Statistics

Advanced HigherStatistical Formulae and Tables

For use in National Qualification Courses leading to the 2016 examinations and beyond.

Publication date: 2015Publication code: BB7179ISBN: 978 1 910180 17 4

Published by the Scottish Qualifications AuthorityThe Optima Building, 58 Robertson Street, Glasgow G2 8DQ Lowden, 24 Wester Shawfair, Dalkeith, Midlothian EH22 1FD

www.sqa.org.uk

The information in this publication may be reproduced in support of SQA qualifications. If it is reproduced, SQA should be clearly acknowledged as

the source. If it is to be used for any other purpose, then written permission must be obtained from SQA. It must not be reproduced for

trade or commercial purposes.

© Scottish Qualifications Authority 2015

Page 02

For an up-to-date list of prices visit the Publication Sales and Downloads section of SQA’s website.

For further details telephone SQA’s Customer Contact Centre on 0845 279 1000.

Page 03

ContentsPage

Statistical Formulae 4

Table 1 Binomial Cumulative Distribution Function 7

Table 2 Poisson Cumulative Distribution Function 10

Table 3 Standard Normal Cumulative Distribution Function 11

Table 4 Percentage Points of the Standard Normal Distribution 12

Table 5 The Student t Distribution 13

Table 6 The Chi-squared Distribution 14

Table 7 The Wilcoxon Signed-Rank Test 15

Table 8 The Mann-Whitney Test 16

Page 04

STATISTICAL FORMULAE

Probability Distributions

Discrete Continuous

Distribution Uniform Binomial Poisson Uniform Normal

Parameters U(k) B(n, p) Po(λ) U(a,b) N(µ, σ 2)

pf/pdf1k ( )1 −− n xn x

xC p p!

− xex

λλ 1−b a

2121

2

−⎛ ⎞− ⎜ ⎟⎝ ⎠

π

x

eμ

σ

σ

Mean1

2+k

np λ2+a b

µ

Variance2 112

−knp(1−p) λ

2( )12−b a

σ 2

Western Electric Company Rules

To determine when a process may be out of statistical control we may use the Western Electric Company Rules:

• Any single data point falls outside a 3σ limit

• Two out of three consecutive points fall beyond the same 2σ limit

• Four out of five consecutive points fall beyond the same 1σ limit

• Eight consecutive points fall on the same side of the centre line

Sums of Squares and Products Sample Standard Deviation

( ) ( )22 2 ∑= ∑ − = ∑ − i

xx i ix

S x x xn 1

=−xxSs

n

( ) ( )22 2 ∑= ∑ − = ∑ − i

yy i iy

S y y yn

( )( ) ∑ ∑= ∑ − − = ∑ − i ixy i i i i

x yS x x y y x yn

Page 05

Correlation

Product moment correlation coefficient = xy

xx yy

Sr

S S

Linear Regression

The linear model is = + +i i iY xα β ε where iε are independent, E( ) 0=iε and 2V( ) =iε σ .

Least squares estimates, a and b, for α and β respectively are given by

= xy

xx

Sb

S and = −a y bx .

The sum of squared residuals is given by 2

= − xyyy

xx

SSSR S

S

and an estimate for 2σ is 2

2=

−SSRsn

.

If additionally 2~ N(0, )iε σ then

a ( )100 1− %α prediction interval for i iY | x is given by ( )2

2 1 211− −

−± + + i

i n ,xx

x xY t s

n Sα / and

a ( )100 1− %α confidence interval for E( | )i iY x is given by ( )2

2 1 21

− −

−± + i

i n ,xx

x xY t s

n Sα / .

Page 06

Hypothesis Test Statistics

z-test for a difference in population means:

1 2 1 22 21 2

1 2

( ) ~ N(0,1)− − −

+

X X

n n

μ μσ σ

t-test for a difference in population means:

1 2

2 221 2 1 2 1 1 2 2

21 2

1 2

( ) ( 1) ( 1)~ where21 1 + −

− − − − + −=+ −

+n n

X X n s n st sn n

sn n

μ μ

z-test for a difference in population proportions:

p p

pqn n

p n p n pn n

1 2

1 2

1 1 2 2

1 21 10 1−

+

= ++

~ ( , )N where

Chi-squared test for goodness-of-fit and contingency tables

22( ) ~−∑ i iv

i

O EE

χ

where at least 80% of the Ei should be at least 5 and none should be less than 1.

To test the null hypothesis that the population product moment correlation coefficient

0=ρ use the test statistic 2

21

−=−

r ntr

and to test the null hypothesis that the slope parameter 0=β use the test statistic = xxb St

s

Page 07

TABLE 1: BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION

The tabulated value is F(x) = P(X ≤ x) where X has the binomial distribution B(n, p).

Omitted entries to the left and right of tabulated values are 1·0000 and 0·0000 respectively,to four decimal places.

p 0·05 0·10 0·15 0·20 0·25 0·30 0·35 0·40 0·45 0·50

n = 4 x = 0123

0·81450·98600·9995

0·65610·94770·99630·9999

0·52200·89050·98800·9995

0·40960·81920·97280·9984

0·31640·73830·94920·9961

0·24010·65170·91630·9919

0·17850·56300·87350·9850

0·12960·47520·82080·9744

0·09150·39100·75850·9590

0·06250·31250·68750·9375

n = 6 x = 012345

0·73510·96720·99780·9999

0·53140·88570·98420·99870·9999

0·37710·77650·95270·99410·9996

0·26210·65540·90110·98300·99840·9999

0·17800·53390·83060·96240·99540·9998

0·11760·42020·74430·92950·98910·9993

0·07540·31910·64710·88260·97770·9982

0·04670·23330·54430·82080·95900·9959

0·02770·16360·44150·74470·93080·9917

0·01560·10940·34380·65630·89060·9844

n = 8 x = 01234567

0·66340·94280·99420·9996

0·43050·81310·96190·99500·9996

0·27250·65720·89480·97860·99710·9998

0·16780·50330·79690·94370·98960·99880·9999

0·10010·36710·67850·88620·97270·99580·9996

0·05760·25530·55180·80590·94200·98870·99870·9999

0·03190·16910·42780·70640·89390·97470·99640·9998

0·01680·10640·31540·59410·82630·95020·99150·9993

0·00840·06320·22010·47700·73960·91150·98190·9983

0·00390·03520·14450·36330·63670·85550·96480·9961

n = 10 x = 0123456789

0·59870·91390·98850·99900·9999

0·34870·73610·92980·98720·99840·9999

0·19690·54430·82020·95000·99010·99860·9999

0·10740·37580·67780·87910·96720·99360·99910·9999

0·05630·24400·52560·77590·92190·98030·99650·9996

0·02820·14930·38280·64960·84970·95270·98940·99840·9999

0·01350·08600·26160·51380·75150·90510·97400·99520·9995

0·00600·04640·16730·38230·63310·83380·94520·98770·99830·9999

0·00250·02330·09960·26600·50440·73840·89800·97260·99550·9997

0·00100·01070·05470·17190·37700·62300·82810·94530·98930·9990

Page 08

TABLE 1: BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION (continued)

p 0·05 0·10 0·15 0·20 0·25 0·30 0·35 0·40 0·45 0·50

n = 12 x = 0123456789

1011

0·54040·88160·98040·99780·9998

0·28240·65900·88910·97440·99570·99950·9999

0·14220·44350·73580·90780·97610·99540·99930·9999

0·06870·27490·55830·79460·92740·98060·99610·99940·9999

0·03170·15840·39070·64880·84240·94560·98570·99720·9996

0·01380·08500·25280·49250·72370·88220·96140·99050·99830·9998

0·00570·04240·15130·34670·58330·78730·91540·97450·99440·99920·9999

0·00220·01960·08340·22530·43820·66520·84180·94270·98470·99720·9997

0·00080·00830·04210·13450·30440·52690·73930·88830·96440·99210·99890·9999

0·00020·00320·01930·07300·19380·38720·61280·80620·92700·98070·99680·9998

n = 14 x = 0123456789

10111213

0·48770·84700·96990·99580·9996

0·22880·58460·84160·95590·99080·99850·9998

0·10280·35670·64790·85350·95330·98850·99780·9997

0·04400·19790·44810·69820·87020·95610·98840·99760·9996

0·01780·10100·28110·52130·74150·88830·96170·98970·99780·9997

0·00680·04750·16080·35520·58420·78050·90670·96850·99170·99830·9998

0·00240·02050·08390·22050·42270·64050·81640·92470·97570·99400·99890·9999

0·00080·00810·03980·12430·27930·48590·69250·84990·94170·98250·99610·99940·9999

0·00020·00290·01700·06320·16720·33730·54610·74140·88110·95740·98860·99780·9997

0·00010·00090·00650·02870·08980·21200·39530·60470·78800·91020·97130·99350·99910·9999

n = 16 x = 0123456789

1011121314

0·44010·81080·95710·99300·99910·9999

0·18530·51470·78920·93160·98300·99670·99950·9999

0·07430·28390·56140·78990·92090·97650·99440·99890·9998

0·02810·14070·35180·59810·79820·91830·97330·99300·99850·9998

0·01000·06350·19710·40500·63020·81030·92040·97290·99250·99840·9997

0·00330·02610·09940·24590·44990·65980·82470·92560·97430·99290·99840·9997

0·00100·00980·04510·13390·28920·49000·68810·84060·93290·97710·99380·99870·9998

0·00030·00330·01830·06510·16660·32880·52720·71610·85770·94170·98090·99510·99910·9999

0·00010·00100·00660·02810·08530·19760·36600·56290·74410·87590·95140·98510·99650·99940·9999

0·00030·00210·01060·03840·10510·22720·40180·59820·77280·89490·96160·98940·99790·9997

Page 09

TABLE 1: BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION (continued)

p 0·05 0·10 0·15 0·20 0·25 0·30 0·35 0·40 0·45 0·50

n = 18 x = 0123456789

10111213141516

0·39720·77350·94190·98910·99850·9998

0·15010·45030·73380·90180·97180·99360·99880·9998

0·05360·22410·47970·72020·87940·95810·98820·99730·99950·9999

0·01800·09910·27130·50100·71640·86710·94870·98370·99570·99910·9998

0·00560·03950·13530·30570·51870·71750·86100·94310·98070·99460·99880·9998

0·00160·01420·06000·16460·33270·53440·72170·85930·94040·97900·99390·99860·9997

0·00040·00460·02360·07830·18860·35500·54910·72830·86090·94030·97880·99380·99860·9997

0·00010·00130·00820·03280·09420·20880·37430·56340·73680·86530·94240·97970·99420·99870·9998

0·00030·00250·01200·04110·10770·22580·39150·57780·74730·87200·94630·98170·99510·99900·9999

0·00010·00070·00380·01540·04810·11890·24030·40730·59270·75970·88110·95190·98460·99620·99930·9999

n = 20 x = 0123456789

1011121314151617

0·35850·73580·92450·98410·99740·9997

0·12160·39170·67690·86700·95680·98870·99760·99960·9999

0·03880·17560·40490·64770·82980·93270·97810·99410·99870·9998

0·01150·06920·20610·41140·62960·80420·91330·96790·99000·99740·99940·9999

0·00320·02430·09130·22520·41480·61720·78580·89820·95910·98610·99610·99910·9998

0·00080·00760·03550·10710·23750·41640·60800·77230·88670·95200·98290·99490·99870·9997

0·00020·00210·01210·04440·11820·24540·41660·60100·76240·87820·94680·98040·99400·99850·9997

0·00050·00360·01600·05100·12560·25000·41590·59560·75530·87250·94350·97900·99350·99840·9997

0·00010·00090·00490·01890·05530·12990·25200·41430·59140·75070·86920·94200·97860·99360·99850·9997

0·00020·00130·00590·02070·05770·13160·25170·41190·58810·74830·86840·94230·97930·99410·99870·9998

Page 10

TABLE 2: POISSON CUMULATIVE DISTRIBUTION FUNCTION

The tabulated value is F(x) = P(X ≤ x) where X has the Poisson distribution Po(λ).

Omitted entries to the left and right of tabulated values are 1·0000 and 0·0000 respectively, to four decimal places.

λ 0·5 1·0 1·5 2·0 2·5 3·0 3·5 4·0 4·5 5·0

x = 0123456789

101112131415

0·60650·90980·98560·99820·9998

0·36790·73580·91970·98100·99630·99940·9999

0·22310·55780·80880·93440·98140·99550·99910·9998

0·13530·40600·67670·85710·94730·98340·99550·99890·9998

0·08210·28730·54380·75760·89120·95800·98580·99580·99890·99970·9999

0·04980·19910·42320·64720·81530·91610·96650·98810·99620·99890·99970·9999

0·03020·13590·32080·53660·72540·85760·93470·97330·99010·99670·99900·99970·9999

0·01830·09160·23810·43350·62880·78510·88930·94890·97860·99190·99720·99910·99970·9999

0·01110·06110·17360·34230·53210·70290·83110·91340·95970·98290·99330·99760·99920·99970·9999

0·00670·04040·12470·26500·44050·61600·76220·86660·93190·96820·98630·99450·99800·99930·99980·9999

λ 5·5 6·0 6·5 7·0 7·5 8·0 8·5 9·0 9·5 10·0

x = 0123456789

1011121314151617181920212223

0·00410·02660·08840·20170·35750·52890·68600·80950·89440·94620·97470·98900·99550·99830·99940·99980·9999

0·00250·01740·06200·15120·28510·44570·60630·74400·84720·91610·95740·97990·99120·99640·99860·99950·99980·9999

0·00150·01130·04300·11180·22370·36900·52650·67280·79160·87740·93320·96610·98400·99290·99700·99880·99960·99980·9999

0·00090·00730·02960·08180·17300·30070·44970·59870·72910·83050·90150·94670·97300·98720·99430·99760·99900·99960·9999

0·00060·00470·02030·05910·13210·24140·37820·52460·66200·77640·86220·92080·95730·97840·98970·99540·99800·99920·99970·9999

0·00030·00300·01380·04240·09960·19120·31340·45300·59250·71660·81590·88810·93620·96580·98270·99180·99630·99840·99930·99970·9999

0·00020·00190·00930·03010·07440·14960·25620·38560·52310·65300·76340·84870·90910·94860·97260·98620·99340·99700·99870·99950·99980·9999

0·00010·00120·00620·02120·05500·11570·20680·32390·45570·58740·70600·80300·87580·92610·95850·97800·98890·99470·99760·99890·99960·99980·9999

0·00010·00080·00420·01490·04030·08850·16490·26870·39180·52180·64530·75200·83640·89810·94000·96650·98230·99110·99570·99800·99910·99960·99990·9999

0·00050·00280·01030·02930·06710·13010·22020·33280·45790·58300·69680·79160·86450·91650·95130·97300·98570·99280·99650·99840·99930·99970·9999

Page 11

TABLE 3: STANDARD NORMAL CUMULATIVE DISTRIBUTION FUNCTION

The tabulated value is P(Z ≤ z) where Z has the standard normal distribution N(0, 1).

z ·00 ·01 ·02 ·03 ·04 ·05 ·06 ·07 ·08 ·09

0·0 0·1 0·2 0·3 0·4

0·50000·53980·57930·61790·6554

0·50400·54380·58320·62170·6591

0·50800·54780·58710·62550·6628

0·51200·55170·59100·62930·6664

0·51600·55570·59480·63310·6700

0·51990·55960·59870·63680·6736

0·52390·56360·60260·64060·6772

0·52790·56750·60640·64430·6808

0·53190·57140·61030·64800·6844

0·53590·57530·61410·65170·6879

0·5 0·6 0·7 0·8 0·9

0·69150·72570·75800·78810·8159

0·69500·72910·76110·79100·8186

0·69850·73240·76420·79390·8212

0·70190·73570·76730·79670·8238

0·70540·73890·77040·79950·8264

0·70880·74220·77340·80230·8289

0·71230·74540·77640·80510·8315

0·71570·74860·77940·80780·8340

0·71900·75170·78230·81060·8365

0·72240·75490·78520·81330·8389

1·0 1·1 1·2 1·3 1·4

0·84130·86430·88490·90320·9192

0·84380·86650·88690·90490·9207

0·84610·86860·88880·90660·9222

0·84850·87080·89070·90820·9236

0·85080·87290·89250·90990·9251

0·85310·87490·89440·91150·9265

0·85540·87700·89620·91310·9279

0·85770·87900·89800·91470·9292

0·85990·88100·89970·91620·9306

0·86210·88300·90150·91770·9319

1·5 1·6 1·7 1·8 1·9

0·93320·94520·95540·96410·9713

0·93450·94630·95640·96490·9719

0·93570·94740·95730·96560·9726

0·93700·94840·95820·96640·9732

0·93820·94950·95910·96710·9738

0·93940·95050·95990·96780·9744

0·94060·95150·96080·96860·9750

0·94180·95250·96160·96930·9756

0·94290·95350·96250·96990·9761

0·94410·95450·96330·97060·9767

2·0 2·1 2·2 2·3 2·4

0·97720·98210·98610·98930·9918

0·97780·98260·98640·98960·9920

0·97830·98300·98680·98980·9922

0·97880·98340·98710·99010·9925

0·97930·98380·98750·99040·9927

0·97980·98420·98780·99060·9929

0·98030·98460·98810·99090·9931

0·98080·98500·98840·99110·9932

0·98120·98540·98870·99130·9934

0·98170·98570·98900·99160·9936

2·5 2·6 2·7 2·8 2·9

0·99380·99530·99650·99740·9981

0·99400·99550·99660·99750·9982

0·99410·99560·99670·99760·9982

0·99430·99570·99680·99770·9983

0·99450·99590·99690·99770·9984

0·99460·99600·99700·99780·9984

0·99480·99610·99710·99790·9985

0·99490·99620·99720·99790·9985

0·99510·99630·99730·99800·9986

0·99520·99640·99740·99810·9986

3·0 3·1 3·2 3·3 3·4 3·5 3·6

0·99870·99900·99930·99950·99970·99980·9998

0·99870·99910·99930·99950·99970·99980·9998

0·99870·99910·99940·99950·99970·99980·9999

0·99880·99910·99940·99960·99970·99980·9999

0·99880·99920·99940·99960·99970·99980·9999

0·99890·99920·99940·99960·99970·99980·9999

0·99890·99920·99940·99960·99970·99980·9999

0·99890·99920·99950·99960·99970·99980·9999

0·99900·99930·99950·99960·99970·99980·9999

0·99900·99930·99950·99970·99980·99980·9999

0 z

Page 12

0 zp

TABLE 4: PERCENTAGE POINTS OF THE STANDARD NORMAL DISTRIBUTION

The entries in the table are such that for the standard normal distribution P(Z > zp) = p.

p zp

0·500 0·00

0·250 0·67

0·100 1·28

0·050 1·64

0·025 1·96

0·010 2·33

0·005 2·58

0·001 3·09

0·0005 3·29

Page 13

TABLE 5: STUDENT’S t DISTRIBUTION

The table entry tν is such that P(T ≤ tν) = q where T has the Student t distribution with ν degrees of freedom.

q 0·900 0·950 0·975 0·990 0·995 0·999 0·9995

ν = 123456789

10

3·078 1·886 1·638 1·533 1·476 1·440 1·415 1·397 1·383 1·372

6·314 2·920 2·353 2·132 2·015 1·943 1·895 1·860 1·833 1·812

12·706 4·303 3·182 2·776 2·571 2·447 2·365 2·306 2·262 2·228

31·821 6·965 4·541 3·747 3·365 3·143 2·998 2·896 2·821 2·764

63·656 9·925 5·841 4·604 4·032 3·707 3·499 3·355 3·250 3·169

318·289 22·328 10·214 7·173 5·894 5·208 4·785 4·501 4·297 4·144

636·578 31·600 12·924 8·610 6·869 5·959 5·408 5·041 4·781 4·587

11121314151617181920

1·363 1·356 1·350 1·345 1·341 1·337 1·333 1·330 1·328 1·325

1·796 1·782 1·771 1·761 1·753 1·746 1·740 1·734 1·729 1·725

2·201 2·179 2·160 2·145 2·131 2·120 2·110 2·101 2·093 2·086

2·718 2·681 2·650 2·624 2·602 2·583 2·567 2·552 2·539 2·528

3·106 3·055 3·012 2·977 2·947 2·921 2·898 2·878 2·861 2·845

4·025 3·930 3·852 3·787 3·733 3·686 3·646 3·610 3·579 3·552

4·437 4·318 4·221 4·140 4·073 4·015 3·965 3·922 3·883 3·850

21222324252627282930

1·323 1·321 1·319 1·318 1·316 1·315 1·314 1·313 1·311 1·310

1·721 1·717 1·714 1·711 1·708 1·706 1·703 1·701 1·699 1·697

2·080 2·074 2·069 2·064 2·060 2·056 2·052 2·048 2·045 2·042

2·518 2·508 2·500 2·492 2·485 2·479 2·473 2·467 2·462 2·457

2·831 2·819 2·807 2·797 2·787 2·779 2·771 2·763 2·756 2·750

3·527 3·505 3·485 3·467 3·450 3·435 3·421 3·408 3·396 3·385

3·819 3·792 3·768 3·745 3·725 3·707 3·689 3·674 3·660 3·646

31323334353637383940

1·309 1·309 1·308 1·307 1·306 1·306 1·305 1·304 1·304 1·303

1·696 1·694 1·692 1·691 1·690 1·688 1·687 1·686 1·685 1·684

2·040 2·037 2·035 2·032 2·030 2·028 2·026 2·024 2·023 2·021

2·453 2·449 2·445 2·441 2·438 2·434 2·431 2·429 2·426 2·423

2·744 2·738 2·733 2·728 2·724 2·719 2·715 2·712 2·708 2·704

3·375 3·365 3·356 3·348 3·340 3·333 3·326 3·319 3·313 3·307

3·633 3·622 3·611 3·601 3·591 3·582 3·574 3·566 3·558 3·551

∞ 1·282 1·645 1·960 2·327 2·576 3·091 3·291

0 tν

Page 14

TABLE 6: THE CHI-SQUARED DISTRIBUTION

The table entry χ2ν is such that P(X2

ν ≤ χ2ν) = q where X2

ν has the chi-squared distribution with ν degrees of freedom.

q 0·900 0·950 0·975 0·990 0·995 0·999 0·9995

ν = 123456789

10111213141516171819202122232425

2·7064·6056·2517·7799·236

10·64512·01713·36214·68415·98717·27518·54919·81221·06422·30723·54224·76925·98927·20428·41229·61530·81332·00733·19634·382

3·8415·9917·8159·488

11·07012·59214·06715·50716·91918·30719·67521·02622·36223·68524·99626·29627·58728·86930·14431·41032·67133·92435·17236·41537·652

5·0247·3789·348

11·14312·83214·44916·01317·53519·02320·48321·92023·33724·73626·11927·48828·84530·19131·52632·85234·17035·47936·78138·07639·36440·646

6·6359·210

11·34513·27715·08616·81218·47520·09021·66623·20924·72526·21727·68829·14130·57832·00033·40934·80536·19137·56638·93240·28941·63842·98044·314

7·87910·59712·83814·86016·75018·54820·27821·95523·58925·18826·75728·30029·81931·31932·80134·26735·71837·15638·58239·99741·40142·79644·18145·55846·928

10·82713·81516·26618·46620·51522·45724·32126·12427·87729·58831·26432·90934·52736·12437·69839·25240·79142·31243·81945·31446·79648·26849·72851·17952·619

12·11515·20117·73119·99822·10624·10226·01827·86729·66731·41933·13834·82136·47738·10939·71741·30842·88144·43445·97447·49849·01050·51051·99953·47854·948

0 χν2

Page 15

TABLE 7: THE WILCOXON SIGNED RANK TEST

The table gives the critical value of W, for a selection of significance levels, where n is the sample size for the test.

1-tail2-tail

n

0·050·10

0·0250·05

0·010·02

0·0050·01

5 0

6 2 0

7 3 2 0

8 5 3 1 0

9 8 5 3 1

10 10 8 5 3

11 13 10 7 5

12 17 13 9 7

13 21 17 12 9

14 25 21 15 12

15 30 25 19 15

16 35 29 23 19

17 41 34 27 23

18 47 40 32 27

19 53 46 37 32

20 60 52 43 37

For larger samples, a normal approximation for W may be employed where

1 1E( ) ( 1) and V( ) ( 1)(2 1)4 24

= + = + +W n n W n n n .

Page 16

TABLE 8: THE MANN-WHITNEY TEST

Samples of sizes m ≤ n have sum of ranks Wm and Wn and W is the smaller of Wm and m(m + n + 1) − Wm

Critical values of W are given for a selection of significance levels.

1-tail 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·012-tail 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02

n m = 3 m = 4 m = 5 m = 6 m = 7 m = 8

3456789101112131415

6 - - 6 - - 7 6 - 8 7 - 8 7 6 9 8 6 10 8 7 10 9 7 11 9 7 11 10 8 12 10 8 13 11 8 13 11 9

11 10 12 11 10 13 12 11 14 13 11 15 14 12 16 14 13 17 15 13 18 16 14 19 17 15 20 18 15 21 19 16 22 20 17

19 17 16 20 18 17 21 20 18 23 21 19 24 22 20 26 23 21 27 24 22 28 26 23 30 27 24 31 28 25 33 29 26

28 26 24 29 27 25 31 29 27 33 31 28 35 32 29 37 34 30 38 35 32 40 37 33 42 38 34 44 40 36

39 36 34 41 38 35 43 40 37 45 42 39 47 44 40 49 46 42 52 48 44 54 50 45 56 52 47

51 49 45 54 51 47 56 53 49 59 55 51 62 58 53 64 60 56 67 62 58 69 65 60

1-tail 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·012-tail 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02

n m = 9 m = 10 m = 11 m = 12 m = 13 m = 14

91011121314151617181920

66 62 59 69 65 61 72 68 63 75 71 66 78 73 68 81 76 71 84 79 73 87 82 76 90 84 78 93 87 80 96 90 83 99 93 85

82 78 74 86 81 77 89 84 79 92 88 82 96 91 85 99 94 88 103 97 91 106 100 93 110 103 96 113 107 99 117 110 102

100 96 91 104 99 94 108 103 97 112 106 100 116 110 103 120 113 107 123 117 110 127 121 113 131 124 116 135 128 119

120 115 109125 119 113129 123 116133 127 120138 131 124142 135 127146 139 131150 143 134155 147 138

142 136 130 147 141 134 152 145 138 156 150 142 161 154 146 166 158 150 171 163 154 175 167 158

166 160 152 171 164 156 176 169 161 182 174 165 187 179 170 192 183 174 197 188 178

1-tail 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·01 0·05 0·025 0·012-tail 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02 0·1 0·05 0·02

n m = 15 m = 16 m = 17 m = 18 m = 19 m = 20

151617181920

192 184 176197 190 181203 195 186208 200 190214 205 195220 210 200

219 211 202225 217 207231 222 212237 228 218243 234 223

249 240 230255 246 235262 252 241268 258 246

280 270 259287 277 265294 283 271

313 303 291320 309 297 348 337 324

For larger samples, a normal approximation for W may be employed where

1 1E( ) ( 1) and V( ) ( 1)2 12

= + + = + +W m m n W mn m n .