Sai s trong ha hc phn tch

1

Chng 1: CC DNG SAI S TRONG HA PHN TCH 1.1. Sai s v cch biu din sai s

Sai s (error) l s sai khc gia cc gi tr thc nghim thu c so vi gi tr mong mun. Tt c cc s liu phn tch thu c t thc nghim u mc sai s. Sai s php o dn n khng chc chn ( khng m bo o) ca s liu phn tch. C hai loi sai s c biu din ch yu trong Ha phn tch l sai s tuyt i v sai s tng i.

1.1.1.Sai s tuyt i (EA) (Absolute error)

L s sai khc gia gi tr o c (xi) vi gi tr tht hay gi tr qui chiu c chp nhn (k hiu l ).

EA = xi - Sai s tuyt i c gi tr m hoc dng, cng th nguyn vi i lng o v

khng cho bit chnh xc ca phng php.

* Gi tr qui chiu c chp nhn: (accepted refrence value): l gi tr c chp nhn lm mc so snh, nhn c t:

a) gi tr l thuyt hoc gi tr c thit lp trn c s cc nguyn l khoa hc; b) gi tr c n nh hoc chng nhn trn c s th nghim ca mt s t chc

quc gia hoc quc t;

c) gi tr tho thun hoc c chng nhn trn c s th nghim phi hp di s bo tr ca mt nhm cc nh khoa hc hoc k thut;

d) k vng ca i lng (o c), ngha l trung bnh ca mt tp hp nht nh cc php o khi cha c a), b) v c).

1.1.2. Sai s tng i (ER) (Relative error)

L t s gia sai s tuyt i v gi tr tht hay gi tr bit trc, c chp nhn.

ER = ix hay ER % =

AE . 100%

* Sai s tng i cng c th biu din di dng phn nghn (parts per thousand-ppt)

ER = AE

. 1000 (ppt)

Sai s tng i cng c gi tr m hoc dng v khng c th nguyn, c dng biu din chnh xc ca phng php phn tch.

2

Th d 1.1: Kt qu xc nh hm lng aspirin trong mt mu chun c biu din hnh 1.1. Hm lng ng ca aspirin trong mu chun l 200 mg. Nh vy, php o mc sai s tuyt i t -4mg n +10mg v sai s tng i t -2% n +5% (hay 20ppt n 50ppt).

195 200 205 210

Sai s tuyt i (EA : mg) -5 0 5 10

Sai s tng i (Er : % ) -2,5 0 2,5 5

Hnh 1.1: Sai s tuyt i v sai s tng i khi phn tch aspirin trong mu chun.

1.2. Phn loi sai s

1.2.1. Sai s h thng hay sai s xc nh (Systematic or determinate error):

L loi sai s do nhng nguyn nhn c nh gy ra, lm cho kt qu phn tch cao hn gi tr thc (sai s h thng dng -positive bias) hoc thp hn gi tr tht (sai s h thng mnegative bias).

Sai s h thng gm:

- Sai s h thng khng i (constant determinate error): loi sai s ny khng ph thuc vo kch thc mu (lng mu nhiu hay t). Do , khi kch thc mu tng th nh hng ca sai s ny hu nh khng ng k v c loi tr bng th nghim vi mu trng (blank sample).

- Sai s h thng bin i (proportional determinate error): loi sai s ny t l vi kch thc mu phn tch, khong cch gia cc tr o lun bin i theo hm lng (nng ), do rt kh pht hin. Sai s h thng bin i rt kh pht hin tr khi bit r thnh phn ho hc ca mu v c cch loi tr ion cn.

Sai s h thng khng i v bin i c biu din trn hnh 1.2.

3

Sai s h thng phn nh chnh xc ca phng php phn tch. Hu ht cc sai s h thng c th nhn bit c v c loi tr bng s hiu chnh nh phn tch mu chun hay loi tr nguyn nhn gy ra sai s.

Cc nguyn nhn gy sai s h thng c th gm:

- Sai s do phng php hay quy trnh phn tch nh: Phn ng ho hc khng hon ton, ch th i mu cha n im tng ng, do ion cn tr php xc nh

- Sai s do dng c nh: dng c cha c chun ho, thit b phn tch sai, mi trng phng th nghim khng sch.

- Sai s do ngi phn tch nh: mt nhn khng chnh xc, cu th trong thc nghim, thiu hiu bit, s dng khong nng phn tch khng ph hp, cch ly mu phin din, dng dung dch chun sai, ho cht khng tinh khit, do nh kin c nhn (nh phn tch kt qu sau da trn kt qu trc) ...

* Cch loi tr sai s h thng:

- Tin hnh th nghim vi mu trng: Mu trng l mu khng c cht phn tch nhng c thnh phn nn ging nh dung dch mu phn tch.

- Phn tch theo phng php thm chun loi tr nh hng ca cc cht cn tr.

- Phn tch mu chun (hay mu chun c chng nhn- mu CRM: Mu chun l mu thc c hm lng cht cn phn tch bit trc, c dng nh gi chnh xc ca phng php.

Khi lng mu (g)

Khi lng cht phn tch (mg)

Gi tr ng

Sai s h thng khng

i Sai s h

thng bin i

Hnh 1.2: Biu din sai s h thng khng i v bin i

4

- Phn tch c lp: khi khng c mu chun th phi gi mu phn tch n phng th nghim (PTN) khc, tin hnh phn tch c lp loi nhng sai s do ngi phn tch v thit b phn tch, i khi c phng php gy nn.

- Thay i kch thc mu: pht hin sai s h thng khng i v bin i.

1.2.2. Sai s ngu nhin hay sai s khng xc nh (random error or indeterminate):

L nhng sai s gy nn bi nhng nguyn nhn khng c nh, khng bit trc.

Sai s ngu nhin thng gy ra do:

- Khch quan: nhit tng t ngt, thay i kh quyn, i lng o c chnh xc gii hn

- Ch quan: thao tc th nghim khng chun xc (c th gy ra gi tr bt thng); thnh phn cht nghin cu khng ng nht

Do sai s ngu nhin khng th bit trc c nn loi tr n cn phi lm nhiu th nghim v tin hnh x l thng k s liu phn tch.

Sai s ngu nhin lm cho kt qu phn tch khng chc chn, cn sai s h thng lm cho kt qu phn tch sai.

1.2.3. Gi tr bt thng (outliers):

Gi tr bt thng l nhng gi tr thu c thng rt cao hoc rt thp so vi gi tr trung bnh. Gi tr bt thng dn n nhng kt qu thu c sai khc nhiu so vi tt c cc s liu lp li ca tp s liu.

Gi tr bt thng do nhng nguyn nhn bt thng xy ra trong qu trnh phn tch gy nn. Do , trc khi x l s liu cn phi loi tr gi tr bt thng.

1.2.4. Sai s tch lu (accumulated error):

Trong mt phng php phn tch, sai s ca s liu phn tch thu c thng bao gm sai s do cc giai on trong qu trnh phn tch ng gp nn. sai s chung l nh th khi phn tch cn phi tm iu kin ti u theo nh lut lan truyn sai s.

Sai s tch lu hay s lan truyn sai s h thng c x l tng t nh sai s h thng. V sai s h thng c du (+) hay (-) nn s dn n s trit tiu sai s v trong mt s trng hp sai s tch lu c th bng khng.

- Khi ch c kt hp tuyn tnh ca php o ngu nhin ( kt qu cui cng ca php cng v tr) th sai s xc nh tuyt i ET l tng cc sai s tuyt i ca php o ring r.

Nu m= A+B +C th Em = EA +EB + EC

- Khi biu din nguyn nhn cc kt qu ( kt qu cui cng l php nhn hoc chia), ngi ta dng sai s xc nh tng i ETR

Nu m= A.B/C th C

E

B

E

A

E

m

ECBA RRRRm ++=

Th d 1.2:

5

a) Khi cn mu trn cn phn tch c chnh xc 0,0002 gam c kt qu nh sau:

mchn +mu= (21,1184 0,0002) gam ; mchn= (15,8465 0,0002) gam

vy khi lng mu s l

mmu= (21,1184 0,002) - (15,8465 0,002) = (5,2719 0,004) gam

b) Khi lng dung dch c tnh theo cng thc m=V.d= (3,430,01).(5,660,01)=?

Ta c:

ERV= 0,01/3,43 ; ERd= 0,01/5,66; ERm= (0,01/3,43)+(0,01/5,66)

Do m=(3,43.5,66) [(0,01/3,43)+(0,01/5,66)]. (3,43.5,66)= 19,41380,0909 Nn m= (19,41 0,09)

1.3. lp li, trng, hi t, phn tn

* lp li (repeatability): Trong phn tch, khi thc hin cc php th nghim

thc hin trn nhng vt liu v trong nhng tnh hung c xem l y ht nhau

thng khng cho cc kt qu ging nhau. iu ny do cc sai s ngu nhin khng

th trnh c vn c trong mi quy trnh phn tch gy ra v khng th kim sot

c hon ton tt c cc yu t nh hng n u ra ca mt php o. Khi bo co

cc d liu o, cn xem xt n nguyn nhn v kt qu s thay i ny.

Nhiu yu t khc nhau (khng k s thay i gia cc mu th c xem l ging nhau) c th ng gp vo s thay i cc kt qu ca mt phng php o, bao gm:

a) ngi thao tc;

b) thit b c s dng;

c) vic hiu chun thit b;

d) mi trng (nhit , m, s nhim ca khng kh ...);

e) khong thi gian gia cc php o

S thay i gia cc php o do c thc hin bi nhng ngi thao tc khc nhau v/hoc vi cc thit b khc nhau s thng ln hn s thay i gia cc php o do cng mt ngi thc hin vi cc thit b nh nhau trong khong thi gian ngn.

* trng (reproducibility): c trng cho mc gn nhau gia gi tr ring l xi ca cng mt mu phn tch, c tin hnh bng mt phng php phn tch, trong iu kin th nghim khc nhau (khc ngi phn tch, trang thit b, phng th nghim, thi gian) (between laboratory precision) .

Vi cng mt phng php phn tch, thng xt n lp li hn l trng.

* hi t (convergence): ch s phn b s liu thc nghim xung quanh gi tr trung bnh. Nu lp li tt th hi t tt.

6

* phn tn (dispersion): ch mc phn tn ca kt qu th nghim sau nhiu ln o lp li. phn tn l nghch o ca lp li. Nu kt qu c lp li cao tc l phn tn cc gi tr xung quanh gi tr trung bnh thp.

1.4. chm v chnh xc

* chm (precision): dng ch mc gn nhau ca cc gi tr ring l xi ca cc php o lp li. Ni cch khc, chm c dng ch s sai khc gia cc gi tr xi so vi gi tr trung bnh x .

Ba khi nim thng k c dng m t chm ca mt tp s liu l lch chun, phng sai v h s bin thin (s xt sau). Tt c cc khi nim ny c lin

quan n lch ca s liu phn tch khi gi tr trung bnh: di = xxi

* ng (trurness): ch mc gn nhau gia gi tr trung bnh ca dy ln cc kt qu th nghim v gi tr qui chiu c chp nhn.

Do , thc o ng thng k hiu bng chch.

* chnh xc (accuracy): l mc gn nhau ca gi tr phn tch (thng l gi tr trung bnh x ) vi gi tr thc hay gi tr c chp nhn xt hay .

Khi khng c sai s h thng th gi tr trung bnh tin ti gi tr thc nu s php o rt ln ( N). V vy, c th ni chnh xc tu thuc vo s php o.

chnh xc c biu din di dng sai s tuyt i hoc sai s tng i.

Trong Ho phn tch, nh gi chnh xc ngi ta pha cc mu t to (synthetic sample) bit trc hm lng (tc l c gi tr bit trc ) v lm th nghim tm ra gi tr trung bnh sau kim tra xem c s sai khc c ngha thng k gia gi tr trung bnh v gi tr thc hay khng. Vn ny s c xt chng 4.

chm v chnh xc l nhng ch tiu quan trng nh gi cht lng ca s liu phn tch. Thng thng, cn nh gi chm trc v nu phng php phn tch mc sai s h thng th ch c dng nh lng khi sai s ngu nhin nh.

7

Chng 2 CC I LNG THNG K (Descriptive statistics)

2.1. Cc i lng trung bnh

* Trung bnh s hc ( x ) (mean, arithmetic mean, average) l i lng dng ch gi tr t c khi chia tng cc kt qu th nghim lp li cho s th nghim lp li.

Gi s c tp s liu th nghim lp li x1, x2,, xN th gi tr trung bnh s hc ca tp s liu gm N th nghim lp li l:

x = N

xxx n+++ ...21 = N

xN

ii

=1 (2.1)

Gi tr trung bnh c tnh cht sau:

- Tng lch gia cc gi tr ring r v gi tr trung bnh bng khng.

= 0)( xxi

- Tng cc bnh phng lch nh hn tng bnh phng ca bt c lch no gia gi tr n l v gi tr a no khng phi gi tr trung bnh.

2)( xxi < 2)( axi ( vi a x )

* Trung bnh bnh phng ( x bp): vi tp s liu gm N s liu lp li x1, x2,,xn ta c:

x bp = N

xxx n22

221 ...+++ (2.2)

* Trung bnh hnh hc hay trung bnh nhn (geometric average) vi cc php o c hm lng cn tm di dng logarit th:

lgx hh= )lg...lg(lg1

21 NnxxxN+++

Do xhh= N Nxxx ..... 21 ( 2.3)

* Trung v (median) : Nu sp xp N gi tr lp li trong tp s liu theo th t tng n hoc gim dn t x1, x2, , xN th s nm gia tp s liu c gi l trung v.

- Nu N l th trung v chnh l s gia dy s.

- Nu N chn th trung v l trung bnh cng ca 2 gi tr nm gia dy s.

Ch : Gi tr trung bnh hay trung v ca tp s liu c gi l cc gi tr trung tm ca tp s liu. Cc tp s liu khc nhau c cng gi tr trung bnh c th rt khc nhau v ga tr ring l v s th nghim. V vy, trung bnh v trung v khng cho ta ci nhn tng qut v s phn b cc s trong tp s liu. Trong trng hp cn xt n phn tn ( lch khi ga tr trung bnh).

* im t phn v (quartile): Nu sp xp cc s liu trong tp s liu t nh n ln th mi tp s liu c 3 im t phn v: 25 % cc s trong tp s liu sp xp

8

c gi tr nh hn hoc bng im t phn v th nht, 75 % cc s trong tp s liu sp xp c gi tr nh hn hoc bng im t phn v th ba, 50% cc s trong tp s liu sp xp c gi tr nh hn hoc bng trung v (im t phn v th hai). Khong gia im t phn v (interquartile) biu th s khc nhau gia im t phn v th nht v th ba.

C th hnh dung im t phn v theo s sau:

Trung v

gi tr 0% 25% 50% 75% 100% gi tr cao

thp im t phn v th nht im t phn v th ba.

* S tri (mode): l s c tn s xut hin l ln nht trong tp s liu lp li.

Ch : Gi tr bt thng c nh hng ng k ti gi tr trung bnh nhng khng nh hng n s trung v. Do vy, vi nhng tp s liu rt nh, (thng N

9

th ( )

kN

xx

S

m

j

k

iiij

=

= =1 1

2

2 (2.5)

vi N l tng tt c cc th nghim N=m.k

(Khi nim ny t dng trong ho hc)

Nu phng sai cng ln th tn mn ca cc gi tr o lp li cng ln hay lp km.

* lch chun (Standard deviation)

- Mu thng k v mu tng th (statistical sample and population).

Trong thng k, mt s xc nh cc quan st thc nghim (hay kt qu php o cc mu phn tch ring r) c gi l mu thng k. Gp tt c nhng mu thng k gi l mu tng th. Nh vy c th xem phn tch mu tng th l nhng php o c th c v v cng ln (N).

Th d: Cn iu tra mc thiu iot trong hc sinh tiu hc thnh ph A. Tin hnh ly mu nc tiu hc sinh mt s trng tiu hc trong thnh ph phn tch hm lng it. Nh vy nc tiu ca mt s hc sinh tiu hc mi trng c ly mu l cc mu thng k. Mu tng th y s l mu nc tiu ca hc sinh tiu hc thnh ph A ni chung.

- Trung bnh mu x v trung bnh tng th .

+ Trung bnh mu ( sampling fluctuation) ( x ) l gi tr trung bnh ca mt mu thng k gii hn c rt ra t tp hp cc s liu v c xc nh theo cng thc:

N

xx

N

ii

== 1 .

+ Trung bnh tng th (population average) () l gi tr trung bnh ca tp hp cc s liu, cng c xc nh theo phng trnh (2.1) nhng vi N rt ln, gn t ti . Khi khng c sai s h thng th trung bnh tng th cng l gi tr tht ca php o.

N

xN

ii

== 1 khi N . Thng thng khi N > 30 c th xem nh x

- lch chun tng th (Population standard deviation): () c trng cho phn tn cc s liu trong tp hp vi gi tr trung bnh v c xc nh theo phng trnh:

( )

N

xxN

ii

=

= 1

2

hay 2 = (2.6)

vi N l s th nghim lp li ca tp hp, thc t thng xem cc tp s liu c N>30 l tp hp.

- lch chun mu c on (Sample estimate standard deviation): (S)

10

( )

11

2

=

=

N

xxS

N

ii

hay 2SS = (2.7)

vi N l s th nghim trong mu thng k c rt ra t tp hp. S bc t do trong trng hp ny l f =N-1.

(Bc t do c th coi l s php o kim tra cn thit c th xc nh c kt qu trong mt tp s liu. Mt cch khc bc t do c hiu l s cc quan st trong mt mu thng k c th t do thay i do bng tng kch thc mu tr i 1 bc t do cho mi trung bnh. Thut ng bc t do cn c dng ch s lch ( )xxi ) c lp dng trong php tnh lch chun)

Nh vy, khi N th x v S . Ni cch khc khi N>30 c th xem S .

So vi phng sai, lch chun thng c dng o lp li hn do c cng th nguyn vi i lng o.

Khi tnh ton ch khng lm trn s liu ca lch chun cho n khi kt thc php tnh ton v ch ghi gi tr cui cng di dng s c ngha.

Nu trng hp c m mu thng k, mi mu lm n th nghim song song th:

( )mnm

xxS

m n

ij

=

.1 1

2

bc t do f=m(n-1) (gi thit Sj khc nhau khng ng k).

i vi tp s liu nh ( N

11

Nu c nhiu dy s liu lp li (nhiu mu thng k), mi dy c N s liu c ly ngu nhin t tp hp s liu th s phn tn ca trung bnh mu c c trng bng sai chun m thay cho lch chun trong tp hp. S phn tn ny gim khi N tng.

m l lch chun trung bnh hay sai chun v c tnh nh sau :

m=N

Dng sai chun m c trng cho sai s ngu nhin cu phng php phn tch. Tuy nhin, i vi tp s liu hu hn (N

12

Th d 2.1 :Cho kt qu phn tch lp li 35 ln hm lng nitrat (g/ml) nh sau : 0.51 0.51 0.49 0.51 0.51 0.51 0.52 0.48 0.51 0.50 0.51 0.53 0.46 0.51 0.50 0.50 0.48 0.49 0.48 0.53 0.51 0.49 0.49 0.50 0.52 0.49 0.50 0.50 0.50 0.53 0.49 0.49 0.51 0.50 0.49

Sv t tnh cc i lng thng k theo cng thc v so snh vi kt qu tnh theo phn mm MINITAB di y, gii thch ngha cc kt qu .

ham luong nitrat ( micogam/ml)0.530.520.510.500.490.480.470.46

Phan bo cac gia tri thuc nghiem theo tan suat

ham luong n

itra

t ( micro

gam/ml)

0.53

0.52

0.51

0.50

0.49

0.48

0.47

0.46

Do thi khoi cac gia tri thuc nghiem

2.3. Bo co kt qu phn tch

2.3.1. S c ngha v cch ly gi tr gn ng

Mt gi tr s hc dng biu din kt qu phn tch s khng c ngha nu khng bit chnh xc ca n. Do vy, khi biu din cn phi ghi r tin cy ca s liu v cc s liu cn c lm trn ch mc khng chc chn ca n (uncertanty). Ni cch khc, s liu ch c cha cc s c ngha.

2.3.1.1. Khi nim s c ngha

S c ngha trong mt dy s l tt c cc s chc chn ng v s khng chc chn ng u tin.

Th d 2.2 : Khi c th tch dung dch ng trong buret 50 ml, chng ta c th thy vch cht lng v tr ln hn 30,2 ml v nh hn 30,3 ml. Nu c th c on v tr vch cht lng cp chia khong + 0,02 ml th c th bo co th tch l

Descriptive Statistics for nitrate concentration Total Count : 35 Mean: 0.50413 SE Mean: 0.00260 StDev : 0.01537 Variance : 0.000236 CoefVar : 3.06 Sum of Squares: 8.80810 Minimum: 0.46 Q1: 0.49 Median : 0.50 Q3 : 0.51 Maximum : 0.53 Range: 0.07 Skewness : -0.20 Kurtosis: 0.50

ham luong nitrat ( microgam/ml)

tan xuat

0.530.520.510.500.490.480.470.46

10

8

6

4

2

0

Bieu do phan bo tan xuat ham luong nitrat

13

30,24 ml (4 s c ngha).

Trong th d ny 3 con s u tin l s chc chn ng, s cui cng l s khng chc chn ng. Nh vy c th vit 30,24 ml hoc 0,03024 lit (4 s c ngha).

S c ngha c qui c nh sau :

+ Gm cc ch s t nhin 1,2,. 9

+ S khng c th l s c ngha hoc khng phi l s c ngha tu thuc vo v tr ca n trong dy s.

- Nu s khng nm gia cc s khc l s c ngha.

- Nu s khng nm cui dy s th ch l s c ngha nu ng sau du phy.

- Nu s khng nm trc du thp phn th khng phi l s c ngha.

* Lm trn s: l loi b cc s khng c ngha trong kt qu. Nu b cc s 6,7,8,9, th tng ga tr trc n ln 1 n v. Nu loi b cc s 1,2,3,4, th khng thay i con s ng trc n. Nu loi b s 5 th lm trn s trc v s chn gn nht. V d: 2,25 lm trn thnh 2,2; 2,35 thnh 2,4.

Th d 2.3 : 25,24 c 4 s c ngha 0,15 c 2 s c ngha

15,00 c 4 s c ngha 1,36 c 3 s c ngha

0,0241 c 3 s c ngha 150,00 c 5 s c ngha

Khi ly V=5,00 ml c ngha l khi tnh nng phi ly 3 s c ngha. (Nh vy c th ghi gi tr nng l 0,0215; 2,15.10-2 hoc 21,5.10-3 hoc 215.10-4M)

Nu ghi th tch bnh l V= 2,0 lit th khi chuyn sang n v ml khng th ghi l 2000 ml (v y ch ghi 1 s c ngha) m phi ghi l 2,0.103ml.

2.3.1.2. Cch ly gi tr gn ng

* i lng o trc tip: gi tr o c phi c hoc o, m c. S liu th nghim c ghi theo nguyn tc s cui cng l s gn ng v s trc s cui cng l s chnh xc.

* i lng o gin tip.

- Php tnh cng v tr : lm trn s thnh s chnh xc v ghi s c ngha theo g tr no c t s c ngha nht.

- Php nhn v chia: kt qu ca php nhn v php chia c lm trn s sao cho n cha s c ngha nh gi tr c t s c ngha nht. (Khi tnh bt n tuyt i khgn tnh n du thp phn)

- Php tnh logrit v ngc logrit:

+ logrit: ly cc ch s sau du phy bng tng cc s c ngha trong s ban u

+ ngc logarit: ly cc s c ngha bng s cc ch s sau du phy.

Th d 2.4: a) 3,4+0,020+7,31=10,73=10,7 y v 3,4 l s ch c 1 s c ngha sau du phy nn trong kt qu ch ghi 1 s c ngha sau du phy.

14

b) %5470578,88%100.1689,1

05300.0.5481,0.63,35 =

Trong dy s trn, khng chc chn ca mi s l 1/3563; 1/5481; 1/5300 v 111689/ Nh vy khng chc chn ca s th nht ln hn so vi khng chc chn ca s th hai v th ba. Do , gi tr c t s c ngha nht l 35,63 nn kt qu cui cng phi c ghi l 88,55%

c) log(9,57.104)=4- log 9,57= 4,981 (gi tr 4 c 1 s c ngha; gi tr9,57 c 3 s c ngha )

log(4,000.10-5)=5- log4,000=-4,397940=-4,3479

Antilog(12,5)=3,162277.1012=3.1012

2.4. Quy lut lan truy n sai s ngu nhin - lch chun ca i lng o gin tip

Tt c cc kt qu phn tch nh lng thu c t thc nghim u c cha sai s ngu nhin. V vy, cc gi tr c bo co thng l gi tr trung bnh vit ng s c ngha km theo sai s ngu nhin ca gi tr . Thng thng chng c vit l Sx , vi S l lch chun.

Th d: Trong tp s liu th tch dung dch chun dng cho qu trnh chun , cc gi tr th tch thu c l 10,09; 10,11; 10,09; 10,10; 10,12 ml. Nh vy, th tch dung dch chun dng s l Sx = 10,10+0,01 (vi N=5 th nghim lp li).

Ngoi ra, khi s th nghim lp li ln, kt qu phn tch cn c trnh by

di dng N

Stx

. v s c xt n trong chng 3.

Tuy nhin, kt qu nh lng thu c t thc nghim trong rt nhiu php o khng phi l kt qu ca php o trc tip m c th c tnh ton t mt hay nhiu php o trc tip. Mt khc, mi s liu thu c trong cc php tnh u c lch chun ring, v vy phi xt n lan truyn sai s gy ra cho kt qu cui cng.

Gi s cc kt qu thc nghim a, b, c, .. l cc s liu thu c t cc php o trc tip M1, M2 , M3. Gi x l gi tr cui cng tnh ton c t cc kt qu ring r a, b, cKhi x l hm ph thuc vo cc tham s a, b, c

Gi cba ,, l lch chun ca cc php o trc tip xc nh a, b, c.. v gi thit l sai s trong cc php o ny c lp ln nhau th lch chun ca i lng x l :

2/12

2

2

2

...][ +

+

= b

b

xa

a

xx

(theo nh lut lan truyn sai s, biu thc

ny ng khi x l hm tuyn tnh ca cc php o a, b,c).

Cch tnh lch chun ca i lng x ny tu thuc vo dng cng thc tnh em s dng.

* lch chun ca tng v hiu:

x = a1. a( Sa) + b 1.b( Sb) c1 .c( Sc) vi a1,b1, c1 l cc hng s th lch chun ca x l

15

...... 22122

122

1 +++= cbax ScSbSaS

* lch chun ca php nhn v chia:

x = 1

11 .c

ba

c

ba th ......

22

1

22

1

22

1 +

+

+

=c

Sc

b

Sbb

a

Sa

x

S cax

Khi , kt qu s c biu din di dng x = 1

11 .c

ba

c

ba Sx .

* lch chun ca php tnh logarit:

x= k.lna th Sx=

a

Sk a.

x= k.loga th Sx=

a

Sk a.30,2

Cc gi tr lch chun trong php o trn c gi l sai s tuyt i ca php

o. i lng

a

Sa gi l sai s tng i.

Th d 2.5: a) Tnh gi tr biu thc:

(65,060,07) +(16,130,01)-(22,680,02)= 58,51?

ta c xS 073,002,001,007,0222 =++= V biu din x = 58,51 0,07

lch chun tng i ca php o l %1,0%100.51,58

07,0 =

b) ?0,356006,0623,4

)2,04,120).(02,067,13(

.=

==

cb

ax

ta c 222

623,4

006,0

4,120

2,0

67,13

02,0

+

+

=x

Sx =0,0026 do vy

Sx=356,0.0,0026=0,93

kt qu cui cng s l x = 356,00,9

Th d 2.6 : Tnh lch chun s mmol Cl- trong 250,0 ml dung dch mu, nu ly 25,00 ml dung dch mu ny chun bng dung dch chun AgNO3 c nng ( 0,11670,0002) M. Th tch dung dch AgNO3 tiu tn sau 3 ln o lp li l 36,78; 36,82 v 36,75 ml.

HD gii : - th tch dung dch chun AgNO3 trung bnh l: 36,78 ml

- p dng cng thc tnh lch chun th tch chun ta c S= 0,035

- Vy 04,078,363

=AgNOV (ml)

- S mmol Cl- c chun trong 250 ml mu :

X= (0,11670,0002).((36,780,04).10= 42,92 ?

16

Ta c : 019,010.78,36

04,0

1167,0

0002,0 222

=

+

=x

Sx

Do Sx= 42,92.0,019=0,082

Kt qu s mmol Cl- trong 250 ml mu l (42,920,08) mmol

Ch : Trong qu trnh tnh ton v c s lan truyn sai s nn cn trnh lm trn s khi vic tnh ton cha kt thc.

17

Chng 3 HM PHN B V CHUN PHN B

3.1. Biu din s liu nh lng Trong phn tch nh lng, s liu thc nghim l cc s liu thu c khi tin

hnh cc php phn tch nh lng. h thng ho nhng s liu ny nhm thu c ci nhn tng qut hn hoc phc v cho nhng nghin cu tip theo, ngi ta biu din chng di dng biu hoc th. Cc dng biu thng gp l biu ct hay biu hnh ch nht (bar chart), biu hnh qut (pie chart), biu tn sut (historgram) hay biu ng gp khc (pylogon). Nu cn biu din gi tr thc nghim ca cc tp s liu khc nhau, th s dng ln ca cc s liu. Trong trng hp cn biu din cc s liu trong cng tp s liu th thng dng tn sut ca gi tr trong tp s liu.

Trong phn trnh by di y ch xt n biu biu din tn s xut hin ca gi tr trong tp s liu di hai dng biu tn sut v biu ng gp khc .

Cch tin hnh: Cc gi tr trong tp s liu c chia thnh cc nhm khc nhau (category) v kim tra tn sut ca gi tr biu din kt qu o di dng im ring bit trn trc s (c chia tuyn tnh 1 chiu) v nhn nh v mt cc im (trng hp ny gi l phn b 1 chiu) hoc biu din dng bc thang (ct) bng cch tp hp cc gi tr ring r thnh k cp c b rng d (5 < k < 20) (k cn bc hai tng cc gi tr o c).

Th d 3.1: Ngi ta xc nh ng thi Al trong mt mu thp 12 phng th nghim (PTN). Mi PTN cho 5 gi tr phn tch thu c trong nhng ngy khc nhau. Cc gi tr ny c h thng ha nh bng 3.1:

Bng 3.1: Kt qu phn tch hm lng Al (%) trong mu thp

STT PTN X1 X2 X3 X4 X5

1 A 0,016 0,015 0,017 0,016 0,019

2 B 0,017 0,016 0,016 0,016 0,018

3 C 0,015 0,014 0,014 0,014 0,015

4 D 0,011 0,007 0,008 0,010 0,009

5 E 0,011 0,011 0,013 0,012 0,012

6 F 0,012 0,014 0,013 0,013 0,015

7 G 0,011 0,009 0,012 0,010 0,012

8 H 0,011 0,011 0,012 0,014 0,013

9 I 0,012 0,014 0,015 0,013 0,014

10 K 0,015 0,018 0,016 0,017 0,016

11 L 0,015 0,014 0,013 0,014 0,014

12 M 0,012 0,014 0,012 0,013 0,012

Gii hn 8 10 12 14 16 18 20 .10-3%

trn ca cp ca

Hnh 3.1: Phn phi tn sut khi xc nh ng

thi hm lng Al trong mu thp ti 12 PTN.

M M M L M L M L L I L K H I K H I K H I I G H F G H C G F C F F B E F B K G E E B K G E C A B D D E C A B D D D C A A A

18

Nh vy c tt c N=60 gi tr. Gi tr thp nht l ca PTN D c 2D

X =0,007%.

Gi tr cao nht ca PTN A l 5AX = 0,019%. Sau khi tp hp cc s liu thnh k= 7

cp vi rng ca cp l d= 0,002 %Al ta c k N . Cp th nht gm cc gi tr 0,007 v 0,008 % Al, cp th hai l 0,009 v 0,010 % Al.... Nh vy ta c phn b tn sut thc nghim c trnh by hnh 3.1 v biu tn sut phn trm hnh 3.2.

Tan

xua

t (%

)

2018161412108

35

30

25

20

15

10

5

0

Hnh 3.2. Biu phn trm tn sut hm lng Al trong kt qu phn tch cc PTN

T dng phn b tn sut c th thy c nh tnh v s xut hin sai s ngu nhin. Khi sai s ngu nhin ln th phn b rng, sai s ngu nhin nh th phn b hp v nhn, nhng trong trng hp ny khng cho bit v sai s h thng v sai s h thng khng lm thay i dng phn b.

3.2. Phn b l thuyt Khi h thng ho cc gi tr o v biu din chng trn th bng cch v tn

sut ca gi tr no vi mt trc l gi tr , ta lun thu c cc phn b dng ct nh trn, c bit khi ch c sai s ngu nhin. Do , cho php gi thit c nhng qui lut ton hc lm c s ca nhng phn b .

3.2.1. Phn b chun (Phn b Gauss) Gi s tin hnh rt nhiu th nghim lp li v thu c rt nhiu cc gi tr (N

) trong c mt s yu t ngu nhin nh hng n cc gi tr ny v cc nguyn nhn gy nh hng c tnh cng tnh, nh hn gi tr o.

Khi rng ca lp nh (d 0) th phn b tn sut c biu din bng hm

mt xc sut sau:

2)(2

1

2

1)(

=

x

exy (3.1)

trong : 3,1416 e 2,7183; l tham s v l lch chun, c trng cho phn tn ca php o (measure of dispersion); l tham s v l gi tr

19

tht hoc gi tr trung bnh, c trng cho php o v tr phn b (measure of location) ; x l to hoc gi tr trn trc honh; Y: tung d, chiu cao ca ng biu din tung ng vi gi tr x.

V tr v dng ng cong c xc nh bi v . Cc i ca ng cong ti y' = 0, tc l im x= . Cc im un l x1= - v x2 = + . Nu cho . th y = f(x). Khi y = 0 th x = . Tuy nhin, trn thc t c th b qua cc gi tr ca trc tung khi x ngoi khong 3 .

Hnh 3.5: Phn b chun vi cc gi tr trung bnh cng khc nhau.

Hnh 3.6 : Biu din hnh hc ca lch chun

Nu k hiu

= xZ th Z l mt bin ngu nhin v hm phn b c dng

2.2

1

2

1)(

ZezY

=

(3.2) khi Z=1 v z=0

Hm phn b Z ny c gi l phn b chun hay phn b Gauss. Phng trnh (3.2) m t mt xc sut ca phn b, l tng din tch gia ng cong v trc x l 1 n v. ng biu din cn c gi l ng cong sai s (error curve).

Nu ly tch phn ca hm phn b chun t - n + th ton b phn din tch gii hn bi ng cong biu din xc sut xut hin cc gi tr xi. Gi tr xc sut ny gn lin vi tin cy thng k P. Ni cch khc, phn din tch gii hn bi ng cong l tin cy thng k xut hin xi trong khong tch phn.

i vi cc tp s liu c cng gi tr thc s c cng din tch ng cong Gauss nhng nu cng nh th ng cong cng hp v cng nhn, chnh xc cng ln. Xc sut gi tr o nm ngoi gii hn trn ca tch phn l =1-P. Phn din tch P cng c biu din theo % so vi tng din tch v gi l tin cy thng k.

Trong khong th mt xc sut chim 68 % din tch ca ng cong. Trong khong 2 th mt xc sut chim 95 % din tch ng cong. C

ngha l c 95 % gi tr trung bnh mu nm trong khong:

mt xc sut

lch chun