Tiêu chuẩn iso ts 17503 2015

© ISO 2015 Statistical methods of uncertainty evaluation — Guidance on evaluation of uncertainty using two factor crossed designs Méthodes statistiques d’évaluation de l’incertitude — Lignes directric[.]

Statistical methods ofuncertainty

ofunc ertainty using two-factor

Méth des statistique s d’é alu tion de l’inc rtitude — Lignes

directrice s p ur l’é alu tion de l’inc rtitude de s modè le s à deu

i © ISO 201 – A ll rights reserved

© ISO 201 , P blshed in Sw itz rlan

A ll rig hts r eserved Unles otherw ise spe ified, nopar of this p blc tion ma y be r eprod c d or utilz d otherwise in an form

or b an me ns, ele tr onic or me hanic l, inclu in p oto opying , or posin on the internet or an intr anet , without prior

written permis ion Permis ion c n be r eq esed fr om either ISO at the ad r es below or ISO’s member bod y in the c u ty of

F r eword i v

Introduction v

1 Sc ope 1

2 Nor mati ve r eferenc es 1

3 Terms and definitio s 1

4 Symbols 2

5 Co duct of e x per iments 4

6 Prelminary r eview of data — Overview 4

7 Var ianc e c ompo ents and unc er tainty estimatio 4

7.1 General considerations for v rianc component an u c rtainty es imation 4

7.2 T wo-wa y la yout without r eplcation 5

7.2.1 Desig n 5

7.2.2 Pr elminary inspe tion 5

7.2.3 Varianc component estimation .5

7.2.4 Stan ard u c rtainty for the mean of al o se v tions 6

7.2.5 Deg r ees of fr eedom for the s an ar d u c rtainty 6

7.3 T wo-wa y b lanc d ex pe iment with r eplcation (b th factorsran om) 7

7.3.1 Desig n 7

7.3.2 Pr elminary inspe tion 7

7.3.3 Varianc component extraction 7

7.3.4 Stan ard u c rtainty for the mean of al o se v tions 8

7.3.5 Deg r ees of fr eedom for the s an ar d u c rtainty 9

7.4 T wo-wa y b lanc d ex pe iment with r eplcation (one factor fixed, one factor ran om) 1

7.4.1 Desig n 1

7.4.2 Pr elminary inspe tion 1

7.4.3 Varianc component extraction 1

7.4.4 Stan ard u c rtainty for the mean of al o se v tions 1

7.4.5 Deg r ees of freedom for thes an ard u c rtainty 1

8 A pplcatio to o ser vatio s o a relati ve s ale 12

9 Use of varianc e c omponents in subsequent me surements 12

10 A lter nati ve treatments 13

1 1 Res ricted (or r esid al) max imum lkelho d es imates 1

1 2 A lte native methods for model r ed ction 1

11 Tr eatment with mis ing values 13

A nne x A (informative)Examples 14

Biblog raphy 19

ISO (he Int ernational Org nization for Stan ardization) is a worldwide fede ation of national s an ards

b dies (ISO membe b dies) The work of pr paring Int ernational Stan ards is normal y car ied out

through ISO t ech ical committ ees Each membe b dy int er st ed in a subje t for w hich a t ech ical

committ ee has be n es a lshed has the right t o be r pr sent ed on that committ ee Int ernational

org nizations, g ove nmental an non-g ove nmental, in laison with ISO, also take p rt in the work

ISO cola orat es closely with the Int ernational Ele trot ech ical C mmis ion (IEC) on al matt ers of

ele trot ech ical s an ardization

The proc d r s used t o develo this document an those int en ed for it furthe maint enanc ar

desc ibed in the ISO/IEC Dir ctives, Part 1 In p rticular the dife ent a pro al c it eria ne ded for the

dife ent ty es of ISO document should be not ed This document was draft ed in ac ordanc with the

edit orial rules of the ISO/IEC Dir ctives, Part 2 ( e www.iso.org dir ctives)

A tt ention is drawn t o the p s ibi ity that some of the element of this document ma be the subje t of

p t ent right ISO shal not be held r sp nsible for identifying any or al such p t ent right Detais of

any p t ent right identified d ring the develo ment of the document wi be in the Introd ction an / r

on the ISO ls of p t ent de larations r c ived ( e www.iso.org p t ent )

Any trade name used in this document is information given for the convenienc of use s an does not

cons itut e an en orsement

F or an ex lanation on the meaning of ISO spe if ic t erms and ex r s ions r lat ed t o conformity

as es ment, as wel as information a out ISO’ s adhe enc t o the WTO principles in the Te h ical

Bar ie s t o Trade (TBT) se the folowing URL: F or word - Sup lementary information

The committ ee r sp nsible for this document is ISO/TC 6 , Ap lc atio s o s tati s tic al meth ds ,

Subcommitt ee SC6, Meas ur ment meth ds a d r s ults

Interntio l Org izatio for Sta ar dizatio

Unc rtainty es imation usualy r q ir s the es imation an subseq ent combination of u c rtainties

arising from ran om v riation Such ran om v riation ma arise within a p rticular ex e iment un e

r peata i ty con itions, or o e a wide rang e of con itions Variation u de r peata i ity con itions

is usual y charact eriz d as r peata i ty s an ard deviation or coeff icient of v riation; pr cision un e

wide chang es in conditions is g ene aly t ermed int ermediat e pr cision or r prod cibi ity

The mos common ex e imental design for es imating the long - an short-t erm component of v rianc

is the clas ical b lanc d nest ed design of the kind used b ISO 5 2 - 2 In this design, a (cons ant)

n mbe of o se v tionsar cole t ed un e r peata i ty con itions for each level of some othe fact or

Whe e this ad itional fact or is ‘ La orat ory’ , the ex e iment is a b lanc d int er-la orat ory s u y, an

can be analy sed t o yield es imat es of within-la orat ory v rianc , σ

r2

, the betwe n-la orat ory

comp nent of v rianc , σ

L2

, an henc the r prod cibi ity v rianc , σ σ σ

= + Es imation of

unc rtainties b sed on such a s u y is conside ed b ISO 2 748 Whe e the ad itional grouping fact or is

anothe con ition of measur ment, howeve , the betwe n-group t erm can usefuly be taken as the

unc rtainty contribution arising from random v riation in that act or F or ex mple, if seve al dife ent

extract ar pr p r d from a homog eneous mat erial an each is measur d seve al times, analy sis of

v rianc can pro ide an es imat e of the efe t of v riations in the extraction proc s Furthe

ela oration is also p s ible b ad ing suc es ive levels of grouping F or ex mple, in an int er-la orat ory

s u y the r peata i ty v rianc , betwe n-da v rianc an betwe n-la orat ory v rianc can be

es imat ed in a single ex e iment b r q iring each la orat ory t o u de take an eq al n mbe of

r plcat ed measur ment on each of two day s

Whie nest ed designs ar among the mos common designs for es imation of ran om v riation, they

ar not the only useful clas of design Conside , for ex mple, an ex e iment int ended t o charact eriz

a r fe enc mat erial, con uct ed b measuring thr e sep rat e u it of the mat erial in thr e sep rat e

ins rument ru s, with ( a ) two o se v tions pe u it pe ru In this ex e iment, unit an ru ar

said t o be ‘c os ed’ ; al u it ar measur d in al ru s This design is oft en used t o inves ig t e v riation

in ‘fix ed’ efe t , b t es ing for chang es w hich ar larg er than ex e t ed from the within-group or

‘ resid al’ t erm This p rticular ex e iment, for ex mple, could easiy t es w hethe the e is evidenc

of significant dife enc s between u it or betwe n ru s Howeve , the unit ar lkely t o have be n

sele t ed ran omly from a much larg er ( if ost ensibly homog eneous) b t ch, an the ru efe t ar also

mos a pro riat ely tr at ed as ran om If the mean of al the o se v tions is taken as the es imat e of

the r fe enc mat erial v lue, it be omes ne es ary t o conside the unc rtainties arising from b th ru

-t o-ru an u it-t o-u it v riation This can be done in much the same wa as for the nest ed designs

desc ibed pr viously, b extracting the v rianc s of int er s using two-wa analy sis of v rianc In the

s atis ical lt eratur , this is g ene al y desc ibed asthe use of a ran om-efe t or ( if one fact or isa f ix ed

efe t) mix ed-efe t model

Varianc comp nent extraction can be achieved b seve al methods F or balanc d designs, eq ating

ex e t ed mean sq ar s from clas ical analy sis of v rianc is s raightorward R es rict ed ( ometimes

also caled r sid al)ma imum l kel ho d es imation (REML)is also widely r commen ed for es imation

of v rianc comp nent , an is a plca le t o both b lanc d an u balanc d designs This Te h ical

Spe ification desc ibes the clas ical ANOV calculations in detai an pe mit the use of REML

Not e that random efe t rar ly inclu e al of the u c rtainties afe ting a p rticular measur ment

r sult If using the mean from a c os ed design as a measur ment r sult, it is g ene aly ne es ary

t o conside u c rtainties arising from p s ible sy st ematic efe t , inclu ing betwe n-la orat ory

efe t , as wel as the ran om v riation visible within the ex e iment, an these othe efe t can be

conside a ly larg er than the v riation visiblewithin a single ex e iment

This pr sent Te h ical Spe if ication desc ibes the es imation an use of u c rtainty contributions

using fact orial designs

Statistic al methods ofuncer tainty e valuation — Guidance on

This Te h ical Spe if ication desc ibes the es imation of u c rtainties on the mean v lue in ex e iment

con uct ed as c os ed designs, an the use of v rianc s extract ed from such ex e iment an a pled t o

the r sult of othe measur ment (for ex mple, single o se v tions)

This Te h ical Spe if ication co e s b lanc d two-fact or designs with any n mbe of levels The

b sic designs co e ed inclu e the two-wa design without r plcation an the two-wa design with

r plcation, with one or b th fact ors conside ed as random C lculations of v rianc comp nent from

ANOV ta les an their use in u c rtainty es imation ar given In ad ition, brief guidanc is given on

the use of r s rict ed ma imum l kelho d es imat es from sof war , an on the tr atment of ex e iment

with smal n mbe sof mis ing data p int

Methods for r view of the data for outle s an a pro imat e normalty ar pro ided

The use of data o tained from the tr atment of r lative o se v tions (for ex mple, a par nt r co e y

in analytical chemis ry) is inclu ed

2 Normati ve referenc es

The folowing document , in w hole or in p rt, ar normatively r fe enc d in this document an ar

in ispensa le for it a pl cation F or dat ed r fe enc s, only the edition cit ed a ples F or un at ed

r fe enc s, the lat es edition of the r fe enc d document ( inclu ing any amen ment )a ples

ISO 35 4-1, Stat ist i cs — Voca ulary a d symb ls — Par t 1: Gene al stat ist i cal ter ms a d ter ms used

i n pro a i li ty

ISO 3 34-3, Stati s tic s — Voc ab lary a d sy mb l s — Part 3 : Des ig o expe iment

3 Terms and definitions

F or the purposes of this document, the t erms an def initions in ISO 3 34-1, ISO 3 34-3 an the

folowing a ply

3.1

factor

pr dict or v ria le that is v ried with theint ent of as es ing it efe t on ther sp nse v ria le

Not e 1to entry: A factor may provide an as igna le cause for the out come of an e periment

Not e 2to entry: T e use of factor here is more specif ic than its g neric use as a sy on m for predict or varia le

Not e 3to entry: A factor may b as ociat ed with the c e tion of blocks

[SOURCE: ISO 3 34-3:2 1 , 3.1.5, modified — c os -r fe enc s within ISO 3 34-3 omit ed from

Notes to entry]

3.2

level

p t ential set ing, v lueor as ignment of a fact or

Not e 1to entry: A sy on m is the value of a predictor varia le

Not e 2 to entry: T e term “level” is normal y as ociated with a q antitative charact eristic However, it also serves

as the term des ribing the ver ion or set ing of q alitative characteristic

Not e 3 t o entry: R esponses o served at the various levels of a fact or provide information for det ermining the

efect of the fact or within the rang of levels of the e periment Extrapolation b yon the rang of these levels is

usualy inap ropriat e without a f irm b sis for as uming model relationships Interpolation within the rang e may

depen on the n mb r of levels an the spacing of these levels It is usualy re sona le to interpolate, although

it is pos ible t o have dis ontin ous or multimodal relationships that cause a ru t chang es within the rang of

the e periment T e levels may b limit ed to certain selected f ix ed values (whether these values are or are not

k own) or they may represent p rely ran om selection over the rang t o b stu ied

E AMP E T e ordinal-s ale levels of a cataly st may b presence an a sence Four levels of a he t tre tment

may b 1 0 °C, 1 0 °C,140 °C an 160°C T e nominal-s ale varia le for a la orat ory can have levels A, B an C,

cor espon ing to thre facilities

[ SOURCE:ISO 3 34-3:2 1 , 3.1.1 ]

3.3

fixed ef ects analy sis of variance

analysis of v rianc in w hich the levels of each fact or ar pr -sele t ed o e the rang e of v lues of the

fact ors

Not e 1 to entry: With f ix ed levels, it is inap ropriate t o comp t e components of variance.This model is sometimes

refer ed t o as a model 1analysis of variance

[ SOURCE:ISO 3 34-3:2 1 , 3.3.9]

3.4

random efects analy sis of variance

analysis of v rianc in w hich each level of each fact or is as umed t o be sampled from the p pulation of

levels of each fact or

Not e 1 t o entry: With ran om levels, the primary interest is usual y t o o tain components of variance estimat es

This model is commonly refer ed to as a model 2 analy sis of variance

E AMP E Consider a situation in which an operation proces es b tches of raw material “Batch” may b

considered a ran om factor in an e periment when a few b tches are ran omly selected from the pop lation

True betwe n-level s an ard deviation for thef irs fact or ( if conside ed a ran om efe t) in a

two-wa fact orial (c os ed)ex e iment

σ

2

True betwe n-level s an ard deviation for these ond fact or ( if conside ed a ran om efe t) in a

two-wa fact orial (c os ed)ex e iment

σ

I

True betwe n-group s an ard deviation for the int eraction t erm in a fact orial ex e iment (w he e

one or mor of the fact ors is conside ed a ran om efe t)

σ

r

True s andard deviation for the r sid al t erm in a clas ical analy sis of v rianc for a two-wa

fact orial (c os ed) ex e iment

d

ij

R esid al cor espon ing t o level i of one fact or an level j of a se on fact or in a two-wa fact orial

ex e iment without r pl cation

Interntio l Org izatio for Sta ar dizatio

1Mean sq ar for the firs fact or in a clas ical analy sis of v rianc for a two-wa fact orial

(c os ed) ex e iment

M

2Mean sq ar for the se on fact or in a clas ical analy sis of v rianc for a two-wa fact orial

(c os ed) ex e iment

M

IMean sq ar for the int eraction t erm in a clas ical analysis of v rianc for a two-wa fact orial

(c os ed) ex e iment with r plcation

M

rMean sq ar for the r sid al t erm in a clas ical analy sis of v rianc for a two-wa fact orial

(c os ed) ex e iment

M

totMean sq ar calculat ed from the “Total” sum of sq ar s in a clas ical analy sis of v rianc for a

two-wa fact orial (c os ed)ex e iment

n The n mbe of r pl cat e o se v tions at each combination of fact or levels (hat is, within each

“c l” in a two-wa fact orial (c os ed) ex e iment with r plcation

p The n mbe of levels for the f irs fact or in a two-wa fact orial (c os ed)ex e iment

q The n mbe of levels for the se on fact or in a fact orial (c os ed)ex e iment

x

ij

Obse v tion cor espon ing t o level i of one fact or an level j of a se on fact or in a two-wa fac

-t orial ex e iment without r pl cation

x

ij

kth

o se v tion cor esp n ing t o level i of one fact or an level j of a se on fact or in a two-wa

fact orial ex e iment with r plcation

Sum of sq ar s for the int eraction t erm in a clas ical analy sis of v rianc for a two-wa fact orial

(c os ed) ex e iment with r plcation

s Stan ard deviation of a set of in epen ent o se v tions

s Es imat ed betwe n-level s an ard deviation for the f irs fact or ( if conside ed a ran om efe t) in

a two-wa fact orial (c os ed) ex e iment

s Es imat ed betwe n-level s an ard deviation for the se on fact or ( if conside ed a ran om efe t)

in a two-wa fact orial (c os ed) ex e iment

s

I

Es imat ed betwe n-group s andard deviation for the int eraction t erm in a fact orial ex e iment

(w he e one or mor of thefact ors isconside ed a ran om efe t)

s

r

Es imat ed s an ard deviation for the r sid al t erm in a clas ical analy sisof v rianc for a

two-wa fact orial (c os ed) ex e iment

s

Es imat ed s an ard e ror as ociat ed with the mean in a two-wa fact orial (c os ed)ex e iment

It should benot ed that as far as p s ible, o se v tions should be cole t ed in ran omiz d orde A ction

should also be taken t o r move confoun ing efe t ; for ex mple, a design int en ed t o inves ig t e the

efe t of chang es in t es mat erial matrix an dife ent analyt e conc ntrations on r co e y in analytical

chemis ry should not ru each dife ent ample ty e in a single run on a dife ent da

6 Prel minary review ofdata — Over view

In g ene al, pr l minary r view should r ly on gra hical inspe tion The g ene al principle is t o form

an f it the a pro riat e lnear model (for b lanc d designs this is adeq at ely done b es imating row,

column an , if ne es ary, c l means in the two-wa layout) an inspe t the r sid als

Man el’ s s atis ics, as pr sented in ISO 5 2 -2, ar a pl ca le to inspe tion of in ivid al data p int

in two-wa designs, b r placing the ‘a oratory’ in ISO 5 2 -2 b the ‘c l ’ in a two-wa design and

ar r commended

Ordinary r sid al plot an normal pro a i ty plot ar also a plca le t o the r sid als

Outl e t es s might ad itional y be sugg est ed, though they would ne d t o be used with car ; the degr es

of fr edom for the r sid als is smal e than for the w hole data set, compromising c itical v lues In

ad ition, in designs for d plcat e measur ment , the r sid als for a c l with a se ious outle ty icaly

a pear as two outle seq idis ant from a common mean R esid als for the ‘ main efe t ’ model as wel

as the model inclu ing c l means (he int eraction t erm) ma usefuly be inspe t ed sep rat ely t o a oid

such an efe t

7 Varianc e c omponents and unc ertainty estimation

7.1 General c onsiderations for var ianc e c omponents and unc ertainty estimation

Basic calculations ar b sed on the two-wa ANOV ta les o tained from clas ical ANOV for the

two-wa la out Detaied proc d r s ar shown below The use of sof war implementations of r s rict ed

ma imum l kel ho d es imation (“ REML” is pe mitt ed w hen normalty isa r als ic as umption for al

ran om efe t

When calculating v rianc es imat es from clas ical ANOV ta les neg tive es imat es ofv rianc can

arise In the folowing calculations (7.2 t o 7.4), it is r commen ed that these es imat es be set t o z ro It

is furthe r commen ed that t erms in the initial, complet e, s atis ical model that ar as ociat ed with

neg tive or z ro es imat es of v rianc ar dro ped from the model and the model r calculat ed w hen

s an ard u c rtainties an as ociat ed efe tive degr es of fr edom ar of int er s

NOTE 1 REML calculations do not return ne ative estimates of variance an it is then u neces ary t o red ce

an re-f it models u les efective de re s of fre dom are of interest

Interntio l Org izatio for Sta ar dizatio

NOTE 2 Variance estimates from smal data sets are highly varia le from one sample t o another For e ample,

estimated variances taken from in epen ent samples of 1 o servations drawn from a normal distribution can

vary by more than a fact or of two ( hat is, either gre t er or smal er) from the true variance Variance estimates

from other distributions can vary more

7.2 T wo-way layout without r epl cation

7.2.1 Desig n

The ex e iment inv lves v riation in two dife ent fact ors (for ex mple, t es it em an ins rument) with

a single o se v tion pe fact or combination Let p be the n mbe of levels for the firs fact or of int er s ,

an q the n mbe of levels for the se on , so that the e ar p o se v tions x

ij, w he e the subsc ipt

denot e level i of F act or 1 an level j of Fact or 2

the mean xfor al data Calculat e the r sid als d

ijfrom

Plot the r sid als in ru orde and inspe t for u ex e t ed tr n s an outlying o se v tions A dditionaly,

pr par a normal pro a i ity plot an inspe t or se ious dep rtur s from normalty Che k an cor e t

any a e rant v lues, b r -measur ment if ne es ary If outlying o se v tions ar foun an can ot

r asona ly be cor e t ed, inspe t othe v lues within the same fact or levels If v lues within the same

level of one fact or al a pear disc epant (for ex mple, if r sult for a p rticular t es mat erial a pear

un sualy impr cise), discard al data from that act or level befor es imating v rianc s If this afe t

mor than one fact or level, discontin e theanaly sis an eithe tr at dife ent fact or levels sep rat ely or

inves ig t e the causean r peat the ex e iment

NOTE A single mis ing value can b removed if it is inconsistent with normal performance of the

me surement, that is, it can b at ributa le t o instrumental or other causes R efer to ‘tre tment with mis ing

values’ b low for further analysis

7.2.3 Varianc e c ompo ent estimatio

Con uct an analy sis of v rianc t o o tain the ANOV ta le of the form shown in Ta le 1

Table 1 — ANOVA table for two-way design with ut repl catio

22

22

σ

r2

tot

=S

1+ S

2+ S

From the ta le, the v rianc es imat es s , s an s

r2

for F act or 1, Fact or 2 an the r peata i ty

v rianc , r spe tively, ar given b

s

q1

Whe e a v arianc component is les than z ro an is of int er s for u c rtainty ev aluation othe than in

the as es ment of the u c rtainty for the mean v alue from the ex e iment, set the es imate eq al to z ro

E AMP E In a ran omized block design used t o det ermine a b tween-u it variance for a reference material,

the b tween-u it variance is of int erest for u certainty evaluation even though the me n of the homog eneity

e periment is of no importance

7.2.4 Standard unc ertainty for the me n of al o servations

Whe e the ex e iment is int en ed t o yield a mean v lue xo e al o se v tions an al v rianc

es imat es ar p sitive, the s an ard unc rtainty arising from r peata i ity, r, an from v riation in the

two ex e imental fact orsF

qs

px

12

2

r

(2)

Whe e one or mor v rianc es imat es ar neg tive or z ro, eithe set the cor esp nding t erm in

F rmula (2) t o z ro if only the s an ard u c rtainty in the mean is of int er s or, if the efe tive degr es

of fr edom is also of int er s , proc ed as in 7.2.5.2

7.2.5 Deg rees of freedom for the standar d unc ertainty

7.2.5.1 A ll var ianc e estimates p siti ve

Whe e al v rianc es imat es ar p sitive:

−+

Interntio l Org izatio for Sta ar dizatio

7.2.5.2 One or mor e var ianc e estimates zer o or negati ve

Whe e oneof thev rianc es imat es s or s is z ro or neg tive ( e 7.2.3):

— r mo e the cor espon ing t erm from the model an r calculat e as a one-wa analy sis of v arianc

(“ red c d model” t o give a single betwe n-group mean sq ar M

bwith degr es of fr edom ν

b

;

NOT T e analy sis of variance wil also provide a within-grou me n sq are M

wwhich is not used

=b

;

— set the n mbe of degr es of fr edom t o the degr es of fr edom as ociat ed with the betwe n-group

mean sq ar in the r d c d model

Whe e the v rianc es imat es for b th of the two ran om fact ors ar z ro or neg tive, tr at the

complet e data set as p in epen ent o se v tions:

— calculat e the s an ard deviation s in the usual wa ;

— calculat e the s an ard e ror s from

s

s

px

=2

;

— set the degr es of fr edom for the s an ard e ror t o p − 1

7.3 T wo-way balanc ed e x periment w ith r epl cation (both factor s random)

7.3.1 Desig n

The ex e iment inv lves v riation in two dife ent fact ors (for ex mple, t es it em an measur ment

ru )with a single o se v tion pe fact or combination Let p be the n mbe of levelsfor thefirs fact or of

int er s , q the n mbe oflevels for the se on , an n the n mbe of o se v tion pe fact or combination,

so that the e ar p n o se v tions

7.3.2 Pr elminar y inspectio

Calculat e c l means, subtract from the data an plot the r sulting r sid als in run orde t o che k for

unex e t ed tr n s or outlying v alues If disc ep nt v lues ar fou d, the disc ep nt v lues should be

che ked an cor e t ed if pos ible If cor e tion is not p s ible, an if the disc epancy can be at ribut ed

t o ins rumental e ror or othe identif ia le cause, r mo e the data p int an r fe t o ‘tr atment with

mis ing v lues

Inspe t a normal pro a i ity plot of the r sid als t o che k for significant dep rtur s from normal ty as

a o e

Optional y, calculat e Man el’ s s atis ics for c ls an plot as in ISO 5 2 - 2 Che k extr me c l means

(Man el’s h) or extr me s an ard deviation (Man el’s k) an if ne es ary cor e t any a e rant data

NOTE In e periments con ucted in d plicat e, in ivid al outlier in d plicate data wil usual y ap e r as

pair of outlying values eq idistant rom the me n for the cel

7.3.3 Varianc e c ompo ent e x tractio

a) Con uct an analy sis of v arianc with int eractions This wi l yield a ta le of the form shown in Ta le 2