtori (Factor) _ damoukidebeli cvladis alternatiuli saxelwodeba.
cdis pirTa Soris erTfaqtoriani sqema (One-factor between subject design): kvlevis sqema, romelSic erTi damoukidebeli cvladis manipulacia xdeba da iqmneba ori an meti jgufi.
damatebiTi cvladebi (Extraneous variables) - damoukidebeli cvladisagan gansxvavebuli nebismieri cvladi, romelsac SeuZlia imoqmedos damokidebul cvladze eqsperimentSi.
Sereuli eqsperimenti (Confounded experiment) aris eqsperimenti, romelSic damatebiTi cvladi sistematurad icvleba damoukidebel cvladTan erTad.
placebos kontroli (Placebo control) - simulirebuli eqsperimentuli
(Carevis) piroba.
damokidebuli cvladi (Dependent variable): aris cvladi eqsperimentSi, romelic damokidebulia damoukidebel cvladze.
saSualoTa Soris sxvaobis SerCeviTi ganawileba (Sampling distribution of the difference between means): SerCevis saSualoebs Soris sxvaobis ganawileba,
ტერმინები 11
roca populaciidan amoRebulia n moculobis SerCevis yvela SesaZlo wyvili.
saSualoTa Soris sxvaobis standartuli Secdoma (Standard eror of the difference berween means): 21 AA XX − mniSvnelobaTa SerCeviTi Teoriuli ganawilebis standartuli gadaxra.
saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba (Estimated standard error of the differnce between means): saSualoTa Soris sxvaobis
standartuli Secdomis Sefaseba miiReba s2-is gamoyenebiT σ2-is SefasebisaTvis.
mdgradoba (Robustness): es termini gamoiyeneba imis aRsaniSnavad, rom statistikuri kriteriumis daSvebebis darRvevas aqvs mcire gavlena pirveli gvaris Secdomaze.
efeqtis siZliere (Strength of effect): damoukidebeli cvladis siZliere
gazomili statistikis gavlenis siZlieriT. cdis pirTa Sida sqema (Within-subject design): kvlevis sqema, romelSic cdis
pirTa erTi jgufia da izomeba damoukidebeli cvladis TiTeuli donis mixedviT. cdis pirTa Sida sqemaSi TiToeul cdis pirs miewodeba eqsperimentis orive piroba.
X Tavi. erTfaqtoriani dispersiuli analizi cdis pirTa Soris
sqemisaTvis
erTfaqtoriani mravaldoniani sqema (One-factor multilevel design): eqsperimenti erTi damoukidebeli cvladiT da am damoukidebeli cvladis sami an meti doniT.
dispersiuli analizi (Analysis of variance ANOVA): statistikuri kriteriumi,
romelic gamoiyeneba mravaldoniani sqemebis analizisaTvis. sruli saSualo (Grand mean): yvela qulis saSualo eqsperimentSi.
kvadratebis saSualo (mean square): dispersiis saxelwodeba, romelic
gamoiyeneba dispersiul analizSi.
ტერმინები 12
jgufSida Secdomis dispersia (Within-groups error variance): monacemTa dispersia jgufSi, romelic jgufis saSualos garSemoa gamoTvlili.
jgufTa Soris dispersia (Between-groups variance): dispersia, romelic
gamoiTvleba sruli saSualos garSemo jgufebis saSualoebis cvalebadobis gamoyenebiT.
a: A faqtoris doneebis raodenoba
nA: qulaTa raodenoba erTfaqtoriani sqemis erT doneze
mravaljeradi Sedarebis kriteriumebi (Multiple comparison tests):
statistikuri kriteriumebi, romlebic gamoiyeneba wyvilTaSoris SedarebebSi imis dasadgenad, Tu romeli saSuloebi gansxvavdeba mniSvnelovnad erTmaneTisagan erTfaqtorian mravaldonian sqemaSi.
wyvilTaSoris Sedarebebi (Pairwise comparisons): statistikuri Sedarebebi, romlebic erTmaneTs adareben or saSualos.
Secdomis xarisxi eqsperimentSi (Error rate in an experiment) aris sul cota
erTi pirveli gvaris daSvebis albaToba mainc statistikur SedarebebSi, romelic Catarebulia eqsperimentSi.
saboloo Sedarebebi (Post hoc comparisons): statistikuri kriteriumebi,
romlebic asruleben yvela SesaZlo wyvilTa Sedarebebs, mas Semdeg, rac statistikurad mniSvnelovani Frealuri dadasturdeba sruli dispersiuli analizisaTvis.
kritikuli gansxvaveba (CD Critical difference): aris or saSualos Soris
raodenobrivi gansxvaveba, romelic statistikurad mniSvnelovania.
ტერმინები 13
terminebi (Tavebi 11, 12)
A faqtoris mTavari efeqtis saSualo (Main effect of factor A): gansxvavebaXA1-sa
daXA2-s Soris, gamoxatuliXA1-XA2. artefaqturi mTavari efeqti (Artifactual main effect): mTavari efeqti, romelic ar iZleva mniSvnelobis mqone informacias damoukidebeli cvladis efeqtis Sesaxeb faqtorul sqemaSi.
B faqtoris mTavari efeqtis saSualo (Main effect of factor B): gansxvavebaXB1-sa
daXB2-s Soris, gamoxatuliXB1-XB2. gavarjiSebis efeqti (Practice effect): Carevis mravaljeradi efeqti, romelic xdeba imis gamo, rom cdis pirebi SeiZleba gavarjiSdnen an daiRalnen eqsperimentis davalebis Sesrulebisas. damoukidebeli cvladis martivi efeqti (Simple effect of an independent variable): faqtorul sqemaSi erTi damoukidebeli cvladis efeqti meore damoukidebeli cvladis mxolod erTi donis dros. mTavari efeqtis saSualo (Main effect mean): faqtorul analizSi damoukidebeli cvladis erTi donisaTvis yvela cdis piris saSualo, meore damoukidebeli cvladis doneebis mixedviT klasifikaciis ignorirebiT. pirobebisa da cdis pirTa urTierTqmedeba (Treatment-by-subject interaction): situacia cdis pirTa Sida erTfaqtorian sqemaSi, rodesac damoukidebeli cvladis efeqti damokidebulia cdis pirze. ujredi an Carevis piroba (Cell or treatment condition): faqtorul eqsperimentSi TiToeuli damoukidebeli cvladis erTi donisagan warmoqmnili kombinacia. ujredis saSualo (Cell mean): ujredSi nAB qulebis saSualo faqtorul sqemaSi. urTierTqmedeba (Interaction): situacia faqtorul sqemaSi, rodesac erTi damoukidebeli cvladis efeqti damokidebulia masTan kombinaciaSi myofi meore damoukidebeli cvladis mniSvnelobaze. faqtoruli analizi (Factorial design): kvlevis iseTi sqema, romelSic erTdroulad icvleba ori an meti damoukidebeli cvladi. cdis pirTa SigniT sqema (Within-subjects design): kvlevis sqema, rodesac cdis pirTa erTi jgufi ganicdis damoukidebeli cvladis zemoqmedebas, rac izomeba am cvladis yoveli donis Sesabamisad. Carevis gadatanis efeqti (Treatment carry-over effect): Carevis mravaljeradi efeqti, romelic xdeba maSin, rodesac damoukidebeli cvladis erTi donis erTi efeqti gadaitaneba, gavlenas axdens Sesrulebaze damoukidebeli cvladis meore doneze.
ტერმინები 14
Carevis mravaljeradi efeqti (Multiple treatment effects): cdis pirTa Soris sqemaSi Sesrulebis cvlilebebi, romlebic xdeba imis gamo, rom mowmdeba damoukidebeli cvladis yovel doneze.
ტერმინები 15
terminebi Tavebi 13-15
Tavi 13
korelaciuri kvlevebi (Correlational Studies) 367: kvlevebi, romlebSic izomeba ori an meti cvladi maTi korelaciis mimarTulebisa da xarisxis dasadgenad. organzomilebiani ganawileba (Bivariate Distribution) 367: ganawileba, romelSic TiToeuli cdis pirisagan miiReba ori monacemi (qula).
wertilTa gabnevis diagrama (Scatterplot) 368: organzomilebiani ganawilebis grafiki, romelSic X cvladi gadazomilia hirizontalur RerZze da Y cvladi gadazomilia vertikalur RerZze. korelacia (Covary) 369: ori cvladi korelirebs, rodesac erTi cvladis cvlileba dakavSirebulia meore cvladis SeTanxmebul cvlilebasTan dadebiTi kavSiri (Positive relationship) 369: or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad izrdeba meore cvladis mniSvnelobac. wrfivi kavSiri (Linear relationship) 369: or cvlads Soris kavSiri, romelic SeiZleba aRiweros wrfiT. uaryofiTi kavSiri (Negative relationship) 370: or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad meore cvladis mniSvneloba mcirdeba. pirsonis korelaciis koeficienti (Pearson correlation coefficient) 374: statistika, romelic aRiniSneba r -iT da asaxavs intervalebis an Sefardebis skalaze or cvlads Soris wrfivi kavSiris siZlieres. X -isa da Y -is jvaredina namravlebi (Cross products of X and Y ) 375: ( )( )∑ −− YYXX
mniSvneloba X da Y cvladebisTvis determinaciis koeficienti (Coefficient of determination) 388: 2r -is mniSvneloba, romelic X da Y cvladebis saerTo cvalebadobas asaxavs. ρ (390): ro, populaciis korelaciis koeficientis aRmniSvneli simbolo. organzomilebiani normaluri ganawileba (Bivariate normal distribuition) 392: X da Y cvladebis iseTi ganawileba, romlisTvisac (1) X qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (2) Y qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (3) TiToeuli X qulisaTvis Y qulebis ganawileba populaciaSi normaluria da (4) TiToeuli Y qulisaTvis X qulebis ganawileba populaciaSi normaluria.
ტერმინები 16
spirmenis ranguli korelaciis koeficienti ( sr ) (Spearman rank-order correlation coefficient) 396: korelaciis koeficienti, romelic rigis skalaze gamoiyeneba.
Tavi 14
wrfivi kavSiri (Linear Relation) 408: kavSiri or cvlads Soris, romlis drosac
X cvladis yoveli erTi erTeuliT cvlilebisas, Y cvladi mudmivi sididiT
icvleba. wrfis gantoleba (General Equation of a straight line) 410: abXY += wrfis daxriloba (Slope of a straight line) 410: wrfis daxriloba mocemulia tolobiT
cvlileba ismniSvnelob is
cvlileba ismniSvnelobis
−−
=XY
b
Y TanakveTa (Y- intercept) 412: abXY += gantolebaSi Y-is mniSvneloba, roca X = 0.
Y ′ (416): wrfis gantolebis meSveobiT, X –is mixedviT gamoTvlili Y –is mniSvneloba. umcires kavadratTa meTodiT miRebuli regresiis wrfe (Least-squares regression line) 420: wrfe, romelic minimumamde amcirebs ( )∑ ′− 2YY -is mniSvnelobas.
naSTi (Residual) 425: YY ′− -is mniSvneloba.
naSTiSS (SSResidual) 426: ( )∑ ′− 2YY -is mniSvneloba.
Sefasebis standartuli Secdoma ( XYs ⋅ ) (Standard error of estimate) 426:
( )2
2
−
′−∑wyviliN
YYgamosaxulebis mniSvneloba
mravlobiTi regresia (Multiple regression) 440: Y qulis mniSvnelobebis winaswarmetyveleba X cvladis ramdenime mniSvnelobis gamoyenebiT.
Tavi 15
araparametruli statistikuri kriteriumi (Nonparametric Statistical Tests) 449: statistikur kriteriums ewodeba araparametruli, Tu hipoTezaSi ar aris gamoTqmuli varaudi populaciis parametris Sesaxeb. xi-kvadrat kriteriumi (Chi-square test) 450: statistikuri kriteriumi, romelic gamoiyeneba saxeldebis skalis monacemebis gasaanalizeblad, roca mocemulia sxvadasxva kategoriis Sesabamisi sixSireebi.
ტერმინები 17
damoukideblobis xi-kvadrat kriteriumi (Chi-square test of independence) 450: xi-kvadrati, roca monacemebis kategorizacia ori damoukidebeli niSnis mixedviT xdeba. niSanTa SeuRlebis cxrili (Contingency Table) 451: striqonebiani da svetebiani cxrili, romelSic warmodgenilia yoveli kategoriis sixSireebi mocemuli striqonisa da svetis cvladebis gadakveTis Sedegad. marginaluri (Sejamebuli) sixSireebi (Marginal Frequencies) 452: niSanTa SeuRlebis cxrilSi sixSireebis jami striqonis an svetis mixedviT. mosalodneli siSireebi (Expected Frequencies) 452: niSanTa SeuRlebis cxrilidan miRebuli sixSireebi im daSvebis safuZvelze, rom cxrilSi warmodgenili ori ganzomileba damoukidebelia.
Tanxmobis χ² kriteriumi (Chi-square test of goodness of fit) 458: xi-kvadrati, roca monacemebis kategorizacia xdeba mxolod erTi ganzomilebis mixedviT man-uitnis U kriteriumi (Mann-Whitney U test) 462: araparametruli kriteriumi cdis pirebs Soris sqemisaTvis, roca gamoyenebulia damoukidebeli cvladis ori done da monacemebi sul mcire, rigis skalazea. uilkoqsonis niSniani rangebis kriteriumi (Wilcoxon signed-ranks test) 473: araparametruli kriteriumi cdis pirebs Soris sqemisaTvis damoukidebeli cvladis ori doniT.
danarTi D statistikuri simboloebi A A faqtori, damoukidebeli cvladi eqsperimentSi. A1, A2, A3 A faqtoris doneebi erTfaqtorian an orfaqtorian sqemaSi. α alfa, pirveli gvaris Secdomis albaToba. A x B A da B faqtorebis urTierTqmedebis aRniSvna faqtorul
sqemaSi. a (1) A faqtoris doneebis raodenoba an (2) wrfis gantolebis Y-
TanakveTa regresiul analizSi. B B faqtori, meore damoukidebeli cvladi faqtorul sqemaSi. B1, B2, B3 B faqtoris doneebi faqtorul sqemaSi. b (1) B faqtoris doneebis raodenoba an (2) wrfis daxriloba
regresiul analizSi.
β beta, meore gvaris Secdomis albaToba.
χ2 xi-kvadrat statistika.
χ2kritikuli χ2-is kritikuli mniSvneloba.
χ2realuri monacemebidan miRebuli χ2-is mniSvneloba. CD kritikuli sxvaoba mravaljeradi Sedarebis kriteriumSi. CPXY X-isa da Y-is jvaredini namravli korelaciur kvlevaSi. cum f an cf qulis dagrovili sixSire. cfL qulebis dagrovili sixSire intervalis qveda realur
sazRvramde sixSireTa dajgufebul ganawilebaSi. cum rf an crf qulis dagrovili fardobiTi sixSire. cum %f an c%f procentebSi gamoxatuli qulis dagrovili sixSiris
ganawileba. D (1) ranJirebuli qulebis wyvilis sxvaoba individisaTvis;
gamoiyeneba spirmenis ranguli korelaciis koeficientSi an (2) cdis pirTa Sida sqemaSi sxvaoba cdis pirisagan miRebul or qulas Soris.
df Tavisuflebis xarisxebi. dispersiul analizSi df-s Cveulebriv indeqsebi moyveba, rogoricaa dfA, dfB, dfAxB, dfSecdoma an dfsruli, raTa miuTiTos, cvalebadobis romel wyaros Seesabameba Tavisuflebis es xarisxi.
η2 eta kvadrati. Erc qulis mosalodneli sixSire xi-kvadrat kriteriumis niSanTa
SeuRlebis cxrilis r striqonsa da c svetSi. f qulis sixSire. fi qulebis sixSire intervalSi sixSireTa dajgufebul
ganawilebaSi. F F kriteriumi dispersiul analizSi. Fkritikuli F-is kritikuli mniSvneloba. Frealuri monacemebidan miRebuli F-is mniSvneloba. H0 nulovani statistikuri hipoTeza. H1 alternatiuli statistikuri hipoTeza.
IQR kvartilTSorisi gabnevis diapazoni i intervalis klasis zoma an sigane sixSireTa dajgufebul
ganawilebaSi. M SerCevis saSualo. simbolo gamoiyeneba publikaciebSi,
romlebic misdeven amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils.
Mdn mediana. MS kvadratis saSualo, dispersiis Sefaseba dispersiul
analizSi. gadaxrebis kvadratebis saSualoebs, rogorc wesi, mohyveba indeqsebi, rogoricaa MSA, MSB, MSAxB, an MSSecdoma, raTa aRiniSnos, cvalebadobis romel wyaros asaxaven isini.
MSE Secdomis kvadratis saSualo. es simbolo gamoiyeneba dispersiul analizSi MSSecdomis aRsaniSnavad iseT publikaciebSi, romlebic misdeven amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils.
µ miu, populaciis saSualo. Xµ saSualos Teoriuli SerCeviTi ganawilebis saSualo.
N SerCevaSi qulebis sruli raodenoba, an eqsperimentSi qulebis sruli raodenoba.
N wyvilebi qulaTa wyvilebis raodenoba.
n qulebis raodenoba SerCevis qvejgufSi.
n A qulebis raodenoba erTi donisaTvis erTfaqtorian sqemaSi.
n AB orfaqtoriani sqemis ujredSi qulebis raodenoba.
rcO xi-kvadrat kriteriumis niSanTa SeuRlebis cxrilSi qulis
realuri sixSire r striqonsa da c svetSi. %f qulis sixSire procentebSi. P proporciiT gamoxatuli procentilis wertili. gamoiyeneba
qulis misaRebad procentilis konkretul wertilze.
xP X-is qulis procentiluri rangi.
p albaToba. q tukis HSD kriteriumSi gamoyenebuli studentis rangobrivi
statistika. r pirsonis korelaciis koeficienti SerCevisaTvis.
kritikulir pirsonis korelaciis koeficientis kritikuli mniSvneloba.
realurir X da Y cvladebs Soris pirsonis korelaciis koeficientis
miRebuli mniSvneloba. rf qulis fardobiTi sixSire. ρ ro, populaciis korelaciis koeficienti.
Sρ populaciis spirmenis ranguli korelaciis koeficienti.
Sr spirmenis ranguli korelaciis koeficienti. r 2 determinaciis koeficienti. σ sigma, populaciis standartuli gadaxra.
σ 2 populaciis dispersia.
Xσ populaciis saSualos standartuli Secdoma.
21 AXAX −σ saSualoebs Soris sxvaobis standartuli Secdoma.
∑=
N
i 1
i=1-dan N-mde Sejamebis aRniSvna.
S SerCevis standartuli gadaxra. s populaciis standartuli gadaxris Sefaseba. SD standartuli gadaxra. simbolo gamoiyeneba publikaciebSi,
romlebic misdeven amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils.
2S SerCevis dispersia. 2s populaciis dispersiis Sefaseba. 2
ulikombinirebs populaciis or saSualos Soris kombinirebuli dispersiis
Sefaseba.
Xs saSualos standartuli Secdomis Sefaseba.
SE saSualos standartuli Secdoma. simbolo gamoiyeneba publikaciebSi, romlebic misdeven amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils.
21 AXAXs−
saSualoebs Soris sxvaobis standartuli Secdomis Sefaseba.
YXs • standartuli Secdomis Sefaseba X-dan Y-is winaswarmetyvelebisas.
SIQR semikvartilTSorisi gabnevis diapazoni.
∑R rangebis jami. SS kvadratebis jami. dispersiul analizSi SS-s Cveulebriv
moyveba indeqsebi, rogoricaa SSA, SSB, SSAxB, SSSecdoma da SSsruli, raTa aRiniSnos, cvalebadobis romel wyaros asaxavs is.
SSnaSTi ∑ ′− 2)( YY mniSvneloba.
XSS korelaciaSi X cvladis kvadratebis jami.
YSS korelaciaSi Y cvladis kvadratebis jami.
t t statistika t kriteriumSi. kritikulit t-s kritikuli mniSvneloba.
elidamoukidebt damoukidebeli jgufebis t kriteriumisaTvis miRebuli t statistika.
realurit monacemebidan miRebuli t-s mniSvneloba.
lidamokidebut damokidebuli jgufebis t kriteriumisaTvis miRebuli t statistika.
T uilkoqsonis T kriteriumi.
kritikuliT T-s kritikuli mniSvneloba.
realuriT monacemebidan miRebuli T-s mniSvneloba.
U man-uitnis U kriteriumi. kritikuliU U-s kritikuli mniSvneloba.
realuriU monacemebidan miRebuli U-s ufro mcire mniSvneloba. X cdis piris qula im cvladis mixedviT, romelic
gansazRvrulia, rogorc X cvladi. gamoyenebuli sqemis mixedviT, qula SeiZleba aRniSnuli iyos Xi, Xij an Xijk.
imaqsimalurX yvelaze didi qula qulaTa ganawilebaSi.
minimaluriX yvelaze mcire qula qulaTa ganawilebaSi.
LX sixSireTa ganawilebaSi im intervalis dabali realuri sazRvari, romelic Seicavs X qulas.
PX ganawilebaSi procentilis P wertilze mdebare qula.
25X qula, romelic Seesabameba ganawilebis 25-e procentils.
50X qula, romelic Seesabameba ganawilebis medianas.
75X qula, romelic Seesabameba ganawilebis 75-e procentils.
X SerCevis saSualo.
1AX , 2AX ,
3AX A faqtoris doneebis mTavari efeqtebi. 1BX , 2BX ,
3BX B faqtoris doneebis mTavari efeqtebi. ABX ujredis saSualo orfaqtorian sqemaSi.
GX sruli saSualo.
SX cdis piris saSualo cdis pirTa Sida sqemaSi.
XX saSualoebis empiriuli SerCeviTi ganawilebis saSualo.
21 AXAXX − saSualoebs Soris sxvaobis empiriuli SerCeviTi ganawilebis saSualo.
Y cdis piris qula im cvladis mixedviT, romelic gansazRvrulia, rogorc Y cvladi.
Y SerCevis saSualo qulebisaTvis, romlebic gansazRvrulia, rogorc Y cvladi.
lY Y-is nawinaswarmetyvelebi mniSvneloba wrfiul regresiul analizSi.
z z gardaqmnis gamoyenebiT miRebuli qulis mniSvneloba.
danarTi E amocanebis pasuxebi aq mocemulia seqciebSi “SeamowmeT Tqveni codna” da “kiTxvebi” moyvanili amocanebis amoxsnebis umetesoba. formulebisa da axsnebisaTvis ixileT Sesabamisi seqciebi teqstSi. Tavi I SeamowmeT Tqveni codna I.1 2. b. SerCevis statistikis safuZvelze (52%) populaciis parametris (52%) Sesaxeb. SeamowmeT Tqveni codna I.2 2. a. micemuli ukukavSiris tipi. b. fexebis muxlebSi moxris xangrZlivoba.
g. kvleva moicavda ori ekvivalenturi jgufis Seqmnas cdis pirTa Soris sqemiT da damoukidebeli cvladis manipulirebas. d. imisaTvis, rom ganisazRvros, SemTxveviTze metia Tu ara fexebis moxris xangrZlivobebis gasxvaveba.
SeamowmeT Tqveni codna I.3 2. a. imis dadgena, aris Tu ara kavSirSi paranormaluri movlenebis
rwmena da arCeuli samecniero kursebis raodenoba. b. paranormaluri movlenebis rwmena; arCeuli samecniero kursebis
raodenoba. kiTxvebi 3. a. korelacia da regresia.
g. mrisxanebis done, sisxlis wneva. 4. a. aRwera da statistikuri hipoTezebis Semowmeba.
b. bavSvisaTvis daniSnuli mkurnalobis tipi. g. bavSvis sxeulis temperatura.
5. a. aRwera. b. procenti.
6. a. nedli monacemebi. b. daskvna. b. statistika. b. nedli monacemebi. b. daskvna. b. statistika.
Tavi II SeamowmeT Tqveni codna II.1
2. a. gamokiTxva. b. eqsperimenti.
g. saarqivo Canawerebi. d. bunebrivi dakvirveba.
SeamowmeT Tqveni codna II.2 3. adamianebi nawildebian imave kategoriaSi, Tumca maT ar aqvT
gazomvis qveS myofi cvladis igive raodenoba. 5. b. ara. SeamowmeT Tqveni codna II.3 2. a. diax, es intervalebis skalaa. b. ara.
g. ara. 3. a. diax, es Sefardebis skalaa. b. ki.
g. ki. SeamowmeT Tqveni codna II.4 2. a. uwyveti. b. uwyveti.
g. wyvetili. d. uwyveti. e. uwyveti. v. wyvetili. z. wyvetili. T wyvetili.
3. a. 152.35-dan 152.45-is CaTvliT sm. b. 17.25-dan 17.35- is CaTvliT wm.
g. 3.5-dan 4.5- is CaTvliT. d. 0.4365-dan 0.4375 is CaTvliT wm. e. 6.85-dan 6.95 is CaTvliT mm. v. 67.805-dan 67.815 is CaTvliT girvanqa.
4. a. 27.44. b. 119.028.
g. 1.44. d. 1263.7592. e. 0.35. v. 13.27.
kiTxvebi 1. a. gamokiTxva. b. saarqivo Canawerebi.
g. eqsperimenti. d. bunebrivi dakvirveba.
3. a. profesiul miRwevaze orientacia. b. rigis. 4. a. sxeulis wona. b. Sefardebis.
g. 134.55-dan 134.65-is CaTvliT girvanqa. d. 127.6 an 127.65 girvanqa, SeTanxmebis Tanaxmad.
5. a. emociuri stabiloba.
b. Sefasebis skala: rigisa da intervalebis skalebs Soris. g. 7.5-dan 8.5 is CaTvliT. d. ara, is unda damrgvaldes 5.8-mde an 5.83-mde, SeTanxmebis Tanaxmad.
6. a. Sefardebis.
b. 96.5-dan 97.5 is CaTvliT wuTi. g. 82.7 an 82.67 wuTi, SeTanxmebis Tanaxmad.
7. rigis. 8. saxeldebis. 9. a. wyvetili.
b. uwyveti. g. wyvetili. d. uwyveti.
10. a. 141.25-dan 141.35-is CaTvliT girvanqa.
b. 33.5-dan 34.5-is CaTvliT wT. g. 175.55-dan 175.65-is CaTvliT iardi. d. 98.15-dan 98.25-is CaTvliT gradusi farengeitiT. e. 108.5-dam 109.5-is CaTvliT mm vercxliswyali.
11. a. 42.26.
b. 1.16. g. 163.65. d. 27.43. e. 13.90. v. 24.62. z. 19.28.
13. a. rigis.
b. Sefardebis. Tavi III SeamowmeT Tqveni codna III.1
2. a.
b. minimaluri = 3; maqsimaluri = 19. g. 8.
SeamowmeT Tqveni codna III.2 2. 10-dan 20-mde. 3. 5. 4. 11. 6. a. 123, 125.
b. 129, 131. g. 122.5, 125.5. d. 128.5-dan 131.5-mde. e. 121. v. 3.
7. a. i = 10.
intervali
realuri sazRvrebi
intervalis Suawertili
Tvlis Sedegi
f
rf
%f
cf
crf
c%f zeda qveda 200-209 190-199 180-189 170-179 160-169 150-159 140-149 130-139 120-129 110-119
199.5 189.5 179.5 169.5 159.5 149.5 139.5 129.5 119.5 109.5
209.5 199.5 189.5 179.5 169.5 159.5 149.5 139.5 129.5 119.5
204.5 194.5 184.5 174.5 164.5 154.5 144.5 134.5 124.5 114.5
/ / / /
1 1 0 0 0 1 1 0 0 0
.02
.02
.00
.00
.00
.02
.02
.00
.00
.00
2 2 0 0 0 2 2 0 0 0
50 49 48 48 48 48 47 46 46 46
1.00 .98 .96 .96 .96 .96 .94 .92 .92 .92
100 98 96 96 96 96 94 92 92 92
qula Tvlis Sedegi
f rf %f
19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2
/ / / / / / / // //// // // // /
1 0 0 0 1 1 1 1 1 1 2 4 0 2 2 2 1 0
.05
.00
.00
.00
.05
.05
.05
.05
.05
.05
.10 .20 .00 .10 .10 .10 .05 .00
5 0 0 0 5 5 5 5 5 5 10 20 0 10 10 10 5 0
100-109 90-99 80-89 70-79 60-69 50-59 40-49 30-39 20-29 10-19
99.5 89.5 79.5 69.5 59.5 49.5 39.5 29.5 19.5 9.5
109.5 99.5 89.5 79.5 69.5 59.5 49.5 39.5 29.5 19.5
104.5 94.5 84.5 74.5 64.5 54.5 44.5 34.5 24.5 14.5
// /// ////// /////////// //////// ////// /////// // /
0 2 3 6 11 8 6 7 2 1
.00
.04
.06 .12 .22 .16 .12 .14 .04 .02
0 4 6 12 22 16 12 14 4 2
46 46 44 41 35 24 16 10 3 1
.92
.92
.88
.82
.70
.48
.32
.20
.06
.02
92 92 88 82 70 48 32 20 6 2
b. distanciebis umetesoba iyo 99 metri an naklebi. yvelaze xSiri iyo distancia 60-dan 69 metramde. arc erTi distancia ar iyo 10 metrze naklebi an 209 metrze meti. g. P68 = 67. d. X50 = 60.4 metrs.
SeamowmeT Tqveni codna III.3 2. a. suraTebi unda iyos
b. ganawilebebi igivea, rac suraTebze, im gamonaklisiT, rom fardobiTi sixSireebi ordinatazea asaxuli. g. distanciebis umetesoba naklebia, vidre 104.5 metri. yvelaze xSiri iyo distancia intervalSi, romlis Suawertilia 64.5 metri. arc erTi distancia ar iyo 9.5 metrze naklebi an 209.5 metrze meti.
3. Rero foToli 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
5 0 4 0 11 133 022256 11234666778 04456789 237899 1377889 59 8
a. 18 metri. b. 205 metri. g. 66 da 72 metri. SeamowmeT Tqveni codna III.4 2. a. simetriuli, unimodaluri.
b. asimetriuli, dadebiTad asimetriuli, unimodaluri. g. asimetriuli, dadebiTad asimetriuli, bimodaluri. d. asimetriuli, uaryofiTad asimetriuli, unimodaluri. kiTxvebi 1. a. intervalis zoma = 20; intervalebis iwyeba dadgenili
sazRvridan 220, pirveli Suawertili 229.5; bolo intervalis zeda dadgenili sazRvari = 499.
b. 400-dan 419-is CaTvliT; Suawertili = 409.5; ara, is ufro moklea. g. mniSvneloba iqneba intervalSi 340-dan 359-is CaTvliT. d. X50 = 356.2. e. P431 = 89. 2. a. intervalis zoma = 5; intervalebis iwyeba dadgenili sazRvridan
0, pirveli Suawertili 2.0; bolo intervalis zeda dadgenili sazRvari = 64.
b. 0-dan 4-is CaTvliT. g. X50 = 6.2. d. P24 = 86. e.
Rero foToli *
6 . *
5 . *
4 . *
3 . *
2 . *
1 . *
0 .
3 8 1 67 7 66 114 5778 0124 5555566666678899 000000001111223333444444
minimaluri qula = 0, maqsimaluri qula = 63. yvelaze xSiri qula = 0.
4. a. gacdenili dReebis
raodenoba
dajildoebis politikis mqone
skola
dajildoebis politikis
armqone skola 25-ze meti
24-25 22-23 20-21 18-19 16-17 14-15 12-13 10-11 8-9 6-7 4-5 2-3
.02
.00
.02
.00
.01 .00 .02 .00 .03 .08 .04 .13 .23
.03
.03 .005 .04 .00 .02 .11 .05 .09 .10 .015 .07 .17
0-1 .42 .27
b. X50 = 2.2 5.2
g. P24 = 68 46 d. dajildoebis politikis mqone skolis SemTxvevaSi moswavleebi 78% acdens 5 dRes an naklebs, sadac gacdenis medianaa 2.2 dRe. dajildoebis politikis armqone skolis SemTxvevaSi moswavleebis mxolod 51% acdens 5 dRes an naklebs, xolo gacdenis mediana aris 5.2 dRe.
5. b. minimaluri qula = 5; maqsimaluri qula = 29. g. 20. 6. a. i = 30. b. 209.5; 239,5. g. 224.5. e. 340 wuTi. v. P264 = 53. 7. a. 40. g. P78 = 59. d. 79.3 wuTi. Tavi IV SeamowmeT Tqveni codna IV.1 3. a. 49. b. unimodaluri. 4. a. 27. b. unimodaluri. 5. a. 24,27,49. b. multimodaluri. 7. 46. 8. 42. 9. 46. SeamowmeT Tqveni codna IV.2
2. a. X = 88.0. g. 262. 3. a. 41.7. b. 39.5. g. 37.7. SeamowmeT Tqveni codna IV.3 1. a. mediana. b. saSualo. g. moda. d. saSualo. 2. igive, rac mediana. 3. dadebiTad asimetriuli. kiTxvebi 1. SesaZlebelia uaryofiTad asimetriuli, radgan saSualo medianaze
naklebia. 2. dadebiTad asimetriuli, radgan saSualo sagrZnoblad metia medianaze. 3. dadebiTad asimetriuli, radgan saSualo sagrZnoblad metia medianaze. 4. jgufi 1
b. moda = 38. Mdn = 38. X = 38.6. g. daaxloebiT simetriuli; centraluri tendencies yvela sazomi
daaxloebiT erTnairia. jgufi 2
b. moda = 28. Mdn = 29. X = 33.5. g. dadebiTad asimetriuli; saSualo medianaze metia.
jgufi 3
b. moda = 52. Mdn = 50. X = 46.7. g. uaryofiTad asimetriuli; saSualo medianaze naklebia. jgufi 4
b. moda = bimodaluri, 26, 52. Mdn = 40. X = 38.6. g. daaxloebiT simetriuli; bimodaluri.
d. radgan ganawileba bimodaluria, arc mediana da arc X ar waradgens kargad cdis piris tipiur qulas.
5.
X mediana moda
roberto dimitri kareni
64 58 71
65 65 67
90 ar aris 56
moda mediana saSualo
6. a. rogorc Cans, zogierTma cdis pirma alkoholis dalevis Zalian
maRali raodenoba miuTiTa kviraSi, ramac ganapiroba dadebiTad asimetriuli ganawileba.
b. kviraSi alkoholis dalevis raodenobis mediana. 7. a. moda an mediana. b. moda = 793. Mdn = 832. Tavi V SeamowmeT Tqveni codna V.1 2. a. diapazoni = 44, IQR = 18, SIQR = 9. 3. a. diapazoni = 287 miliwami, IQR = 85 miliwami, SIQR = 42.5 miliwami. SeamowmeT Tqveni codna V.2 2. a. da b.
g. jgufSi, romelsac maswavlebeli ukiTxavda. d. 10.23. 3. a.
kiTxulobdnen
TviTon
ukiTxavdaT
maswavlebeli
X s s2
59.0 5.54 30.69
57.0 10.23 104.65
mamrobiTi mdedrobiTi
X s s2
15.55 7.41 54.91
7.05 2.59 6.71
b. mamrobiTSi. g. 7.41 wuTi. d. gogonebisaTvis; s qalebisaTvis naklebia, vidre mamakacebisaTvis.
SeamowmeT Tqveni codna V.3 1. a. Mdn, SIQR. b. Mdn, SIQR. g. X , s. d. Mdn, SIQR. e. X , s. v. Mdn, SIQR. z. Mdn, SIQR. kiTxvebi
1. a. Mdn = 50, X = 49.60, IQR = 10, s = 7.26. b. orive erTnairad kargia.
2. a. Mdn = 50, X = 49.45, IQR = 10, s = 11.56. b. orive erTnairad kargia. g. IQR-ze ar axdens gavlenas qulebi ganawilebis kudebidan. 3. a. diapazoni. b. IQR. g. SIQR. d. s2. e. s. v. IQR. z. diapazoni. T. S2. i. s2. k. S. l. s. m. SIQR. n. σ. o. σ2. 4. a. 15.4 alkoholis miReba kviraSi. b. did cvalebadobaze.
g. didi standartuli gadaxrebis migviTiTebs, rom ganawilebebi dadebiTad iyo asimetriuli. zogma cdis pirma mogvca dalevis Zalian didi raodenoba kviraSi da gazarda s-is mniSvneloba. es Sedegi gviCvenebs, rom mediana am qulebisaTvis centraluri tendenciis ukeTesi sazomi iqneboda.
5. IQR da SIQR, radgan isini ar moiTxoven mecxre pacientis qulis ricxviTi mniSvnelobis codnas.
Tavi VI SeamowmeT Tqveni codna VI.1
4. a. .3413. b. .4772. g. .4987. d. .3413. e. .4772. v. .4987. z. .8400. T. .9544. i. .9974. k. .1359. l. .1574. m. .0215. SeamowmeT Tqveni codna VI.2 2.
z farTobi 0-dan z-mde farTobi z-is zemoT a. b. g. d. e. v.
+1.90 -0.52 +0.77 -1.83 +1.75 -1.26
.4713
.1985 .2794 .4664 .4599 .3962
.0287
..3015
.2206
.0336
.0401
.1038
3. a. .0401. b. .1465. g. .3345. d. daaxloebiT 58.8-dan 85.2-mde, rodesac z = + an -1.65. SeamowmeT Tqveni codna VI.3 3. .0014. 4.
z p a. b. g. d. e. v.
+1.90 -0.52 +0.77 -1.83 +1.75 -1.26
.0287
..3015
.2206
.0336
.0401
.1038
5. a. .0668. b. .1865. g. .2426. kiTxvebi 3. a. .1587. b. .0228. g. .0038. d. .4082. e. .4525. v. daaxloebiT 80.0-dan 119.2-mde, rodesac z = + an -1.28.
z. .6568. T. .0918. i. .0475. k. .0038. 4. a. .016. b. .018. 5. a. p = .06. b. p = .08. g. p = .54. 6. a. .0968. b. .0968. g. .8064. 7. a. .0668. b. .0401. g. .9544. d. .8664. e. .2266. v. .3085. 8. a.
videoTamaSebi semestruli niSnebis saSualo
X S
8.0 5.86
2.7 0.87
aleqsi lela ia levani daviTi ivane
-0.34 0.00 -1.37 +1.19 -0.85 +1.37
-0.23 -0.80 +1.38 +0.80 +0.46 -1.61
b. lela jgufis saSualoa videoTamaSebisaTvis daxarjuli drois mixedviT. g. ia aris 1.38 standartuli gadaxriT maRla jgufis semestruli niSnebis saSualosTan SedarebiT. d. adamianis semestruli niSnebis saSualo jgufis saSualoze qvemoT aris. e. adamianis mier videoTamaSebze daxarjuli dro jgufis saSualoze maRalia.
9.
literatura istoria z = +0.83 z = +1.00
istoriaSi, radganac Tqveni qula klasis saSualoze metia erTi standartuli gadaxriT. 10. a. p = .206. b. p = .045. 11. a. p = .458. b. p = .267. Tavi VII SeamowmeT Tqveni codna VII.1 2.
(a) proporcia (b) raodenoba generalizebuli SfoTviTi aSliloba agorafobia martivi fobia
.27 .23 .19
27 23 19
panikuri aSliloba obsesiur-kompulsiuri
.22
.09
22 9
SeamowmeT Tqveni codna VII.2 2. Zalmosili; ufro naklebad afasebs. 3. Caunacvlebeli. 4. Zalmosili; odnav naklebad afasebs. 7. a. .6826. b. .1587. g. .0228. d. 0475. e. .4525. 8. a. 9, 6, 3 wami, Sesabamisad.
b. N-is zrdasTan erTad mcirdeba Xs .
9. a. s = 10, Xs = 2.
s = 15, Xs = 3.
s = 20, Xs = 4.
b. s-is zrdasTan erTad aseve izrdeba Xs
10. populaciidan A2, radganac mas σ-is ufro didi mniSvneloba aqvs. SeamowmeT Tqveni codna VII.3 2. a. 73.0. b. 69.1-dan 76.9-mde. g. 67.8-dan 78.2-mde. d. 73.0. e. 71.0-dan 75.0-mde. v. 70.4-dan 75.6-mde.
z. Xσ -is mniSvneloba mcirdeba N-is zrdasTan erTad. kiTxvebi
1. a. σ. b. µ. g. 2σ . d. Xσ .
2. N = 24, radgan X aris Zalmosili Sefaseba. 3. stratificirebuli; axsnisaTvis ixileT teqsti. 5. a. 9.0 sigareti dReSi. b. 7.5-idan 10.5-mde sigareti dReSi. 6. a. 0.3. b. .0013. g. .1587. d. .9544. 8.
SerCeva a b g
a
b
g
d
e
2.00
1.33
1.00
0.80
0.40
.1587
.0668
.0228
.0062
<.0001
.3413
.4332
.4772
.5938
>.4999
Tavi VIII SeamowmeT Tqveni codna VIII.1
3. a. .100:0 =µH
.100:1 ≠µH
b. .4.4:0 =µH saaTi.
.4.4:1 ≠µH saaTi.
g. 0.423:0 =µH miliwami.
0.423:1 ≠µH miliwami. 4.
realuriz
a. b. g. d.
-2.13
-1.20
+2.25
+1.72
xvdeba uaryofis areSi; uarvyofT
oH ; viRebT 1H .
ar xvdeba uaryofis areSi; ar
uarvyofT oH ; ar viRebT 1H .
xvdeba uaryofis areSi; uarvyofT
oH ; viRebT 1H .
ar xvdeba uaryofis areSi; ar
uarvyofT oH ; ar viRebT 1H .
5. a. ( )XXz σµ /−=
b. .100:0 =µH .100:1 ≠µH
g. α = .05. d. kritikuliz =1.96.
e. realurisz ufro eqstremaluri mniSvnelobebi, vidre z = - 1.96 an + 1.96.
v. realuriz = + 2.50.
z. uarvyofT oH ; viRebT 1H .
T. SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis saSualosagan.
SeamowmeT Tqveni codna VIII.2 3. a. 7. b. 6. g. 6. 4. a. 9. b. 15. g. 6. d. 53. e. 119. v. 29. 5. ganawileba 13 df-iT. 6. rodesac df = ∞. SeamowmeT Tqveni codna VIII.3
1. a. 493:0 =µH ; .493:1 ≠µH
b. 1.36:0 =µH ; .1.36:1 ≠µH
g. 100:0 =µH ; .100:1 ≠µH
2. a. 2.447. b. 2.898. g. 2.069. d. 2.042. e. 2.042. v. 2.660. z. 2.000. T. 1.980. 3.
xvdeba uaryofis
areSi
gadawyvetileba
a. ki uarvyofT oH
viRebT 1H
b. ara ver uarvyofT oH
ver viRebT 1H
g. ki uarvyofT oH
viRebT 1H
d. ara ver uarvyofT oH
ver viRebT 1H
e. ki uarvyofT oH
viRebT 1H
v. ki uarvyofT oH
viRebT 1H
z. ara ver uarvyofT oH
ver viRebT 1H
T. ki uarvyofT oH
viRebT 1H
i. ki uarvyofT oH
viRebT 1H
k. ki uarvyofT oH
viRebT 1H
l. ki uarvyofT oH
viRebT 1H
5. a. ( )XsXt /µ−=
b. ixileT teqsti.
g. 0.7:0 =µH ; .0.7:1 ≠µH
d. α = .05. e. kritikulit (10) = 2.228.
v. realurist ufro eqstremaluri mniSvnelobebi, vidre –2.228 an +2.228.
z. realurit = -4 979.
T. uarvyofT oH , viRebT 1H .
i. SerCevis saSualo 5.8 wuTi mniSvnelovnad gansxvavdeba populaciis saSualosagan – 7.0 wuTi.
6. angariSi 1 a. 70.4 kg.
b. 7.6 kg. g. t erTi SerCevisaTvis. d. 75:0 =µH kg, 75:1 ≠µH kg.
e. 6. v. –1.48. z. 5. T. 2.571.
i. ver uarvyofT oH .
k. ver viRebT 1H . l. SerCevis saSualo mniSvnelovnad ar gansxvavdeba populaciis saSualosagan.
angariSi 2
a. 10.4 dRe. b. 2.3 dRe. g. t erTi SerCevisaTvis.
d. 7.8:0 =µH dRe; 7.8:1 ≠µH dRe.
e. 25. v. +3.70. z. 24. T. 2.064.
i. uarvyofT oH .
k. viRebT 1H . l. SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis saSualosagan; 10.4 dRe mniSvnelovnad ufro grZeli periodia, vidre hipoTeturi saSualo 8.7 dRe.
SeamowmeT Tqveni codna VIII.4
2. b. p = α g. pirveli gvaris Secdoma ver moxdeba, Tu oH ar aris WeSmariti.
d. ara. 4. 80.1-dan 113.9 wamamde. Tavis mimoxilva
5. a. uarvyofT oH ; viRebT 1H .
b. ver uarvyofT oH ; ver viRebT 1H .
7. a. SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi saSualosagan.
b. SerCevis saSualo mniSvnelovnad ar gansxvavdeba populaciis hipoTeturi saSualosagan.
8. uaryofis are Sedgeba realurisz ufro ukiduresi mniSvnelobebisagan,
vidre z = -2.58 an +2.58.
realuriz xvdeba uaryofis
areSi
gadawyvetileba
+3.71
-2.60
+1.74
-3.05
-1.96
+2.58
ki
ki
ara
ki
ara
ki
uarvyofT oH ; viRebT 1H .
uarvyofT oH ; viRebT 1H .
ver uarvyofT oH ; ver
viRebT 1H .
uarvyofT oH ; viRebT 1H .
ver uarvyofT oH ; ver
viRebT 1H .
uarvyofT oH ; viRebT 1H .
9. realuriz = -1.52; aramniSvnelovani; ver uarvyofT oH .
11. a. SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi saSualosagan.
b. SerCevis saSualo mniSvnelovnad ar gansxvavdeba populaciis hipoTeturi saSualosagan.
xvdeba uaryofis
areSi
gadawyvetileba
a. ki uarvyofT oH
viRebT 1H
b. ara ver uarvyofT oH
ver viRebT 1H
g. ki uarvyofT oH
viRebT 1H
d. ki uarvyofT oH
viRebT 1H
e. ara ver uarvyofT oH
ver viRebT 1H
v. ara ver uarvyofT oH
ver viRebT 1H
13. a. realurit (8) = -2.924; aramniSvnelovani .05 doneze; uarvyofT oH ,
viRebT 1H . g. ki. d. ara.
e. 6.81-dan 7.59-mde saaTi; µ = 7.7 ar xvdeba am intervalSi.
14. a. realurit (11) = +3.139; mniSvnelovani .05 doneze; uarvyofT oH , viRebT
1H . b. ara. 15. a. pirveli gvaris Secdoma. b. meore gvaris Secdoma. g. swori gadawyvetileba. d. swori gadawyvetileba.
Tavi IX SeamowmeT Tqveni codna IX.1 6. a. damatebiTi cvladi = asaki; jgufebi sistematurad gansxvavdebian
saSualo asakisa da xmauris pirobis mixedviT. b. damatebiTi cvladi = eqsperimentatoris waxaliseba; jgufebi
sistematurad gansxvavdebian waxalisebisa da xmauris pirobis mixedviT.
g. damatebiTi cvladi = genderi; jgufebi sistematurad gansxvavdebian genderisa da xmauris pirobis mixedviT.
d. damatebiTi cvladi = kolejSi aRebuli wamyvani sagani; jgufebi sistematurad gansxvavdebian kolejSi aRebuli wamyvani sagnisa da xmauris pirobis mixedviT.
e. damatebiTi cvladi = dRis monakveTi, rodesac eqsperimenti Catarda; jgufebi sistematurad gansxvavdebian dRis monakveTisa da xmauris pirobis mixedviT.
SeamowmeT Tqveni codna IX.2 3. a. 3.77. b. 3.56. g. 3.81. d. 2.25. SeamowmeT Tqveni codna IX.3 6.
SerCeva df kritikulit xvdeba
uaryofis areSi
gadawyvetileba
1. 18 2.101 ki uarvyofT oH
viRebT 1H
2. 18 2.101 ara ver uarvyofT oH
ver viRebT 1H
3. 28 2.048 ki uarvyofT oH
viRebT 1H
4. 28 2.048 ki uarvyofT oH
viRebT 1H
5. 68 2.000 ki uarvyofT oH
viRebT 1H
6. 33 2.042 ara ver uarvyofT oH
ver viRebT 1H
7. 127 1.980 ki uarvyofT oH
viRebT 1H
8. 148 1.980 ki uarvyofT oH
viRebT 1H
9. amocana # 1
a. 210 : AAH µµ = ; 211 : AAH µµ ≠ .
b. realurit (34) = -3.656.
g. 34. d. 2.042.
e. realurist ufro ukiduresi mniSvnelobebi, vidre –2.042 an +2.042.
v. ki.
z. uarvyofT oH ; viRebT 1H .
T. gansxvavebuli populaciebidan. i. ki. k. ki; sakontrolo jgufSi statistikurad mniSvnelovani ufro
mokle drois intervali gamovlinda, vidre eqsperimentul jgufSi. amocana # 2 a. 210 : AAH µµ = ; 211 : AAH µµ ≠ .
b. realurit (28) = -1.742.
g. 28. d. 2.048.
e. realurist ufro ukiduresi mniSvnelobebi, vidre –2.048 an +2.048.
v. ara.
z. ver uarvyofT oH ; ver viRebT 1H . T. ar arsebobs imis TvalsaCinoeba, rom isini erTi da imave populaciidan arian.
i. ara; aramniSvnelovania. k. ara; gansxvaveba jgufebis saSualoebs Soris unda ganvixiloT,
rogorc SemTxveviTi gansxvaveba. SeamowmeT Tqveni codna IX.4 2. a. jgufi 2. b. jgufi 2. g. jgufi 1.
3. mosalodnelia, rom 2021 == AA nn eqneba ufro mcire 21 AXAXs−
da
migviyvans ufro mZlavr statistikur kriteriumamde. 4. a. .42. b. .27. g. .19. d. .23. e. .08. 7. angariSi 1 a. 2. b. 62. g. 79.7; 73.0. d. 8.57; 5.85. e. 3.60. v. 2.000.
z. uarvyofT oH . 1H .
T. viRebT 1H . i. statistikurad mniSvnelovani. k. Carevis efeqts.
angariSi 2 a. 2. b. 30. g. 22.78; 24.33.
d. 5.23; 5.14. e. –0.82. v. 2.048.
z. ver uarvyofT oH . 1H .
T. ver viRebT 1H . i. aramniSvnelovani. k. SerCevis Secdomas. SeamowmeT Tqveni codna IX.5
2. a. 210 : AAH µµ = ; 211 : AAH µµ ≠ .
b. realurit (10) = -3.148.
g. 10. d. 2.228.
e. realurist ufro eqstremaluri mniSvnelobebi, vidre –2.228 an +2.228.
v. ki.
z. uarvyofT oH ; viRebT 1H . 1H .
T. gansxvavebuli populaciebidan. i. ki. k. ki; sakontrolo jgufma statistikurad mniSvnelovani reaqciis
droiT upasuxa, gansxvavebiT eqsperimentuli jgufisagan. l. .50. m. - 41.8-dan - 7.2 miliwamamde. kiTxvebi 1.
jgufi uxmauro xmauriani
X s
122.64 8.76
131.27 10.45
a. realurit (20) = -2.099, p<.05. uxmauro jgufis saSualo sisxlis wneva
mniSvnelovnad dabalia, vidre xmauriani jgufisa.
b. 18.2 =η ; damokidebuli cvladis dispersiis 18%. g. 0.05-dan 17.21-mde. 2.
jgufi uxmauro xmauriani
X s
113.09 4.55
115.27 6.48
realurit (20) = -0.913, p>.05. ori jgufis saSualo sisxlis wneva
mniSvnelovnad ar gansxvavdeba erTmaneTisagan. 3.
sqesi mamr. mdedr.
X s
17.77 4.99
18.59 5.32
realurit (28) = -0.429, p>.05. mamakacebi da qalebi mniSvnelovnad ar
gansxvavdebian Sefasebuli ganmartoebis moTxovnilebis mixedviT. 4. a.
instruqciis piroba instruqciamde instruqciis
Semdeg
X s
2.45 .604
2.52 .677
b. alfas .05 doniTa da ormxrivi kriteriumiT, instruqciamde gakeTebuli Sefasebebis saSualo Secdoma (M = 2.45 mm, SD = .604 mm) mniSvnelovnad ar gansxvavdeboda instruqciis Semdeg gakeTebuli Sefasebebis saSualo Secdomisagan (M = 2.52 mm, SD = .677 mm), t(11) = - 1.261, p > .05.
5. a. realurit (88) = +3.170, p<.01. kontrolis lokusis saSualo qula
naSvilebi bavSvebis mSoblebis jgufSi mniSvnelovnad maRali iyo, vidre lokusis kontrolis saSualo qula biologiuri mSoblebis jgufSi.
b. 10.2 =η g. 1.29-dan 5.69-mde.
6. a. realurit (5398) = -25.003, p<.001; ori jgufis saSualo qulebis
mniSvnelovnad gansxvavdeba. wesrigis damrRvevi studentebis saSualo qula mniSvnelovnad dabalia, vidre im studentebisa, romlebic wesrigs ar arRvevdnen.
b. 10.2 =η g. 0.47-dan 0.55-mde.
7. realurit (32) = -0.818, p>.05. aramniSvnelovani t miuTiTebs, rom jgufis
saSualoebi mniSvnelovnad ar gansxvavdeba.05 doneze.
8. a. realurit (24) = - 3.744, p<.05. im mosaubris saSualo sandoobis
reitingi, romelic uSualo qcevas avlens, mniSvnelovnad metia, vidre im mosaubris reitingi, romelic ar aris uSualo qcevis pirobaSi.
b. η2 = .37. g. - 3.223-dan - 0.931-mde.
d. alfas .05 doniTa da ormxrivi kriteriumiT, saSualo sandoobis reitingi mosaubrisa, romelic uSualo qcevas avlens, (M = 6.9, SD = 1.41) mniSvnelovnad meti iyo vidre im mosaubris reitingi, romelic ar aris uSualo qcevis pirobaSi (M = 4.8, SD = 1.42), t(24) = - 1.261, p > .01, η2 = .37.
Tavi X SeamowmeT Tqveni codna X.1
6. a = 4, An = 11, N = 44. 11. 150.000 12. 25.00 13. 79. 14. 25.00. 15. 10.00.
16. An = 3, Secdomadf = 40, srulidf = 43.
17.
wyaro SS df MS F A faqtori Secdoma sul
24.00 16.00 40.00
1 4 5
24.00 4.00
6.00
18.
wyaro SS df MS F A faqtori Secdoma sul
42.00 24.00 66.00
2 6 8
21.00 4.00
5.25
19. cxrili # 1
wyaro SS df MS F A faqtori Secdoma sul
50.00 260.00 310.00
1 26 27
50.00 10.00
5.00
a. 2. b. 14. g. 28. cxrili # 2
wyaro SS df MS F A faqtori Secdoma sul
18.00 152.00 170.00
3 76 79
6.00 2.00
3.00
a. 4. b. 20. g. 80. SeamowmeT Tqveni codna X.2 6. daaxloebiT 1.00 7. is 1.00-ze meti xdeba.
8. a. 210 : AAH µµ =
1H : Aµ -ebi ar aris toli.
b. 43210 : AAAAH µµµµ ===
1H : yvela Aµ ar aris erTmaneTis toli.
g. 6543210 : AAAAAAH µµµµµµ =====
1H : yvela Aµ ar aris erTmaneTis toli. 9. damoukidebel cvlads ar aqvs efeqti. 10. damoukidebel cvlads aqvs efeqti. 11. nuli. 12. zeda sazRvari ar aris. 13.
eqsperi-menti
kritikuliF xvdeba uaryofis
areSi
gadawyvetileba
1. 4.96 ki uarvyofT oH
viRebT 1H
2. 3.40 ara ver uarvyofT oH
ver viRebT 1H
3. 4.26 ki uarvyofT oH
viRebT 1H
4. 3.15 ki uarvyofT oH
viRebT 1H
5. 3.88 ki ver uarvyofT oH
ver viRebT 1H
6. 3.68 ara ver uarvyofT oH
ver viRebT 1H
7. 2.61 ara ver uarvyofT oH
ver viRebT 1H
8. 3.10 ara ver uarvyofT oH
ver viRebT 1H
9. 2.25 ki uarvyofT oH
viRebT 1H
10. 2.53 ara ver uarvyofT oH
ver viRebT 1H
SeamowmeT Tqveni codna X.3 1.
jgufi sakontrolo eqsperimentuli
X s
7.09 1.30
8.82 1.47
a. 210 : AAH µµ =
1H : Aµ -ebi ar aris toli.
b. realuriF (1,20) = 8.51.
g. Adf = 1, Secdomadf =20.
d. kritikuliF (1,20) = 4.35.
e. realurisF mniSvnelobebi ≥ 4.35. v. ki.
z. uarvyofT oH ; viRebT 1H .
T. gansxvavebuli populaciebidan. i. ki. k. isini gansxvavdeba; sakontrolo jgufma ufro mokled drois
intervali gviCvena. l. ara, radganac icvleboda damoukidebeli cvladis mxolod ori
done.
m. 30.2 =η n. 30 procenti. 2.
klasi meore meoTxe meeqvse
X s
565.80 32.410
530.60 29.99
501.00 24.42
a. 3210 : AAAH µµµ ==
1H : yvela Aµ ar aris erTmaneTis toli.
b. realuriF (2, 39) = 17.76.
g. Adf = 2, Secdomadf = 39.
d. kritikuliF (2, 39) = 3.32. (kritikuli mniSvneloba 2, 30 df-isaTvis)
e. realurisF mniSvnelobebi, romlebis metia an tolia 3.32.
v. ki.
z. uarvyofT oH ; viRebT 1H .
T. saSualoebs Soris aris sul mcire erTi mniSvnelovani sxvaoba.
i. ki, radganac realuriF statistikurad mniSvnelovania da
eqsperimentSi Sedardeba sami saSualo. k. CD = 26.86. l. TiToeuli jgufi mniSvnelovnad gansxvavdeba yvela sxva
jgufisagan. m. Zilis periodebi yvelaze grZelidan yvelaze moklesaken aris me-
2, me-4 da me-6 klasis moswavleebi.
n. η2 = .48. o. 48 procenti. 3.1. a. 2. b. 62. g. 14.15. d. 4.00
e. uarvyofT oH .
v. viRebT 1H . z. ki. T. nacnobi da ucnobi leqsis jgufebis saSualoebi. 3.2. a. 4. b. 40. g. 1.13. d. 2.92.
e. ver uarvyofT oH .
v. ver viRebT 1H . z. ara. T saSualoebs Soris ar aris mniSvnelovani gansxvaveba. 3. a. 3. b. 39. g. 33.94. d. 3.32.
e. uarvyofT oH .
v. viRebT 1H . z. ki. T. tukis CD gviCvenebs, rom mZime rokis saSualo (29.9)
mniSvnelovnad maRali iyo rogorc klasikuri musikis (18.5), aseve musikis gareSe sakontrolo jgufis (17.5) saSualoebze. klasikuri musikisa da musikis gareSe sakontrolo jgufebis saSualoebi ar gasxvavdeboda mniSvnelovnad.
kiTxvebi 1. a.
eqsperimentuli jgufi musikis gareSe
Zveli musika mZime roki
X s
36.0 3.70
35.6 3.50
22.9 3.60
realuriF (2,21) = 34.39, p < .01. tukis CD = 4.6. mZime rokis jgufma
mniSvnelovnad naklebi Secdomebi aRmoaCina, vidre danarCenma orma jgufma. musikis gareSe da Zveli musikis jgufebis saSualoebi ar gansxvavdeboda mniSvnelovnad.
b. η2 = .77. 2. a.
instruqciis piroba videofiris
gareSe fexburTis videofiri
srolis, daWeris
videofiri
X s
45.22 5.63
44.33 5.74
57.11 6.68
realuriF (2,24) = 12.57, p < .01. tukis CD = 7.1. srolis, daWeris
videofiris yurebis saSualo mniSvnelovnad gansxvavdeba rogorc videofiris gareSe pirobis, iseve fexburTis videofiris pirobis saSualoebisagan. videofiris gareSe pirobisa da fexburTis videofiris pirobis saSualoebi mniSvnelovnad ar gansxvavdeba.
b. η2 = .51. 3. a.
jgufi uxmauro xmauriani
X s
122.64 8.76
131.27 10.45
realuriF (1,20) = 4.41, p < .05. uxmuro jgufis saSualo mniSvnelovnad
dabalia xmauriani jgufis saSualoze.
b. ki, t = 41.4=F = 2.10.
4. a. a = 6, An = 7, N = 42. b. Adf = 5, Secdomadf = 36, srulidf = 41.
5. cxrili # 1
wyaro SS df MS F A faqtori Secdoma sul
50.00 270.00 320.00
2 27 29
25.00 10.00
2.50
a. 3. b. 10. g. 30.
d. kritikuliF (2,27) = 3.35; realuriF aramniSvnelovania.
e. ver uarvyofT oH ; ver viRebT 1H .
cxrili # 2
wyaro SS df MS F A faqtori Secdoma sul
60.00 216.00 276.00
5 72 77
12.00 3.00
4.00
a. 6. b. 13. g. 78.
d. kritikuliF (5,72) = 2.37; realuriF statistikurad mniSvnelovania.
e. uarvyofT oH ; viRebT 1H .
cxrili # 3
wyaro SS df MS F A faqtori Secdoma sul
45.00 320.00 365.00
3 64 67
15.00 5.00
3.00
a. 4.
b. 17. g. 68.
d. kritikuliF (3,64) = 2.76; realuriF statistikurad mniSvnelovania.
e. uarvyofT oH ; viRebT 1H .
7. realuriF (1,18) = 4.00.
8. realurit (22) = 3.00.
10. statistikurad mniSvnelovani realuriF (3,64) = 21.64 gviCvenebs, rom
arsebobs sul mcire erTi mniSvnelovani gansxvaveba jgufebs Soris. Tu wyvilTaSorisi SedarebebisaTvis CD = 5.0 wuTs gamoviyenebT, aRmoCndeba, rom qcevis Secvlas da sigaretebis anonimur klubs mniSvnelovnad didi saSualo drois periodebi aqvT sigaretebis mowevas Soris, vidre rogorc sakotrolo, aseve mowevis Sewyvetis saReWi rezinis jgufs. qcevis Secvlisa da sigaretebis anonimuri klubis saSualoebi erTmaneTisagan mniSvnelovnad ar gansxvavdeba, xolo sakontrolo jgufis da mowevis Sewyvetis saReWi rezinis jgufis saSualoebic aseve erTmaneTisagan mniSvnelovnad ar gansxvavdeba.
Tavi XI SeamowmeT Tqveni codna XI.1 2.
damoukidebeli cvladebis raodenoba
doneebi saWiro cdis pirebis
raodenoba a. b. g. d. e. v. z. T.
2 2 2 2 2 3 3 3
A = 3; B = 3 A = 3; B = 2 A = 2; B = 3 A = 6; B = 2 A = 4; B = 4
A = 2; B = 2, C = 2 A = 2; B = 4, C = 3 A = 3; B = 3, C = 2
90 60 60 120 160 80 240 180
3. cxrili # 1 faqtori A
A1 A2
mTavari efeqtis saSualoebi B faqtorisaTvis
B1 faqtori B
B2
13.3 20.2 16.8 22.8
16.8
19.8 mTavari efeqtis
saSualoebi A faqtorisaTvis
15.1 21.5
3.18=GX
cxrili # 2
faqtori A
A1 A2
mTavari efeqtis saSualoebi B faqtorisaTvis
B1 faqtori B
B2
18.9 30.1 31.3 33.9
24.5
32.6 mTavari efeqtis
saSualoebi A faqtorisaTvis
25.1 32.0
6.28=GX
SeamowmeT Tqveni codna XI.2
8. 240.00. 9. 45.00. 10. 55.00. 11. 60.00
12. .35,30,2,2,1 ===== sruliSecdoma dfdfdfdfdf AxBBA
13. .77,71,2,1,2 ===== sruliSecdoma dfdfdfdfdf AxBBA
14. 79.
15. 00.860/00.480
;00.222/00.44;00.132/00.26;00.141/00.14==
======
SecdomaMSMSMSMS AxBBA
16. a. wyaro SS df MS F A faqtori B faqtori A x B Secdoma sul
108.00 0.00 12.00 32.00 152.00
1 1 1 8 11
108.00 0.00 12.00 4.00
27.00 0.00 3.00
b. B faqtoris orive mTavari efeqtis saSualo udris 32.0.
17. cxrili # 1
wyaro SS df MS F A faqtori B faqtori A x B Secdoma sul
32.00 24.00 24.00 480.00 560.00
2 1 2 60 65
16.00 24.00 12.00 8.00
2.00 3.00 1.50
a. 3 b. 2. g. 11. d. 66.
cxrili # 2
wyaro SS df MS F A faqtori B faqtori A x B Secdoma sul
36.00 24.00 160.00 840.00 1060.00
2 4 8 210 224
18.00 6.00 20.00 4.00
4.50 1.50 5.00
a. 3
b. 5. g. 15. d. 225.
SeamowmeT Tqveni codna XI.3
6. realuriF 1.00-ze meti xdeba.
7. realuriF 1.00-ze meti xdeba.
9. a. F Sefardeba: statistikuri hipoTezebi
A faqtorisaTvis H0 : µ A1 = µ A2;
H1 : µ A.ebi ar aris toli
B faqtorisaTvis H0 : µ B1 = µ B2 = µ B3 = µ B4 ;
H1 : µ B-ebi ar aris toli
A x B H0 : yvela (µAB - µ A - µ B + µ G) = 0 H1 : yvela (µAB - µ A - µ B + µ G) ar udris 0
b. F Sefardeba: statistikuri hipoTezebi
A faqtorisaTvis H0 : µ A1 = µ A2 = µ A3;
H1 : µ A.ebi ar aris toli
B faqtorisaTvis H0 : µ B1 = µ B2;
H1 : µ B-ebi ar aris toli
A x B H0 : yvela (µAB - µ A - µ B + µ G) = 0 H1 : yvela (µAB - µ A - µ B + µ G) ar udris 0
g. F Sefardeba: statistikuri hipoTezebi
A faqtorisaTvis H0 : µ A1 = µ A2 = µ A3;
H1 : µ A.ebi ar aris toli
B faqtorisaTvis H0 : µ B1 = µ B2 = µ B3;
H1 : µ B-ebi ar aris toli
A x B H0 : yvela (µAB - µ A - µ B + µ G) = 0
H1 : yvela (µAB - µ A - µ B + µ G) ar udris 0
10. cxrili # 1
wyaro df realuriF kritikuliF realuriF
xvdeba uaryofis areSi
gadawyvetileba
A faqtori B faqtori A x B
1 3 3
4.63
2.11
3.97
4.08
2.84
2.84
ki
ara ki
uarvyofT oH
viRebT 1H
ver uarvyofT oH
ver viRebT 1H
uarvyofT oH
viRebT 1H
cxrili # 2
wyaro df realuriF kritikuliF realuriF
xvdeba uaryofis areSi
gadawyvetileba
A faqtori B faqtori A x B
3 2 6
1.42
3.51
2.37
2.68
3.07
2.17
ara ki ki
ver uarvyofT oH
ver viRebT 1H
uarvyofT oH
viRebT 1H
uarvyofT oH
viRebT 1H
cxrili # 3
wyaro df realuriF kritikuliF realuriF
xvdeba uaryofis areSi
gadawyvetileba
A faqtori B faqtori A x B
2 4 8
4.12
2.83
1.87
3.07
2.45
2.02
ki ki
ara
uarvyofT oH
viRebT 1H
uarvyofT oH
viRebT 1H
ver uarvyofT oH
ver viRebT 1H
SeamowmeT Tqveni codna XI.4
suraTi # 11.2 a. ki. b. ara. g. ki.
d. statistikurad mniSvnelovani urTierTqmedebis gamo swavlebis meTodis tipis efeqti damokidebulia masalis tipze, B faqtorze. maTematikuri masalis SemTxvevaSi (B1) swavlebis kompiuterizebul meTods (A1) mivyavarT testis mniSvnelovnad maRal qulebamde, vidre arakompiuterirebul meTods (A2). socialuri Sinaarsis masalis SemTxvevaSi ki (B2) swavlebis meTodebis erTmaneTisagan ar gansxvavdebian. e. statistikurad mniSvnelovani urTierTqmedeba miuTiTebs, rom masalis tipis efeqti damokidebulia swavlebis meTodze, A faqtorze. swavlebis kompiuterizebuli meTodis SemTxvevaSi (A1) maTematikuri Sinaarsis masala (B1) iZleva testis mniSvnelovnad maRal qulebs socialuri Sinaarsis masalasTan SedarebiT (B2). swavlebis arakompiuterizebuli meTodis SemTxvevaSi (A2) masalis tipebs Soris gansxvaveba ar aris. v. ara; A faqtoris mTavari efeqti miuTiTebs swavlebis kompiuterizebuli meTodis sayovelTao upiratesobaze, magram swavlebis kompiuterizebuli meTodi testis mniSvnelovnad maRal qulebs iZleva mxolod maTematikuri Sinaarsis masalis SemTxvevaSi. amdenad, A faqtoris mTavari efeqti urTierTqmedebis artefaqtia; is ar iZleva mniSvnelovan informacias A faqtoris efeqtis Sesaxeb testis qulebze.
suraTi # 11.2 a. ara. b. ki. g. ki.
d. statistikurad mniSvnelovani urTierTqmedeba miuTiTebs, rom swavlebis meTodis efeqti damokidebulia masalis tipze, B faqtorze. maTematikuri Sinaarsis masalisaTvis (B1) swavlebis meTodebi mniSvnelovnad ar gansxvavdebian. socialuri Sinaarsis masalisaTvis (B2) swavlebis arakompiuterizebuli meTodi (A2) iZleva testis mniSvnelovnad maRal qulebs, vidre swavlebis kompiuterizebuli meTodi (A1). e. statistikurad mniSvnelovani urTierTqmedeba miuTiTebs, rom masalis tipis efeqti damokidebulia swavlebis meTodze, A faqtorze. swavlebis orive meTodi testis ufro maRal qulebs iZleva maTematikuri Sinaarsis masalisaTvis, magram gansxvaveba maTematikuri da socialuri Sinaarsis masalebs Soris ufro didia swavlebis kompiuterizebuli meTodis SemTxvevaSi, vidre swavlebis arakompiuterizebuli meTodis SemTxvevaSi. v. ki; B faqtoris mTavari efeqti miuTiTebs maTematikuri Sinaarsis masalis sayovelTao upiratesobaze da maTematikuri Sinaarsis masala testis mniSvnelovnad maRal qulebs iZleva swavlebis orive meTodisaTvis. Tumca, urTierTqmedeba migviTiTebs, rom
gansxvaveba maTematikuri da socialuri Sinaarsis masalebs Soris damokidebulia swavlebis meTodis tipze.
2. cxrili # 1 a. ki. b. ara. g. ara. d. swavlebis kompiuterizebuli meTodi (A1) testis mniSvnelovnad
maRal qulebs iZleva, vidre arakompiuterizebuli swavleba (A2). e. efeqti ar aris. v. ki, A da B faqtorebs Soris ar aris urTierTqmedeba. cxrili # 2 a. ara. b. ki. g. ara. d. efeqti ar aris. e. maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal
qulebs iZleva, vidre socialuri Sinaarsis masala (B2). v. ki, A da B faqtorebs Soris ar aris urTierTqmedeba. cxrili # 3 a. ki. b. ki. g. ara. d. swavlebis arakompiuterizebuli meTodi (A2) testis
mniSvnelovnad maRal qulebs iZleva, vidre kompiuterizebuli swavleba (A1).
e. maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal qulebs iZleva, vidre socialuri Sinaarsis masala (B2).
v. ki, A da B faqtorebs Soris ar aris urTierTqmedeba. cxrili # 4 a. ara. b. ara. g. ki. d. statistikurad mniSvnelovani urTierTqmedebis gamo swavlebis
meTodis efeqti damokidebulia masalis tipze, B faqtorze. maTematikuri Sinaarsis masalisaTvis (B1) swavlebis kompiuterizebuli meTodi (A1) testis mniSvnelovnad maRal qulebs iZleva, vidre swavlebis arakompiuterizebuli meTodi (A2). Tumca, socialuri Sinaarsis masalisaTvis (B2) arakompiuterizebuli meTodi (A2) testis mniSvnelovnad maRal qulebs iZleva, vidre kompiuterizebuli meTodi (A1).
e. statistikurad mniSvnelovani urTierTqmedebis gamo masalis tipis efeqti damokidebulia swavlebis meTodze, A faqtorze. v. ki, A da B faqtorebs Soris ar aris urTierTqmedeba. swavlebis kompiuterizebuli meTodisTvis (A1) maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal qulebs iZleva, vidre socialuri Sinaarsis masala. swavlebis arakompiuterizebuli meTodisTvis (A2)
socialuri Sinaarsis masala (B2) testis mniSvnelovnad maRal qulebs iZleva, vidre maTematikuri Sinaarsis masala. v. ara; TiToeuli damoukidebeli cvladis mTavari efeqti ar miuTiTebs damoukidebeli cvladis sayovelTao efeqtze. miuxedavad imisa, rom orive damoukidebel cvlads aqvs efeqti, TiToeuli damoukidebeli cvladis efeqti damokidebulia meore damoukidebeli cvladis doneze.
cxrili # 5 a. ki. b. ki. g. ki. d. statistikurad mniSvnelovani urTierTqmedebis gamo swavlebis
meTodis tipis efeqti damokidebulia masalis tipze. maTematikuri Sinaarsis masalisaTvis (B1) gansxvaveba kompiuterizebul (A1) da arakompiuterizebul (A2) swavlebis meTodebs Soris aramniSvnelovania. Tumca, socialuri Sinaarsis masalisaTvis arakompiuterizebuli swavlebis meTodi (A2) testis mniSvnelovnad maRal qulebs iZleva, vidre kompiuterizebuli swavlebis meTodi (A1).
e. statistikurad mniSvnelovani urTierTqmedebis gamo masalis tipis efeqti damokidebulia swavlebis meTodze. swavlebis rogorc kompiuterizebuli (A1), ise arakompiuterizebuli (A2) meTodebisaTvis maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal qulebs iZleva, vidre socialuri Sinaarsis masala (B2). Tumca, gansxvaveba maTematikuri da socialuri Sinaarsis masalebis qulebs Soris ufro didia kompiuterizebuli meTodis SemTxvevaSi (A1), vidre arakompiuterizebuli meTodis SemTxvevaSi (A2). v. A faqtoris - swavlebis meTodis - mTavari efeqti artefaqturia, radganac is miuTiTebs swavlebis arakompiuterizebuli meTodis sayovelTao upiratesobaze. magram mwavlebis arakompiuterizebuli meTodi mniSvnelovnad ukeTes Sesrulebas iZleva mxolod socialuri Sinaarsis masalis SemTxvevaSi. B faqtoris - masalis tipis - mTavari efeqti mniSvnelovania; maTematikuri Sinaarsis masala mniSvnelovnad ukeTes Sesrulebas iZleva rogorc kompiuterizebuli, iseve arakompiuterizebuli swavlebis meTodis SemTxvevaSi.
cxrili # 6 a. ki. b. ki. g. ki. d. statistikurad mniSvnelovani urTierTqmedebis gamo swavlebis
meTodis tipis efeqti damokidebulia masalis tipze. maTematikuri Sinaarsis masalisaTvis (B1) swavlebis kompiuterizebuli meTodi (A1) testis mniSvnelovnad maRal qulebs iZleva, vidre swavlebis arakompiuterizebuli meTodi (A2). xolo socialuri Sinaarsis masalisaTvis ar arsebobs swavlebis meTodis efeqti.
e. masalis tipis efeqti damokidebulia swavlebis meTodze. swavlebis kompiuterizebuli meTodisaTvis (A1) maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal qulebs iZleva, vidre socialuri Sinaarsis masala (B2). swavlebis arakompiuterizebuli meTodisaTvis (A2) gansxvaveba maTematikuri (B1) da socialuri (B2) Sinaarsis masalebis mier mocemul qulebs Soris aramniSvnelovania. v. rogorc A faqtoris - swavlebis meTodis, iseve B faqtoris - masalis tipis mTavari efeqtebi artefaqturia.
SeamowmeT Tqveni codna XI.5 1. dispersiuli analizis Sejameba: wyaro SS df MS F lokus-kontroli (A) instruqciebi (B) AxB Secdoma sul
704.167 1,472.667 46,112.666 12,233.000 60,522.500
1 1 1 20 23
704.167 1,472.667 46,112.666 611.650
1.15 2.41 75.39*
* p < .01 a. (1) arc erTi damoukidebeli cvladisaTvis ar arsebobs
mniSvnelovani mTavari efeqti. (2) urTierTqmedeba mniSvnelovania. b. (CD = 36.15) statistikurad mniSvnelovani urTierTqmedeba
miuTiTebs, rom instruqciebis efeqti damokidebulia kontrolis lokusze. kontrolis Sida lokusis mqone cdis pirebisaTvis unarebis instruqtirebam mogvca mniSvnelovnad meti dro, romelic daixarja davalebis Sesrulebaze, vidre SemTxveviTobis instruqtirebam. xolo kontrolis gare lokusis mqone cdis pirebisaTvis SemTxveviTobis instruqtirebam mogvca mniSvnelovnad meti dro, romelic daixarja davalebis Sesrulebaze, vidre unarebis instruqtirebam.
g. urTierTqmedebis gamo aramniSvnelovani mTavari efeqtebi ar gvawvdian gamosadeg informacias Sedegebis axsnisaTvis.
d. η2 = .76 A da B faqtorebis urTierTqmedebisaTvis. 2. dispersiuli analizis Sejameba: wyaro SS df MS F instruqcia (A) danaSauli (B) AxB Secdoma sul
191.361 1667.361 210.250 259.333 2328.305
1 1 1 32 35
191.361 1667.361 210.250 8.104
23.61* 205.75* 25.94*
* p < .01 a. (1) arsebobs mniSvnelovani mTavari efeqti rogorc A, iseve B
faqtorisaTvis. (2) urTierTqmedeba mniSvnelovania. b. (CD = 3.31) statistikurad mniSvnelovani urTierTqmedeba
miuTiTebs, rom instruqciebis tipis efeqti damokidebulia
danaSaulis tipze; gansxvaveba folis saxiT micemul da normalur instruqciebs Soris manqanis moparvisaTvis aramniSvnelovania, magram fizikuri Zaladobis gamoyenebiT TavdasxmisaTvis mniSvnelovnad grZeli patimrobis vada aris miCenili filmis saxiT instruqciis SemTxvevaSi, vidre normaluri instruqciis SemTxvevaSi.
g. statistikurad mniSvnelovani urTierTqmedeba miuTiTebs, rom danaSaulis tipis efeqti damokidebulia instruqciebis tipze. orive tipis instruqciisaTvis Tavdasxma fizikuri Zaladobis gamoyenebiT iZleva miCenili patimrobis mniSvnelovnad grZel vadas, vidre manqanis moparva. magram gansxvaveba patimrobis miCenil vadebSi manqanis moparvasa da Tavdasxmas Soris ufro didia filmis saxiT instruqtaJis SemTxvevaSi, vidre normaluri instruqtaJis SemTxvevaSi.
d. danaSaulis tipis statistikurad mniSvnelovani mTavari efeqti mniSvnelobis mqonea; mniSvnelovnad ufro grZeli patimrobebia misjili TavdaxsmisaTvis rogorc video, iseve normaluri instruqtaJis pirobebSi. instruqciis tipis mTavari efeqti artefqturia; ar arsebobs gansxvaveba video da normalur instruqciebs Soris manqanis moparvisaTvis; magram mniSvnelovnad ufro grZeli patimrobaa misjili Tavdasxmis videoinstruqtaJis SemTxvevaSi.
e. η2 = .72 danaSaulis tipisaTvis.
v. η2 = .09 A da B faqtorebis urTierTqmedebisaTvis. 3.1. a. 3. b. 2. g. 42. d.
kritikuliF
A B
AxB
3.32 4.17 73.32
e. ki. v. ki. z. ara. 3.2. a. 2. b. 4. g. 96. d.
kritikuliF
A B
AxB
4.00 2.76 2.76
e. ki. v. ara. z. ki. 3.3. a. 2. b. 2.
g. 60. d.
kritikuliF
A B
AxB
4.08 4.08 4.08
e. ara. v. ara. z. ki. kiTxvebi 1. dispersiuli analizis Sejameba. wyaro SS df MS F mxedvelobiTi kontaqti (A) emocia (B) AxB Secdoma sul
24.500 780.125 325.125 903.750 2033.500
1 1 1 28 31
24.500 780.125 325.125 32.277
0.76 24.17* 10.07*
* p < .01 a. ki; xdeba A da B faqtorebis mniSvnelovan urTierTqmedeba (tukis
CD = 7.1). dadebiTi emociis pirobisaTvis mxedvelobiTi kontaqtis ararsebobis saSualo mniSvnelovnad maRalia, vidre mxedvelobiTi kontaqtis damyarebis saSualo. uaryofiTi emociis pirobisaTvis mxedvelobiTi kontaqtis pirobis saSualoebi mniSvnelovnad ar gansxvavdeba.
b. mxedvelobiTi kontaqtis ararsebobis pirobisaTvis ar arsebobs mniSvnelovani gansxvaveba dadebiTi da uaryofiTi emociebis pirobebs Soris. xolo mxedvelobiTi kontaqtis damyarebis pirobisaTvis uaryofiTi emocia iZleva mniSvnelovnad ufro maRal guliscemis sixSires, vidre dadebiTi emocia.
g. emociis pirobis statistikurad mniSvnelovani mTavari efeqti artefaqturia; ar arsebobs mniSvnelovani gansxvaveba emociis pirobebs Soris mxedvelobiTi kontaqtis ararsebobis pirobaSi, magram mxedvelobiTi kontaqtis damyarebis pirobaSi uaryofiTi emociis ujredis saSualo mniSvnelovnad maRalia, vidre dadebiTi emociis ujredis saSualo.
d. η2 = .16 A da B faqtorebis urTierTqmedebisaTvis. 2. dispersiuli analizis Sejameba. wyaro SS df MS F pirovnebis tipi (A) xmauris foni (B) AxB Secdoma sul
36.750 30.083 27.000 245.833 339.666
1 1 1 44 47
36.750 30.083 27.000 5.587
6.58* 5.38* 4.83*
* p < .05 a. dispersiuli analizi avlens statistikurad mniSvnelovan
urTierTqmedebas; ase rom, saWiroa tukis CD martivi
efeqtebisaTvis. es CD 3,44 df-isaTvis aris 2.35. martivi efeqtebis analizi gviCvenebs, rom introvertebis saSualo xmamaRali fonis pirobaSi (M = 8.2 swor debulebas) mniSvnelovnad dabalia, vidre introvertebis saSualo xmadabali fonis pirobaSi (M = 11.3 swor debulebas) da aseve mniSvnelovnad dabalia, vidre eqstravertebis saSualo xmamaRali fonis pirobaSi (M = 11.4 swor debulebas). introvertebis saSualo xmadabali fonis pirobaSi mniSvnelovnad ar gansxvavdeboda eqstravertebis saSualosagan xmadabali fonis pirobaSi (M = 11.5 swor debulebas) da eqstravertebis saSualo xmadabali fonis pirobaSi mniSvnelovnad ar gansxvavdeboda eqstravertebis saSualosagan xmamaRali fonis pirobaSi.
b. η2 = .08. g. orive mTavari efeqti statistikurad mniSvnelovania, magram
artefaqturia da maTi mniSvnelovnad axsna SeuZlebelia. 3. dispersiuli analizis Sejameba. wyaro SS df MS F instruqciis meTodi (A) varjiSi (B) AxB Secdoma sul
436.594 60.168 69.777 219.332 785.871
2 1 2 48 53
436.594 60.168 43.888 4.569
47.78* 13.17* 7.64*
* p < .01
a. xdeba statistikurad mniSvnelovani urTierTqmedeba; amdenad, instruqciis meTodis efeqti damokidebulia varjiSis raodenobaze. tradiciuli meTodisTvis erTkviriani varjiSis saSualo (M = 13.1 swor sityvas) mniSvnelovnad naklebi iyo, vidre eqvskviriani varjiSis saSualo (M = 17.9 swor sityvas; CD = 2.9). miuxedavad amisa, ar aRmoCnda mniSvnelovani gansxvaveba erTkvirian da eqvskvirian varjiSis pirobebs Soris rogorc sityvebis yuTis (Sesabamisad, M = 18.7 swor sityvas; M = 21.0 swor sityvas), aseve sityvebis dajgufebis meTodisTvis (Sesabamisad, M = 22.8 swor sityvas; M = 22.0 swor sityvas). garda amisa, gansxvavebebi instruqciis meTodebs Soris damokidebulia varjiSis raodenobaze. erTkviriani varjiSis pirobisaTvis intruqciis meTodis yoveli jgufi sxva jgufebisagan mniSvnelovnad gansxvavdeboda; sityvebis dajgufebis meTodis cdis pirebma yvelaze maRali qulebi miiRes, sityvebis yuTis meTodis cdis pirebma - momdevno maRali qulebi da tradiciuli meTodis cdis pirebma - yvelaze dabali qulebi. eqvskviriani varjiSis pirobisaTvis, tradiciuli meTodis saSualo mniSvnelovnad naklebi iyo rogorc sityvebis yuTis, iseve sityvebis dajgufebis saSualoebze. Tumca, sityvebis yuTis da sityvebis dajgufebis saSualoebi mniSvnelovnad ar gansxvavdeboda.
b. η2 = .09. g. orive mTavari efeqti artefaqturia.
4. a = 2, b = 3, .47,42,2,2,1,48,8 ======= sruliSecdoma dfdfdfdfdfNn AxBBAAB
5. cxrili # 1
wyaro SS df MS F davalebis tipi (A) xmauris done (B) AxB Secdoma sul
15.00 16.00 36.00 120.00 187.00
3 2 6 60 71
5.00 8.00 6.00 2.00
2.50 4.00* 3.00*
* p < .05. a. 4. b. 3. g. 6. d. 72.
e. realuriF B faqtorisaTvis da A da B faqtorebis urTierTqmedebisaTvis statistikurad mniSvnelovania.
v.
A faqtori: B faqtori A x B
ver uarvyofT oH
ver viRebT 1H
uarvyofT oH
viRebT 1H
uarvyofT oH
viRebT 1H
cxrili # 2 wyaro SS df MS F treningis xangrZlivoba (A) unaris done (B) AxB Secdoma sul
40.00 120.00 960.00 3200.00 4320.00
4 3 12 160 179
10.00 40.00 80.00 20.00
0.50 2.00 4.00*
* p < .01. a. 5. b. 4. g. 9. d. 180.
e. realuriF A da B faqtorebis urTierTqmedebisaTvis statistikurad
mniSvnelovania. v.
A faqtori: B faqtori A x B
ver uarvyofT oH
ver viRebT 1H
ver uarvyofT oH
ver viRebT 1H
uarvyofT oH
viRebT 1H
cxrili # 3 wyaro SS df MS F ukukavSiris tipi (A) davalebis sirTule (B) AxB Secdoma sul
200.00 160.00 240.00 2700.00 3300.00
4 2 8 135 149
50.00 80.00 30.00 20.00
2.50* 4.00* 1.50
* p < .05. a. 5. b. 3. g. 10. d. 150.
e. realuriF A faqtorisaTvis da B faqtorisaTvis statistikurad mniSvnelovania.
v.
A faqtori: B faqtori A x B
uarvyofT oH
viRebT 1H
uarvyofT oH
viRebT 1H
ver uarvyofT oH
ver viRebT 1H
6. a. statistikurad mniSvnelovani urTierTqmedebis gamo xatis tipis efeqti damokidebulia damuSavebis doneze. damuSavebis dabali donisaTvis xatis Cveulebrivi tipi iwvevs aRdgenis mniSvnelovnad ukeTes Sedegebs, vidre xatis ucnauri tipi. xolo damuSavebis maRali donisaTvis xatis Cveulebrivi tipi iwvevs mniSvnelovnad uares aRdgenas, vidre xatis ucnauri tipi. xatis tipis mTavari efeqti artefaqturia.
b. aqac, statistikurad mniSvnelovani urTierTqmedebis gamo, damuSavebis donis efeqti damokidebulia xatebis tipze. Cveulebrivi xatebisaTvis dabali damuSaveba iwvevs mniSvnelovnad ukeTes aRdgenas, vidre maRali damuSaveba. ucnauri xatebisaTvis aris sawinaaRmdego kavSiri; dabali damuSaveba iwvevs mniSvnelovnad ufro dabal aRdgenas, vidre maRali damuSaveba.
7. a. statistikurad mniSvnelovani urTierTqmedeba migviTiTebs, rom
xelsawyos tipis efeqti damokidebulia Zilis deprivaciis raodenobaze. rodesac Zilis deprivacia ar xdeba, sxvaoba swor wakiTxvebs Soris analoguri da cifruli xelsawyoebisaTvis aramniSvnelovania. Zilis 24-saaTiani deprivaciisaTvis cifruli xelsawyo iZleva mniSvnelovnad met swor wakiTxvas, vidre analoguri xelsawyo. xelsawyos tipis mTavari efeqti artefaqturia.
b. aqac, statistikurad mniSvnelovani urTierTqmedeba migviTiTebs, rom Zilis deprivaciis raodenobis efeqti damokidebulia
xelsawyos tipze. rogorc analoguri, iseve cifruli xelsawyoebisaTvis Zilis 24-saaTiani deprivacia Sedegad iZleva ufro nakleb swor wakiTxvas, vidre Zilis deprivaciis ararseboba. Tumca, sxvaoba swori wakiTxvebis raodenobebs Soris deprivaciis ararsebobasa da 24-saaTiani deprivaciis pirobebisaTvis ufro didia analoguri xelsawyosaTvis, vidre cifruli xelsawyosaTvis. radganac Zilis 24-saaTiani deprivacia Sedegad iZleva ufro nakleb swor wakiTxvebs rogorc analoguri, iseve cifruli xelsawyosaTvis, Zilis deprivaciis raodenobis mTavari efeqti mniSvnelovania.
Tavi XII SeamowmeT Tqveni codna XII.1 4. 199.00. 5. 33.00. 6. 25.00. 7. 96.00.
8. .64,48,12,4 ==== srulidfdfdfdf AxSSA
9. 59.
10. .00.2,00.2,00.11 === AxSSA MSMSMS
11. cxrili # 1 wyaro SS df MS F A faqtori S faqtori AxS sul
48.00 27.00 72.00 147.00
4 9 36 49
12.00 3.00 2.00
6.00
a. 5. b. 10. g. 10. cxrili # 2 wyaro SS df MS F A faqtori S faqtori AxS sul
6.00 36.00 36.00 78.00
2 12 24 38
3.00 3.00 1.50
2.00
a. 3. b. 13. g. 13. SeamowmeT Tqveni codna XII.2
3. ;543210 AAAAAH µµµµµ =====
=1H yvela Aµ -ebi ar aris erTmaneTis toli.
SeamowmeT Tqveni codna XII.3 2.
cdis piri
qalaqi (A) dalasi tihuana tripoli
X s
30.8 4.14
21.6 3.15
22.6 3.55
a. ;3210 AAAH µµµ ===
=1H yvela Aµ -ebi ar aris erTmaneTis toli. b.
wyaro SS df MS F qalaqi (A) cdis pirebi (S) AxS sul
709.000 269.333 245.667 1224.000
2 13 26 41
354.500 20.718 9.449
37.52*
* p <.01. g. 2,26.
d. .37.3)26,2( =kritikuliF
e. realurisF mniSvnelobebi meti an toli 3.37-ze.
v. ki.
z. uarvyofT oH ; viRebT 1H .
T. gansxvavebuli populaciebidan. i. ki.
k. ki, radganac realuriF statistikurad mniSvnelovania da icvleba
damoukidebeli cvladis sami done. l. tukis CD = 2.90. m. dadgenili ganedis saSualo dalasisaTvis mniSvnelovnad metia,
vidre saSualoebis tihuanasaTvis da tripolisTvis. tihuanas da tripolis saSualoebi mniSvnelovnad ar gansxvavdeba.
n. .74.2 =η 3. a. 2. b. 19. g. 4.76. d. 4.41.
e. uarvyofT oH .
v. viRebT 1H . z. ki. T. martivi da rTuli Carevis pirobebi. 4. a. 3. b. 12. g. 2.81. d. 3.44.
e. ver uarvyofT oH .
v. ver viRebT 1H .
z. ara. T. Carevis pirobebi mniSvnelovnad ar gansxvavdeba. kiTxvebi 1. a.
cdis piri
sityvis tipi bednieri neitraluri mowyenili
X s
4.5 1.41
3.0 1.20
2.5 1.20
wyaro SS df MS F sityvis tipi (A) cdis pirebi (S) AxS sul
17.333 16.667 17.333 51.333
2 7 14 23
8.666 2.381 1.238
7.00*
* p <.05.
tukis CD = 1.5. bednieri sityvebis saSualo mniSvnelovnad didia, vidre rogorc neitraluri, ise mowyenili sityvebis saSualoebi. neitraluri da mowyenili sityvebis saSualoebi erTmaneTisagan mniSvnloevnad ar gansxvavdeba.
b. .50.2 =η 2.
instruqciis piroba instruqciamde instruqciis
Semdeg
X s
2.45 0.60
2.52 0.68
a. wyaro SS df MS F instruqciebi (A) cdis pirebi (S) AxS sul
0.027 8.843 0.203 9.073
1 11 11 23
0.027 0.804 0.018
1.50
realuriF aramniSvnelovania .05 doneze. ar arsebobs imis
TvalsaCinoeba, rom instruqciebma gavlena moaxdina aRqmul iluziaze.
b. .255.150.1.261.1)11( === Ftkritikuli (mniSvnelobebis mcire gansxvaveba
t-sa da F-s Soris gamoTvlebSi damrgvalebebis Secdomis gamoa.) g. erTsa da imave daskvnebs.
3. a. ;543210 AAAAAH µµµµµ =====
=1H yvela Aµ -ebi ar aris erTmaneTis toli.
b. .59,44,11,4 ==== srulidfdfdfdf AxSSA
g. .61.2)44,4( =kritikuliF
d. uarvyofT oH ; viRebT 1H .
e. arsebobs sul mcire erTi mniSvnelovani sxvaoba saSualoebs Soris. tukis CD kriteriumi aucilebelia imisaTvis, raTa ganisazRvros specifikuri mniSvnelovani sxvaobebi.
4. cxrili # 1
wyaro SS df MS F A faqtori S faqtori AxS sul
10.668 69.834 27.332 107.834
1 11 11 23
10.668 6.349 2.485
4.29
a. 3. b. 24. g. 24. d. ara.
e. ver uarvyofT oH ; ar viRebT 1H .
cxrili # 2
wyaro SS df MS F A faqtori S faqtori AxS sul
60.00 133.00 380.00 573.00
5 19 95 119
12.0 7.0 4.0
3.00
a. 6. b. 20. g. 20. d. ki.
e. uarvyofT oH ; viRebT 1H .
5. statistikurad mniSvnelovani realuriF (2,26) = 15.99 migviTiTebs, rom
arsebobs sul mcire erTi mniSvnelovani sxvaoba saSualoebs Soris. wyvilTaSorisi Sedarebebi tukis CD-s gamoyenebiT gviCvenebs, rom yoveli saSualo mniSvnelovnad gansxvavdeba yoveli sxva saSualosagan. aqedan gamomdinare, Cvilebi yvelaze met dros nacnob saxeebs akvirdebodnen. ucnob saxeze fiqsaciis xangrZlivoba mniSvnelovnad naklebi iyo, vidre nacnob saxeze, xolo cariel ovalebze fiqsaciis xangrZlivoba mniSvnelovnad naklebi iyo, vidre nacnob da ucnob saxeebze. Tavi XIII SeamowmeT Tqveni codna XIII.1 2. ara, or qulas Soris damokidebuleba ar icvleba. 3. cxrili # 1 a. dadebiTi kavSiri. b. srulyofilze naklebi. g. ki. martoobis qulebis codna depresiis ukeTesad
winaswarmetyvelebis saSualebas iZleva, vidre am qulebis arcodna. cxrili # 2 a. uaryofiTi kavSiri. b. srulyofilze naklebi. g. ki. 4. a. susti uaryofiTi kavSiri.
b. srulyofilze naklebi. g. ara, es ar gazrdis wianswarmetyvelebis sizustes. SeamowmeT Tqveni codna XIII.2
3. a. .870.+=realurir
4. a. .764.−=realurir
5. a. .277.−=realurir
SeamowmeT Tqveni codna XIII.3 4.
2r a. b. g. d. e. v.
.0144
.5476 .3481 .1296 .0529 .0361
SeamowmeT Tqveni codna 13.4
2. .0;0:0 ≠= ρρH
3. kritikuli mniSvneloba df .05 .01 a. b. g. d. e. v.
6 13 18 30 41 73
.707 .514 .444 .349 .304 .232
.834
.641
.561 .449 .393 .302
4.
statistikuri hipoTezebi
kritikuli mniSvneloba
xvdeba uaryofis
areSi
gadawyvetileba
a.
0:0:
1
0
≠=
ρρ
HH
.576 ki
0H : uarvyofT
1H : viRebT
b.
0:0:
1
0
≠=
ρρ
HH
.576 ara
0H : ver uarvyofT
1H : ver viRebT
g.
0:0:
1
0
≠=
ρρ
HH
.444 ki
0H : uarvyofT
1H : viRebT
d.
0:0:
1
0
≠=
ρρ
HH
.423 ara
0H : ver uarvyofT
1H : ver viRebT
e.
0:0:
1
0
≠=
ρρ
HH
.232 ki
0H : uarvyofT
1H : viRebT
v.
0:0:
1
0
≠=
ρρ
HH
.666 ki
0H : uarvyofT
1H : viRebT
5. a. 550.−=realurir
b. realurikritikuli rr ;576.)10( = aramniSvnelovania.
g. ar dasturdeba hipoTeza, rom im populaciaSi, romlidanac aRebulia es qulebi, arsebobs kavSiri gacdenebsa da testis qulebs Soris.
6. statistikuri
hipoTezebi kritikuli mniSvneloba
xvdeba uaryofis
areSi
gadawyvetileba
a.
0:0:
1
0
≠=
SHH
ρρ
.490 ara
0H : ver uarvyofT
1H : ar viRebT
b.
0:0:
1
0
≠=
SHH
ρρ
.591 ki
0H : uarvyofT
1H : viRebT
g.
0:0:
1
0
≠=
SHH
ρρ
.409 ki
0H : uarvyofT
1H : viRebT
d.
0:0:
1
0
≠=
SHH
ρρ
.648 ara
0H : ver uarvyofT
1H : ver viRebT
e.
0:0:
1
0
≠=
SHH
ρρ
.438 ki
0H : uarvyofT
1H : viRebT
v.
0:0:
1
0
≠=
SHH
ρρ
.364 ki
0H : uarvyofT
1H : viRebT
kiTxvebi 1. b. dadebiTi.
g. realurikritikulirealuri rrr ;497.)14(;655. =+= statistikurad
mniSvnelovania. d. arsebobs dadebiTi kavSiri sibrazis donesa da sistolur
arteriul wnevas Soris. rac ufro maRalia sibrazis done, miT ufro maRalia sistoluri arteriuli wneva.
e. .43.2 =r
2. a. 12.−=realurir gogonebisaTvis; realurikritikuli rr ;273.)51( =
aramniSvnelovania. 32.+=realurir vaJebisaTvis;
realurikritikuli rr ;304.)40( = statistikurad mniSvnelovania.
b. 01.2 =r gogonebisaTvis. 10.2 =r vaJebisaTvis. g. gogonebisaTvis ar arsebobs kavSiri. vaJebisaTvis kavSiri dadebiTia; maRal biWebs aqvT tendencia, miiRon ufro maRali qulebi angariSis baraTebze.
3. b. dadebiTi.
g. realurikritikulirealuri rrr ;632.)8(;781. =+= statistikurad mniSvnelovania.
d. isini dadebiTad aris dakavSirebuli; rac ufro maRalia niSani semestris SuaSi, miT ufro maRali iqneba saboloo gamocdis niSani.
4. a. realurikritikulirealuri ss rrr ;545.)14(;937. =+= statistikurad
mniSvnelovania. b. popularoba da fizikuri mimzidveloba dadebiT kavSirSia im
gogonebis populaciaSi, saidanac aRebuli iyo SerCeva.
g. realurikritikulirealuri ss rrr ;545.)14(;134. =−= aramniSvnelovania.
d. ar arsebobs kavSiri vaJebis popularobasa da fizikur mimzidvelobas Soris im populaciaSi, saidanac aRebuli iyo SerCeva.
5. kolejis studentebis inteleqtis koeficienti moicavs cvladis SezRudul diapazons. inteleqtis koeficientis Zalian dabali da Zalian maRali qulebi ar Sedioda SerCevaSi. SesaZlebelia aseve, rom niSnebis saSualoebi msgavsad aris SezRuduli diapazonSi. 6. b. ar aris kavSiri.
g. realurikritikulirealuri ss rrr ;553.)11(;001. =−= aramniSvnelovania.
d. kompiuteris gamoyenebis saaTebis raodenoba da siRrmis aRqmis Secdoma ar aris dakavSirebuli gazomil SerCevaSi.
7. b. kavSiri, rogorc Cans, uaryofiTia.
g. realurikritikulirealuri ss rrr ;514.)13(;807. =−= statistikurad
mniSvnelovania. d. dRis uaryofiTi movlenebis raodenoba da skolisgan miRebuli
kmayofileba uaryofiTadaa dakavSirebuli. rac ufro metia yoveldRiuri uaryofiTi movlenebi, miT ufro dabalia skolisgan miRebuli kmayofileba.
8. a. .735.−=realurir
b. realurikritikuli rr ;482.)15( = statistikurad mniSvnelovania.
g. uaryofiTi kavSiri gviCvenebs, rom X cvladis ufro maRali qulebi dakavSirebulia Y cvladis ufro dabal qulebTan. amdenad, Tqven ufro dabal niSnebs elodebiT im megobrisagan, romelic kviraSi 15 saaTis ganmavlobaSi uyurebs televizors, vidre megobrisagan, romelic 4 saaTs uyurebs televizors kviris manZilze.
9. a. .847.+=realurir
b. realurikritikuli rr ;507.)16( = statistikurad mniSvnelovania.
g. arsebobs dadebiTi korelacia mowonebis rangsa da monawileobis rangs Soris. rac ufro maRalia mowonebis rangi, miT ufro maRalia monawileobis rangi.
Tavi XIV SeamowmeT Tqveni codna XIV.1 4. a. b = +2; a = 0; Y = +2X.
b. b = -1; a = +2; Y = -1X+2. g. b = +1.5; a = +1; Y = +1.5X+1. d. b = -2; a = +4; Y = -2X+4. SeamowmeT Tqveni codna XIV.2
3. a. 98.0 ++= XY l b.
X lY 60 65 80 85 90
57 61 73 77 81
g. Y =69.8. regresiis wrfe gadis wertilebs X = 0, Y = 9; .8.69,76 == YX
4. a. 234.1 +−= XY l b.
X lY 5 8 11 14 17
16.0 11.8 7.6 3.4 -0.8
g. Y =7.6. regresiis wrfe gadis wertilebs X = 0, Y = 23; .6.7,11 == YX
SeamowmeT Tqveni codna XIV.3 5. 4.98. 6. 4.80. 7. 4.30. SeamowmeT Tqveni codna XIV.4 1. a. +11.507. b. 202.215.
g. .215.202507.11 ++= XY l d. regresiis wrfe gadis wertilebs X = 0, Y = 202.215;
.7.261,17.5 == YX e.
X lY 6 3 9 1 4 2 3
271.3 236.7 305.8 213.7 248.2 225.2 236.7
5 8 7 8 6
259.8 294.3 282.8 294.3 271.3
v.
cdis piri
naSTi
(Y- lY ) 1 2 3 4 5 6 7 8 9 10 11 12
-54.3 -6.7 +9.2 +10.3 +22.8 -27.2 +19.3 +3.2 +26.7 +6.2 -29.3 +19.7
z. 6824.43. T. 26.12.
i. ki; XYs • naklebia Ys -ze. 2. a. b = -0.43. b. a = 24.6. g. lY =-0.43X+24.6. d. 41 skanireba, lY = 7.0.
e. 18 skanireba, lY = 16.9. kiTxvebi 1. a. +0.884. b. +0.884.
g. lY =+0.884X + 10.96. d. 79.0. e. 5.52.
2. a. lY =-0.116X + 3.23. b. 2.88. g. 2.64.
3. a. lY =+-4.3X + 128, sadac X = dReSi moweuli sigaretebis raodenobas. g.
sigaretebi dReSi
nawinaswarmetyvelebi wona dabadebisas (oz)
0 1 5 8 10 13
128.0 123.7 106.5 93.6 85.0 72.1
20 42.0
4. a. lY =+0.786X + 107.16. b.
cdis piri
nawinaswarmetyvelebi arteriuli wneva
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
157.5 130.7 136.2 137.8 120.5 130.0 135.5 133.9 141.7 132.3 136.2 144.1 146.5 139.4 143.3 155.1
d. XYs • aris Ys -is .78. 5. a. –0.562. b. 28.33.
g. lY =-0.562X + 28.33. e. 2.90. 6.
aplikanti nawinaswarmetyvelebi Sesruleba
samsaxurSi: lY
diana rodrigo kolini maria rubeni luki holi bliTi
55.2 61.2 54.0 67.2 64.8 52.8 50.4 70.8
rodrigos, marias, rubenis da bliTis samsaxurSi Sesrulebis nawinaswarmetyvelebi qulebi 60-ze metia. amdenad, am aplikantebs samsaxurSi aiyvanen.
7. a. lY = -1.317X + 89.17.
b. lY 4 saaTisTvis = 83.9; lY 15 saaTisTvis = 69.4.
g. XYs • =8.01; XYs • aris Ys -is .70. Tavi XV
SeamowmeT Tqveni codna XV.1 7.
2realuriχ
df 2kritikuliχ xvdeba
uaryofis areSi
gadawyvetilebebi
7.26
8.33
14.19
26.94
2 4 7 26
5.99
9.49
14.1
31.4
ki
ara
ki
ara
oH : uarvyofT
1H : viRebT
oH : ver uarvyofT
1H : ar viRebT
oH : uarvyofT
1H : viRebT
oH : ver uarvyofT
1H : ar viRebT
8. TiToeul ujredSi mocemuli mniSvnelobebi mosalodneli sixSireebia. a.
sveti 1 2
1 12.6 31.4 striqoni
2 15.4 38.6
2realuriχ (1) = 0.073, p>.05;
aramniSvnelovani. b.
sveti 1 2
1 19.2 29.8 striqoni
2 24.8 38.2
2realuriχ (1) = 9.256, p<.01;
statistikurad mniSvnelovani. g.
sveti 1 2
1 42.4 56.6 striqoni
2 17.6 23.4
2realuriχ (1) = 12.442, p<.01;
statistikurad mniSvnelovani.
9. 2realuriχ (2) = 0.168. es 2
realuriχ aramniSvnelovania; am magaliTSi ar
arsebobs kavSiris kviris dResa da dRis periods Soris im manqanebisaTvis, romlebic gadian SeCerebis niSanTan.
10. 2realuriχ (1) = 70.914, p<.01; hipoTeturi kavSiri arsebobs.
SeamowmeT Tqveni codna XV.2 1.
qulebi, dalagebuli umciresidan udidesamde
7 10 14 15 16 19 23 26
jgufis identoba 2 1 2 1 2 1 1 2 rangi 1 2 3 4 5 6 7 8 raodenoba SemTxvevebisa, rodesac A1 qula win uswrebs A2 qulas
3 2 1 1
raodenoba SemTxvevebisa, rodesac A2 qula win uswrebs A1 qulas
4 3 2
UA1 = 7; UA2 = 9. a. UA1. b. ara. 2. UA1 = 7; UA2 = 9.
a. ki. 4. UA1 daaxloebiT unda utoldebodes UA2-s. 5. isini sakmaod unda gansxvavdebodnen. 6.
Ukritikuli xvdeba uaryofis areSi
gadawyvetileba
a. 0 ara oH : ver uarvyofT
1H : ar viRebT
b. 16 ki oH : uarvyofT
1H : viRebT
g. 27 ara oH : ver uarvyofT
1H : ar viRebT
d. 22 ki oH : uarvyofT
1H : viRebT
e. 56 ki oH : uarvyofT
1H : viRebT
v. 87 ara oH : ver uarvyofT
1H : ar viRebT
7. ufro mcire U = UA1 = 54.5; Ukritikuli = 41; UA1 ar xvdeba uaryofis areSi. ar dasturdeba, rom jgufebi gansxvavdeba SfoTvis SefasebebSi. SeamowmeT Tqveni codna XV.3
4. Tkritikuli xvdeba uaryofis
areSi gadawyvetileba
a. 4 ki oH : uarvyofT
1H : viRebT
b. 8 ara oH : ver uarvyofT
1H : ar viRebT
g. 17 ki oH : uarvyofT
1H : viRebT
d. 46 ki oH : uarvyofT
1H : viRebT
e. 90 ara oH : ver uarvyofT
1H : ar viRebT
5. Trealuri (14) = 27; Tkritikuli (14) = 21; Trealuri ar aris statistikurad mniSvnelovani. am SerCevaSi SfoTvis done ar Seicvala pirveli TamaSidan meore TamaSamde. kiTxvebi
1. a. 2realuriχ (1) = 1.235, p > .05. skolidan garicxvis sixSire ar aris
kavSirSi sqesTan.
b. 2realuriχ (2) = 28.990, p > .01. skolidan garicxvis sixSire kavSirSia
problemis seriozulobasTan. iricxeba saSualo problemebis mqone individebis mosalodnelze ufro meti raodenoba; aseve, iricxeba seriozuli problemebis mqone individebis mosalodnelze naklebi raodenoba.
2. ∑ − )( R = 50, ∑ + )( R = 5; amdenad, Trealuri (11) = 5, p < .01. uarvyofT oH ;
davalebis tipi gavlenas axdens mTqnarebaze. a. uintereso videofilmis pirobaSi qulebi dadebiTadaa
asimetriuli. b. damTqnarebebis raodenobis ganawilebebi gansxvavebulia. ufro
meti mTqnareba xdeba, rodesac stimuli uinteresoa, vidre rodesac stimuli sainteresoa.
3. .05.,5.382 >= pU A ganawilebebi mniSvnelovnad ar gansxvavdeba.
4. .05.,5.442 >= pU A ganawilebebi mniSvnelovnad ar gansxvavdeba.
5. realurikritikulirealuri TTT .3)8(;3 == xvdeba uaryofis areSi; ase rom
“ 0H :dakavSirebuli A1 da A2 qulebis populaciis ganawilebebi
identuria” uaryofilia.
danarTi F xSirad gamoyenebuli formulebi
sixSireTa ganawilebebi daujgufebel sixSireTa ganawilebebi
qulis rf = N
rfqulis
qulis %f = (qulis rf) x 100
dajgufebel sixSireTa ganawilebebi
klasis intervalis zoma = i = raodenoba isintervaleb
minimaluriimaqsimalur − XX
intervalSi qulebis rf = N
f qulebis intervalSi
intervalSi qulebis %f = N
f qulebis intervalSix 100
procentilis rangi
XP = ( )[ ]
−+
NfiiXXcf LL / x 100
procentilis wertili
pX = +LX ( )
−
i
L
fcfNP i
centraluri tendenciis sazomebi
mediana
Mdn = 50X = +LX ( )
−
i
L
fcfN50. I
saSualo
X = N
X∑
cvalebadobis sazomebi
rangi
rangi = minimaluriimaqsimalur XX −
kvartilTSorisi rangi
IQR = 2575 XX −
semikvartilTSorisi rangi
ISQR = 2
IQR =2
2575 XX −
dispersia
populaciis dispersia
σ 2 = populaciaN
X∑ − )( 2µ
SerCevis dispersia
2S =N
XX∑ − 2)(
populaciis dispersiis Sefaseba
ganmartebiTi formula: 2s =1
)( 2
−
−∑N
XX
kvadratebis jamebis formula: 2s =1−N
SS
gamoTvliTi formula: 2s =[ ]
1/)( 22
−
−∑ ∑N
NXX
standartuli gadaxra populaciis standartuli gadaxra
σ = 2σ
SerCevis standartuli gadaxra
S = 2S = N
XX∑ − 2)(
populaciis standartuli gadaxris Sefaseba
ganmartebiTi formula: s = 2s = 1
)( 2
−
−∑N
XX
kvadratebis jamis formula: s = 1−N
SS
gamoTvliTi formula: s = ( )[ ]
1/22
−
−∑ ∑N
NXX
saSualos standartuli Secdoma
σ cnobili
σ X = Nσ
σ ucnobi
s X = Ns
saSualoebs Soris sxvaobis standartuli Secdoma
σ cnobili
σ 21 AXAX − =
2
22
1
21
A
A
A
A
nnσσ
+
σ ucnobi
s 21 AXAX − =
( ) ( )
+
−+
−+−
2121
222
211 11
211
AAAA
AAAA
nnnnsnsn
albaToba xdomilebis albaToba (p)
p (xdomileba) = ipopulaciaS raodenoba isxdomilebeb iSesaZlebel
ipopulaciaS raodenoba xdomilebis
z qulebi da standartuli qulebi
erTeuli qula
z = σ
µ−X
standartuli qula
z = S
XX − an z =
( )∑ −
−
NXX
XX
/2
SerCevis saSualo
z = X
Xσ
µ−
t kriteriumebi t kriteriumi erTi SerCevisaTvis
t = σ
µ−X an t =
NsX/
µ−, df = N – 1
t kriteriumi damoukidebeli jgufebisaTvis
ganmartebiTi formula: elidamoukidebt = 21
21
AXAX
AA
sXX
−
−
gamoTvliTi formula: elidamoukidebt = ( ) ( )
+
−+
−+−
−
2121
222
211
21
112
11
AAAA
AAAA
AA
nnnnsnsn
XX,
df = N – 2 t kriteriumi damokidebuli jgufebisaTvis
lidamokidebut =
( )1/)( 2221
−
−
−
∑ ∑wyvilebiwyvilebi
wyvilebi
NNNDD
XX AA, df = 1−wyvilebiN
dispersiuli analizi cdis pirTa Soris erTfaqtoriani sqema wyaro SS dfa MS F A faqtori Secdoma sul
( )∑∑ −2
GA XX
( )∑∑ −2
AXX
( )∑∑ −2
GXX
a – 1 N – a N - 1
AA dfSS /
SecdomaSecdoma dfSS /
ar gamoiTvleba
SecdomaMSMS A /
a a = A faqtoris doneebis raodenoba, N = qulebis sruli raodenoba.
cdis pirTa Soris orfaqtoriani sqema wyaro SS dfa MS F A faqtori B faqtori A da B faqtorebis urTierTqme- deba Secdoma sul
ASS
BSS
AxBSS
SecdomaSS
sruliSS
a – 1 b – 1 (a – 1)( b – 1) ab( ABn -1) N - 1
A
A
dfSS
B
B
dfSS
AxB
AxB
dfSS
Secdoma
Secdoma
dfSS
ar
gamoiTvleba
SecdomaMSMSA
SecdomaMSMSA
SecdomaMSMS AxB
a a = A faqtoris doneebis raodenoba; b = B faqtoris doneebis raodenoba;
ABn = qulebis raodenoba TiToeul ujredSi; N = qulebis sruli
raodenoba. cdis pirTa SigniT erTfaqtoriani sqema wyaro SS dfa MS F A faqtori S faqtori AxS sul
ASS
SSS
AxSSS
sruliSS
a – 1
An – 1
(a – 1)( An – 1)
N - 1
A
A
dfSS
S
S
dfSS
AxS
AxS
dfSS
ar gamoiTvleba
AxSMSMS A
a a = A faqtoris doneebis raodenoba; An = qulebis raodenoba Carevis
pirobaSi, anu, Sesabamisad, cdis pirebis raodenoba; N = qulebis sruli raodenoba.
tukis HSD mravaljeradi Sedarebis kriteriumi cdis pirTa SigniT erTfaqtoriani sqemisaTvis
CD = qAn
SecdomaMS
cdis pirTa Soris erTfaqtoriani sqemisaTvis
CD = qA
AxS
nMS
ubralo efeqtebisaTvis cdis pirTa SigniT erTfaqtorian sqemaSi
CD = qABn
SecdomaMS
efeqtis siZlieris sazomebi eta kvadrati t kriteriumisaTvis
dftt
+= 2
22
realuri
realuriη
eta kvadrati cdis pirTa Soris erTfaqtoriani disperiuli analizisaTvis
sruliSSSS A=2η
an
( )( ) Secdomarealuri
realuri
dfFdfFdf
A
A
+=
))((2η
eta kvadrati cdis pirTa SigniT erTfaqtoriani disperiuli analizisaTvis
AxSA
A
SSSSSS+
=2η
eta kvadrati cdis pirTa Soris orfaqtoriani disperiuli analizisaTvis
sruliSSSS A=2η A faqtorisaTvis
sruliSSSSB=2η B faqtorisaTvis
sruliSSSS AxB=2η A da B faqtorebis urTierTqmedebisaTvis
korelacia pirsonis korelaciis koeficienti
ganmartebiTi formula: ( )( )
( )[ ] ( )[ ]∑∑∑
−−
−−=
22YYXX
YYXXr
jvaredini namravlis formula: ( )( )YX
XY
SSSSCPr =
gamoTvliTi formula:
( )( ) ( )( )( )( ) ( )[ ]( )( ) ( )[ ]2222 ∑∑∑∑
∑∑∑−−
−=
YYNXXN
YXXYNr
wyvilebiwyvilebi
wyvilebi
standartuli qulis formula: ( )wyvilebiN
zzr YX∑= ,
df = Nwyvilebi – 2 yvela formulisaTvis. determinaciis koeficienti r2 spirmenis ranguli korelaciis koeficienti
( )( )
−−= ∑
16
1 2
2
wyvilebiwyvilebi NND
rs
regresia
wrfis gantoleba
Y = bX + a
wrfis daxriloba
aSimniSvnelob isX cvlileba
aSimniSvnelob is cvlileba
−−
=Y
b = 12
12
XXYY
−−
umcires kvadratTa meTodiT miRebuli regresiis wrfis gantoleba
abXY +=′ umcires kvadratTa meTodiT miRebuli regresiis wrfis daxriloba
gadaxraTa namravlebis jamis formula: ( )( )
( )∑∑
−
−−= 2
XX
YYXXb
jvaredini namravlis formula: X
XY
SSCPb =
gamoTvliTi formula: ( )( ) ( )( )
( )( )( )22 ∑∑∑∑∑ −
=YXN
YXXYNb
wyvilebi
wyvilebi
korelaciuri formula:
=
Y
X
ssrb
umcires kvadratTa meTodiT miRebuli regresiis wrfis Y- TanakveTa
a = XbY − Sefasebis standartuli Secdoma
ganmartebiTi formula: 2−
=•wyvilebi
naSTi
NSSs XY
an
( )2
2
−
−= ∑
•wyvilebiN
YYs
l
XY
gamoTvliTi formula:
( ) ( ) ( ) ( )( )[ ]( )
−
−−−
−= ∑
∑∑∑∑∑∑• 22
222
21
XXN
YXXYNYYN
NNs XY
wyvilebi
wyvilebi
wyvilebi
wyvilebiwyvilebi
korelaciuri formula: ( )2121
realuri
wyvilebi
wyvilebi rNN
ss YXY −
−
−=•
araparametruli kriteriumebi
xi-kvadrat damoukideblobis kriteriumi
( )∑∑ −
=rc
rcrc
EEO 2
2χ , df = (r - 1)(c - 1)
=
sixSire
imosalodnel
ujredis
= ( )( )
raodenoba sruli pasuxebis
sveti imarginalur isujredisaTvstriqoni imarginalur isujredisaTv
Tanxmobis xi-kvadrat kriteriumi
( )∑ −
=c
cc
EEO 2
2χ , df = c – 1
man-uitnis U kriteriumi
( ) ∑−+
+= 111
211 21
AAA
AAA RnnnnU
( ) ∑−
++= 2
22212 2
1A
AAAAA RnnnnU
an
1212 AAAA UnnU −= uilkoqsonis niSniani rangebis kriteriumi
T = ufro mcires ( )∑ − R da ( )∑ + R -dan