ear 11 mathematics ias 1 - nulake 1.10 sample.pdf · • achievement standard ... which is for ncea...

• AchievementStandard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2• TheStatisticalEnquiryCycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3• TheComparativeStatisticalProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4• ThePlan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7• TheData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 – CensusAtSchool’sDataViewer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 – CleaningtheData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11• TheAnalysis–Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 – MeasuresoftheCentre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 – Quartiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 – MeasuresofSpread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21• TheAnalysis–TheDataDisplay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 – DotPlots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 – StemandLeafPlots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 – Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 – BoxandWhiskerPlots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27• DataDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35• CensusAtSchool’sData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40• SamplesandthePopulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 – MakinganInference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46• PracticeInternalAssessment1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51• PracticeInternalAssessment2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55• Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59• OrderForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Year 11Mathematics

ContentsRobert Lakeland & Carl Nugent

Multivariate Data

uLake Ltdu a e tduLake LtdInnovative Publisher of Mathematics Texts

IAS 1.10

IAS 1.10 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent

2 IAS 1.10 – Multivariate Data

◆ Usingthestatisticalenquirycycleinvolvesusingeachcomponentofthestatisticalenquirycycletomakecomparisons .

◆ Usingthestatisticalenquirycyclewithjustificationinvolveslinkingaspectsofthestatisticalenquirycycletothecontextandthepopulationandmakingsupportingstatementswhichrefertoevidencesuchassummarystatistics,datavalues,trendsorfeaturesofvisualdisplays .

◆ Usingthestatisticalenquirycyclewithstatisticalinsightinvolvesintegratingstatisticalandcontextualknowledgethroughoutthestatisticalenquirycycle,andmayinvolvereflectingontheprocessorconsideringotherexplanationsforthefindings .

◆ Studentsneedtobefamiliarwiththestatisticalenquirycycletoinvestigateagivenmultivariatedataset,whichinvolves:

❖ investigatingdatathathasbeencollectedfromasurveysituation

❖ posinganappropriatecomparisonquestionusingagivenmultivariatedataset

❖ selectingandusingappropriatedisplay(s)

❖ givingsummarystatisticssuchasthefivesummaryvalues(minimum,maximum,median,quartiles)

❖ discussingfeaturesofdistributionscomparatively,suchasshape,middle50%,shift,overlap,spread,unusualorinterestingfeatures

❖ communicatingfindings,suchasinformalinferenceandsupportingevidence,inaconclusion .

NCEA 1 Internal Achievement Standard 1.10 – Multivariate DataThisachievementstandardinvolvesinvestigatingagivenmultivariatedatasetusingthestatisticalenquirycycle .

◆ ThisachievementstandardisderivedfromLevel6ofTheNewZealandCurriculum,Learning Media .Theachievementstandardisalignedtothefollowingachievementobjectivestakenfrom theStatisticalInvestigationthreadoftheMathematicsandStatisticslearningarea:

◆ Planandconductsurveysandexperimentsusingthestatisticalenquirycycle

❖ determiningappropriatevariables

❖ cleaningdata

❖ usingmultipledisplays,andre-categorisingdatatofindpatterns,variations,inmultivariatedatasets

❖ comparingsampledistributionsvisually,usingmeasuresofcentre,spread,andproportion

❖ presentingareportoffindings .

◆ Planandconductinvestigationsusingthestatisticalenquirycycle

❖ justifyingthevariablesused

❖ identifyingandcommunicatingfeaturesincontext(differenceswithinandbetweendistributions),usingmultipledisplays

❖ makinginformalinferencesaboutpopulationsfromsampledata

❖ justifyingfindings,usingdisplaysandmeasures .

Achievement Achievement with Merit Achievement with Excellence• Investigateagiven

multivariatedatasetusingthestatisticalenquirycycle .

• Investigateagivenmultivariatedatasetusingthestatisticalenquirycyclewithjustification .

• Investigateagivenmultivariatedatasetusingthestatisticalenquirycyclewithstatisticalinsight .



The Comparative Statistical Problem

Asking the QuestionWhatisitthatyouwouldlikemoreinformationon?Ifyouhadthecapacitytocollectanydatafromallthestudentsinyourclassyoumaydecidetolookinto,forexample:

◆ thestrengthofstudents◆ differentreactiontimes◆ howmanytextmessagesweresentor

receivedetc .Allthisdataiscalledquantitativeornumericdata,itistheresultofmeasuringorcounting .Itisthemeasureofaquantityhencethetermquantitative .Theinformationbecomesinterestingifyouareabletoshowonegroupwithinyourpopulationisdifferentfromanothergroupinsomeway .Groupsthatwelooktocomparewithinthepopulationarecalledqualitativegroups .Theyaredifferentdependinguponsomequalitythatdefinesthem .Possibleexamplesofqualitativegroups(orsubgroups)are:

◆ gender–maleorfemale◆ transport–bike,walk,carorbus◆ yearlevel–Year9,10,11,12or13◆ phone–nil,basicorsmart .

Possiblythegroupofstudentsthatbiketoschoolisdifferentfromthegroupthatwalkbysomecharacteristicthatyoucanmeasure .If,whenlookingatthequalitativegroupsyouaskyourself‘I wonder if ...’thencompletingthequestionoftenidentifiesthevariableyouwillusetocompareandthequestionyouwillinvestigate .Remembertofitthecriteriaforaninvestigationofmultivariatedata,youmustidentify

◆ twoormoresubgroupsofthepopulation .◆ variablesthatcanbemeasuredwhichare

likelytoshowdifferences .Ifweweregivenasamplefromthe‘SuperRugby’ofNewZealandandSouthAfricanplayers .Welookatthequalitativegroups(countryorposition)andstarttowonderif,forexample,thereisadifferencebetweentherugbyplayersofthetwocountries .Ourquestioncouldbe:DoSouthAfricanrugbyplayersinSuperRugbytendtobeheavierthantheNewZealandSuperRugbyplayersfromthepopulationofSuperRugbyplayersplayinginthecurrentyear?

Statistical LiteracyThepopulationisthegroupyouwishtoinvestigate .

Asampleispartofthepopulationthatyouwillusetoseeifyoucanconcludethatdifferencesexistinthepopulation .

Aquantitativevariableistheresultofmeasuringorcounting .Itisameasureofhowmuch(quantity)wehave .

Aqualitativevariableisacategorythatdiffersbysomequality .Forexample,genderoryearlevelarequalitativevariables .

Weusequalitativevariablestoidentifysubgroupsofthepopulationtocompareandquantitativevariablestoseeiftherearedifferencesinthesequalitativegroups .

Aquestionmustidentify◆ thepopulationthesamplewastakenfrom .◆ thequalitativegroupstobecompared◆ thequantitativevariablewhichwillbeused

tocomparethetwogroups◆ thedirectionofanydifference .

Cntry. Ht. Wt. PositionNZ 197 112 Forward

NZ 192 104 ForwardSA 178 109 ForwardSA 190 94 BackNZ 185 98 BackSA 199 114 ForwardNZ 180 90 BackNZ 188 100 ForwardSA 183 114 ForwardSA 183 83 BackSA 188 96 ForwardNZ 192 109 ForwardNZ 187 101 BackSA 191 110 BackNZ 180 103 ForwardNZ 174 92 BackSA 183 91 BackSA 180 88 BackNZ 183 91 BackSA 192 120 ForwardNZ 184 91 Back

Asampleof‘SuperRugby’players’statistics .

©uLake LtduLake Ltd Innovative Publisher of Mathematics Texts

9IAS 1.10 – Multivariate Data


Census At School’s Data Viewer

TherearetwoapproacheswithCensusatSchooltogettingyoursample .Wewillexplainbothbutrecommendthesecondapproachofdownloadingthedatasoyoucan‘clean’thedatapriortoanalysingit .

Thedataviewerisdesignedtotakeasampleanddisplaythisasadotplotoverlaidontopofaboxplot .

Select‘Usethedataviewer’ .

Thenextpageistoconfirmthatyouagreewiththeconditionswhichmeansyoumustusethedataforeducationandneedpermissiontousethedataforanythingelse .

Agreetotheconditionsandyouwillbeaskedsomequestionsaboutwhatdatayouwanttoanalyse .Itisdesignedtohelpyouconsideryourstatisticalquestionbutassumingyouhavealreadycompletedthis,selectyourpopulationandsamplesize .

InthisexamplethepopulationistheCensusAtSchool2013dataandoursamplesizeis100 .Donotselect‘Iamayear12student’asitwilladdaninformalconfidenceintervaltoyourdisplay,whichisforNCEA2AchievementStandardIAS2 .9 .

Nowselect‘GetData’ .Yoursamplewillbedrawnandyouwillbeaskedfortwovariablestoanalyse .

Fromyourstatisticalquestionyouselectyourqualitativevariable(forthisexamplewehaveselectedgender)andyourquantitativevariable(inthisexampletheresultsofthememoryexercisestudentscompletedaspartofthequestionnaire) .

Nowselect‘Addsummaries’and‘DoAnalysis’ .Thedotplotandboxplotwillbedrawnforthetwocomponentsofyourqualitativevariable(inthiscasegender) .Youwillbeabletocomparetheresultsforyourquantitativevariable .

Notethatalthoughweselected100inthesampleforthisexampleonlytheresultsfor98ofoursamplearedisplayed .Twostudentsfromthesampleleftoneofthetwovariablesblankandthesewere‘cleaned’fromtheanalysis .

TheadvantageoftheDataViewerisitisquicktouseandgivesyouagooddisplay .Youcancontrolorrightclickontheimagetosaveorcopytheoutput .

Adisadvantageistheremaybeotherdatavaluesyoushouldremove(seeCleaningtheDataonPage11) .




39. Acleaningcompanyislookingatthetotaltimerequiredtocleantwoschoolsofsimilarsizeandarea .Thetotalcleaningtime(inhoursperday)afterfiveweeksare:

School1: 36,39,42,29,45,29,35,42,38,29,42,34,42,37,36,96,78,32,28,34,41,42,30,38and42 .

School2: 28,35,43,52,27,67,61,42,28,64,31,29,63,59,28,29,57,32,61,29,48,73,36,56and62 .

40. Apharmaceuticalcompanyiscomparingthetimeforpatientstorecoverfromillnessusingtheirdrug‘A’andacompetitor’sdrug‘B’ .Timesareindaysandtheshortertimesarebetter .

DrugA:3,4,5,11,13,7,6,na,8,13,6,5,8,11,3,6,9,12,6,6,9,46,5,8,8,11,18,5,7,5and9days .

DrugB:7,8,9,8,5,10,11,12,7,8,6,7,9,21,8,7,12,11,6,14,9,8,21,9,22,15,8,18,9,8and31days .

Merit/Excellence–Investigateeachtwinsetofdatabycalculatingrelevantstatistics(averages,quartiles,minimumandmaximum)foreachsampleandthenusethesestatisticstodescribeanysimilaritiesanddifferences .




Dot plot of the half-days absent from school

0 1 52 3 4 6 7

10

8

6

4

2

8 9 121110

16

14

12

Half-days absent

Freq

uenc

y

The standard deviation is a numeric measure of

spread. In this case 2.2 half-days.

2.22.2

We can use a graphics calculator to find the range, inter-quartile range and the standard deviation.

Measures of SpreadOftenwewantafigurethatdescribeshowspreadoutourdatais .

The RangeTheRangeisthedifferencebetweenthemaximumandminimumvalues .Therangeisveryeasytocalculateandiseasilyunderstood,butifthereisanextremedatavalue,thenthemaximumminustheminimumdoesnotreflecthowspreadoutthemajorityofthedatavaluesare .

The Inter-quartile RangeTheInter-quartile Rangeisthedifferencebetweentheupperquartileandthelowerquartileandgivesustherangeofthemiddle50%ofthedata .Becauseextremevaluesaregoingtobeoutsidethemiddle50%ofthedata,theinter-quartilerangegivesaclearindicationofthespread .

The Standard DeviationAcalculatororcomputerwillalsocalculatetheStandard Deviation .Thisisanumericalmeasureofspreadwhichisnotaseasytointerpretbutyoucansafelyconcludethatasetofdatawithalargerstandarddeviationismorespreadoutthanasetofdatawithasmallerstandarddeviation .Forthedotplotabovethestandarddeviationisabout2 .2halfdays .

Withasymmetrical‘bellshaped’distributionofdata,onestandarddeviationisthedistancefromthemeantoapproximatelywherethecurvestopsgettingsteeperandstartstolevelouttothepeakofthemean .Therearetwostandarddeviationstothecorrespondingpointontheothersideofthecurve .Ifwegoonestandarddeviationeithersideofthemeanitincludesapproximatelytwothirdsofthedatavalues .

To calculate the range we find the maximum and minimum and subtract them. Similarly with the inter-quartile range, we will subtract the lower quartile from the upper quartile.

The symbol for standard deviation is usually σx.Dot plot of the half-days absent from school

0 1 52 3 4 6 7

10

8

642

8 9

The inter-quartile range measures the spread of the middle 50%. In this case the inter-quartile range is 3.

121110

16

14

12

Half-days absent

The range measures the spread of all the data, that is the maximum minus the minimum. In this case the range is 12.

Freq

uenc

y

If we go one standard deviation either side of the mean it approximately includes two thirds of the data values.




Dstandsfordatadistribution .Itistoremindyoutolookatyoursampletoseehowthedataisdistributed .Lookattwotypesofgraphs .

i includingthefollowing

SShift .Lookatstatisticsandgraphsofyourcategories .Hastherebeenashiftindatafromonetotheother?Areresultstendingtoincreaseordecrease?

CCentre .Lookatthemediansasthebestmeasureofthecentre .Whatdotheytellyouaboutyouabouttypicalresults?

Uunusualfeatures .Arethereanyextremevalues?Isthegraphskewedtowardsoneend?Group2hereisskewedleft(directionofitstail)andhasan

extremevalueat2 .5 .

SShape .Weoftenexpecta‘bellshaped’distributionasperGroup2 .Howwouldyoudescribetheshapeofyourdistribution?Doyouhaveabimodaldistribution?

S Spreadofyourresults .Howdoestheinter-quartilerangecompareforyourtwocategories?

Ifthedistributionisnotsymmetricalbutispredominantlyatoneendthenwedescribethedistributionasskewed .Itcouldbeskewedtotheleftortotheright .Aleftskewedresulthasthetailontheviewer’sleft .Arightskewedresulthasthetailontheright .Themedianwillstillhavehalfthedataoneithersideofitbutthemeanwillbeonthetailsideofthemedian .

Askewedresultonaboxandwhiskerplotisshownbelow .Againitisleftskewedasthelongtailisontheviewer’sleft .

Ifourdistributionhastwodistinctpeakswedescribethedistributionasbimodal(morethanonemode) .

Intheexampleaboveofabimodaldistribution,wecanseetheshapeofthedistributionwiththehistogrambutthisinformationislostintheboxandwhiskerplotofthesamedata .Itshowstheimportanceofpresentingahistogram(oradotplot)inadditiontotheboxandwhiskerplot .

20 m 40 m 60 m

Box and Whisker plot of Distances Thrown

0 m

01020304050607080

20 m 40 m 60 m

Histogram of Distances Thrown

A left skewed graph has the tail on the left side.

Distribution Terms cont...Use the Mnemonic DiSCUSS

Acknowledgment: This mnemonic was suggested to us by Claire Laverty of Bayfield High School.

010203040

20 m 40 m 60 m

Distances Thrown

A bimodal graph has two peaks

f

020 m 40 m 60 m

0 208 16124

Grp. 1

Grp 2

0 208 16124Grp. 1

Grp 2

0 208 16124

Grp. 1

Grp 2

0 208 16124Grp. 1

Grp 2

0 208 16124

Grp. 1

Grp 2

0 208 16124Grp. 1

Grp 2

0 208 16124

Grp. 1

Grp 2

0 208 16124Grp. 1

Grp 2




Making an InferenceAboxandwhiskerplothastheadvantagethatyoucaneasilyseetheminimum,quartiles,medianandmaximum .Wewillconcentrateonthisformofgraphbutotherforms(dotplots,histogramsetc .)shouldalsobeused .InIAS1 .10youshoulddisplayyourdatausingatleasttwodifferenttypesofgraphs .Inexplaininghowtodecideifthedifferencesyouseeinapairofboxandwhiskerplotsarelikelytobesimilarinthepopulationwewillconcentrateonthespreadbetweenthetwoquartileswhichwewillcallthesample’s‘middlespread’ .Thespreadofthesamplemiddle50%ofresults .

Wehavetodecidewhenadifferencebetweensamplesislikelytoindicatethatadifferencealsoexistsinthepopulation .Thefirsttwogeneralisationsareforsamplesofsize20to40 .1. Nooverlapbetweenthetwosamples‘middle

spreads’ .Wecanconcludethereisadifferenceinthepopulation .Withnooverlapofthe‘middlespreads’thenthewholeboxor‘middlespread’ofonesubgroup(females)isbelowthewholeboxoftheothersubgroup(males)orviceversa .

2. Thesample‘middlespread’ofonesubgroupisabovethemedianofthesecondsubgroup .

10 20 30 40 50 60

Throwing competition

Male

(m)

75% is below 50%

Female

10 20 30 40 50 60


(m)

From lower quartile to upper quartile = ‘middle spread’

10 20 30 40 50 60


Male

(m)

75% is below 75%

Female

Wecanagainconcludethatthereislikelytobeadifferenceinthepopulation .Withthemedianofonegroupbeingabovetheupperquartilethen75%ofthatsamplemustbeabove50%oftheothersample .

Wherethereisanoverlapbetweenthetwosamples‘middlespreads’wecanstillidentifywhetheradifferenceislikelytoexistinthepopulationifwehaveasufficientlylargesample .

3. Withasamplesizeofatleast30,lookatthedifferencesinthetwomedians .Visuallymultiplythisbythree .Inthiscasethedifferenceis7times3=21 .

Wecanconcludethatthereislikelytobeadifferenceinthepopulationifthis‘threetimesspread’isgreaterthanthedifferencefromthehighestupperquartiletothelowestlowerquartile .Wecallthisthe‘Spread of the middle spreads’ .

Thesethreecaseswillaccountformostoftheexampleswherewecanconcludethereisadifferenceinthepopulation .Ifthereisalargersamplesizethenthemultiplierofthedifferencebetweenthemeansincreases .

4. Wherethereisasamplesizeofatleast100lookatthedifferencesinthetwomedians .Visuallymultiplythisbyfive .

Wecanagainconcludethatthereislikelytobeadifferenceinthepopulationifthis‘timesfivespread’isgreaterthanthedifferencefromthehighestupperquartiletothelowestlowerquartile,the‘Spread of the middle spreads’ .

10 20 30 40 50 60

Throwing competition (n ≥ 30)

Male

(m)

Female

Median differenceTimes three

10 20 30 40 50 60


Male

(m)

Female

Median differenceTimes five

(n ≥ 100)




Page 15 cont...32. Bestmeasureismedianweight

of3 .5kgashalftheclasshavebagslessthanorgreaterthanthisfigure .Ifmostofthebagsabovethemedianarecloseto3 .5kgandmostofthebagsunderthemedianarecloseto0kgthenthemeanwillbelowerthanthemedian .

Page 1833.

Theaveragetimespentonhomeworkis75minutesandhalfthestudentsspendbetween30and100minutesonhomework .

34.

ThespendingonChristmaspresentsaveraged$72 .50(mean$82 .30)withhalfthestudentsspendingbetween$35and$100 .Themaximumspentwas$275 .

35.

Onequarterofthestudentsreceivedatleast450textslastmonth .Themediannumberoftextsreceivedwas260andhalfthestudentsreceivedbetween125and450texts .

Page 12 cont...

19. • SexneedssameunitssochangeboytoMandgirltoF .

• Facebook .StudentAisfine .StudentBat5lookswrongandcouldbeminutes,check .

• Facebook .StudentFat0maynothaveaFacebookpageinwhichcasenaisbetter .

• UnitsneedtobethesamesochangeallofFacebooktominutes .StudentGlookslike1hour20minutessocheck .

•Texts .CheckStudentB .•Timetogethome .StudentCeitherhasajoborthereissomethingsuspiciousinthetime .Itisanoutlier .StudentFlivesverycloseortheunitsarehours,check .

20. • Ageneeds<18or≥18socorrectShopperCandShopperG .

• TimeinMallmaybesuspicious .Iftakenwhentheyleavecouldbefinebutiftakeninsidetheymayhavejustarrived .

• NumberofparcelslooksapoorchoiceofvariableparticularlyasChas0buthasspent$25whileGhas11butonlyspent$30 .

• Amountspentshouldbelimitedtotypesofpurchase .CheckifFintendstoshopandifnecessaryreplacewithna .

Page 14

21. Mean=14 .75Median=15Mode=8,15and21

22. Mean=28 .3Median=19Mode=0

Statistics TimeMinimum 0min .LowerQ . 30min .Mean 79min .Median 75min .UpperQ . 100min .Maximum 250min .

Statistics DollarsMinimum $15LowerQ . $35Mean $82 .30Median $72 .50UpperQ . $100Maximum $275

Statistics TextsMinimum 32LowerQ . 125Mean 351Median 260UpperQ . 450Maximum 1250

Page 14 cont...

23.

Year11appeartypicallytosendtwiceasmanytextsonaSaturdaycomparedtoaMonday .

24.

Year13appeartypicallytosendovertwiceasmanytextsonaSaturdaycomparedtoaMonday .

25.

Year11andYear13typicallyappeartosendasimilarnumberoftextsonaMonday .

26.

Year13typicallyappeartosendapproximately20%moretextsthanYear11onaSaturday .

Page 1527. a) Totalofclass=1100

Newmean=57%(0dp) b) Newstudent=10%28. a) Totalofclass=850 b) Classmean=56 .7%

Therearemoregirlsandtheirmeanmarkishigher .

29. a) Mean=1 .35hours Median=1 .5hours

b) Mean=81minutes Median=90minutes

30. a) Mean=37minutes Median=30minutes

b) Mean=397minutes Median=390minutes

31. Bestmeasureismediandistanceof2 .3kmashalftheclasstraveleachsideofthisfigure .Afewstudentsabovethemediantravellingalongwaywillraisethemeanbutnotaffectthemedian .

Y11Texts Mon . Sat .Mean 14 .7 29 .2Median 11 28 .5

Y13Texts Mon . Sat .Mean 15 .2 34 .9Median 15 35

Monday Y11 Y13Mean 14 .7 15 .2Median 11 15

Saturday Y11 Y13Mean 29 .2 34 .9Median 28 .5 35


ear 11 mathematics ias 1 - nulake 1.10 sample.pdf · • achievement standard ... which is for ncea...

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