remaus.

NATIONAL BUREAU OF STANDARDS REPORT

18^9

PROBABILITY TABLES FOR THE WILOOXON TEST

WHEN THERE ARE TIES

Statistical Engineering Laboratory

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103- 11-1107 AUG 1 3 1952 1659

PROBABILITY TABLES FOR THE WILCOXON TEST

WHEN THERE ARE TIES

Statistical Engineering Laboratory

The publication, rep

unless permission is

25, 0, C, Such perr

cal ly prepared If th

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

in part. Is prohibited

andards, Washington

port has been speclfiiport for its own use,
FOREWORD

This report was prepared as part of a continuing program

of research on mathematical ‘s tatis tics and its applications

carried out at the National Bureau of Standards under the

general supervision of Dr„ Churchill Eisenhart* Chief of the

Statistical Engineering Laboratory, The Statistical Engineering

Laboratory is Section 11,3 of the National Applied Mathematics

Laboratories (Division 11 * National Bureau of Standards* and

is concerned with the development and application of modern

statistical methods in the physical sciences and engineering,

J, Ho Curtiss

Chief* National Applied

Mathematics Laboratories

A, Vo Astin

Director

National Bureau of Standards
PROBABILITY TABLES FOR THE WILCOXON TEST

WHEN THERE ARE TIES

Statistical Engineering Laboratory

Summary : Tables are given for the exact distribution of

the Wilcoxon statistic in five selected cases where both

of the samples contain five observations, some of which

are tied.

Introduction : William H. Kruskal and W. Allen Wallis in

"Use of Ranks in One-Criterion Variance Analysis" [SRC20609KW] suggest "the average-rank method" for handling the

Wilcoxon statistic when some of the observations are tied.

They point out that this method of treating ties does not

affect the mean value of the Wilcoxon statistic* Their

formula (3*6) gives the variance of the Wilcoxon statistic

when ties are treated according to the average-rank method.

Unfortunately, it would be expensive to make complete

tables of the exact distribution of the Wilcoxon statistic

when there are ties, since these ties may occur in a good

many ways. However, when specific problems arise, if the

samples are small, one can make the necessary tables as

needed
- 2 -

The Statistical Engineering Laboratory (National Bureau

of Standards* Division 11 o 3K in using the Wilcoxon test in

a specific problem* treated the ties as suggested by Kruskal

and Wallis o The tables of this report give the exact distribution of the Wilcoxon statistic for the cases that arose

.

It is realized that these probability tables as such

will not be found useful* for it is unlikely that others

will have samples which have the same ties occurring in therm

But it was felt that these tables might be helpful in indi=

eating the nature of the exact distributions of the Wilcoxon

statistic that do arise when ties occur
Observed Ranks

- 3

-

TABLE I,

1, 2 , 3, k , 6* &p 6* 8 s 9 c, 10

Score ^2] Frequency

Cumulative

Probability

!

; Score Frequency Cumulative

Probability

16 3 ,012 28 21 o*Q3

16 ,5 0 ,012 28.5 0 0 9 j

17 0 o 012 29 18 O J 17.5 0 ,012 29.5 0 A u

0 « ^ 18 ll ,028 30 19 0 ; JiV 18,5 0 ,028 30.5 0 o730

19 4 •OJ4I4

. 31 15 o790

19.5 0 •044 31.5 0 0 790

20 7 o071 32 15 ,849

20,5 0 ,071 32.5 0 0 8k9

21 10 olll 33 10 0689

21,6 0 .111 33.5 0 ,889

22 10 ,151 34 10 .929

22,5 0 <,151 54 0 5 0 ,929

23 15 ,210 35 7 0-956

23.5 0 ,210 35.5 0 o956

24 15 ,270 36 4 o97': 24.5 0 ,270 36,5 0 0 1 51 25 19 .345 37 4 .9?-e

25 .5 0 .345 37.5 0 o' 58

26 18 .417 \

• 38 0 0 26,5 0 .417 j 38.5 0 o98<-

2? 21 ,500 i 39 3 lo OC

27,5 0 ,500

f-^M = 27 c 5

0 [ 6 ) v - 22<,36? [7] a = 4.73

M ± 1„96o * 36,77 or 18,23

/

- 3

-

TABLE I,

Observed Ranks W? 1, 2, 3, 4» 6, 6, 6, 8, 9# 10

Scored ^ ] Frequency^ 33 Cumulative^ 4J

Probability

Score Frequency Cumulative

Probability

16 3 .012 28 21 = 544

16.

5

0 .012 28.5 0 .544

17 0 .012 29 18 .615

17.5 0 .012 29.5 0 .615

18 4 .028 30 19 .690

18.5 0 .028 30.5 0 .690

19 4 .044 31 15 .790

19.5 0 .044 31.5 0 .790

20 7 .071 32 15 .849

20,5 0 .071 32.5 0 • 849

21 10 .111 33 10 .889

21,5 0 .111 33.5 0 .889

22 10 .111 34 10 = 929

22.5 0 .111 34.5 0 = 929

23 15 .171 35 7 •956

23.5 0 .171 35.5 0 .956

24 15 .230 36 4 *972

2^.5 0 .230 36.5 0 *972

25 19 .306 37 4 .988

25.5 0 .306 37.5 0 .988

26 18 *377 38 0 .988

26 .5 0 *377 38.5 0 *988

27 21 olj.60 39 3 1.000

27.5 0 a 460

= 2?o 55

[6 V = 22o36; C7] a = 4*73

M ± 1o96o = 36o 77 or 18.23
/

- 4 -

TABLE II.

Observed Ranks^^s 1-i, 1 ig s 3l?p 3ls> 6-£, 6^, ^>e 9 &s» 9? 10

Scored 2] Frequency^ 3 3 Cumulative^ 4]

Probability

Score Frequency Cumulative

Probability

I6e5» 4 o 0l6 28 24 o603

17 0 o0l6 28.5 4 o6l9

17=5 0 o016 29 6 .643

18 0 .016 29=5 2 o65l

l8o5 0 0 OI6 30 14 o706

19 1 o020 30.5 16 o770

19=5 12 o06? 31 8 .802

20 1 .071 31.5 0 .802

20.5 0 o071 32 8 =833

21 0 o071 32=5 4 .849

21 .5 12 oll9 33 8 088I

22 8 ol5l 33=5 12 o929

22.5 4 0 I 67 34 0 <.929

23 8 ol98 34=5 0 o929

23=5 0 ol98 35 1 o933

24 8 o230 35 = 5 12 .980

21+ = 5 16 o294 36 1 =981+

25 14 • 349 36=5 0 =981+

25=5 2 = 357 37 0 = 981+

26 6 o38l 37.5 0 =981+

26.5 4 o397 38 0 .984

27 24 o492 38=5 4 1.000

27=5 4 o508

^ 3 M = 27o 5 so[6J V = 21o25l t73 cf s /V = 4 0 6l

M ± 1<»96g - 36.54 or l8o 4^
- 5

-

TABLE IIIo

Observed Ranks l s 3s 3 s 3s 6 S 6 , 6^ 8^, 8^- s 10

Scored 2] Frequency^- 3\ Cumulative^ ^ Probability

Score Frequency Cumulative

Probability

15.5 0 o000 27.5 4 .508

16 3 0Q12 28 9 °544

16o5 0 .012 28.5 18 0615

17 0 o012 29 10 o655

17.5 0 o012 29.5 6 .679

18 0 o012 30 3 o690

18.5 2 .020 30.5 18 .762

19 9 o056 31 6 .786

19.5 0 .056 31.5 6 .810

20 1 0O60 32 9 .845

20.5 0 .060 32.5 0 .845

21 3 0O71 33 3 .857

21 .5 18 oi43 33.5 18 *929

22 3 .155 34 3 o940

22 .5 0 .155 34.5 0 .940

23 9 d90 35 1 .944

23.5 6 .214 35.5 0 »944

24 6 .238 36 9 .980

24 o5 18 .310 36 .5 2 .988

25 3 .321 37 0 .988

25.5 6 .345 37.5 0 .988

26 10 .385 36- 0 .988

26.5 18 .456 38.5 0 .988

27 9 °492 39 3 loOOO

[53 M = 2? ,*

[63 V = 21

r 7 1 4* L ,67;

1 M o = lf^74

M + 1o96o = 36083 or I80I7
- 6 -

TABLE IV o

Observed Ranks^ 1 ^ Ip 2\ 9 2^9 4las 4l^# &2® 6"i® 8, 9® 10

Score £2! Frequency^ 3 3 Cumulative^ 3 ProbabilityScore Frequency Cumulative

Probability

15 1 o004 28 14 .563

15-5 0 o004 28,5 8 o593

16 0 *004 29 13 .647

16,

£

0 o004 29.5 4 0663

17 4 o020 30 9 o698

17.5 0 o020 30.5 14 • 754

18 0 o020 31 6 <.778

l8o5 2 .028 31.5 8 *810

19 5 0O48 32 3 0821

19-5 2 o056 32.5 14 .877

20 0 o056 33 1 ,881

20o5 8 .087 33.5 4 .897

21 4 ol03 34 4 ,912

21 0 5 4 .119 34.

5

8 ,944

22 l ol23 35 0 ,944

22,5 14 ol79 35.5 2 • 952

23 3 ol90 36 5 .972

23=5 8 o222 36.5 2 ,980

24 6 *246 37 0 <>980

24o5 14 <,302 37.5 0 ,980

25 9 o337 38 4 • 996

25.5 4 .353 38.5 0 .996

26 13 .405 39 0 .996

26 o5 8 •437 39.5 0 • 996

27 14 .492 40 1 1,000

27 .5 4 *508

[£3 m = 27c5®o[6 ^V = 22, 5S [7 ^o = 4.74

M + 1 0 96o = 36.79 or 18*21
- 7

-

TABLE V.

Observed Ranks 1> 24, 24, 5, 6, 7? 8, 9* 10

Score 1 2 J Frequency^ ^ Cuimilative[ k]

Probability

Score Frequency Cumulative

Probability

15 1 .004 28 8 •556

15.5 0 .004 28.5 12 .603

16 1 ,008 29 8 .635

16.5 0 .008 29.5 10 .675

17 2 .016 JO 7 .702

17.5 0 .016 30.5 12 • 750

18 2 .024 31 6 . 77 b

18.5 2 .032 31.5 8 .805

19 3 o044 32 6 .829

19.5 2 .052 32.5 8 .861

20 4 .067 33 4 .877

20.5 4 ,083 33.5 6 .901

21 4 o099 34 4 .917

21.5 6 .123 34.5 4 • 933

22 4 .139 35 4 .948

22.5 8 .171 35.5 2 .956

23 6 .194 36 3 ,968

23.5 8 .226 36.5 2 • 976

24 6 .250 37 2 • 984

24.5 12 .298 37.5 0 .984

25 7 .325 38 2 •992

25.5 10 .365 38.5 0 •992

26 8 *397 39 1 .996

26,5 12 •444 39-5 0 .996

27 8 .476 40 1 1.000

27.? 12 .524

= 27.5; f6] V = 22,67; [7 'I a = 4.76

M ± 1.96a = 36.83 or 18,17
- 8 -

FOOTNOTES ON TABLES,

[1] The "Observed Ranks" are the ranks given to the observeTM

tions from two samples of five each, using the averagerank method*

[2J The "Score" column contains all of the possible sums of

five of the ranks from the observed ranks*

[3] The "Frequency" column gives the number of ways that a

particular sum can be formed from the observed ranks*

[43 The "Cumulative Probability" column gives the partial

sums of the "Frequency" column divided by 252 (252 is

the number of possible sums that can be formed from the

"observed ranks" when repetitions are permitted — 252

being the number of possible ways of selecting five

things from ten things)*

[53 "M" is the average of the tabulated distribution* For

this distribution the average is exactly the theoretical

value and hence serves as a check that the distribution

was tabulated correctly*

[63 "V" is the variance of the tabulated distribution* For

this distribution the variance agrees exactly with the

theoretical value given by equation (3*6) in Kruskal and

Wallis* The fact that the variances are correct serves

as another check on the tabulation of the distributions*

[73 "a" is the standard deviation of the tabulated distribution*
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