New Cambridge Statistical Tables

NEW CAMBRIDGE STATISTICAL TABLES D. V. LINDLEY & W. E SCOTT Second Edition CAMBRIDGE UNIVERSITY PRESS CONTENTS PREF...

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NEW CAMBRIDGE STATISTICAL TABLES D. V. LINDLEY & W. E SCOTT

Second Edition

CAMBRIDGE UNIVERSITY PRESS

CONTENTS PREFACES

page 3

TABLES: 1 The Binomial Distribution Function 2 The Poisson Distribution Function 3 Binomial Coefficients 4 The Normal Distribution Function 5 Percentage Points of the Normal Distribution 6 Logarithms of Factorials 7 The )(2-Distribution Function 8 Percentage Points of the )(2-Distribution 9 The t-Distribution Function 10 Percentage Points of the t-Distribution 11 Percentage Points of Behrens' Distribution 12 Percentage Points of the F-Distribution 13 Percentage Points of the Correlation Coefficient r when p = 0 14 Percentage Points of Spearman's S 15 Percentage Points of Kendall's K 16 The z-Transformation of the Correlation Coefficient 17 The Inverse of the z-Transformation Percentage Points of the Distribution of the Number of Runs 18 19 Upper Percentage Points of the Two-Sample Kolmogorov—Smirnov Distribution 20 Percentage Points of Wilcoxon's Signed-Rank Distribution 21 Percentage Points of the Mann—Whitney Distribution 22A Expected Values of Normal Order Statistics (Normal Scores) 22B Sums of Squares of Normal Scores 23 Upper Percentage Points of the One-Sample Kolmogorov—Smirnov Distribution 24 Upper Percentage Points of Friedman's Distribution 25 Upper Percentage Points of the Kruskal—Wallis Distribution 26 Hypergeometric Probabilities 27 Random Sampling Numbers 28 Random Normal Deviates 29 Bayesian Confidence Limits for a Binomial Parameter 30 Bayesian Confidence Limits for a Poisson Mean Bayesian Confidence Limits for the Square of a Multiple Correlation 31 Coefficient A NOTE ON INTERPOLATION CONSTANTS

4 24 33 34 35 36 37 40 42 45 46 50 56 57 57 58 59 60 62 65 66 68 70 70 71 72 74 78 79 80 88 89 96 96

CONVENTION. To prevent the tables becoming too dense with figures, the convention has been adopted of omitting the leading figure when this does not change too often, only including it at the beginning of a set of five entries, or when it changes. (Table 23 provides an example.)

PREFACE TO THE FIRST EDITION The raison d'etre of this set of tables is the same as that of the set it replaces, the Cambridge Elementary Statistical Tables (Lindley and Miller, 1953), and is described in the first paragraph of their preface. This set of tables is concerned only with the commoner and more familiar and elementary of the many statistical functions and tests of significance now available. It is hoped that the values provided will meet the majority of the needs of many users of statistical methods in scientific research, technology and industry in a compact and handy form, and that the collection will provide a convenient set of tables for the teaching and study of statistics in schools and universities. The concept of what constitutes a familiar or elementary statistical procedure has changed in 30 years and, as a result, many statistical tables not in the earlier set have been included, together with tables of the binomial, hypergeometric and Poisson distributions. A large part of the earlier set of tables consisted of functions of the integers. These are now readily available elsewhere, or can be found using even the simplest of pocket calculators, and have therefore been omitted. The binomial, Poisson, hypergeometric, normal, X2 and t distributions have been fully tabulated so that all values within the ranges of the arguments chosen can be found. Linear, and in some cases quadratic or harmonic, interpolation will sometimes be necessary and a note on this has been provided. Most of the other tables give only the percentage points of distributions, sufficient to carry out significance tests at the usual 5 per cent and i per cent levels, both one- and two-sided, and there are also some io per cent, 2.5 per cent and 0•1 per cent points. Limitation of space has forced the number of levels to be reduced in some cases. Besides distributions, there are tables of binomial coefficients, random sampling numbers, random normal deviates and logarithms of factorials. Each table is accompanied by a brief description of what is tabulated and, where the table is for a specific usage, a description of that is given. With the exception of Table 26, no attempt has been made to provide accounts of other statistical procedures that use the tables or to illustate their use with numerical examples, it being felt that these are more appropriate in an accompanying text or otherwise provided by the teacher. The choice of which tables to include has been influenced by the student's need to follow prescribed syllabuses and to pass the associated examinations. The inclusion of a table does not therefore imply the authors' endorsement of the technique associated with it. This is true of some significance tests, which could be more informatively replaced by robust estimates of the parameter being tested, together with a standard error.

All significance tests are dubious because the interpretation to be placed on the phrase 'significant at 5%' depends on the sample size: it is more indicative of the falsity of the null hypothesis with a small sample than with a large one. In addition, any test of the hypothesis that a parameter takes a specified value is dubious because significance at a prescribed level can generally be achieved by taking a large enough sample (cf. M. H. DeGroot, Probability and Statistics (1975), Addison-Wesley, p. 421). All the values here are exact to the number of places given, except that in Table 14 the values for n > 17 were calculated by an Edgeworth series approximation described in 'Critical values of the coefficient of rank correlation for testing the hypothesis of independence' by G. J. Glasser and R. F. Winter, Biometrika 48 (1961), pp. 444-8. Nearly all the tables have been newly computed for this publication and compared with existing compilations: the exceptions, in which we have used material from other sources, are listed below: Table 14, n = 12 to 16, is taken from 'The null distribution of Spearman's S when n = 13(1)16', by A. Otten, Statistica Neerlandica, 27 (1973), pp. 19-20, by permission of the editor. Table 24, k = 6, n = 5 and 6, is taken from 'Extended tables of the distribution of Friedman's S-statistic in the two-way layout', by Robert E. Odeh, Commun. Statist. — Simula Computa., B6 (I), 29-48 (1977), by permission of Marcel Dekker, Inc., and from Table 39 of The Pocket Book of Statistical Tables, by Robert E. Odeh, Donald B. Owen, Z. W. Birnbaum and Lloyd Fisher, Marcel Dekker (1977), by permission of Marcel Dekker, Inc. Table 25, k = 3, 4, 5, is partly taken from 'Exact probability levels for the Kruskal—Wallis test', by Ronald L. Iman, Dana Quade and Douglas A. Alexander, Selected Tables in Mathematical Statistics, Vol. 3 (1975), by permission of the American Mathematical Society; k = 3 is also partly taken from the MS thesis of Douglas A. Alexander, University of North Carolina at Chapel Hill (1968), by permission of Douglas A. Alexander. We should like to thank the staff of the University Press for their helpful advice and co-operation during the printing of the tables. We should also like to thank the staff of Heriot-Watt University's Computer Centre and Mr Ian Sweeney for help with some computing aspects. 10

January 1984

PREFACE TO THE SECOND EDITION The only change from the first edition is the inclusion of tables of Bayesian confidence intervals for the binomial and Poisson distributions and for the square of a multiple correlation coefficient.

D. V Lindley Periton Lane, Minehead Somerset, TA24 8AQ, U.K.

2

W. F Scott Department of Actuarial Mathematics and Statistics, Heriot-Watt University Riccarton, Edinburgh EHI4 4AS, U.K.

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION

n= 2

r =o

1

n= 3 r =o p=

i

P = 0.01 0.9801

0.9999

0 '9997

*9996

0•01 •02

0.9703

'9604

*9412

'9988

'9409 •9216

.9991 '9984

.03 '04

.9127 .8847

'9974 .9953

'02 '03 '04 0'05

0.9025 •8836

0 '9975

0'05

0'8574

•8306

0.9928 •9896

•8044

'9860

'7787 '7536

•9818 '9772

'07

'8649

•9964 '9951

.68

'8464

'993 6

•06 •07 .08

•09

•8281

•9919

'09

0. zo

0•8roo

•I I •I2

'7921

0'10 0'7290 0'9720 ' II '7050 '9664

'7744

'13 •14

'7569 .7396

0.9900 '9879 '9856 '9831 .9804

'12 '13 •x4

•6815 '6585 •6361

0'15

0'7225

•7056

0.6141 .5927

•6889

0.9775 '9744 '9711

0'15

•16 •17

-6724

•9676

'17 •18

•6561

•9639

'19

0'20 •21 '22 '23

o'6400 •6241

0.9600 '9559

0'5120 '4930

'6084 '5929

'9516

'24

'5776

'9471 '9424

0'20 '21 '22 '23

'24

0'25

•5625

0 '9375

•26

'5476 '5329

•9324 •9271

0'25 '26

•5184

•9216

'5041

'9159

0'30

0.4900

0.3430

'4761

0.9100 '9039

0'30

'31

.31

•3285

'32

'4624

'8976

'32

'3144

'33 .34

'4489

•8911 .8844

'33

.3008

'4356

'34

'2875

•66

•18 •19

z

The function tabulated is

0.9999 .9998 .9997 '9995 '9993

0.9990 '9987 .9983 '9603 '9537 '9978 .9467 .9973

0.9393 .9314 '9231 .9145 '9054

0.9966 '9959 .9951 .9942 '9931

0'8960

0'9920

'4390

•8862 •8761 •8656 '8548

.9907 '9894 •9878 •9862

0'4219

0.8438

0 '9844

'8324 '8207

'9824 '9803

•28

'4052 '3890 '3732

'29

'3579

•8087 '7965

'9780 '9756

0.7840 '7713 '7583 .7452 .7318

0 '9730

•x 6

'5718

•5514 '5314

F(r1n, p) = E (n) pt(i_p)n-t

0.9999

Pr {X < r} = F(rIn, p). Note that Pr {X 3 r} = 1 -Pr {X < r= I F(r iln, p). F(nin, p) = I, and the values for p > 0.5 may be -

•27 '28 '29

6.35

•36 '37 '38 .39

0.4225 •4096 .3969 '3844 .3721

'43 •44

0'2746

0.7182

•36

•2621

'7045

•37 .38

•2500

•6966

.9493

.2383

.6765

. 9451

'39

'2270

•662,3

'9407

0.8400 •8319 '3364 •82 36 •8151 '3249 •3136 •8064

0'40

0•2160

•41

'2054

0.6480 .6335

0.9360 .9311

'42

'1951

'6190

'9259

'43 •44

•1852 •1756

•6043

•9205 '9148

6.1664 '1575 '1489

0'5748

•1406 .1327

'5300 '5150

0.3025

0 '7975

0.45

•46

•2916 '2809

•7884 .7791

•46

'7696

•49

.2704 •26or

0'50

0'2500

.48

0.9571 '9533

0'35

•8704 •8631 .8556 .8479

0 '45

.47

'9702 '9672 .9641 •9607

0'8775

040 0.3600 .41 .3481 '42

•27

'7599

'47 '48 '49

0'7500

0'50

•5896

.5599 '5449

0.9089 .9027 .8962 •8894 •8824

F(rIn, p) =

0'5000

0'8750

4

-

F(n r -

-

'In, x

-

p).

The probability of exactly r occurrences, Pr {X = r}, is equal to -

F(r

-

1ln, p ) = (nr) Pr(I

Linear interpolation in p is satisfactory over much of the table but there are places where quadratic interpolation is necessary for high accuracy. When r = o, x or n-1 a direct calculation is to be preferred:

F(o1n, p) = (i - p)n , F(1 In, p) = (1 -p)"-1[1 + (n- 1)p]

F(n- iln, p) = 1 -p".

and

For n > 20 the number of occurrences X is approximately normally distributed with mean np and variance np(1 p); hence, including for continuity, we have -

F(rin, p) * (1)(s)

npand 0(s) is the normal distribution A/np(i p)

where s = r+

-

-

function (see Table 4). The approximation can usually be improved by using the formula

F(rin, p) * 0(s) where y

I -

Y e-is (s2 - x) 6 A/27/

2P

Vnp(i -p)

An alternative approximation for n > zo when p is small and np is of moderate size is to use the Poisson distribution: F(rIn, p) * F(rI#) where # = np and F(r1,a) is the Poisson distribution function (see Table 2). If 1 p is small and n(i p) is of moderate size a similar approximation gives -

F(rin, p) *

-

-

F(n r -

-

11,a)

where u = n(i p). Omitted entries to the left and right of tabulated values are o and I respectively, to four decimal places. -

0'1250

-

found using the result

F(r1n, p)

'4746 '4565

t

t

for r = o, r, n- 1, n < 20 and p 5 O. 5 ; n is sometimes referred to as the index and p as the parameter of the distribution. F(rin, p) is the probability that X, the number of occurrences in n independent trials of an event with probability p of occurrence in each trial, is less than or equal to r; that is,

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n =4 p

= o•ox '02

.03 .04 0.05

7= 0

0.9606 -9224 .8853 •8493

I

I

0•0I '02

0'9510

0'9990

'9039

.8587

'9998

.81 54

.9962 .9915 '9852

0 '9999

-03 •04

0 '9995

0'05

0 '7738

0 '9774

'9992 '9987 •9981 '9973

-o6 •o8

'7339 '6957 '6 591

.9681 '9575 '9456

0.9988 •998o •9969 '9955

0 '9999

.09

'6240

.9326

'9937

'9999 '9998 '9997

0.9963 '9951 '9937 .9921 '9902

0'9999 '9999 '9998 '9997 '9996

0•10 'II

0.5905

0.9185 '9035 -8875 .8708 .8533

0 '9914

0 '9995

.9888 .9857 '982, .9780

'9993 '9991 '9987 '9983

0'9995

(Yr5 •i6

0'4437

0 '9734

0 '9999

'17

0.9978 '9971 .9964 '9955 '9945 0 '9933

0 '9997

'9919 •9903 •9886 •9866

'9996 '9995 '9994 '9992

p

.07 •o8

.7164

'09

•6857

0.9860 •9801 '9733 '9656 '9570

o•io

0.6561

0 '9477

'II •12

•6z74 '5997

•13 '14

.5729 '5470

'9376 '9268 '9153 '9032

0'15

0.5220

•i6

'4979

•17

'4746

0.8905 '8772 .8634

•18 •19

.4521 '4305

'8344

0.9880 .9856 .9829 '9798 .9765

0'20 •21

0.4096

0.8192

0.9728

'3895

'22

•3702 •3515

'8037 .7878 '7715 '7550

•9688 '9644 '9597 '9547

•8491

.23 .24

'3336

0'25

0 '3164

•26

'2999

0.7383 .7213

•27

•2840

*7041

•28 '29

•2687 '2541

•6868 '6693

0.9492 '9434 '9372 •9306 '9237

0'30

0.2401

0.6517

0.9163

'31 '32

-2267 •2138

'6340 '6163

'9085 •9004

'33 '34

•2015 '1897

•5985 •5807

0.35 •36 '37

0.1785 •1678 '1 575

0.5630

•38

'1478

'39

•1385

0'40

0•1296

'41

•1212 •1132 •1056

'42 '43 '44

0'45 -46 '47 '48 •49

o.5o

'5453

.5276 -5100 '4925 0.4752 .4580 '4410 '4241

=

0 '9999

0.8145 •7807 '7481

•o6

3

0 '9994

'9977 '9948 •9909

n =5

7' = 0

2

'07

•12 •x3

'14

'5584 •5277

'4984 '4704

0.9984

0'20

0.3277

0 '7373

•9981 '9977 '9972 .9967

•21

'7167 '6959

.23

'3077 •2887 *2707

'24

.2536

.6539

0.9421 '9341 .9256 '9164 .9067

0 '9961

0'25

0 '2373

0.6328

0.8965

0 '9844

0 '9990

'9954 '9947 '9939 '9929

•26

•2219

•8857

*9819

'27

*2073

•6117 •5907

'5697 '5489

'8743 •8624 '8499

'9792 '9762 '9728

.9988 •9986 '9983 '9979

0•5282 '5077 '4875 '4675 '4478

0•8369 •8234 '8095 '7950 •7801

0.9692 '9653 .9610 '9564 '9514

0 '9976

0.7648 '7491 '7330 '7165 '6997

0.9460 .9402 '9340 '9274 '9204

0 '9947

0.6826 •665, '6475 •6295 •6114

0.9130 •8967 •8879 '8786

0.9898 •9884 •9869 '9853 '9835

0 '5931

0.9815 '9794 '9771 '9745

'22

'6749

0'30

*9908

'31

0•1681 '1564

'32

'1454

•8918 '8829

'9895 •9881 •9866

'33 '34

'1350 -1252

0.8735 -8638 '8536 •8431 .8321

0.9850 -9832 '9813 '9791 '9769

0.35

0.1160

•36 •38

•1074 '0992 •09,6

'39

.0845

0.4284 '4094 '3907 '3724 '3545

0•82.08 •8091 '7970 •7845 '7717

0 '9744

0.40

0.0778

0'3370

'9717 •9689 •9658 •9625

'41 '42 '43 '44

.0715 •o6o2 •0551

'3199 '3033 •2871

0 '9590

0'45

0.0503

'9552 '9512 '9469 '9424

'46 '47 '48 '49

'0459

.0731

'3431

•0677

•3276

0.0625

0.3125

0.6875

0'9999

'3487

0'9919

0.7585 '7450 .7311 '7169 •7023

0 '9999

•i8

'1935

0.3910 '3748 '3588

'9997 '9994

'19

•1804

0.0915 •0850 '0789

4

'3939 '3707

-4182

.28

'4074

3

0.8352 •8165 '7973 '7776 '7576

'9993 '9992 '9990 .9987

'29

'0983

2

'37

-0656

'2714

•9051

'9999 '9999 '9998 '9998

'9971 •9966 - 9961 '9955

'9940 '9931 '9921 '9910

-2272

'5561

•2135 •2002

'5375

0.8688 '8585 '8478 .8365

'5187

'8248

'9718

0•50 0'0313 0'1875 0'9375 See page 4 for explanation of the use of this table.

0'5000

0.8125

0.9688

5

'0418 '0380

'0345

0•2562 '2415

•9682 '9625 '9563 '9495

'5747

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n =6

r

= 0

p = 0•01

0.9415

'02

'8858 •8330

.03 •04 0'05

•06 .07 •o8 .09

0•25

•26 .27 -28 •29

n =7

r=o

I

2

3

p = 0•01

0'9321

•o2 -03 .04

•8681 -8080

0'9980 •9921 •9829 •9706

0.9997 '9991 •9980

0.9999

0.9556 •9382 •9187 •8974 •8745

0.9962 '9937 •9903 .9860 •9807

0.9998 '9996 '9993 '9988 •9982

0 '9743

0'9973

'3773 '3479

0.8503 •825o .7988 .7719 '7444

•9669 •9584 '9487 •9380

•9961 '9946 .9928 •9906

•z6

0.3206 •2951

0.7166 •6885

0.9262 .9134 .8995 .8846 •8687

0.9879 .9847 •9811 .9769 •9721

'4702

0•8520 .8343 •8159 '7967 *7769

0'9667 •9606 '9539 '9464 .9383

0'4449

0'7564

0'9294

•4204 •3965 '3734 •3510

'7354 .7139 •6919 •6696

•9198 .9095 .8984 •8866

0'6471

•6243 •6013 •5783 '5553

0.8740 •8606 •8466 •8318 .8163

0.9978 .9962 '9942 .9915 •9882

0.9999 '9998 '9997 '9995 '9992

0'9842 '9794 '9739 .9676 .9605

0.9987 .9982 '9975 -9966 '9955

0'9999 '9999 '9999 '9998 '9997

0:::

'4046

0'8857 .8655 .8444 .8224 '7997

0.3771 .3513 •3269 •3040 •2824

0-7765 .7528 •7287 •7044 .6799

0'9527 .9440 '9345 '9241 .9130

0.9941 .9925 .9906 •9884 .9859

0.9996 '9995 '9993 '9990 '9987

0'15

0'2621 •2431

0'9011 •8885 •8750 •8609 .8461

0•9830 '9798 *9761 .9720 .9674

0 '9984

'2252 '2084 '1927

0'6554 •6308 '6063 •5820 '5578

.9980 '9975 .9969 '9962

0.1780 •1642

0.5339 •5104

•1513 •1393 •1281

'4872 '4420

0.8306 •8144 '7977 .7804 •7626

o•9624 .9569 .9508 '9443 '9372

0'7443 •7256 '7064 .687o •6672

0.7351 -6899 . 6470 -6064 '5679

'4970

•23 .24

5

4

0.9672 '9541 '9392 •9227 *9048

•7828

0.5314

0'20 '21 '22

3

0'9998 '9995 -9988

'II -12 '13 '14

•i6 •17 •18 •19

2

0.9985 '9943 .9875 .9784

010

0'15

I

'4644 '4336

'4644

0'05

•06 .07 •o8 •09

*7514

0.6983 .6485 •6017

.5578 •5168 0:4783 4423 *4087

'12 •13 '14

•17

'2714

'6604

•18 •19

'2493 •2288

.6323 •6044

0'9999 '9999 '9999 '9999 '9998

0'20

0'2097

•2X .22 •23 •24

•1920 '1757 '1605 *1465

0'5767 '5494 •5225 *4960

0'9954 '9944 '9933 .9921 •9907

0.9998 '9997 '9996 '9995 '9994

0.25 •26 •27 •28 .29

0'1335 '1215 '1105

0.9295 •9213 •9125 .9031 •8931

0.9891 .9873 '9852 '9830 •9805

0.9993 '9991 .9989 .9987 .9985

0'30

0'0824

.31 •32 •33 •34

•0745 •0606 .0546

0.3294 '3086 •2887 •2696 .2513

'1003 •0910

0.30 •31 -32 .33 •34

0.1176 .1079 .0905 •0827

0.4202 •3988 '3780 '3578 •3381

0.35 •36 •37 •38 -39

0.0754 •0687 •0625 0568 .0515

0.3191 •3006 •2828 •2657 •2492

0.6471 •6268 -6063 .5857 *5650

0.8826 -8714 .8596 .8473 '8343

0.9777 '9746 '9712 .9675 '9635

0.9982 '9978 '9974 '9970 '9965

0.35 •36 '37 •38 •39

0'0490 •0440 '0394 •0352 -0314

0'2338 '2172 •2013 •1863 '1721

0'5323 '5094 '4866 '4641 '4419

0'8002 .7833 .7659 '7479 •7293

0'40

0-0467

0.8208 '8067 •7920 .7768 •7610

0.9590 •9542 '9490 '9434 '9373

0'0280

0'1586

0'4199

'9952 '9945 '9937 '9927

•41 -42 .43 •44

.0249

'0343 •0308

0.5443 •5236 •5029 .4823 •4618

0'40

'0422 '0381

0'2333 •2181 -2035 •1895 •1762

0'9959

'41 •42 '43 •44

'0195 -0173

.1459 '1340 •1228 •1123

'3983 '3771 '3564 •3362

0'7102 -6906 -6706 •6502 •6294

'46 •47 •48 •49

0.0277 •0248 '0222 0198 •0176

0.1636 .1515 •1401 •1293 •1190

0.4415 '4214 •4015 •3820 '3627

0.7447 •7279 '7107 •6930 *6748

0.9308 •9238 •9163 •9083 '8997

0.9917 •9905 '9892 •9878 *9862

0.45 -46 '47 •48 '49

0'0152 '0134 •0117 •0103 •0090

0'1024 '0932 •o847 •0767 •o693

0'3164 •2973 .2787 •2607 '2433

0'6083 '5869 . 5654 '5437 '5219

0'50

0.0156

0.1094

0'3438

0.6562

0.8906

0.9844

0'50

0.0078

0.0625

0•2266

0•5000

0'45

'0989

'0672

•0221

See page 4 for explanation of the use of this table.

6

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n= 7 p

r=4

5

6

n= 8

r

=o

1

2

3

4

5

= O'OI

p = 0'01

0.9227

0'9973

0 '9999

•02 •03

'02

'8508

'03 •04

'7837

'9897 '9777

'9996 '9987

0'9999

'7214

'9619

.9969

'9998

0.6634 .6096 .5596 •5132 - 4703

0.9428 •9208 .8965 •8702 '8423

0.9942 .9904 '9853 .9789 '9711

0 '9996

.9993 .9987 .9978 '9966

0.9999 .9999 '9997

0.9619 0'9950 '9929 '9513 .9903 .9392 .9871 9257 9109 •9832

0.9996 '9993 .9990 '9985 *9979

0.9999 .9999 '9998

'04

0.05 •66 •o7 •68 '09

cros •o6 '07

6

0.9999 '9999

•38 •09

0.10 'II •I2

0.9998 '9997

'II

0.4305 '3937

'9996

•12

'3596

•X3 •14

'9994 '9991

•13 '14

•3282 •2992

0.8131 '7829 .7520 •7206 •6889

0'15

0.9988 .9983 '9978 -9971 '9963

0.9999 .9999 '9999 .9998 '9997

0'/5

0.2725 .2479 -2252 •2044 .1853

0.6572 •6256 '5943 . 5634 '5330

0.8948 .8774 •8588 '8392 •8185

0'9786

0.9971

0 '9998

.16 '17 .18 '19

'9733 -9672 •9603 .9524

'9962 -9950 '9935 .9917

'9997 .9995 '9993 -9991

0 '9999

0'20

0.9953

'9942

0.9996 '9995

'22 '23

'9928 '9912

'9994

0'20 '21 '22

'24

'9893

.9992 '9989

.23 •24

0'1678 '1517 '1370 '1236 '1113

0.5033 '4743 '4462 '4189 .3925

0'7969

•21

0.9437 '9341 *9235 '9120 .8996

0.9896 .9871 '9842 '9809 '9770

0.9988 '9984 '9979 '9973 .9966

0.9999 '9999 '9998 .9998 .9997

0'25

0.9871 *9847 '981 9 *9787

0•6785 - 6535 .6282 '6027

0.9727 '9678 '9623 .9562 '9495

0'9958 *9948 '9936 '9922 '9906

0.9996 '9995 '9994 .9992 '9990

.26

'27 '28 •29

'9752

•10

•i6 •17 •19

'7745 '7514 '7276 -7033

'0722

0.3671 '3427 . 3193 '2969

•29

•0646

'2756

'5772

0•8862 *8719 •8567 '8466 '8237

0'9998 '9997 '9997 '9996 '9995

0'30

0.0576 .0514

0'2553

•0406 -0360

'2360 •2178 •2006 •1844

0.5518 .5264 •5013 *4764 .4519

0.8059 .7874 •7681 *7481 .7276

0.9420 -9339 '9250 *9154 .9051

0.9887 '9866 .9841 '9813 -9782

0.9987 .9984 •9980 '9976 •9970

0'9910 '9895 '9877 •9858 .9836

0 '9994

0.35 •36 '37 •38 .39

0.0319 -0281 '0248 -0218 -0192

0.1691 '1548 '1414 •1289 '1172

0.4278 '4042 •3811 •3585 '3366

0.7064 '6847 -6626 '6401 '6172

0.8939 '8820 '8693 '8557 •8414

0'9747

0'9964

'9992 '9991 .9989 '9986

'9707 '9664 '9615 '9561

-9957 '9949 '9939 '9928

0.8263 '8105 '7938 '7765 '7584

0.9502 '9437 '9366 •9289 •9206

0.9915 .9900 •9883 '9864 .9843

0'9115 •9018 •8914 -8802 •868z

0'9819 .9792 •9761 .9728 •9690

0.8555

0.9648

0 '9987

0 '9999

0'25

0' IOOI

'9983 '9979 '9974 .9969

'9999 '9999 '9999 .9998

•26

•0899 •0806

'27 •28

•31 •32

0'9712 - 9668 •9620

0'9962 '9954

'33

'9566

'34

'9508

'9935 '9923

0.35 '36 '37 -38 -39

0 '9444

0'40 '41 '42 - 43 '44

0 '9037

0.9812 '9784 '9754 '9721 '9684

0'9984

0'40

'8937 .8831 '8718 '8598

.9981

0.0168 '0147 -0128 '011 1 '0097

0.1064 *0963 •0870 '0784 '0705

0 '3154 .2948

0 '5941 . 5708

'9977 '9973 •9968

'41 '42 '43 •44

' 2750 '2560 '2376

'5473 '5238 •5004

0'45 '46 '47 .48 '49

0'8471 '8337 '8197 '8049 '7895

0'9643 '9598 '9549 '9496 '9438

0'9963 .9956 '9949 '994' '9932

0'45 •46 '47 •48 '49

0.0084 •0072 -0062 -0053 '0046

0.0632 •0565 •0504 .0448 '0398

0.2201 .2034 '1875 .1724 -1581

0.4770 -4537 '4306 '3854

0.7396 '7202 •7001 '6795 •6584

0•50

0.7734

0'9375

0.9922

0•50

0.0039

0.0352

0.1445

0.3633

0.6367

0'30

'9375 '9299 •9218 .9131

'9945

'31 .32 '33 -34

'0457

'4078

See page 4 for explanation of the use of this table.

7

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION

n =8

r= 7

n= 9

r

=o

I

2

0.9966 •9869 •9718

0.9999 '9994 •9980

'9522

'9955

0.9916 •9862 '9791

3

4

5

6

7

P = o•ox

p = O'OI

0.9135

•02 •04

'02 •03 '04

•8337 '7602 *6925

0'05

0'05

•o6

•o6

0.6302 '573 0

.07

'07

'5204

0.9288 •9022 •8729

•o8

•o8

•4722

•8417

'9702

•09

'09

•4279

•8088

'9595

o•xo

o•xo

0.3874

0 '7748

0 '9470

'II •I2

'3504 '3165

'7401

•x3 '14

-x3

'14

•2855 '2573

•6696 •6343

.9328 .9167 .8991 •8798

0.9917 '9883 •9842 '9791 '9731

0.9991 .9986 '9979 '9970 '9959

0 '9999

'II •I2

0'15

0'15

0.2316

0.5995

0.8591

0.9661

•x6 •17 •x8

•2082 •1869 •1676

'5652

•8371

•9580

.19

.19

'1501

•4988 '4670

.8139 .7895 •7643

'9488 '9385 •9270

'9991 .9987 .9983 '9977

0 '9999

'531 5

0.9944 •9925 .9902 .9875 '9842

0 '9994

•x6 •17 •x8

0'20 '21 '22

0'1342

0'4362

0'7382

0'9144

0'9804

0 '9969

0 '9997

'1199 •1069 '0952 'o846

•4066 •3782

'24

0'20 •2I '22 '23 '24

•7115 •6842 •6566 •6287

•9006 •8856 •8696 '8525

-9760 '9709 •9650 .9584

.9960 '9949 '9935 '9919

'9996 '9994 '9992 '9990

0'25 •26 '27

0'25 '26 •27

0.075x

0.3003 •2770 •2548

0.8343 .8151 '7950 '7740 .7522

0.9511 '9429 '9338 •9238 '9130

0.9900 •9878 '9851 •9821 .9787

0.9987 .9983 '9978 '9972 .9965

0 '9999

'0665 '0589

•03

.23

•7049

'3509

•3250

8

0.9999 '9997 0.9994 '9987 '9977 '9963 '9943

0 '9999

'9998 '9997 '9995

'9999 '9998 '9997 '9996

'9999 '9998 '9998

0 '9999

'9999

-28

•0520

•2340

•29

0'9999

'29

'0458

'2144

0.6007 '5727 . 5448 •5171 '4898

0'30

0 '9999

0'30

0'0404

0'1960

0'4628

0'7297

'31

.0355

•4364

'9999

'32

'0311

'1788 •1628

'33 '34

'9999 '9998

.33

'0272

'1478

'4106 •3854

'34

.0238

'1339

•3610

•7065 •6827 .6585 •6338

0.9747 '9702 .9652 '9596 '9533

0 '9996

'9999

0.9012 •8885 '8748 •86oz '8447

0 '9957

'31 .32

'9947 '9936 '9922 .9906

'9994 '9993 '9991 '9989

0 '9999

0'35

0 '9998

0'35

0.0207

.36

'9997

•36

'37 '38

'9996 .9996

•0135

0. 1 zi 1 •1092 •0983 •0882

0.3373 '3144 •2924 '2 713

'39

'9995

•37 -38 .39

•0i8o •0156 '0117

'0790

'2511

0.6089 •5837 •5584 '5331 '5078

0.8283 .8110 '7928 '7738 '7540

0.9464 •9388 .9304 •9213 '9114

0.9888 .9867 .9843 •9816 •9785

0.9986 •9983 '9979 '9974 '9969

0.9999 '9999 '9999 '9998 '9998

0'40

0'9993 '9992 '9990 '9988

0'40

0.0101

0'0705

0'2318

0'4826

'41

•0087

•0628

•2134

'4576

0.7334 -7122 •6903 •6678 '6449

0.9006 •8891 •8767 •8634 '8492

0.9750 •9710 •9666 '9617 '9563

0.9962 '9954 '9945 '9935 '9923

0.9997 '9997 '9996 '9995 '9994

0.6214 '5976 '5735 '5491

0.8342 •8183 '80,5 '7839 •7654

0.9909 '9893 '9875 '9855 '9831

0 '9992

'5246

0.9502 '9436 '9363 •9283 '9196

0.5000

0.7461

0.9102

0'9805

0.9980

•28

'41 '42 '43 •44

'42

'0074

•0558

'1961

.4330

•oo64

'0495

•9986

'43 •44

•0054

'0437

•1796 .1641

•4087 '3848

0'45

0.9983

0.45

0.0046

0.0385

0.1495

'46 '47

•9980

•0039

'0338

'1358

•0033

•0296

•1231

0.3614 '3386 '3164

•0028

'49

'9976 '9972 •9967

•46 '47 '48 '49

'0023

'0259 '0225

'III, •I00I

'2948 '2740

0'50

0.9961

0•50

0'0020

0'0195

0.0898

0.2539

'48

See page 4 for explanation of the use of this table.

8

'9999 '9998 '9997 '9997

'999! •9989 '9986 '9984

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n

p

I0

r =0

1

2

= 0•0i '02 '03

0.9044

0 '9957

0 '9999

'8171

•9838 9655 '9418

'9991 '9972 '9938 0.9885 '9812 '9717 '9599 '9460

0 '9990

0 '9999

'9980 '9964 '9942 •9912

'9998 '9997 '9994 '9990

0 '9999

0.9298 •9116 '8913 •8692 '8455

0'9872 •9822 '9761 •9687 '9600

0 '9984

0 '9999

'9975 •9963 '9947 •9927

'9997 '9996 '9994 '9990

0.8202 '7936

0.9500 '9386

0.9901 .9870

0.9986 .9980

7659

9259

983 2

9973

9997

. 7372

•7078

.9117 •8961

.9787 '9734

.9963 '9951

'9996 '9994

0 '9999

'7374 •6648

-04

3

4

5

6

7

8

9

0 '9999

'9996

•o6 .07 •o8

0.5987 .5386 '4840 '4344

'09

- 3894

0.9139 •8824 .8483 .81z1 '7746

0.10

0.3487 •3118 ' 2785 •2484

0.7361 •6972 '6583 •6196

'2213

'5816

0.15 •16 '17 •18 •19

0.1969 •1749

0 '5443

:1135752 '1374 •12.16

.5080 '4730 '4392 -4068

0'20 •21 •22

0'1074

0'3758

.23 •24

'0733 •0643

0.8791 •8609 '8413 •8206 .7988

0'9672 '9601 .9521 '9431 '9330

0'9936 •9918 .9896 '9870 '9839

0 '9999

*3464 -3185 •2921 .2673

0.6778 '6474 . 6169 •5863 •5558

0 '9991

'0947 •o834

.9988 '9984 '9979 '9973

'9999 *9998 '9998 '9997

0'25

0'0563

0'2440

0.5256

'2222

'4958

'28 '29

'0374 •0326

•2019 .1830

'4665 '4378 '4099

'7274 '7021 •6761

0.9219 .9096 .8963 •8819 •8663

0.9803 .9761 '9713 '9658 '9596

0 '9996

'0492 '0430

0 '7759 . 7521

0 '9965

•26 •27

'9955 '9944 '9930 '9913

'9994 '9993 '9990 '9988

0'6496 •6228

o'8497 •8321

4 0'9894

0 '9984

0 '9999

'9871

'595 6 5684 4

'8133

'7936 '7730

0'9527 '9449 '9363 '9268 •9164

.9815 '9780

•9980 '9975 .9968 '9961

.9998 '9997 '9997 '9996

0.5138 '4868 '4600 '4336 '4077

0.7515 '7292 •7061 •6823 .658o

0.9051 •8928 '8795 •8652 •8500

0.9740 '9695 '9644 '9587 '9523

0'9952 '9941 '9929 '9914 '9897

0 '9995

0.6331 •6078 '5822 '5564 '5304

0.8338 •8166 '7984 '7793 '7593

0'9452 '9374 '9288 '9194 '9092

0'9877 '9854 '9828 '9798 '9764

0'9983 '9979 '9975 '9969 '9963

0 '9999

0.5044 '4784 .4526

0'7384

0.8980 •8859 '8729

'9726 '9683 '9634

0 '9955

0'9997

'9996 '9995 '9994 '9992 0 '9990

0'05

•II •I2

•13 '14

0'30

'1655

0.1493 '1344

0'9999

'9999 0 '9999

'9998

0 '9999

'9999 .9999

'0211

•Iz(36

•0182.

•1o8o

•0157

'0965

0.3828 '3566 '3313 •3070 .2838

0.35 •36 '37 -38 '39

0.0135 •0115 •0098 -0084

0.0860 '0764 '0677 •0598

0.2616 •2405 •2206 '2017

•0071

'0527

'1840

0.40 '41

0.0060 '005! - 0043 •0036 •0030

0.0464 •0406 '0355 •0309

0.1673 '1517 '1372 •1236

'0269

'III!

0.3823 '3575 '3335 •3102 •2877

0'45 •46 '47 -48 '49

0.0025

0.0233

'0021 •0017 •0014 '0012

'0201 '0173 •0148 '0126

0.0996 •0889

0.2660 '2453

'0791 •0702 '0621

'2255 •2067

'4270

'6712

'8590

'9580

•1888

•4018

'6474

'8440

'9520

'9946 '9935 '9923 '9909

0'50

0.0ozo

0.0107

0'0547

O'1719

0'3770

0.6230

0.8281

0 '9453

0 '9893

'31 •32 '33 '34

•42

'43 '44

0

4 85 2

•4

'7168 '6943

See page 4 for explanation of the use of this table.

9

'9993 '9991 '9989 '9986

0 '9999

'9999

'9999 '9998 '9998 '9997

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n = xx p = 0•01

6

8

I

2

.03 •04

•8007 -7153

0'9948 •9805 .9587

0.9998 •9988 .9963

'6382

'9308

'9917

0.9998 .9993

0.05 •o6

0.5688 •5063 '4501

•o8 '09

.3996 '3544

0.9848 .9752 •963o •9481 '9305

0.9984 '9970 '9947 '9915 .9871

0'9999

'07

0.8981 •8618 •8228 •7819 '7399

0'10

0.3138 *2775

0.6974 '6548

0'9104

'2451 '2161

•1903

.5311

'8985

0.9815 '9744 .9659 '9558 '9440

0.9972 '9958 '9939 •9913 •9881

0.9997 '9995 '9992 .9988 •9982

0 '9999

•5714

•888o •8634 •8368

o•xs •16 .17 -18 •19

0.1673 - 1469 -1288 •1127 .0985

0'4922 '4547 -4189 •3849 -3526

0'7788 '7479 •7161 •6836 •6506

0.9306 '9154 .8987 •8803 •8603

0.9841 '9793 '9734 .9666 -9587

0'9973

0'9997

'9963 '9949 .9932 -9910

'9995 '9993 '9990 .9986

0'20

0.0859 0748 .0650 •0489

0.3221 '2935 •2667 •2418 -2186

0.6174 '5842 '5512 .5186 •4866

0.8389 •8160 '7919 •7667 .7404

0.9496 '9393 '9277 . 9149 •9008

0.9883 '9852 '9814 '9769 '9717

0.9980 '9973 '9965 '9954 '9941

0.9998 '9997 '9995 '9993 '9991

0'25

0'0422

0'1971

0'4552

'1773

-4247

'27

'0314

'1590

'395 1

•28 •29

•0270

- 1423

•3665

0'9657 '9588 - 9510 . 9423

0'9924 '9905 -9881 .9854

'9984 '9979 '9973

'0231

•1270

'3390

'6570 -6281 -5989

0'8854 *8687 '8507 -8315 •8112

0 '9999

'0364

0'7133 .6854

0 '9988

•26

'9326

'9821

'9966

'9998 .9998 '9997 .9996

0.1130 •1003 .0888

0.3127 •2877 . 2639

'0784 '0690

'2413 '2201

0.5696 '5402 .5110 '4821 '4536

0.7897 '7672 '7437 '7193 •6941

0.9218 '9099 .8969 '8829 -8676

0.9784 '9740 .9691 '9634 .9570

0.9957 '9946

'33 '34

0.0198 •0169 -0144 •0122 •0104

0'35

0'0088

0'0606

0'2001

•0074 '0062 -0052 '0044

.0530 •0463 .0403 •0350

•1814 "1640 *1478 •1328

0.4256 *3981 . 3714 '3455 •3204

0.6683 '6419 '6150 '5878 '5603

0.8513 '8339 '8153 '7957 '7751

0 '9499

•36 '37 -38 '39 0.40 41 •42 '43 '44

0.0036 .0030 -0025 •0021 •0017

0.0302 •0261 -0224

0.1189 •./062

0.2963 -2731

0.5328 -5052

0 '7535

2510

'4777

0. 45

46 '47 '48 '49 0'50

'02

•II •I2

•13 •14.

.21 -22 •23 -24

0'30

-31 -32

.

-

r

o

0'8953

-

'0564

'6127

3

5

4

'9997 '9995 '9990 .9983

7

9

0 '9999

.9998

.9999 .9998

0 '9999

'9999 .9998

0 '9999

'9999

0 '9994 0 '9999

'9918 '9899

'9992 '9990 '9987 '9984

'9419 '9330 '9232 '9124

0'9878 '9852 '9823 . 9790 '9751

0.9980 '9974 - 9968 '9961 '9952

0.9998 '9997 - 9996 '9995 '9994

0.9006 -8879 .8740 -8592 - 8432

0.9707 .9657 •9601 '9539 - 9468

0.9941 -9928 *9913 '9896 '9875

0 '9993

.7310 '7076 .6834 .6586

.9991 - 9988 '9986 '9982

0.9999 '9999 '9999 '9999 0.9998 '9998 '9998 .9997 .9996 0'9995

'9933

'9999 '9999 . 9998

'0945

.

'0192

'0838

'2300

'4505

•0164

. 0740

•2100

.4236

0'0014

0'0139

0'0652

0'1911

0'3971

•0011 •0009 •0008 •0006

•0118 .0100 -0084 •007o

-0572 •0501 .0436 •0378

'1734 •1567 .1412 •1267

'3712 '3459 . 3213 '2974

0.6331 .6071 •5807 . 5540 '5271

0•8262 -8081 -7890 •7688 '7477

0.9390 '9304 '9209 .9105 -8991

0.9852 '9825 '9794 .9759 '9718

0.9978 '9973 '9967 .9960 .9951

0.0005

0.0059

0.0327

0'1133

0 '2744

0.5000

0.7256

0.8867

0.9673

0.9941

See page 4 for explanation of the use of this table. I0

xo

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION

n = 12

r

p=

(•oi

0•8864

•02 '03 •04

'7847

0.05 •o6

=o

I

•6938 '6127

'9191

0.9998 '9985 '9952 •9893

0'5404

0'8816

0'9804

0 '9978

'4759

'9684 -9532 .9348 '9134

'9957 '9925 -988o .9820

0.8891 •8623 .8333 •8923 -7697

0 '9744 - 9649

0.4435 '4055 •3696 '3359 -3043

0'7358

'47 '49

0'9957

0'9995

0 '9999

0'50

0.9998

.9536 '9403 -9250

'9935 .9905 '9867 •9819

'9991 .9986 '9978 '9967

'9999 '9998 '9997 .9996

0.9078 •8886 •8676 .8448 '8205

0.9761 -9690 •9607 •9511 '9400

0.9954 .9935 .9912 -9884 '9849

0 '9993

0'9999

.7010 •6656 *6298 '5940

'9990

'9999 '9998 '9997

0.2749 -2476 •2224 '1991

0 '5583

0 '7946

•1778

'4222

*6795

0.9806 '9755 '9696 •9626 '9547

0.9961 '9948 '9932 •9911 '9887

0 '9999

-7674 - 7390 •7096

0'9274 '9134 '8979 •8808 '8623

0 '9994

.5232 •4886 -4550

'9992 '9989 '9984 '9979

'9999 '9999 '9998 '9997

0'9456

0 ' 9857

0'9972

0'9996

'9995

0 '9999

:9 9324 504

'9953 '9940 '9924

'9993 '9990 '9987

'9999 '9999 '9998

0'9983 - 9978

0'9998 '9997 .9996 '9995 '9993

0'9999

0.9999 '9999 '9999 -9998 '9998

0.2824

0.6590

•II •I2

'2470

*6133 •5686

-0924

.0798

0'20 '2I '22 '23

0.0687 •0591 '0507

•5252 .4834

'46 '48

'9985

.9979 '9971

.9996

.24

'0434 '0371

0'25

0.0317

0.1584

0'3907

0.6488

0.8424

•26 -27 •28

'0270 •0229

'1406 .1245

'3603

•6176

•8210

•0194

. 1 Ioo

'29

'0164

'0968

'3313 •3037 ' 2 775

.5863 '5548 •5235

'7984 '7746 '7496

'9113 '8974

.• '9733 .9678

0'30

0'0138 'oi 16 '0098

0'0850

0'2528 .2296

0'4925

0.7237 -6968 .6692 •6410 •6124

0.8822 •8657 '8479 •8289 •8087

0.9614 '9542 '9460 '9368 •9266

0 '9905

0.5833 '5541 '5249 '4957 •4668

0.7873 .7648 '7412 -7167 •6913

0'9154

0.9745

.9030 •8894 -8747 '8589

-9696

0'4382

0.6652 •6384

•2078 •1876 •1687

-4619 .4319 '4027 .3742

0.1513 •1352 •1205

0.3467 •3201 .2947

'1069 '0946

*2704

•9682 •0068

'0744 •0650 •0565 •0491

0.0057

0.0424

•0047 •0039 -0032 •0027

'0366 •0315 '0270 '0230

0'0022 •ooi8 •0014 •oolz •0010

0'0196 •0166 '0140 •or 18 •0099

0'0834

'41 •42 '43 '44

'0642 •0560 '0487

•1853

•3825

•1671

•3557

'1502

'3296

'5552

0.45 '46

00008 •0006

0.0083 •oo69

0.0421 •0363

0.1345 •1199

0.3044 •2802

0.5269 •4986

'47 '48 '49

•0005 •0004 •0003

•0057 '0047 •0039

•0312 '0267 '0227

•1066 *0943 '0832

•2570

'4703

-2348

.4423

•2138

0'50

0'0002

0'0032

0'0193

0.0730

0.1938

'31 '32

'33 '34

0.35 •36 '37

•38 '39 0'40

r=x

'9998 '9997

0•10

•18

n = 12

zo

0 '9999

'7052

'19

9

0'9999 '9999 '9999 '9999 '9998

•3225

0.1422 •1234 •1069

8

0'45

•09

•z6 •z7

7

0.9998 .9996 '9991 .9984 '9973

'4186

•1637

6

0'9999

•3677

•1880

'9997 '9990

5

p = 0.44

•o8

•2157

0 '9999

4

0 '9999

•07

0'15

3

0.9938 '9769 '9514

'8405 '7967 .7513

•z3 '14

2

'0733

•2472 0.2253 '2047

'4101

.9882 *9856 '9824 .9787

.9578 '9507

'9915 .9896 .9873

0'9992 '9989 •9986 .9982 '9978

0.8418 .8235

0.9427 .9338

0.9847 -9817

0'9972 •9965

'6111

•8041

'9240

'9782

'9957

'5833

'7836 •7620

'9131 •9012

'9742 '9696

'9947 '9935

0.9997 '9996 .9995 '9993 '9991

0.8883 '8742 •8589 .8425 •8249

0'9644

0.9921

0'9989

'4145

0.7393 .7157 •6911 .6657 '6396

.9585 -9519 '9445 '9362

-9905 •9886 .9863 .9837

.9986 .9983 '9979 '9974

0.3872

0.6128

0.8062

0.9270

0.9807

0.9968

.9641

See page 4 for explanation of the use of this table. II

'9972 '9964 .9955 0 '9944 .9930

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n = x3

r=o

i

2

3

4

p = 0.01

0.8775

0'9928

'02

•7690 •6730 •5882

'9730

'9436 •9068

0.9997 •9980 '9938 •9865

0.9999 '9995 •9986

0.9999

•06

0'5133 '4474

.07

'3893

.08

.09

•3383 '2935

0.8646 .8186 .7702 •7206 •6707

0.9755 •9608 '9422 •9201 .8946

0.9969 '9940 .9897 '9837 '9758

0.9997 '9993 .9987 '9976 '9959

0•10

0.2542

0.6213 '5730

0.8661 .8349 •8015 •7663 /296

0.9658 '9536 '9391 •9224 '9033

0.9935 .9903 '9861 •9807 '9740

0'9991

.11

o•8820 .8586 •8333 •8061 '7774

0.9658 '9562 '9449 .9319 .9173

0.9925 .9896 .9861 .9817 .9763

0'9987

0 '9998

•9981 '9973 '9962 '9948

'9997 '9996 '9994 '9991

0.7473 .7161 •6839 '6511 •6178

0.9009 •8827 •8629 ' 8415 •8184

0.9700 •9625 '9538 '9438 '9325

0.9930 '9907 •9880 .9846 .9805

0.9988 .9983 '9976 .9968 '9957

0.9998 '9998 '9996 '9995 '9993

0.5843 •5507 .5174 '4845 .4522

0 '7940

0'9198 •9056 .8901 •8730 .8545

0'9757

0.9944 .9927 .9907 •9882 '9853

0'9990

0'9999

•7681 '7411 '7130 •6840

'9987 '9982 '9976 .9969

'9998 '9997 '9996 '9995

0.4206 •3899 •3602 '3317 .3043

0.6543 .624o '5933 •5624 '5314

0.8346 •8133 -7907 •7669 '7419

0.9376 .9267 •9146

0.9960 '9948 '9935 .9918 •9898

0.9993 '9991 '9988 .9985 •9980

0 '9999

•8865

0.9818 '9777 .9729 '9674 •9610

0'5005 '4699

0'7159

0.8705

0 '9538

0 '9874

'2536 '2302 '2083

'4397 '4101

'6889 •6612

•8532 •8346

'9456 •9365

0.9975 '9968 .9960 '9949 '9937

0.9997 '9995 '9994 '9992 '9990

.03 '04 0'05

•12

•2198 '1898

•13 '14

•1636 '1408

0'15

0.1209

•16 •17 •x8

•1037

•0887 .0758

.2920

•19

•0646

•2616

0.6920 .6537 •6152 '5769 .5389

0'20 '21 '22 '23

0'0550 '0467

0'2336

0'5017

•2080 •1846 •1633

'4653 •43ox .3961 •3636

.0396

•5262 •4814 .4386 0'3983 •3604 •3249

5

6

7

8

9

10

0 '9999

'9999 '9997 '9995

.9985 '9976 '9964 '9947

0'9999

0.9999 '9998 '9997 '9995 '9992

0 '9999

'9999

0 '9999

'9999

•24

'0334 •0282

'1 441

0'25

0.0238

0.1267

•26

•0200

•27 •28 •29

'0167 '0140

•0117

•1 1 11 '0971 '0846 *0735

0•30

'31 •32

0.0097 •oo80 •oo66

o'o637 •0550 .0473

0.2025 •1815 •1621

'33 '34

•0055 •0045

•0406 *0347

.1280

0.35 •36

0'0037

•6327

•8147

'9262

'39

•0025 '0020 •0016

O'1132 *0997 '0875 '0765 •0667

0'2783

'37 •38

0'0296 '0251 '0213 '0179 •0151

•1877

•3812

•6038

'7935

•9149

.9846 •9813 '9775 '9730

0'40

0.0013

o•o126

0.0579

0.1686

0.3530

0.5744

0'9023

0'9679

0'9922

0'9987

'41 '42 '43 '44

•0010 •0008 •0007 •0005

•0105

'0501

•I508

'3258

'5448

•0088

.0431

.2997

'5151

•8886 •8736 .8574 •8400

•9621 '9554 '9480 '9395

.9904 .9883 .9859 .9830

.9983 '9979 '9973 .9967

0.45

•0030

0.3326 •3032 '2755 '2495

.2251

'1443

'9701 .9635 •9560 '9473

'9012

0'9999

'9999

0 '9999

'9999 '9999 '9998 '9997

'0072

'0370

'1344 '1193

'2746

'4854

•oo60

•0316

•1055

' 2507

'4559

0.7712 '7476 •7230 -6975 .6710

0'0049

0'0269

0'0929

0'2279

0'4268

0•6437

0'8212

•0040

•o228

•0815

•2065

•3981

•6158

'0033 •0026 •0021

'0192 •002 •0135

'0712 •0619 '0536

•1863 •1674

'49

0•0004 •0003 •0003 -0002 •0002

0.9302 '9197 •9082 •8955 •8817

0.9797 '9758 '9713 •9662 •9604

0.9959 '9949 '9937 •9923 .9907

0.50

0.0001

0.0017

0.0'12

0.0461

0.8666

0.9539

0.9888

'46 '47

'48

'3701

•5873

'1498

'3427 •3162.

'5585 .5293

•80'2 •7800 '7576 '7341

0.1334

0.2905

0.5000

0.7095

See page 4 for explanation of the use of this table. 12

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n = 13

r = II

12

p=

n = i4 p = o•ox

•02

'02

•03 •04

•03 .04

0.05 •06 .07 •o8 •09

0'05

0•10

r=

0

0.8687 '7536 '6528

'5647

I

2

0.9916 .9690 '9355 '8941

o•9997 '9975 '9923 •9833

3

4

5

6

7

0 '9999

'9994 '9981

0.9998

0-4877 •4205

0.8470

0'9699

0 '9958

0 '9996

-o6

'7963

'07

'362o

'7436

.9522 •9302

•9920 .9864

'9990 •9980

•o8 •09

'3112

'9042

'9786

'9965

'2670

•6900 '6368

. 8745

•9685

'9941

'9998 '9996 '9992

o•ro

0•2288 •1956 •1670

'13 •14.

'1 423

0.5846 '5342 '4859 '4401

'1211

•3969

0.8416 •8061 .7685 '7292 •6889

0.9559 •9406 •9226 '9021 •8790

0.9908 .9863 '9804 '9731 '9641

0.9985 '9976 '9962 '9943 '9918

0'9998

'II •I2

0.15 •x6 •x7 •x8 .x9

0'15

o•1o28 •0871 .0736 •0621 •0523

0.3567 •3193 .2848 .2531 - 2242

0.6479 •6068 .5659 .5256

0.8535 •8258 .7962 . 7649

0.9533 '9406 '9259 •9093

0.9885 '9843 '9791 '9727

0'9978 .9968 '9954 '9936

'4862

•7321

'8907

'9651

'9913

0.9997 '9995 '9992 •9988 .9983

0'20 '2I '22 '23 '24

0•20 '21 '22

0'0440

0'1979

0.4481

'0369

•1741

0•6982 '66 34

0.8702 - 8477

'0309

'1527

'4113 '3761

'6281

'8235

-23 •24

•oz58 '0214

'1335

.3426

'5924

'7977

'1163

'3109

•5568

•7703

0.9561 '9457 '9338 '9203 '9051

0.9884 .9848 .9804 '9752 . 9690

0.9976 •99 67 '9955 '9940 '9921

0.25 •26 .27 •28 .29

0'25

0'0178

0•1010

0'2811

0'5213

0'7415

0'8883

0'9617

0'9897

'26

'0148 •0122 '0101 '0083

•o874 '0754 •0556

'2533 ' 2273 •2033 •1812

'4864 '45 21 '4187 •3863

•7116 •6807 •6491 •6168

•8699 '8498 •8282 •8051

'9533 '9437 '9327 '9204

•9868 .9833 '9792 '9743

0.30 '31 •32 '33 '34

0'30

0•oo68 •0055 •0045 .0037

o.o475

0.1608

0.3552

'0404

'1423

'3253

'0343 •0290

•1254 •1101

•2968 •2699

•0030

'0244

'0963

'2444

0.5842 '5514 •5187 .4862 '4542

0.7805 '7546 .7276 •6994 .6703

0.9067 '8916 '8750 '8569 .8374

0.9685 .9619 '9542 '9455 '9357

0.35 •36 '37 '38 '39

0.0024 •0019 •0016 •0012 •0010

0.0205

0'9999

0.35 •36 .37 •38 '39

•0172 •0143 •0119

0.0839 •o729 •0630 .0543

0.2205 •1982 •1774 •1582

0.4227 •3920 .3622 '3334

'0098

'0466

'1405

•3057

0.6405 •6101 '5792 '5481 •5169

0.8164 '7941 •7704 '7455 '7195

0.9247 '9124 •8988 .8838 •8675

0.40 '41 '42 '43 '44

0'9999 '9998 '9998 '9997 '9996

0.40

o-0008 •0006

o.0081 •oo66

0.0398 •0339

0.1243 .1095

0'2793 '2541

0'4859 '4550

-42 •43 •44

'0005

•0054 •0044 '0036

'0287 '0242 •0203

•0961 '0839 '0730

'2303 •2078 •1868

'4246 '3948

•3656

0.6925 •6645 -6357 •6063 '5764

0.8499 •8308 •8104 •7887 •7656

0'45

0 '9995

0.0002 •0002

0.0170

0'5461 .5157

•0001 •0001 '0001

'0142 '0117 '0097 '0079

0.3373 •3100

'9999 '9999

'0023 '0019 •0015 '0012

0.0632 '0545

0'1672

'9993 '9991 '9989 '9986

0.45 '46 '47 '48 •49

0.0029

'46 '47 '48 '49 0'50

0'9983

0 '9999

0'50

0•000I

0'0009

0'0065

•I2 •I3 '14

•16 .17 •18 •19

•27 •28 •29

•31 •32

-33 -34

'9999 '9999

'41

0'9999

'0004

•0003

'0648

3

0 '9999

'9997 '9994 '9991 '9985

0'9999

'9999 '9998

•0468

'1322

'2837

'48 52

•0399 '0339

•1167 •io26

•2585 '2346

'4549 '4249

0.7414 •7160 -6895 •6620 .6337

0'0287

0'0898

0'2120

0.3953

0.6047

•1490

See page 4 for explanation of the use of this table. 1

0 '9999

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION t/ = 14

71 = 15

r= 0

I

2

p = o•ox '02 '03

p = 0.01

o•86oi

r= 8

9

xo

II

12

13

3

.02

'7386

'03

'6333

0.9904 '9647 '9270

'04

.04

.5421

'8809

0.9996 .9970 .9906 '9797

0.05 •06 .07 -438 •09

0.05 -o6 .07 •o8 •09

0.4633 '3953 '3367 •2863

0.8290 '7738 .7168 .6597

0.9638 '9429 •9171 •887o

0.9945 '9896 •9825 .9727

'2430

•6035

•8531

•9601

0•I0 'II •I2 •I3

crro •x x

0'2059

0.5490 '4969 '4476 '4013 .3583

0.8159 /762 '7346 •6916 •6480

0.9444 .9258 •9041 •8796 •8524

0.6042 •5608 •5181

0'8227

'14

0.15 •x 6 '17 •x8 '19

.9999 '9999 .9998 '9997

0'20 •2I •22

0.9996

.23 '24

.9989 '9984

0'25

0'9978

.26 '27 .28 .29

'9971

'9994 '9992

'9962 -9950 '9935

0 '9999

.9999 .9998 '9998 0.9997 .9995 '9993 '9991 .9988

0.9999 '9999 .9999 -9998

•12

'1470

•1238 '1041

0'15

0.0874

•x6 •17 •i 8

•0731 •o611

0.3186 -2821 •2489

'0510 - 0424

'2187

'4766

'7218

•19

'1915

'4365

•6854

0'20 '21 '22

0'0352 •0291 •0241

0'1671

0'3980

0'6482

•1453

.3615

•6105

'1259

'3269

'5726

•23 •24

•0198 •0163

•1087 .0935

'2945 .2642

'5350 '4978

0•25 '26 '27

0'0134 •0109 •0089 •0072 •0059

0'0802 '0685 '0583 '0495

0.2361 '2101 •1863

0'4613 •4258

'0419

•1447

•3914 •3584 .3268

0.0047 •0038

0.0353 •0296

o•,268 •ii07

0.2969 •2686

•28 •29

'9963 '9952

0.9999 '9999 '9999

'31 •32 '33 '34

0 '9757

0 '9940

0 '9989

0'9999

o•35

'9706 '9647 .9580 '9503

'9924 '9905 .9883 '9856

'9986 '9981 .9976 '9969

'9998 '9997 .9997 '9995

•36

0.9417 '9320 '9211 .9090 '8957

0.9825 '9788 '9745

0.9961 '9951 '9939

0'9994

0'9999

'41 '42 '43 '44

*9696

'9924

'9639

'9907

'9992 '9990 '9987 '9983

'9999 '9999 '9999 '9998

0.45 '46 '47 '48 '49

o•88ii •8652 '480 '8293 '8094

0'9574

0.9886 •9861 •9832 '9798 '9759

0.9978 '9973 •9966 .9958 '9947

0 '9997

•9500 '9417 '9323 '9218

0'50

0.7880

0.9102

0.9713

0.9935

0'9991

0.9917 '9895

0'9983

'9869

'9971

'9837 '9800

0.35 .36 '37 .38 '39 040

'9978

0'30

•0031

'0248

'0962

'2420

-0206

•0833

•2171

'0171

'0719

•1940

0.0016 -00I2 .00 I 0 •0008 •0006

0'0142 '0117 '0096

0'0617 •0528 •0450

0'1727 •1531 •I351

•0078 •oo64

•o382 •0322

•1187 •I039

0'0005 '0004 •0003 0002 0002

0'0052 •0042 •0034 '0027 0021

0•0271 •0227 •0189 *0157

-0130

0'0905 '0785 '0678 ' 0583 '0498

0'0001

0'0017 •0013 '0010 '0008

0.0107

0.0424

- 0087

'0071 '0057

*0359 •0303 '0254

'49

•0006

•0046

-0212

0'50

0'0005

0'0037

0.0176

-38

'39 0'40 •41 •42

'43 '44 0.45 '46 '47 •48

0.9999

See page 4 for explanation of the use of this table. 1

4

.1645

•7908 '7571

•0025 .0020

'37

'9997 '9996 '9994 '9993

'9992 '9976

•13 '14

0.9998 '9997 .9995 '9994 '9992

0.30 '31 .32 '33 '34

•1741

0 '9998

•0001 •0001 .000 I

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION Y/

= 15

r= 4

5

6

8

7

9

xo

II

12

13

p = 0. oi '02 '03

0 '9999

'04

'9998

0.05 •36

0'9994

0 '9999

'9986

.07

'9972

'9999 '9997 '9993 '9987

-o8

'9950

.09

•9918

0'10 'II 'I2

0.9873

'13

•9639

'14 0.15 •16

•17 •18

•19 0'20 '21 '22

.23 '24 0.25 •26 '27 •28

.29 0.30

'31 '32 '33 '34

.9522

0.9997 '9994 '9990 '9985 '9976

0.9383

0.9832

0'9999

'9773

0.9964 '9948 -9926 •9898 .9863

0 '9994

•9222 .9039

'9990 .9986 '9979 '9970

'9999 '9998 '9997 '9995

0'9999

0.9819 *9766 .9702 •9626 .9537

0.9958 '9942 •9922 '9896 '9865

0.9992 '9989 '9984 '9977 '9969

0.9999 '9998 '9997 '9996 '9994

0.9434 .9316 -9183 '9035 •8870

0'9827 .9781 '9726 '9662 '9587

0.9958 '9944 •9927 .9906 .9879

0'9992 '9989 .9985 '9979 '9972

0 '9999

'9998 '9998 '9997 '9995

0 '9999

0.8689 •8491 •8278 •8049 •7806

0.9500 '9401 .9289 .9163 •9023

0.9848 •9810 '9764 .9711 '9649

0.9963 .9952 '9938 .9921 '9901

0 '9993

0'9999

-9991 .9988 '9984 '9978

'9999 '9998 '9997 '9996

0.7548 •7278 •6997 '6705 •6405

0.8868 •8698 '8513 '8313 -8098

0.9578 '9496 '9403 .9298 •9180

0.9876 .9846 .9810 •9768 '9719

0.9972 '9963 '9953 '9941 .9925

0 '9995

0'9999

'9994 '9991 .9989 '9985

'9999 '9999 '9998 '9998

0'6098

0.7869 -7626 '7370 .7102 •6824

0.9050 8905 .8746 •8573 .8385

0.9662 '9596 '9521 '9435 '9339

0.9907 .9884 -9857 .9826 '9789

0.9981 '9975 '9968 -9960 '9949

0 '9997

0.8182 7966 '7735

0 '9745

0 '9937

0'9989

0 '9999

3 4 29 30

0.9231 •9110 .8976 •88 7 69 62

'9695 -9637 -9570 '9494

'9921 .9903 •9881 '9855

.9986 .9982 '9977 '9971

'9998 '9998 '9997 '9996

0.6964

0.8491

0.9408

0.9824

0'9963

0'9995

'9735

•8833 •8606 0.8358 '8090 '7805 '7505

'7190 0.6865 •6531 •6190 .5846 '5500

0.5155 '4813 '4477 '41 48 •3829

.9700 '9613 .9510 0.9389 .9252 •9095

•8921 •8728 0.8516 •8287 •8042 .7780 '7505

0.7216 •6916 '6607 •6291 •5968

•38 '39

•2413

'4989 •4665 '4346

0'40

0'2173 '1948

0'4032 '3726

•1739

•36 '37

'9998

0.9978 '9963 '9943 '9916 '9879

'9813

0.3519 •3222 '2938 •2668

0'35

0 '9999

0'5643 '5316

0 '9999

'9999 '9998 '9996

'43 '44

'1546

'3430 '3144

•1367

•2869

•5786 '5470 '5153 '4836

o•45

0•1204

'46

0.2608 '2359

0.4522 '4211

•2125

'3905

'48

•1055 '0920 .o799

.1905

•3606

0.6535 •6238 '5935 •5626

49

0690

'1699

3316

5314

0'50

0.0592

0.1509

0.3036

0.5000

'41 .42

'47

.77

0'9999

'9999

See page 4 for explanation of the use of this table.

15

'9996 '9995 '9993 '9991

0 '9999

'9999

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n = i6 P=

1' = 0

I

2

3

4

•or

0•8515

0'9891

•02

'7238

•9601

•03 '04

•6143

-9182 '8673

0.9995 .9963 '9887 '9758

0.9998 *9989 '9968

0 '9999

.5204

5

6

0.9999 '9999 *9997

7

8

9

10

'9997

•07

'3131

•o8

•2634

0.81(38 .7511 .6902 '6299

•09

'221 I

'5711

0.9571 .9327 '9031 •8689 •8306

0.10

43.1853

0.7892 '7455 -7001 '6539 •6074

0.9316 '9093 •8838 .8552 '8237

0.9830 '9752 •9652 .9529 '9382

0.9967 '9947 •9918 •9880 -9829

0'9999

'1 550

0.5147 '4614 •4115 .3653 -3227

0 '9995

•xx •x2 •13 •14

'9991 .9985 .9976 '9962

'9999 .9998 .9996 '9993

0'5614 *5162 .4723 '4302 .3899

0.9209

0 '9765

0 '9944

•9012 '8789 '8542 •8273

'9920 •9888 '9847 '9796

0'9989 '9984 '9976 '9964 '9949

0 '9998

'7540 '7164 '6777 '6381

'9685

'19

0'2839 *2487 .2170 •1885 -1632

0'7899

•0614 .0507 •0418 .0343

0'20 •21 '22

0'0281 '0230 '0188

0'1407 '1209 '1035

0'3518 '3161

0'5981

0'7982 .7673

o'9183 •9008

'7348

•8812

•23

.0153

.0883

.2517

'24

'0124

'0750

'2232

*4797 •4417

'7009 •6659

'8595 *8359

'9979 '9970 '9959 '9944

.9996 *9994 '9992 '9988

0 '9999

•5186

0'9930 '9905 '9873 '9834 '9786

0 '9998

'2827

0. 9733 -9658 •9568 '9464 '9343

0 '9985

•5582

0'25

0'0100

0'0635

0'1971

0'4050

.0535 -0450

'0052 '0042

'0377 '0314

'1733 '1518 .1323

'3697 •3360 •3041

'1149

'2740

0.8103 '7831 '7542 '7239 .6923

0'9204 '9049 '8875 •8683 '8474

0.9729 •9660 '9580 *9486 '9379

0.9925 '9902 '9873 '9837 '9794

0 '9997

•0081 •0065

o•63o2 '5940 '5575 •5212 '4853

0'9984

•26

'9977 '9969 '9959 '9945

•9996 '9994 *9992 .9989

0.0033 •oo26

o.o261 •0216

0.0994 •0856

0 '2459

'0021

•0178

'0734

'33 '34

•0016

•0146

•0626

'3819 '3496

0.8247 •8003 '7743 . 7469

'001 3

'0120

'0533

'1525

'3 187

'5241

'7181

0'9256 •9119 '8965 '8795 '8609

0 '9984

'32

0.6598 *6264 •5926 '5584

0 '9929

•2196 '1953 •1730

0 '4499 .4154

0 '9743

•31

•9683 •9612 '9530 '9436

'9908 .9883 .9852 '9815

'9979 .9972 .9963 '9952

0.35 •36

0.00 10 •0008

0'0451

'37

-0006

0-1339 •1170 . 10 1 8 •0881

0.4900 •4562 .4230

0.6881 •6572 '62 54

'3906

•5930

'3592

•5602

0.8406 •8187 '7952 .7702 '7438

0.9329 •9209 '9074 -8924 '8758

0.9771 .9720 '9659 '9589 '9509

0'9938

-0380 •0319 •0266

0.05 •o6

0.4401

0.15 •16 .17 •18

•27 -28 •29 0.30

.3716

•1293 .1077 •0895

0.0743

0'9930

•9868 .9779 '9658 .9504

0'9991 .9981 -9962 . 9932 '9889

0'9999 .9998 '9995 '9990 *9981

'9588 '9473 '9338

0'9999

'9999

'9997 '9996 '9993 '9990

0 '9999

'9999 '9998

'9999 '9999 .9998

•38

.0005

0.0098 •0079 •0064 -0052

'39

'0004

'0041

'0222

'0759

o'2892 •2613 '2351 •2105 '1877

0.40

0.0003

0.0033

'41 •42

'0002

'0026 '0021

o'o183 •0151 •0101 •oo82

0.0651 .0556 '0473 •0400 •0336

0.1666 . 1471 '1293 •1131 •o985

o'3288 '2997 •2720 . 2457 •2208

0.5272 '4942 •4613 .4289 '3971

0'7161 •6872 .6572 '6264 '5949

0.8577 •8381 •8168 '7940 .7698

0.9417 -9313 '9195 '9064 '8919

0.9809 .9766 .9716 .9658 .9591

o'o281 '0234 •0194 - 0160 •0131

0.0853 '0735 •0630 '0537 •0456

0.1976 '1759 '1559 '1374 •1205

0.3660 '3359 •3068 '2790 '2524

0.5629 '5306 •4981 '4657 '4335

0.7441 •7171 •6889 '6595 '6293

0'8759 •8584 *8393 •8,86 *7964

0 '9514

0'0106

0'0384

0'1051

0'2272

0'4018

0.5982

0.7728

0 '8949

'43 '44

•0002 •000i

•0016

'0001

'0013

0.0001 •0001

0.00,0 -oo08 •0006

'0124

•48

•0005

'49

•0003

0•0066 •0053 •0042 •0034 •0027

0•50

0'0003

0'0021

0'45

'46 '47

See page 4 for explanation of the use of this table. 16

'9921 '9900 '9875 '9845

.9426 '9326 '9214 .9089

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n = 16

7= 11

12

13

14

n = 17 p=

P = o•ox '02 •03 '04

7= 0

0.0I '02

0.8429 '7093

.03 '04

'5958

I

2

0 '9877

0.9994 '9956 •9866 '9714

'4996

'9554 •9091 .8535

0.4181 '3493

0.7922 .7283

'07

'2912

•o8 •09

•2423

•6638 '6005

'2012

'5396

0.4818 - 4277

3

4

5

0 '9997

•9986 •9960

0.9999 '9996

•8073

0.9912 .9836 '9727 '9581 '9397

0.9988 '9974 '9949 •9911 '9855

0.9999 '9997 '9993 '9985 '9973

0.9174 .8913 •8617 •8290

0.9779 '9679 '9554 .9402

0.9953 .9925 •9886

0.9497 .9218 •8882

0.05 'o6 '07 •o8 •og

0'05

0•I0 •II - 12 •I3

0•10 'II •12

0.1668

.0937 .0770

.3318

'14

'13 '14

0.7618 '7142 •6655 •6164

'2901

'5676

'7935

'9222

0.15 •x6 •x.7 •x8 •19

0.15 •16 •17 •18 •19

0.0631 •0516 •0421 .0343 •0278

0'2525

0'5198

0'7556

0'9013

•2187 •1887 •1621 -1387

'4734 •4289 •3867 '3468

•7159 '6749 '6331 •5909

'8776 .8513 •8225 '7913

0.9681 '9577 '9452 '9305 '9136

0'20 •2I '22

0•20 '21 '22

0'0225 •0182

0'1182 •1004

0'3096

0'5489 •5073

0'7582

0.8943

'2751

'0146

•0849

'2433

'4667

'7234 •6872

'8727 •8490

'23 '24

•23

•0118 '0094

•0715 •0600

•2141

'4272

•6500

'8230

'24

•1877

•3893

'6121

'7951

0.0075 .0060

0.0501 •0417

0.1637 '1422

0.3530

0'5739

'3186

•0047

'0346 •oz86 '0235

'1229

•Io58

•2863 •2560

'5357 '4977 -4604

0.7653 '7339 .7011 •6671

*0907

'2279

'4240

-6323

0'0193 '0157 '0128 '0104 •0083

0'0774 '0657 •0556 '0468 '0392

0'2019 -1781 •1563 '1366 •1188

0'3887

0'5968

'3547

.5610

•2622

'4895 '4542

0'25 •26

.27 '28

-29

•o6

0'25

0.9999 '9999 '9999 '9998

•26

0'30

0'0023

•0018

0'0007 •0005

0'0067

0'0327

0'1028

0'2348

0'4197

•0054

•0272

•0885

•0004

•0043 •0034

'0225 •0185

-3861 '3535

-1640

•0027

•0151

•0759 -0648 '0550

•2094 .1858 ' 1441

•3222 '2923

0'0021

0'0123

0'0464

0•12,60

•0016 •0013

•0100 •oo8o

•0390 •0326

•1096 •0949

'0010

'0065

•0271

- 0817

'2121 '1887

•0008

•0052

•0224

•0699

•1670

0.0041 •0032 •0025 •0020

0.0184 •0151 '0123 •0099

0.0596 •0505 '0425 •0356

0'1471 •iz88 'I 122 •0972

•0015

•oo8o

-0296

-0838

0'0012

0.0064

0'0245

0.0717

0'9999

'9999 '9999

0.35 .36 '37 -38 '39

0'9987

0 '9998

0'35

'9983 '9977 '9970 '9962

'9997 '9996 '9995 '9993

'36

0•40

0.9951 '9938

'9999

'37 •38 '39

0 '9991

0 '9999

0'40

.9988

'41 '42 '43 '44

0'9999

- 0038

•0014 •0011 •0009

'0003 '0002 0'0002 '0001

'9922

'9984

'9902 '9879

'9979 '9973

.9998 .9998 '9997 '9996

0.45 '46 '47 '48 '49

0.9851 '981 7 '9778 '9732 .9678

0.9965 '9956 '9945 '9931 .9914

0.9994 '9993 '9990 '9987 '9984

0.9999 '9999 '9999 '9999 '9998

0.45

0'0006

'46 '47 '48 '49

•0004 •0003

0•50

0'9616

0.9894

0.9979

0 '9997

0•50

O'COOI

'41 '42 '43 '44

*9834 '9766

'0030

'31 •32 '33 '34

'31 .32 '33 '34

'3777

'27 '28 •29

0.9997 '9996 '9995 '9993 '9990

0'30

'1379 '11 3 8

*8497

•0001 •000x •000x

•0002 •0002

See page 4 for explanation of the use of this table.

17

'3222 •2913

'5251

0.2639 '2372

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION 7Z=17

r=6

7

8

0.9999 '9998 '9996 '9993 .9989

0'9999

0.9997 '9995 '9992 .9988 -9982

9

xo

II

12

13

14

15

p = o•ox '02 '03 '04

0.05 •o6 .07 •o8

0 '9999

'9998

.09

'9996

0'10

0.9992

•II •12 •13

'9986

.9963

'14

'9944

0'15

0.9917

•16

-9882

•17 •x8 •19

.9837 .9709

0.9983 '9973 '9961 '9943 .9920

0'20 '21 •22

0.9623

0.9891

0 '9974

0 '9995

0 '9999

'9853

.23 •24

'9521 •9402 '9264 •9106

•9680

.9963 '9949 '9930 .9906

'9993 •9989 '9984 '9978

'9999 '9998 '9997 '9996

0 '9999

0•25

0.8929

0.9598

0.9876

0 '9999

•8732

•9501

.9839

•8515

'9389

•28

•8279

•9261

'9794 '9739

•29

•8024

•9116

•9674

0.9969 '9958 '9943 .9925 .9902

0 '9994

•26 .27

'9991 '9987 •9982 '9976

'9998 '9998 '9997 '9995

0 '9999

0'30

0.7752

0 '9993

0'9999

•32

'7162

'8 574

'33 '34

•6847 •6521

•8358

.9508 '9405 •9288

•8123

'9155

0'9873 .9838 '9796 '9746 •9686

0 '9968

'7464

0.8954 .8773

0 '9597

'31

'9957 '9943 .9926 •9905

'9991 .9987 .9983 '9977

'9998 '9998 '9997 '9996

0 '9999

0.35 .36

0.6188 .5848

0.9617 '9536 '9443 '9336 •9216

0.9880 . 9849 •9811 •9766 '9714

0.9970 •9960 '9949 '9934 .9916

0 '9999

•5505

0.9006 '8841 .8659 .8459 •8243

0 '9994

'37

0•7872 •76o5 '7324 •7029 •6722

'9992 .9989 .9985 .9981

'9999 '9998 '9998 '9997

0.6405 •6080 '5750 '5415 •5079

0.801 x .7762 '7498 .7220 •6928

o•9o81 •8930 . 8764 .8581 •8382

o.9652 •9580 '9497 '9403 '9295

0.9894 .9867 .9835 '9797 '9752

0 '9975

0 '9995

0'9999

.9967 '9958 '9946 '9931

'9994 '9992 .9989 .9986

'9999 '9999 '9998 '9998

0 '4743

'49

•1878

.3448

0.6626 •6313 '5992 .5665 '5333

0.8166 '7934 •7686 '7423 '7145

0.9174 .9038 •8888 •8721 .8538

0.9699 .9637 •9566 '9483 .9389

0.9914 .9892 •9866 .9835 '9798

0.9981 '9976 .9969 .9960 '9950

0 '9997

'48

0.2902 •2623 .2359 •2110

0.50

o•1662

0.3145

0.5000

0.6855

0.8338

0.9283

0.9755

0.9936

0.9988

'9977

-978o

•38

•5161

'39

'4818

0'40

0.4478

'41 '42

'4144 .3818 '3501 •3195

'43 '44 0'45

'46

'47

•9806 '9749

'4410 '4082

•3761

'9999 '9998

0 '9999

'9999 '9998 '9997

See page 4 for explanation of the use of this table. 18

'9996 '9995 '9993 '9991

0 '9999

'9999 '9999

0'9999

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n = 18

r=0

x

2

3

4

5

p = c•in •02 .03 .04

0•8345

0.9862

'6951

•9505

-578o '4796

•8997 •8393

0.9993 •9948 '9843 .9667

0.9996 .9982 '9950

0.9998 '9994

0 '9999

0.05

0.3972

0.7735

•o6

•3283

.7055

0.9419 •9102 .8725 .8298 .7832

0.9891 '9799 '9667 '9494 .9277

0-9985 .9966 '9933 '9884 .9814

0.9998 '9995 '9990 '9979 .9962

0 '9999

'9997 '9994

0 '9999

0.9018 .8718 •8382 •8014 .7618

0.9718 '9595 '9442 '9257 .9041

0.9936 .9898 .9846 '9778 •9690

0.9988 '9979 .9966 '9946 .9919

0.9998 '9997 '9994 '9989 •9983

6

7

8

9

xo

•07

•2708

•6378

•o8

'2229

•5719

•09

•1831

.5091

0•10 'II

0.1501

0.4503

•12

•1227 •I002

•395 8 •3460

•13

•0815

•14

•o66z

•3008 •26oz

0.7338 .6827 •6310 '5794 •5287

0.15

0.0536

'17 •i8 •19

.0349 -0225

0.4797 '4327 •3881 •3462 •3073

0.7202 '6771 -6331 •5888 '5446

0.8794 -8518 •8213 •7884 '7533

0.9581 '9449 '9292 •9111 •8903

0.9882 '9833 '9771 .9694 •9600

0.9973 '9959 '9940 '9914 •988o

0 '9999

'0434

0.2241 '1920 '1638 •1391 •1176

0 '9995

•x6

'9992 •9987 •9980 '9971

'9999 '9998 '9996 '9994

0'20 '21 •22

0'0180 •0144 '0114 '0091 '0072

0'0991 '0831 '0694 '0577 '0478

0'2713 •2384

0'5010 '45 86

0'7164 '6780

0'8671 •8414

0'9487

0.9991

0 '9998

-2084 .1813

'4175 •3782

•1570

.3409

•6387 '5988 '5586

•8134 •7832 •7512

0.9837 '9783 '9717 '9637 '9542

0'9957

'9355 •9201 -9026 •8829

'9940 '9917 '9888 .9852

'9986 '9980 '9972 •9961

'9997 '9996 '9994 '9991

0.25

0'0056

•26 •27 •28 •29

0.0044

0.0395 •0324

0.1353 •1161

0.3057 •2728

0.5187 '4792

0.7175 •6824

0.8610 -837o

0.9431 '9301

•0035

'0265

'0991

'2422

'4406

•6462

•8109

'9153

•oo27 •0021

•0216

•0842

•2140

•4032

•6093

'0712

•1881

•3671

•5719

.8986 •8800

0-9946 .9927 •9903 .9873

'0176

'7829 '7531

0.9807 '9751 -9684 '9605 •9512

0.9988 '9982 '9975 '9966 '9954

0.30 '31 •32

0.0016 •0013 •0010

0.0142 •0114 •0092

o.o600 •0502 •0419

0.1646 '1432

0.3327 '2999

0.5344 '4971

0'7217 •6889

0.8593 •8367

0.9404 •9280

0 '9939

'0073

'0348

'4602 •4241

•6550 •6203

.91 39

•0007 -0006

•2691 '2402

•8122

•33 '34

'1241 '1069

•0058

•0287

.0917

•2134

'3889

•5849

•7859 '7579

•8981 •8804

0.9790 '9736 .9671 '9595 -9506

0.35

0'0004

0.3550

0.5491

'0665

•1659

'3224

. 5 1 33

•0561

•1451

•2914

'4776

•0002

0'0236 •0193 '0157 '0127

0.1886

-0003 •0002

0'0046 '0036 '0028 •0022

0.0783

•36

•0472

•1263

'2621

.4424

'39

•0001

•0017

•0103

'0394

•1093

- 2345

'4079

0.7283 .6973 •6651 •6319 '5979

0.8609 .8396 •8165 '7916 •7650

0.9403 •9286 .9153 •9003 .8837

40.9788 '9736 .9675 •9603 '95 20

0.40

0.0001

0.0013

0.0082

•0001 •0001

•0010 '0008

'0066

•0052 •0041

0.0328 •0271 -0223 •0182

0.0942 •0807 •0687 •0582

0.2088 '1849 •1628 •1427

0.3743 •3418 '3105 •2807

0.5634 '5287 '4938 '4592

0.7368 .7072 •6764 '6444

'0032

'0148

'0490

•1243

•2524

•423 0

'6115

0.8653 •8451 •8232 '7996 '7742

0 '9424

'41

0'2258 '2009

0-3915

•1778 -1564

.3272 '2968

0.5778 '5438 '5094 '4751

•1368

'2678

.4409

0.7473 .7188 •6890 -6579 •6258

0.8720 •8530 •8323 •8098 •7856

0.1189

0.2403

0.4073

0.5927

0.7597

•23 -24

'37 •38

'42

•oz81

'43 '44

•0006 •0004

0'45

0'0003

'49

0•0120 •0096 '0077 •0061 '0048

0.1077

•0002 •0002 •0001 •0001

0.0025 '0019 '0015 •0011 '0009

0.0411

'46

'0342 •0283 '0233 •0190

'0928 •0795 '0676 '0572

0.50

0.0001

0.0007

0.0038

0.0154

0.0481

'47

'48

•3588

See page 4 for explanation of the use of this table.

19

0'9999

'9998 '9997

'9836

0 '9999

'9999

•9920 •9896 .9867 •9831

-9314 •9189 .9049 '8893

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION

n= ig

r

P = o•ox

P= o•oz

'02 '03

'02

•03 •04

•5606

•8900

•4604

0'05 •o6 •07 •08

' 20 5 1

'5440

'8092

•09

'1666

'4798

.7585

•1() •II •12

0.1351

0'4203

0'7054

•1092

'3658

'6512

•0881

'0709

•3165 '2723

-5968 .5432

'0569

*2331

•4911

0'0456 -0364 -0290 •0230 •0182

0•1985 -1682 •1419 •1191 -0996

0'4413

0'20 '21 '22 '23 •24

0.0144

o'o829 -0687

0.2369 •2058

0'25

n = 18

r = II

12

13

14

xs

x6

'04 0'05

'o6 .07 •o8 •09 0'10 •I I •I2 •I3 'I4

•13 '14

0.15

0'15

•16 •x7 •x8

•x6 •x7 •x8 •19

•19 0'20 '21 '22

0 '9999

'23 '24

'9999 '9998

0'25

0 '9998 '9997

•26 '27 •28 .29

'9995 -9993 *9990

0-9999 '9999 '9999 '9998

=

I

2

0.8262

0•9847

'6812

'9454 '8249

0'9991 '9939 -9817 *96 I 6

0 '3774

0 '7547

0 '9335

•3086

-6829 •6i2x

.8979 •8561

0

'2519

'0113

.3941 '3500 -3090 -2713

'0089

•0566

•I 778

'0070

*0465

*1529

'0054

'0381

•I 308

0.0042 •0033

0.0310 •0251 '0203 '0163

•0667

'29

'0025 *00 I 9 •0015

o•iii3 '0943 .0795

•0131

'0557

•26 '27

-28

0.9986 .998o '9973 '9964 '9953

0 '9997

0'30

0'0011

0'0104

0'0462

.31 '32 '33 '34

'9996 '9995 '9992 '9989

0'9999

•3 I

'0009

'0083

'9999 '9999 '9998

•32 •33 '34

-0007

•0065

•0382 •0314

'0005

•0051

'0257

'0004

•0040

'0209

0'35

0 '9938

•36 •37 .38 '39

.9920 .9898 -9870 '9837

0'9986 •9981 '9974 .9966 '9956

o•9997 '9996 '9995 '9993 '9990

0.40 '41 '42 '43 '44

0 '9797

'9750 '9693 •9628 '9551

0'9942 '9926 '9906 •9882 -9853

0.45 '46 '47 '48 '49

0.9463 .9362 '9247 •9117 '8972

0'50

o•8811

0'30

0.35

0'0003

0'0031

0'0170

0'9999

•36

•0002

'9999 '9999 '9998

•37 •38 -39

•0002

'0024 '0019

' 0137 '0110

'0001

•0014

'0087

•000x

'0011

'0069

0 '9987

0'9998

0'40

o•000i

'9983 -9978 '9971 .9962

'9997 '9996 '9994 '9993

0.0008 •0006 •0005 •0004 •0003

0•0055 '0043 •0033

'9999 '9999

'41 '42 '43 '44

0.9817 '9775 .9725 •9666 '9598

0 '9951

0'9990 '9987 .9983 '9977 '9971

0 '9999

0'45

0'0002

'9937 '9921 '9900 '9875

'9998 '9997 '9996 '9995

•000x •0001

0.0015 •0012

0 '9999

'46 '47 48 '49

0.9519

0.9846

0'9962

0 '9993

0 '9999

0•50

0 '9999

See page 4 for explanation of the use of this table. 20

'0026 '0020

'0009

'000!

'0007

'0001

•0005 0'0004

TABLE I. THE BINOMIAL DISTRIBUTION FUNCTION =1

9

r=

3

4

5

6

8

7

10

9

II

12

13

p = 0'01 '02 •03 •04

0 '9995

-9978 *9939

0.9998 '9993

0'05

•o6

0.9868 '9757

0.9980 '9956

•07

'9602

'9915

•98

.09

.9398 '9147

-9853 *9765

o•io

0•885o

•II •I2

'8510 '8133

0.9648 '9498

.13

-7725

.14

-7292

o•15 •16 •17

0.6841 •638o .5915

'18 -19

'5451 '4995

0'20 '21 '22

0.4551

.23 -24

.3329

0 '9999

0.9998 '9994 .9986 .9971 '9949

0 '9999

.9998 .9996 *9991

0.9999 '9999

-9096 .8842

0'9914 •9865 -9798 •9710 '9599

0.9983 '9970 '9952 '9924 .9887

0.9997 '9995 '9991 '9984 '9974

0.8556 •8238 -7893 '7524 *7136

0.9463 .9300 .9109 '8890 -8643

0'9837 .9772 -9690 '9589 '9468

0 '9959

0.8369 .8071

0.9324 .9157

0.9767 '9693

'7749 - 7408 .7050

•8966

-9604

'2968

0.6733 .6319 '5900 -5480 .5064

'8752 '8513

'9497 '9371

0.25

0.2631

0.4654

0.6678

0•8251

0.9225

-26 •27 -28 •29

'2320 '2035

'4256

'6295

'7968

'9059

-5907

.7664

.1776

-3871 . 3502

'5516

•1542

'3152

•5125

0.30

0.1332

•32

.1144 •0978

'33 '34

'0831 '0703

0'2822 . 2514 '2227 ' 1963

0'4739

•31

0.35

'4123 '3715

.9315

'4359 '3990

.9939 .9911 '9874 .9827

0 '9999

'9998 '9997 '9995

0 '9999

0.9992 .9986 '9979 '9968 '9953

0.9999 -9998 '9996 '9993 '9990

0 '9933

0-9984 '9977 *9966

'9907 '9873 '9831 '9778

'9953

'9934

0 '9999

'9999 '9998 0 '9997

'9995 '9993 '9989 '9984

0'9999

'9997

0 '9999

0'9977

0.9995

0 '9999

'9968 .9956 '9940 '9920

'9993 .9990 '9985 '9980

'9999 .9998 '9997 '9996

0 '9999

'9999 . 9998

'7343

.8871 .8662

0.9713 -9634 .9541 .9432

0.9911 .9881 -9844 '9798

•7005

'8432

'9306

'9742

0.6655 '6295 '5927 '5555 -5182

0.8180 '7909 •7619 '7312 •6990

0.9161 '8997 -8814 '8611 •8388

0 '9674

0-9895 •9863 .9824 '9777 '9720

0.9972 .9962 '9949 '9932 '9911

0 '9994

0 '9999

'9595 •9501 '9392 •9267

.9991 '9988 '9983 '9977

-9998 -9998 '9997 '9995

0.6656 •6310 '5957 '5599 *5238

0.8145 .7884 '7605 '7309 '6998

0.9125 .8965 .8787 '8590 '8374

0.9653 '9574 .9482 '9375 '9253

0.9886 '9854 '9815 '9769 '9713

0.9969 '9959 '9946 '9930 '9909

0 '9993

0.9884 '9854 -9817 '9773 •9720

0.9969 •9960 9948 '993 3 '9914

.1720

'3634 .3293

0.0591 •0495

0- I 500 •1301

0.2968 -2661

'39

•0412 '0341 •0281

•1122 '0962 •0821

'2373 '2105 •1857

0.4812 '4446 '4087 •3739 •3403

0'40

0'0230

0'0696

0'1629

0'3081

-0587 -0492 •0410 •0340

•1421 - 1233 •1063 - 0912

'2774 •2485 •2213 •1961

0.6675 *6340 '5997 .5647 - 5294

0.8139 •7886 •7615 .7328 •7026

0.9648

'0187 - 0151

0.4878 .4520 -4168 •3824 .3491

0'9115

'V

•896o -8787 .8596 -8387

'9571 '9482 '9379 •9262

0.0777 •0658 •0554 •0463

0.1727 .15 r 2 -1316 •1138

0.3169 •2862 -2570 •2294

•0978

'2036

0.6710 '6383 - 6046 '5701 - 5352

0.8159 '7913 - 7649 -7369 .7072

0'9129

'0385

0.4940 '4587 •4238 •3895 '3561

'8979 - 8813 -8628 .8425

0.9658 '9585 •9500 .9403 . 9291

0.9891 -9863 •9829 .9788 '9739

0'0318

0.0835

0.1796

0.3238

0.5000

0.6762

0.8204

0.9165

0.9682

.36 '37

•38

- 42

'43 '44

'0122

0.45

0.0280 •0229 •0186

•49

0'0077 •006r '0048 '0037 •0029

0'50

0'0022

0'0096

•46 '47

'48

'0097

-0150 '0121

See page 4 for explanation of the use of this table. 21

'9991 .9987 '9983 '9977

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n = 19 P

r = 14

15

16

n = 20

r

= 0

I

2

0.9990 '9929 '9790 .9561

O• 0 I •02 '03 '04

p = 0•0i

0.8179

•02

'6676

.03

'5438

0.9831 •9401 •8802

•04

'4420

'8103

0'05

0'05

•o6 -07 •o8 •09

•o6 •07

0.3585 •290I

0.7358 •6605

'2342

•5869

•o8

•1887

•09

•1516

•5169 •4516

&If) •I I •12 •X3

0•10 •I I '12

0'1216 '0972 '0776

0.3917

•13

•0617

•2461

•14

'14

•0490

&IS

0'15

•x6 •x7 •/8 •19

•16 •17 •x8 •19

0'20 '21 '22 '23 •24

0'20 '21 '22

•23 -24

0•25 '26 '27 '28 '29

0'25 •26

•0014

•0123

•0526

'29

•0011

'0097

0'30

0'30

.31

'31 '32

o•0008 .0006 •0004

'33 '34

•0003

'32 '33 '34

'9999

.36 '37 '38 '39

0.9999 .9998 '9998 '9997 '9995

0.40 'V

0-35

4

5

6

0 '9994

'9973 .9926

0 '9997

'9990

0 '9999

0.9245 •885o •8390 .7879 '7334

0.9841 '9710 '9529 •9294 -9007

0.9974 '9944 '9893 .9817 .9710

0.9997 '9991 •9981 .9962 '9932

0.6769 •6198

'2084

•5080 '4550

0.8670 •8290 .7873 '7427 .6959

0.9568 '9390 .9173 •8917 •8625

0.9887 .9825 '9740 -9630 '9493

0.9976 '9959 '9933 .9897 .9847

0.0388

0.1756

0-4049

0.6477

-0306

•1471

•3580

'5990

•0241

•1227

'3146

•5504

•o189 -0148

•1018

•2748

•5026

'0841

'2386

•4561

0.8298 '7941 '7557 •7151 .6729

0.9327 .9130 .8902 -8644 .8357

0.9781 -9696 '9591 '9463 '9311

0'0115 •0090 •0069

0'0692 '0566 •0461

0'2061 •1770 •1512

04114

•0054

.0374

•1284

-2915

0.6296 •5858 '5420 .4986

0.8042 •7703 '7343 •6965

'0041

'0302

'1085

'2569

'4561

'6573

0.9133 -8929 .8699 '8443 •8162

0'0032 '0024 •0018

0'0243 •0195 •0155

0'0913 '0763 •o635

0'2252

0'4148

0'6172

0•7858

'0433

•1962 •1700 •1466 •1256

•3752 '3375 •3019 •2,685

.5765 '5357 '4952 '4553

'7533 •7190 •6831 •6460

0.0076 •oo6o •0047

0.0355 •0289 •0235

0.1071 •0908 •0765

0.2375 •2089 •1827

0.4164 •3787 .3426

0.6080 -5695 •5307

'0036 '0028

'0189

•3083

'4921

•0152

'0642 '0535

'1589

'0002

-1374

.2758

'4540

0.35

0'0002

0'0021

0'0121

0.0444

•0001

•0016

-0096

-0366

0.1182 •101 z

0'2454

•36

0.4166 •3803

•27 •28

0 '9999

3

'3376 •2891

•5631

-3690 '3289

•2171

0'9999

'9997 '9994 .9987

'37

•000 I

•0012

•0076

'0300

'0859

•1 9 10

. 3453

•38

•0001

•0009

•oo6o

•0245

•0726

•1671

0 '9999

'39

•0001

'0007

•0047

•0198

'0610

'1453

•3118 •2800

0 '9994

0'9999

0'40

0'0005

0'0036

0.1256

'41

•0004 •0003

-0028 -0021

•0423 '0349

1079

'43 '44

'9999 '9998 '9997 '9996

o•o16o -0128

0.0510

'9991 '9988 '9984 '9979

0 '9999

0.45

0.9972

0'9995

'46 '47 '48 '49

'9964

'9954 '9940 '9924

'9993 '9990 '9987 '9983

0.50

0.9904

0'9978

0 '9996

. 42

42

'0783

•oo63

.0286 •0233

•0660

0.2500 2220 21925 1959 -1719 •1499

0.0009

0.0049

0.0189

0.0553

0.1299

'0007 '0005

'0152 •0121 •0096 '0076

'0461 •0381 •0313 '0255

•I I I 9 '0958 'o688

0.0059

0.0207

0.0577

•0102

.0080

'44

'0002 0002

0012

0 '9999

0'45

0.000!

'9999 '9999 '9998 '9997

'46

•0001

'47

•000I

43

'49

•0004 •0003

'0038 •0029 •0023 . 0017

0'50

0.0002

0.0013

•48

See page 4 for explanation of the use of this table. 22

.0922

•0814

TABLE 1. THE BINOMIAL DISTRIBUTION FUNCTION n - 20 p=

r = 7

8

9

I0

I1

0.9999 '9999 .9998 '9996

0 '9999

12

13

14

15

16

0.01 '02 •03 '04

0.05 •o6 .07 •o8 '09

0.9999 '9998

0. 10 •II •I2

0.9996

•13 •14

'9976 .9962

0' 15

0.9941

.16 '17 •18 '19

.9912

'9992 '9986

0.9999 .9999 .9998 *9995 '9992

0 '9999

'9999

'9759

0'9987 .9979 '9967 .9951 '9929

0'9998 .9996 '9993 -9989 '9983

0'20 •2I

0.9679

0.9900

0 '9974

•9862

'22

'23 •24

*9464 '9325 •9165

'9814 '9754 -9680

•9962 *9946 *9925 '9897

0.9994 '9991 '9987 •9981 '9972

0 '9999

'9581

'9998 '9997 .9996 '9994

0.9999 '9999

0'25 '26

0.8982 '8775

.27 •28 -29

'8 545 •8293 '8018

0.9591 '9485 '9360

0.9861 '9817 .9762 '9695 •9615

0.9961 '9945 .9926 .9900 •9868

0.9991 '9986 .9981 .9973 •9962

0.9998 '9997 *9996 '9994 '9991

0'9999

0'30

0.7723 -7409

0.8867 •866o

0'9987

0 '9997

•6732 •6376

-8182 •7913

'9909 -9881 '9846

-9982 '9975 •9966 '9955

'9996 '9994 '9992 '9989

0 '9999

'8432

0.9829 .9780 *9721 '9650 .9566

0 '9949 .9931

'7078

0.9520 .9409 '9281 .9134 •8968

0.35 '36 -37 •38 '39

0•6oio '5639 .5265 -4892 '4522

0.7624 '7317 '6995 •6659 '6312

o•8782 '8576 '8350 •8103 '7837

0.9468 '9355 '9225 •9077 '8910

0.9804 '9753 '9692 •9619 '9534

0.9940 '9921 '9898 •9868 *9833

0.9985 '9979 '9972 .9963 '9951

0 '9997

'9996 '9994 '9991 '9988

0'9999

0.40 '41 '42 '43 '44

0.4159 '3804 '3461 '31 32 '2817

0.5956 '5594 '5229 '4864 •4501

0 '7553

0'8725 .852o '8295 •8o5i •7788

0 '9435

0'9984

0'9997

'9321 '9190 •9042 •8877

0'9790 .9738 .9676 •9603 '9518

0 '9935

'7252 '6936 '6606 •6264

•9916 '9893 •9864 •9828

'9978 '9971 .9962 '9950

'9996 '9994 .9992 '9989

0 '9999

0.45 '46 '47 48 '49

0.2520 '2241 '1 980 '1739 '1518

0.4143 *3793 •3454 '3127 •2814

0.5914 '5557 '5196 '4834 '4475

0.7507 *7209 •6896 •6568 *6229

0.8692 . 8489 -8266 •8o23 •7762

0'9420 - 9306 '9177 •9031 •8867

0'9786 '9735 .9674 '9603 '9520

0 '9936

0 '9985

0 '9997

'9917 '9895 •9867 '9834

'9980 '9973 '9965 '9954

'9996 '9995 '9993 *9990

0 '9999

0'50

0'1316

0.2517

0.4119

o'5881

0.7483

0.8684

0.9423

0 '9793

0 '9941

0'9987

0 '9998

.31 .32 '33 '34

'9873 -9823

'9216

•9052

'9999 '9998

'9999 '9999 '9998

See page 4 for explanation of the use of this table. 23

'9999 '9998 '9998

'9999 .9999 '9998

'9999 '9999 '9999

TABLE 2. THE POISSON DISTRIBUTION FUNCTION r=

0

I

3

4

5

6 The function tabulated is

0'00 '02 •04

1'0000 0'9802

•06 •o8

0.9418 0•9231

.9992 -9983 •9970

0.10

0.9048

0 '9953

•12

'8869

•14 •x6 •x8

'8694 •8521 .8353

'9934 '9911 •9885 .9856

0'20 '22

0.8187

0.9825

'8025

'9791

•24

•7866

0.9608

2

0'9998

F(r

for r = o, I, 2, ... and ft < 2o. If R is a random variable with a Poisson distribution with mean it, F(r1/1) is the r; that is, probability that R

0 '9999

0.9998 '9997 '9996 '9994 '9992

r} = F(rl,u).

Pr {R Note that Pr {R r} =

=I

'26

'7711

'28

.7558

'9754 .9715 '9674

0'30

0.7408

0.9631

•32 '34 .36 -38

'7261

*9585

'7118 .6977 .6839

'9538 '9488 '9437

0.9989 .9985 .9981 -9976 '9970

0'9999

0.9964 .9957 '9949 '9940 '9931

0 '9997

'9999 '9999 '9998 '9998

-

F(r

F(r1/1)- Ffr -

-

r - I} ilp).

=

r!

Linear interpolation in is is satisfactory over much of the table, but there are places where quadratic inter-

.9997 '9996 '9995 '9994

polation is necessary for high accuracy. Even quadratic interpolation may be unsatisfactory when r = o or I and

a direct calculation is to be preferred: F(olp) = e1 and

F(il#) =

0.6703 .6570 '6440 '6313 •6188

0'9384 .9330 '9274 •9217 •9158

0.9921 •9910 •9898 .9885 '9871

0.9992 '9991 '9989 '9987 '9985

0.9999 '9999 '9999 '9999 '9999

0'50

'52

o•6o65 '5945

'54

'5827

-56

.5712

-58

'5599

0.9098 '9037 '8974 •8911 '8846

0.9856 '9841 '9824 *9807 '9788

0.9982 '9980 '9977 '9974 '9970

0.9998 '9998 .9998 '9997 '9997

o•6o •62 .64 -66 -68

0.5488

0.8781 '8715 '8648

0.9769 '9749 '9727

-8580 .851r

•9705

0-9966 .9962 '9958 '9953 '9948

0.9996 .9995 '9995 '9994 '9993

0•70 •72

0.4966

•9682

Pr {R

equal to

0.40 .42 '44 '46 '48

'5273 •5169 •5o66

-

The probability of exactly r occurrences, Pr {R = r}, is

+p). R is approximately normally distributed with mean it and variance p; hence, including 4 for For # >

.5379

li:

=t

20,

continuity, we have F(rlis)

Co(s)

where s = (r+4 p)Alp and 0(s) is the normal distribution function (see Table 4). The approximation can usually be improved by using the formula -

F(rip)

0:10(s) -

I 2rt

e-"I

(s2 - I) (s5 -7s2 +6s)1

6,/,7

72#

For certain values of r and > 20 use may be made of the following relation between the Poisson and X2distributions : F(rlit) = I -F2(r+i) (210

0'9999

where Fv(x) is the x2-distribution function (see Table 7). Omitted entries to the left and right of tabulated values

'9999 '9999

are o and I respectively, to four decimal places.

'74

'4771

.76 •78

.4677

.4584

0•8442 •8372 •83o2 '8231 .8160

o•8o - 82 .84

0.4493 '4404 .4317

o•8o88 •8o16 '7943

0.9526 '9497 '9467

-86 -88

.4232 -4148

.7871

.9436

'7798

'9404

0•90

0.4066

'92

'3985

0.7725 '7652

0-9371 '9338

'94

'3906

'7578

'9304

'96

'3829

'98

'3753

'7505 '7431

roo

0.3679

0-7358

'4868

j•

0'9659 0'9942 0'9992 0 '9999 '9634 '9991 '9999 '9937 -9608 .9930 '9990 '9999 '9582 '9924 '9989 '9999 '9998 '9917 '9554 '9987 0.9909 '9901 '9893 '9884 '9875

0.9986 '9984 '9983 '9981 '9979

0.9998 .9998 '9998 '9997 '9997

0'9977 '9974 '9972 '9969 .9966

0'9997

'9269 . 9233

0-9865 '9855 '9845 '9834 .9822

0.9197

0.9810

0.9963

0'9994

'9996 '9996 '9995 '9995

0'9999

'9999 '9999 0 '9999

24

TABLE 2. THE POISSON DISTRIBUTION FUNCTION ii

r

=

0

I

2

3

4

5

6

0.9197 '9103 '9004 •8901 '8795

0.9810 '9778 '9743 •9704 .9662

0.9963 '9955 '9946 .9935 '9923

0 '9994

0'9999

'9992 '9990 .9988 '9985

'9999 '9999 .9998 *9997

0.9617 .9569 '9518 '9463 '9405

0'9909 .9893 .9876 '9857 '9837

0'9982 '9978 '9973 '9968 '9962

0 '9997

•6092 '59 x 8 '5747

0.8685 •8571 •8454 '8335 '8213

'9996 '9995 '9994 '9992

0.9999 '9999 '9999 '9999

0 '9955

0'9991 .9989 '9987 *9984 '9981

0 '9998

'9948 '9940 '9930 '9920

8

7

9

10

II

I t.

r = 12

3'40

0'9999

'45

'9999

3.50

0-9999

1.00 '05 •xo •15

0.3679 '3499 -3329 •3166

0.7358 *7174 •6990 •6808

'20

'3012

'6626

1.25 -30 '35 '40 '45

o'2865

0.6446

'2725 .2592

'6268

1.50 '55 •60 •65 •70

0.2231 •2422 •2019 •1920 •1827

0.5578

0.8088

'5412

'7962

0 '9344 '9279

•5249

*7834 .7704 . 7572

'9212 *9141 •9068

0.9814 '9790 '9763 '9735 '9704

1.75 -8o .85 '90 '95

0.1738 •1653 -1572 '1496 '1423

0 '4779

0'7440 •7306 .7172 '7037 •6902

0.8992 -8913 .883 x '8747 •866o

0.9671 *9636 '9599 '9559 •9517

0.9909 *9896 '9883 '9868 '9852

0.9978 '9974 '9970 '9966 .9960

0 '9995

0 '9999

•4628 .4481 '4337 •4197

'9994 '9993 '9992 .9991

'9999 '9999 '9998 .9998

2'00

0.1353 •1287 •1225 •1165 •x 108

0.4060 •3926 '3796 .3669 .3546

0.6767 •6631 •6496 •6361 •6227

0.8571 •848o •8386 *8291 •8194

0 '9473

0'9989 '9987 '9985 '9983 '9980

0'9998

•9427 '9379 *9328 '9275

0'9834 - 9816 '9796 '9774 '9751

0'9955

•05 •10 •15 •20

'9997 '9997 '9996 '9995

0 '9999

2'25

0'1054

0.61293 '5960 •5828 '5697 .5567

0.8094 '7993 •7891 '7787 •7682

0.9220 '9162 •9103 '9041 •8978

0.9726 •9700 '9673 '9643 •9612

0.9916 •9906 *9896 '9884

0 '9994

0 '9999

•1003 '0954 '0907 •0863

0.3425 *3309 '3195 '3084 -2977

0'9977

•30 '35 '40 •45

'9974 '9971 '9967

'9872

'9962

'9994 '9993 '9991 '9990

'9999 .9998 .9998 .9998

2'50

o.o821 '0781 •0743 .0707 •0672

o•z873 '2772 .2674 '2579 '2487

0'5438 '5311 .5184 '5060 '4936

0.7576 '7468 . 7360 •7251 .7141

o'8912 '8844 '8774 '8703 •8629

0.9580 '9546 '9510 '9472 '9433

0.9858 '9844 '9828 •9812 '9794

0.9958 '9952 '9947 .9940 '9934

0.9989 '9987 *9985 .9983 '9981

0'9997

2.75 •8o •85 .90 '95

0.0639 •0608 •o578

0.2397 -2311 •2227

0.4815 •4695 '4576

0.7030

'2146

'4460

•6696

'0523

'2067

'4345

'6584

0'9392 '9349 '9304 '9258 '9210

0'9776 '9756 '9735 '9713 '9689

0.9927 '9919 '9910 .9901 '9891

0.9978 '9976 '9973 .9969 '9966

0 '9994

'0550

0.8554 '8477 •8398 •8318 '8236

'9993 '9992 '9991 '9990

0'9999 '9998 '9998 '9998 '9997

3'00

0.1991 •1918 •1847 . 1778 '1712

0.4232 •4121 •4012 -3904 '3799

0.6472 •6360 •6248 '6137 '6025

o'8153 •8o68 '7982 . 7895 •7806

0.9161 •9110 '9057 '9002 .8946

0.9665 •9639 '9612 '9584 '9554

0.9881 '9870 '9858 '9845 '9832

0.9962 '9958 '9953 '9948 '9943

0.9989 '9988 '9986 '9984 '9982

0'9997 '9997 '9996 '9996 '9995

0'9999

.15 •20

0.0498 .0474 •0450 '0429 •0408

3.25 '30 '35 '40 '45

0.0388 -0369 '035 1 '0334 '0317

0.1648 •1586 •1526 '1468 '1413

0.3696 '3594 '3495 '3397 '3302

0.5914 '5803 '5693 '5584 '5475

0.7717 •7626 '7534 '7442 '7349

0.8888 '8829 '8768 '8705 '8642

0.9523 '9490 '9457 '9421 '9385

0.9817 •9802

0 '9937

'9786 '9769 '9751

0.9980 '9978 '9976 '9973 '9970

0 '9994 '9994 '9993 '9992 '9991

0 '9999

.9931 '9924 '9917 '9909

3'50

0.0302

0.1359

0.3208

0.5366

0.7254

0.8576

0 '9347

0 '9733

0'9901

0'9967

0.9990

0 '9997

'55 •6o .65 .70

'05 •10

'2466 '2346

•5089

.4932

.6919

•68o8

'9948 '9941 '9934 '9925

.9998 '9997 '9997 '9996

0 '9999

'9999

See page 24 for explanation of the use of this table.

25

'9999 '9999 '9999

'9997 '9996 '9996 '9995

0.9999 0'9999 '9999 '9999 '9999 '9999

0 '9999

'9999

'9999 '9999 '9999 '9999

'9998 '9998 '9998 '9997

TABLE 2. THE POISSON DISTRIBUTION FUNCTION IL

r

= 0

/

2

3

4

5

6

7

8

9

10

0.3208 •3117

0.5366 •5259

0.7254 •7160

0'8576 •8509

0 '9347

0 '9733

•9308

.9713

0'9901 .9893

0.9967 •9963

0.9990 -9989

3.50 .55 •6o .65 •7o

0'0302

0'1359

•0287 •0273 •0260 •0247

•1307 •1257

•3027

•5152

'7064

'8441

'9267

'9692

'9883

'9960

'9987

•1209 •ii62

'2940 •2854

•5046 '4942

.6969 .6872

.8372 •8301

.9225 •9182

.9670 •9648

.9873 •9863

'9956 '9952

•9986 '9984

3'75 •8o

0 ' 0235

0'2771 '2689 '2609 '2531

0'4838

0.6775 •6678 •6581 •6484 .6386

0•8229 •8156 •8o81 •8006 .7929

0.9I37 .9091 '9044 .8995 .8945

0.9624 '9599 '9573 '9546 .9518

0.9852 '9840 •9828 -9815 -9801

0'9947

'9942 '9937 '9931 .9925

0.9983 •9981 '9979 '9977 '9974

0•7851 '7773 •7693 •7613 '7531

0.8893 .8841 .8786 •8731 .8675

0'9489 '9458 '9427 '9394 .9361

0'9786 '9771 '9755 '9738 '9721

0.9919 .9912 .9905 .9897 .9889

0.9972 .9969 .9966 .9963 '9959

0.8617 .8558

0.9702 .9683 •9663 .9642 .9620

0.9880 .9871 •9861 .9851 .9840

0.9956 '9952 '9948 '9943 '9938

•85

•0213

•90 .95

'0202

0•I I17 •1074 •1032 •0992

.0193

.0953

- 2455

4'00

o.o183 - 0174 •o166 •0158

0.0916 •42880 - o845 •0812

0.2381 •2309 •2238 •2169

0'4335

•05 •zo •15 '20

'0150

'0780

•2102

.3954

o•6288 •6191 •6093 '5996 .5898

4'25 •30 •35 '40 '45

0 '0143 '0136

0 '0749

0.2037 •1974

0.3862 '3772

0.5801 '5704

0'7449

•1912 •1851

'3682

•5608

•7283

'8498

.3594

•0117

'0663 '0636

•1793

'3508

'5512 '5416

•7199 .7114

•8436 '8374

0.9326 •9290 '9253 -9214 '9175

4'50 •55 •6o •65 .70

0.01 11 •0'06 .0 1 0 1 •0096 •0091

o.o6 i 1 •0586 -0563 '0540 -o51/3

0.1736 -1680 •i626 '1574 .1523

0.3423 '3339 •3257 •3176 •3097

0.5321 •5226 .5132 .5039 '4946

0.7029 .6944 •6858 •6771 •6684

0.831! •8246 •8180 •8114 •8o46

0.9134 .9092 '9049 -9005 •896o

0'9597

0'9829 .9817 .9805 '9792 '9778

0 '9933

'9574 '9549 '9524 '9497

4'75 •8o -85 -90 -95

0•0087 •008z •0078 •0074

0.0497 '0477 '0458 '0439

0.1473 '1425 •1379 '1333

0.3019 '2942 •2867 .2793

0.4854 '4763 •4672 '4582

'0071

'0421

•1289

•2721

'4493

0'6597 •6510 •6423 '6335 .6247

0.7978 •7908 -7838 .7767 .7695

0-8914 •8867 •8818 -8769 .8718

0.9470 '9442 '9413 .9382 '9351

0.9764 '9749 '9733 .9717 •9699

0.9903 .9896 •9888 •9880 .9872

5.00 •05 •zo •z5

0•oo67

0.0404

•0064

'0388

•oo61 •0058

•0372 -0357

0.1247 •1205 •1165 -1126

0.2650 •2581 .2513 '2446

0.4405 '4318 '4231 .4146

0.616o •6072 '5984 '5897

0.7622 '7548 '7474 •7399

'20

'0055

'0342

•I088

'2381

'4061

'5809

•7324

0.8666 •8614 •856o -8505 '8449

0.9319 •9286 .9252 •9217 •9181

0.9682 •9663 . 9644 •9624 •9603

0.9863 . 9854 '9844 '9834 •9823

5'25

0'0052

0'0328

0'1051

0'2317

0.3978

0'5722

0'7248

•30 '35 •40 '45

'0050

•0314 •0302 •0289 -0277

-ior6 •0981 . 0948 •0915

-2254 •2193 -2133 - 2074

'3895 . 3814 •3733 '3654

•5635 '5548 '5461 '5375

'7171 '7094 •7017 •6939

0.8392 -8335 •8276 •8217 -8156

0'9144 •9106 .9067 •9027 •8986

0.9582 .9559 .9536 .9512 .9488

0.9812 •9800 •9788 '9775 .9761

5'50 .55 •6o •65

0'0041

0'0266 '0255

0'0884 '0853

0'2017 '1961

•70

'0033

•1906 -1853 •z 800

0.6860 •6782 '6703 '6623 '6544

0.8095 •8033 '7970 •7906 '7841

-89o1 '8857 •8812 •8766

0.9462 '9436 '9409 •9381 .9352

0.9747

•0824 '0795 •0768

0.5289 '5204 .5119 '5034 .4950

0 '8944

•0244 •0234 -0224

0. 3575 '3498 '3422 '3346 .3272

5'75 •8o •85 •90 •95

0.0°32

0.0215

0'0741

0'1749

0.3199

0.4866

•0030 •0029

'0206

'0715

•1700

•3127

•4783

•0197

'0027

'0189

•0026

•0181

•0690 •o666 -0642

•1651 •1604 •1557

•3056 •2987 .2918

'4701 '4619 '4537

0.6464 •6384 •6304 •6224 . 6143

0.7776 •7710 •7644 '7576 -7508

0.8719 •8672 •8623 .8574 •8524

0.9322 •9292 •9260 •9228 '9195

0.9669 •9651 •9633 •9614 '9594

6•oo

0.0025

0.0174

o.o620

0.1512

0.2851

0'4457

0.6063

0.7440

0.8472

0.9161

0 '9574

•0224

•0129 '0123

'0047 •0045 '0043

•0039 •0037 •0035

.0719 -0691

'41735 '4633

'4532 '4433

'4238 .4142 '4047

'7367

See page 24 for explanation of the use of this table.

26

-9928 .9922 •9916 •9910

'9733 .9718 •9702 •9686

TABLE 2. THE POISSON DISTRIBUTION FUNCTION = II

12

3.50 '55 '6o •65 .70

0.9997 '9997 '9996 '9996 '9995

0 '9999

3'75 •8o •85 •90 '95

0 '9995

'9994 '9993 '9993 '9992

4'00

0.9991 '9990 '9989 -9988 '9986

4'25 .30 '35 '40 '45 4•50 '55 -6o .65 .70

0.9976 '9974 '9971 '9969

4.75 '8o '85

0 '9963

x3

x4

x5

x6

17

= 0

I

2

6'0 'I '2

0'0025

0'0174

0'0620

'0022

'0159

'0577

'0020

'0146

'0536

•3 •4

•0018

'0134

•0017

'0123

•0498 '0463

0 '9999

6'5

0'0015

0'0113

0'0430

'9998 '9998 '9998 '9998

'0014

'0103

'0400

•0012

'0095

'0371

'9999 '9999

•6 •7 •8 '9

0 '9997

0'9999

'9997 '9997 '9996 '9996

'9999 '9999 '9999 '9999

0 '9985

0 '9995

0 '9999

'9983

'9995 '9994 '9993 '9993

'9998 '9998 '9998 '9998

0.9992 '9991 '9990 .9989 '9988

'9999 '9999 '9999 '9999

0'9999

'0011

'0087

'0344

•oolo

•oo8o

•0320

7.0

0'0009

0'0073

0'0296

'I

•0008

•0067

•0275

'2

'0007

'0061

'0255

•3 •4

'0007

'0056

'0236

•0006

•0051

•0219

0.0006

0.0047

0.0203

'0005

•0043

'0188

'0005

'0039

•0174

'9999 '9999

7.5 •6 .7 •8 •9

'0004

'0036

•0004

•0033

•0161 •0149

0'9997

0 '9999

8•o

0'0003

0'0030

0'0138

'9997 '9997 '9997 '9996

'9999 '9999 '9999 '9999

•I '2

'0003

•0028

•0127

'0003

•0025

'0118

•3 .4

'0002

'0023

'0109

'0002

'0021

'0100

0.9987 '9986 .9984 .9983 '9981

0.9996 '9995 '9995 '9994 '9994

0-9999 '9999 '9998 '9998 '9998

8'5

0'0002

0'0019

'0002

•0018

0.0093 •0086

'0002

'0016

'0079

'0002

'0015

'0073

'9999

•6 .7 •8 •9

'000I

'0014

•0068

0.9993 '9992 '9992 '9991 '9990

0.9998 '9997 '9997 '9997 '9997

0 '9999

9.0

o-000i

0'0012

o.0062

'9999 '9999 '9999 '9999

'I

'0001

'0011

'0058

'2

'0001

'0010

'9932 '9927

0.9980 '9978 '9976 '9974 .9972

'3 '4

'000!

'0009

' 00 53 '0049

'000!

'0009

'0045

5.25 •30 '35 '40 '45

0'9922

0'9970

0•9989

0'9996

0'0042

'9988 '9987 '9986 '9984

'9996 '9995 '9995 '9995

'000I

'0007

'0038

'0001

'0007

'0035

•000r •0001

-0006

•0033

'9999

9'5 •6 '7 •8 •9

0'0008

'9967 '9964 .9962 '9959

0'9999 '9999 '9999 '9998 '9998

o'000z

'9916 '9910 '9904 '9897

'0005

'0030

5•50 '55 •6o -65

0.9955 '9952 '9949 '9945 '9941

0 '9983

0 '9994

0 '9998

0 '9999

I0•0

0'0005

0'0028

-9982 '9980 '9979 '9977

'9993 '9993 '9992 '9991

'9998 '9998 '9997 '9997

'9999 '9999 '9999 '9999

'0005

'0026

'2

'0004

'0023

'3 '4

•0004

'0022

'70

0.9890 '9883 '9875 .9867 '9859

'0003

'0020

5.75 •8o .85

0.9850 '9841 '9831

0 '9937

0.9975 '9973 '9971 '9969 '9966

0.9991 '9990 '9989 '9988 '9987

0 '9997

0 '9999

'9996 '9996 '9996 '9995

'9999 '9999 '9999 '9998

0.0003 -0003 •0003

0 '9999

10.5 •6 .7 .8 .9

•0002

0.0018 •0017 •0016 •0014 •0013

0'9964

0 '9986

0 '9995

0'9998

0'9999

I I•0

0'0002

0'0012

.05 •IO

'15 •20

'90

'95 5'00

'05 •10 '15 '20

'9982 .9980 '9978

'9966

'9960 '9957 '9953 '9949 0.9945 '9941 '9937

'90

'9821

'95

'9810

'9932 '9927 '9922 '9917

6•oo

0.9799

0.9912

0 '9999

0 '9999

0'9999

See page 24 for explanation of the use of this table.

27

'0002

TABLE 2. THE POISSON DISTRIBUTION FUNCTION IL

r= 3

4

5

6

8

7

9

10

II

12

13

0.9964 '9958 '9952 '9945 '9937

0 '4457

0.6063

0 '7440

0•8472

0.9161

0 '9574

0 '9799

'4298

•5902

'7301

'8367

'9090

'9531

'4141 .3988

. 5742

.3 •4

'1342 .1264 .1189

0'2851 '2719 '2592 -2469 '235 1

•3837

'5423

•7160 .7017 •6873

•82.59 .8148 •8033

•9016 •8939 •8858

•9486 •9437 .9386

'9776 '9750 .9723 .9693

0.9912 .9900 •9887 .9873 '9857

6.5

0•1118

0.2237

0.3690

•6 .7 •8 •9

•1052 •0988 •9928

•2127 •2022

'3547 •3406

-0871

•1920 •1823

•3270 •3137

0.5265 •5108 '4953 '4799 '4647

0.6728 •6581 •6433 •6285 .6136

0.7916 '7796 .7673 '7548 .7420

0.8774 •8686 .8596 •8502 •8405

0.9332 '9274 '9214 •9151 •9084

0.9661 .9627 '9591 '9552 •9510

0.9840 •9821 •9801 '9779 '9755

0.9929 -9920 .9909 .9898 -9885

7.0

0•0818 •o767

0'1730 •1641

0 '5987

'0719

.3 .4

.9674 •0632

'1555 .1473

' 1395

'4349 •4204 •4060 '3920

•5838 •5689 '5541 '5393

0.7291 -716o •7027 .6892 '6757

0.8305 •82o2 •8096 '7988 .7877

0.9015 .8942 •8867 •8788 '8707

0 '9467

•2

0.3007 •2881 .2759 •2640 .2526

0 '4497

•1

0.9730 .9703 .9673 •9642 •9609

0.9872 .9857 '9841 •9824 •9805

7'5 •6 .7 •8

0'1321

0. 2414

•2307

•1181

o.66zo •6482 .6343 •62o4 •6o65

0.7764 .7649 '7531 '7411 •7290

0•8622 .8535 .8445 •8352 •8257

0.9208 '9148 .9085 •9020 •8952

0'9573

•1249

'9

0'0591 .0554 •0518 •9485 '0453

'9536 '9496 '9454 '9409

0.9784 •9762 '9739 '9714 •9687

8.43

0'5925

0.7166

0'8159

'I

.5786 .5647 •5507 '5369

-7041 '6915 •6788 -6659

•8058 '7955 •7850 '7743

0.8881 •8807 .8731 •8652 .8571

0.9362 •9313 •9261 •9207 .9150

0.9658 •9628 '9595 -9561 '9524

0.5231 •5094 '4958 •4823 .4689

0.6530 •6400 •6269 .6137 •6006

0.7634 •7522 '7409 -7294 •7178

0.8487 -8400 -8311 •82.20 •8126

0.9091 •9029 .8965 •8898 •8829

0.9486 '9445 '9403 '9358 •93"

6•o

0'1512

•I

*1425

'2

•5582

-2203

0.3782 •3646 '3514

•1117

•2103

•3384

•1055

•2006

'3257

0.5246 •5100 '4956 •4812 '4670

0.0424

0'0996

•0396

*0940

O'1912 '1822

01134 •3013

0'4530 '4391

'4254 •4119 '3987

.9420 '9371 '9319 •9265

-2

•0370

•o887

•1736

-2896

-3 '4

.0346 '0323

•9837

•1653

•2781

'0789

•1573

•2670

8.5 •6 -7 •8 •9

0'0301 •9281 •oz62 •9244 •9228

0'0744 •0701

0'1496 •1422

0'2562 '2457

•0660

-1352

.2355

•9621 •9584

•1284

'2256

0.3856 •3728 •3602 •3478

•1210

•2160

'3357

TO

0'0212

0.0550

0'1157

.0517

'2

•0108 •0184

'0486

•1098 •1041

•3 •4

•0172 •0160

'0456 .0429

'0986 •0935

o.2o68 •1978 •1892 •1808 •1727

0.3239 •3123 •3010 •2900 '2792

0 '4557

•I

'4426 '4296 •4168 '4042

0.5874 '5742 •5611 '5479 '5349

0.7060 .6941 -682o .6699 -6576

0.8030 '7932 -7832 '7730 •7626

0.8758 •8684 •8607 •8529 '8448

0.9261 •9210 •9156 '9100 •9042

9.5 •6 •7

0 . 0149 -0138 '0129

0.0403

0'0885

0'1649

0:22 : 26 2844 5 898 5 7

0'3918

0'5218

0'6453

0'7520

•0378

•0120

.0333

'1574 •1502 •1433

•2388

•3796 •3676 '3558

•5089

8

•0338 .0793 .0750

•9

-0111

- 0312

•07 10

•1

.6329 •6205 .6080 '5955

.7412 -7303 •7193 •7081

0.8364 •8279 •8191 •81o, •8009

0.8981 •8919 •8853 •8786 -8716

10•0

0.0103

•I

0'0293 •0274

0'0671 •0634

'2

•0096 '0089

•3 •4

•0083 •0077

'0257 •0241

0.5830 .5705 '5580 '5456 '5331

0.6968 '6853 .6738 •6622 -6505

0.7916 -7820 •7722 •7623 .7522

0.8645 •8571 '8494 •8416 •8336

10'5

•6 •7 •8 •9

0.6387 •6269 •6150

0.7420 •7316 -7210

0.8253 •8169 •8083

11*0

'0355

366

0.1301

'3442

'4960

•4832 '4705

0'2202 '2113

0'3328 '3217

0 '4579

'1240

'0599

-xi8o

•2027

•0566

•1123

•1944

•0225

'0534

•1069

•1863

•3108 •3001 .2896

'4332 •4210 '4090

0.0071 -9966 •0062 -0057 •0053

0'0211

0'0504

0.1016

0'1785

0 . 2794

0'3971

•0197

.0475

•0966

•1710

•0185 •0173 '0162

'0448 '0423 •0398

'0918 '0872 •o82.8

•1636 '1566

•2694 '2597

'3854 •3739

0.5207 •5084 '4961

'2502

'3626

•4840

•6031

'7104

'7995

'1498

.2410

•3515

'4719

'5912

.6996

'7905

0.0049

0'0151

0'0375

0.0786

0'1432

0'2320

0.3405

0 '4599

0 '5793

0.6887

0.7813

'4455

See page 24 for explanation of the use of this table. 28

TABLE 2. THE POISSON DISTRIBUTION FUNCTION

6•o 'I

r = 14

15

i6

17

0.9986 '9984

0.9995 '9994 '9993 '9992 '9990

0.9998 '9998 '9997 '9997 '9996

0.9999 '9999 '9999 '9999 '9999

0.9996 -9995 '9994 '9993 '9992

0.9998 -9998 '9998 '9997 '9997

0.9999 .9999 '9999 '9999 '9999

i8

19

20

21

22

23

24

'2

'9981

'3 '4

'9978 '9974

6'5 .6 '7 '8 '9

0.9970 '9966 '9961 '9956 '9950

0 '9988

7.0 'I

0'9943 '9935 '9927 '9918 '9908

0'9976 '9972 '9969 '9964 '9959

0 '9990

0-9996 '9996 '9995 '9994 '9993

0 '9999

'9989 '9987 '9985 '9983

'9998 '9998 '9998 '9997

0 '9999

7'5 •6 '7 .8 '9

0 '9897

0 '9954

'9948 '9941 '9934 '9926

0'9980 '9978 '9974 '9971 '9967

0'9992 '9991 '9989 '9988 '9986

0'9997 '9996 '9996 '9995 '9994

0 '9999

'9886 '9873 '9859 '9844

8•o 'I

0'9918 '9908 '9898 '9887 '9875

0'9963 '9958 '9953 '9947 '9941

0'9984 .9982 '9979 '9977 '9973

0'9993 '9992 '9991 '9990 '9989

0 '9997

0 '9999

'3 '4

0.9827 '9810 '9791 '9771 '9749

'9997 '9997 '9996 '9995

'9999 '9999 '9998 '9998

8'5

0 '9726

0 '9934 .9926

0'9987 '9985 '9983 '9981 '9978

0'9995 '9994 '9993 '9992 '9991

0'9998 '9998 '9997 '9997 '9996

0 '9999

-9701 '9675 '9647 '9617

0.9862 '9848 '9832 '9816 '9798

0 '9970

'6 '7 .8 '9

'9999 '9999 '9999 '9998

0'9999

9.0 -1 -2 '3 '4

0.9585 '9552 '9517 '9480 '9441

0*9780 '9760 '9738 '9715 '9691

0.9889 '9878 '9865 '9852 '9838

0 '9947 - 9941

0'9976 '9973 '9969 '9966 '9962

0'9989

0'9996 '9995 '9994 '9993 '9992

0'9998

0 '9999

'9988 '9986 '9985 '9983

'9998 '9998 '9997 '9997

'9999 '9999 '9999 '9999

9'5 .6 '7 .8 '9

0'9400 .9357 '9312 '9265 '9216

0.9665 '9638 '9609 '9579 '9546

0'9823 '9806 '9789 '9770 '9751

0'9911 '9902 '9892 .9881 '9870

0 '9957

0'9980 '9978 '9975 '9972 '9969

0.9991 '9990 '9989 '9987 '9986

0 '9996

0'9999

'9952 '9947 .9941 '9935

'9996 '9995 '9995 '9994

'9998 '9998 '9998 '9997

0'9999 .9999 '9999 '9999 '9999

IWO

0 '9513

0'9857 '9844 '9830 -9815 '9799

0 '9965

'9921 '9913 '9904 '9895

'9962 '9957 '9953 '9948

0.9984 .9982 '9980 '9978 '9975

0 '9993

'9477 '9440 '9400 '9359

0'9730 '9707 •9684 •9658 '9632

0 '9928

'3 '4

0.9165 '9112 '9057 '9000 '8940

'9992 '9991 '9990 '9989

0'9997 '9997 '9996 '9996 '9995

0'9999 '9999 '9998 '9998 '9998

10.5 •6 '7 •8 '9

0.8879 •8815 .8750 •8682 '8612

0'9317 '9272 '9225 '91 77 '9126

0'9604 '9574 '9543 '9511 '9477

0'9781 '9763 '9744 '9723 '9701

0.9885 '9874 '9863 '9850 '9837

0.9942 '9936 '9930 '9923 '9915

0'9972 '9969 '9966 .9962 '9958

0 '9987

0 '9994

'9994 '9993 '9992 '9991

0'9998 '9997 '9997 '9996 '9996

0 '9999

'9986 '9984 '9982 '9980

i•o

0.8540

0'9074

0 '9441

0'9678

0'9823

0 '9907

0 '9953

0 '9977

0'9990

0'9995

0'9998

'2

'3 '4

'2

I

'2

-9986 '9984 '9982 '9979

'9918 '9909 -9899

'9966 '9962 '9957 '9952

'9934 '9927 - 9919

r = 25 101

'8 '9

0.9999 '9999 '9999 0 '9999

'9999 '9999 '9999

'9999 '9998 '9998 '9998

0'9999

'9999 '9999

0 '9999

'9999

See page 24 for explanation of the use of this table. 29

0 '9999

'9999 '9999 '9999

'9999 '9999 '9998 '9998

TABLE 2. THE POISSON DISTRIBUTION FUNCTION P,

r =

2

3

4

5

6

7

8

9

xo

II

12

11•0

0'0012

0'0049

0.0151

0'0375

0'0786

0'1432

0'2320

0.3405

0 '4599

0'5793

'2

'0010

'0042

•0132

'0333

-0708

'4

•0009

•0036

•0115

•0295

•0636

•6

•0007

•0261 '0230

-0571 •0512

'3192 •2987 .2791 '2603

'5554 •5316 •5080

•0006

•oioo •oo87

-2147 •1984 •1830 •1686

'4362 -4131 .3905

•8

-0031 •0027

-1307 •1192 •1085 '0986

'3685

'4847

0.6887 •6666 •6442 •6216 '5988

12'0

0'0005

0'0023

0'0458

0'0895

0'1550

0 '2424

•0004 '0004

'0020

0'0076 •oo66

0'0203

'2 •4

-0017

•0057

•0179 •0158

•0014

•0050

•0139

•8

•0003 '0003

•081x '0734 •0664

'1424 .1305

•6

'04ro •0366 •0326

•0012

•0043

'0122

•0291

*0599

•2254 •2092 •1939 •1794

0'3472 '3266 •3067 .2876 '2693

0.4616 '4389 •4167 '3950 '3738

0.5760 '5531 '5303 .5077 '4853

13•0

0.0002

0•00I I

0'0259

0'0540

0.2517

0'3532

0'4631

'0009

•0094

'0230

•0487

0'0998 •0910

0. 1658

•0002 •000z

0'0037 •0032

0'0107

'2 •4

' 1530

. 2349

•0008

•0028

-0083

•0204

•6

•0001

•0007

-0024

'0072

•0,8i

•000i

•0006

•oo2i

•0063

•ox6i

•0828 .0753 •o684

•1410 •1297 '1192

-2189 .2037 •1893

'4413 '4199 .3989

.8

•0438 '0393 .0353

'3332 •3139

14.0

0.0001

0'0005

0'0018

0'0055

0'0142

0'0316

0•062I

0.1094

0'1757

0'2600

0•3585

'2

•0001 •0001

'0004

•0048

.0126

•0283

'0003

'0016 •0013

•0111

•0253

•0003

-0012

•0098

•0226

•1003 -0918 •0839

•1628 •1507

•0001

'1392

•2435 •2277 •2127

'3391 •3203 •3021

•0002

•0010

'0042 •0037 •0032

-0562 •0509 •0460

•0087

•0202

•0415

'0766

•1285

' 1984

•2845

15'0

0'0002

0'0009

0'0028

0'0076

0'0180

0'0374

'2

•0002 •0002

•0007

•0024

•0067

•0160

'0337

0.0699 •40636

•0006

•0021

•0059

'0143

.0005

.0018

-0052

•0127

'0304 •0273

'0579

•0001

•8

•0001

•0005

•0016

•0046

•0113

- 0245

•0478

0.1185 •1091 •1003 •0921 -0845

0.1848 -1718 •1596 •1481 •1372

0.2676 - 2514 '2358 '2209 •2067

16•o

0.000i

0.0004

0'0014

0'0220

.0089

.0197

'4 •6

•0001

•0003

'0012 •0010

.0079

0176

0355

0'0774 .0708 .0647

0.1931

'0003

0'0433 .0392

o.1270

*0001

0'0040 *0035

0•0I 00

*2

'1174 .1084

568: :116

•000I

'4 •6 •8

'4 -6

•0003

•0009

.8

0002

-0008

170

0'0002

0'0007

'2

'0002

.0006

'4 •6

•0001

'1195 •1093

.0526

'2952 '2773

'3784

•I8o2

'0070

•0158

•0321

•oo6I

•0141

•0290

•0591 .0539

•0920

'1454

0•0021

0'0054

0'0126

0.0261

0•0491

0'0847

0'1350

•0018

'0048

'0112

•0235

•0447

•0778

•1252

'0005

•00'6

-0042

-0I00

-0212

•0406

•0714

•I 160

•0004

•0014

•0037

•0004

•0012

•0033

•0089 -0079

•0191 •0171

•0369 .0335

•0655 •0600

•1074

-8

•000! •0007

18•o

0'0001

0'0003

0'0010

0'0029

0'0071

0'0154

•2

•0001 •0001

•0033 '0002

'0009 -0008

•0025

'0063

'0138

0'0304 '0275

0.0549 •0502

0.0917 '0846

'0022

•0056

'0124

'0002

'0007

*0020

'01 1 1

'0458 '0418

'0002

-0006

•0017

'0049 - 0044

' 0249 '0225

'0779

•0001

•0099

-0203

•0381

•0659

0.0039 •0034

o'oo89 •0079

0.0183 -0165

0.0347 •0315

0.0606 •0556

'4 •6

r =

.:0000002:4 71

.0999

'0993

'0717

X X•0

0'0002

'2

•0002

'4 •6

•000I

0'0002

0'0005

0'0015

•00ox

•8

•0001

-0005 •0004

•0013

'4 .6

•0001 •0007

•0012

'0030

•0071

•0149

•0287

•0509

•0001

'0003

•0010

•0027

'0063

-0260

•0467

-8

12'0

0.0001

•0001

•0003

•0009

•0024

•0056

•0134 •0120

'0236

'0427

'2

'0001

'4

•0001

0• 0 00 I

0•000 3

0.0008

0•0021

0 ' 00 50

0.0108

0 ' 02 I 4

0•0 3 9 0

.8 19•0 '2

20'0

See page 24 for explanation of the use of this table.

30

TABLE 2. THE POISSON DISTRIBUTION FUNCTION ti,

Y = 13

14

15

i6

17

i8

19

zo

21

22

23

XII)

0.7813

•7025

0.9074 .8963 •8845 *8719 .8585

0.9441 .9364 •9280 '9190 .9092

0.9678 •9628 '9572 '9511 '9444

0.9823 '9792 '9757 .9718 '9674

0'9907 '9889 '9868 '9845 '9818

0'9977 '9972 '9966 '0958 '9950

0'9990 '9987 '9984 .9980 '9975

0'9995

-7624 '7430 •7230

0.8540 . 8391 •8234 '8069 .7898

0'9953

•2

12•0 '2

0'6815

0.7720

'4 •6 '8

'6387

*7347 '7153 '6954

•8875 '8755 •8629 '8495

0'9370 .9290 '9204 •9111 '9011

0'9626 .9572 '9513 - 9448 '9378

0'9787 *9753 '9715 .9672 '9625

0.9884 '9863 '9840 .9813 '9783

0.9939

'7536

0'8444 •8296 '8140 *7978 *7810

0'8987

•6603

.9927 '9914 .9898 '9880

0•9970 .9963 '9955 '9946 '9936

0.9985 .9982 '9978 '9973 '9967

13'0

0.5730 *5511

0.6751 '6546 '6338 •6128 .5916

0.7636 *7456 .7272 •7083 •6890

0.8355 •8208 .8054 *7895 .7730

0.8905 '8791 .8671 '8545 . 8411

0.9302 '9219 .9130 '9035 '8934

0 '9573

'9516 '9454 *9387 '9314

0'9750 '9713 '9671 '9626 '9576

0.9859 '9836 •9810 '9780 *9748

0.9924 '9910 •9894 '9876 .9856

0.9960 '9952 '9943 '9933 .9921

0.5704 '5492 '5281 '5071

0'6694 '6494 •6293 '6090

0 '7559

•4863

•5886

'7384 '7204 •7020 •6832

0.8272 •8126 '7975 •7818 . 7656

0.8826 •8712 . 8592 •8466 '8333

0.9235 .9150 •9060 •8963 •8861

0.9521 •9461 .9396 '9326 •9251

0.9712 •9671 '9627 '9579 '9526

0.9833 •9807 '9779 '9747 .9711

0.9907 •9891 '9873 '9853 .9831

0.4657 '4453 '4253

0.5681 '5476 .5272

0.6641 '6448 .6253

0.7489 '7317 '7141

0.8195 •8051 .7901

0.9170 .9084 .8992 '8894 •8791

0.9469 .9407 .9340 •9268 *9190

0.9673 '9630 *9583 .9532 '9477

0.9805 '9777 '9746 .9712 '9674

'4 •6 •8

.2 '4 •6 -8

•6169

*5950

'5292

.5074 .4858

14.0

0.4644

'2

'4434

'4 •6 •8

'4227 •4024 •3826

15.0 .2

0.3632

'9943 .9932 '9918 '9902

'9994 *9992 .9991 '9988

•4056

•5069

•6056

•6962

'7747

•3864

'4867

'5858

'6779

'7587

0.8752 •8638 .8517 •8391 •8260

0.2745

0'3675 '3492

0.5660 .5461 •5263 •5067 '4871

0.6593 •6406 •6216 •6025 '5833

0'7423 '7255 •7084 •6908 •6730

0.8122 '7980 . 7833 •7681 •7524

0.8682 •8567 .8447 •8321 •8191

0.9108 •9020 '8927 •8828 .8724

0.9418 '9353 '9284 •9210 .9131

0'9633

'25 85

0 '5640

*5448 •5256 •5065 '4875

0.6550 '6367 •6182 '5996 •5810

0.7363 '7199 •7031 '6859 •6685

0.8055 '7914 •7769 .7619 •7465

0.8615 •85oo •838o -8255 -8126

0.9047 •8958 •8864 '8765 •866o

0.9367 •9301 •9230 '9154 '9074

'4 •6 -8

'3444 '3260 -3083 •2911

16 0

'2

•6 •8

•2285 •2144

'2971

0.4667 '4470 .4276 •4085 •3898

17•0

0.2009

0.2808

0.3715

'2

'1880

'2651

'3535

'4 .6 •8

'1758 .1641 •1531

2500 ' 2354 . 2215

•3361 '3191 '3026

0.4677 '4486 .4297 '4112 '3929

18.0

0.2081 .1953 •183o '1714 '1603

0.2867 •2712 .2563 '2419 *2281

0.3751 .3576 '3405 .3239 *3077

0.4686 .4500 '4317 .4136 '3958

0.5622 '5435 *5249 '5063 .4878

0.6509 '6331 •6151 '5970 .5788

0.7307 '7146 •6981 '6814 .6644

0'7991 .7852 •7709 •7561 •7410

0.8551 .8436 •8317 •8193 •8065

0.8989 *8899 '8804 '8704 •8600

0.1497 '1397

0'3784 '3613 •3446 '3283 '3124

0'4695 *4514 '4335 •4158 '3985

0'5606 *5424 '5242 •5061 '4881

0'6472 '6298 •6122 *5946 •5769

0.7255 '7097 '6935 '6772 '660 5

0.7931 '7794 '7653 '7507 '7358

0.8490 '8376 '8257 '8134 •8007

0.2970

0.3814

0.4703

0'5591

0'6437

0.7206

0 '7875

'4

13 2432 '33 •3139

4

0.1426 .1327 '1233

.6 •8

•1145 •xo62

10'0

0.0984

'2

'0911

'4 .6 •8

•0842 .0778 •0717

-1213

0'2148 '2021 •1899 .1782

•1128

-1671

0'2920 '2768 •2621 ' 2479 •2342

20.0

0.0661

0.1049

0.1565

0.2211

'2

-1303

See page 24 for explanation of the use of this table.

31

'9588 '9539 •9486 .9429

TABLE 2. THE POISSON DISTRIBUTION FUNCTION

ii.

r = 24

25

mo

0.9998

0 '9999

'2

'9997

'4 '6 '8

'9997 '9996 '9995

'9999 '9999 '9998 '9998

36

37

38

39

fl•

1= 35

IT2

'4 '6 '8

0.9999 '9999 '9999 '9999

x8.0

0.9999

'2

'9999

0 '9999

'4 '6 .8

'9998 '9998 '9997

0.9999 '9999 '9999 '9999 '9999

'9999 '9999

19•0

.2 '4 .6 .8

0.9997 '9996 '9995 '9994 '9993

0.9998 '9998 '9998 '9997 '9996

0.9999 '9999 '9999 '9999 '9998

0 '9999

20'0

0'9992

0'9996

0.9998

0'9999

0'9999

30

31

32

33

34

26

27

0 '9999

'9999

12'0 *2

0.9993

0 '9997

0'9999

'9991

'4 .6 '8

'9989 .9987 '9984

'9996 '9995 '9994 '9992

'9998 '9998 '9997 '9996

13'0

0.9980 '9976 '9971 .9965 '9958

0.9990 '9988 .9985 .9982 '9978

0'9995

'9994 '9993 '9991 '9989

0.9998 '9997 '9997 '9996 '9995

0 '9999

.2 '4 .6 .8 14.0 .2 '4 '6 .8

0.9950 '9941 '9930 .9918 '9904

0 '9974

0 '9987

0 '9994

0 '9997

0 '9999

0 '9999

•9969 '9963 '9956 '9947

'9984 •9981 '9977 '9972

'9992 '9990 '9988 •9986

'9996 '9995 '9994 .9993

'9998 '9998 '9997 '9997

'9999 '9999 '9999 '9998

15.0

0.9888

0.9996 '9995 '9994 '9992 '9991

'9998 '9997 '9996 '9995

'9999 '9999 '9998 '9998

0 '9999

'9851 •9829 •9804

0.9991 '9990 .9987 .9985 '9982

0'9999

'4 .6 -8

0.9983 '9979 '9975 '9971 '9965

0 '9998

'9871

0.9938 .9928 .9915 •9902 '9886

0'9967

'2

16•o

0 '9777 '9747

0'9989 '9986 '9984 '9981 '9977

'9993 '9992 '9990 '9988

0'9997 '9997 '9996 '9995 '9994

0 '9999

'9952 '9944 '9934 '9924

0'9978 '9974 '9969 '9964 '9957

0 '9999

'9713 .9677 '9637

0.9925 •9913 .9900 -9884 '9867

0 '9994

'4 '6 '8

0'9869 .9849 •9828 '9804 '9777

0 '9959

'2

'9998 '9998 '9998 '9997

'9999 '9999 '9999 '9999

17.0

0'9594 '9546

0'9748

'9968 •9962 '9956 '9949

'9983 .9980 '9976 •9972

'9991 '9989 '9987 -9985

0'9996 '9995 '9994 '9993 '9992

0'9998 '9998 '9997 '9997 '9996

0'9999

'9495 .9440 '9381

0.9950 '9942 '9933 -9922 •9910

0 '9993

'4 •6 .8

0.9912 '9898 .9883 •9866 .9848

0 '9986

.9715 '9680 .9641 '9599

0.9848 •9827 '9804 '9778 '9749

0 '9973

'2

x8•o

0.9317

0'9554

'9249

'4 •6 •8

'9177 .9100 •9019

.9505 '9452 .9395 '9334

0.9718 .9683 - 9646 •9606 '9562

0.9827 -9804 '9779 •9751 '9720

0.9897 '9882 •9866 .9847 •9827

0.9941 '9931 '9921 '9909 '9896

0.9967 •9961 '9955 '9948 '9939

0.9982 '9979 '9975 '9971 '9966

0.9990 '9989 '9986 '9984 .9981

0'9995

'2

0.9998 '9997 '9996 '9996 '9995

19.0

0.8933 '8842

'4 •6 •8

'8746 •8646 8541

0.9269 '9199 '9126 '9048 •8965

0.9514 '9463 '9409 '9350 •9288

0.9687 .9651 '9612 '9570 '9524

0.9805 .9780 '9753 .9724 '9692

0.9882 •9865 '9847 .9828 •9806

0.9930 '9920 '9908 '9895 •9881

0'9960 '9954 '9946 '9938 .9929

0.9978 '9974 '9970 '9965 '9959

0'9988 '9986 '9983 •9980 '9977

0 '9994

'2

20'0

0.8432

0.8878

0.9221

0 '9475

0.9657

0'9782

0.9865

0.9919

0'9953

0 '9973

0'9985

.

.996; '9954 '9945 '9936

0.9999 '9999 '9999 '9999 '9998

z8

'9999 '9999

0 '9999

'9999

'9999 '9999 '9998 '9998

r

=

29

0'9999

'9999 '9999

0 '9999

'9999

See page 24 for explanation of the use of this table.

32

'9999 '9999 '9999

'9994 '9993 '9991 '9990

0 '9999

'9999

'9999 '9999 '9998 '9998

'9992 '9991 .9989 '9987

TABLE 3. BINOMIAL COEFFICIENTS This table gives values of

n! n(n — 1). .(n — r + 1) (n — r)! r! — r!

(r ) =nC,

when r> in use (1 = n )• (n) is the number of r n r r ways of selecting r objects from n, the order of choice being immaterial. (See also Table 6, which gives values of log10 n! for n < 300.) —

2

n

=x

I

3

4

2

I

I 2

3

I I

3 4

3 6

I 4

5

I 1

10 15

7 8

I x

21

9

I

5 6 7 8 9

10

6

35 56 84

xo II 12

I I I

12

13

I

13

14

I

15 16 17 18

I 1 I 1

19

I

20 21 22 23 24

4

Jo II

I

28 36 45 55 66

6

5

I

5 15 35 70 126

20

6

1

21

7 28 84

56 126

120

210

252

210

120

330 495

462 792

462 924

33o 792

1716

1716

3003

3432

14

715 1001

1287 2002

15

105

455

1365

16 17

120

560

1820

18 19

136 153 171

68o 816 969

2380 3060 3876

3003 4368 6188 8568 11628

5005 8008 12376 18564 27132

I I I I I

20 21 22

190 210 231

I I 4o 1330 1540

23

253

1771

4845 5985 7315 8855

15504 20349 26334 33649

38760 54264 74613 100947

24

276

2024

10626

42504

134596

25 26 27

I I I

300

28 29

x I

25 26 27 28 29

325 351 378 406

2300 2600 2925 3276 3654

12650 14950 17550 20475 23751

53130 65780 80730 98280 118755

230230 296010 376740 475020

30

I

30

435

4060

27405

142506

593775

?I

II

12

= 20

184756

167960

125970

21 22 23

352716

352716

293930

646646 1144066 1961256

705432 1352078 2496144

646646 1352078 2704156

3268760 5311735

4457400 7726160

24

26 27 28 29

8436285

13037895

13123110

21474180

2.0030010

34597290

5200300 9657700 17383860 30421755 51895935

30

30045015

54627300

86493225

25

I

8 36

165 220 286 364

10

9

I

78 91

r

8

7

33

177100

13

77520 203490 497420

6435

1

9

1

45 165 495

xo 55 220

1287 3003

715 2002

31824 50388

6435 12870 24310 43758 75582

24310 48620 92378

77520 x 1628o 170544 245157 346104

125970 203490 319770 490314 735471

167960 293930 497420 817190 13.07504

480700 657800 888030 1184040 1560780

1081575 1562275 2220075 3108105 4292145

2042975 3124550 4686825 6906900 10015005

2035800

5852925

14307150

11440

19448

14

5005 11440

15

2496144

3876o 116280 319770 817190 1961256

15504 54264 170544 490314 1307504

5200300 10400600 20058300 37442160 67863915

4457400 9657700 20058300 40116600 77558760

3268760 7726160 17383860 37442160 77558760

119759850

145422675

155117520

1144066

TABLE 4. THE NORMAL DISTRIBUTION FUNCTION The function tabulated is 0(x) =

fx .V2it

dt. 0(x) is

-00

the probability that a random variable, normally distributed with zero mean and unit variance, will be less than or equal to x. When x < o use 40(x) = i -0( - x), as the normal distribution with zero mean and unit variance is symmetric about zero.

x

0(x)

x

(I)(x)

x

(11(x)

x

I(x)

x

(13( x)

x

0'00

0'5000 '5040 '5080

0'40 '41 '42

0*6554 '6591

0'80 •81

1'20

0'8849 •8869

2'00

0.97725

.9463 *9474

•5120 •5160

•82 •83

x.6o •61 •62

0'9452

•8907 '8925

'63

'9484

- 44

•6628 •6664 •67oo

0'7881 •7910 '7939

.64

'9495

•0, •02 •03 •04

'97778 *97831 '97882 *97932

0.8944 •8962 •898o .8997

1.65 •66 •67 -68 '69

0.9505 •9515 •9525 '9535 '9545

2'05

0'97982

•o6 •07 •o8 •09

•98030

1•70

0.9554 .9564 '9573 .9582 '9591

2•10

•01 •02

•03 •04

'43

'84

.7967 '7995

•21 •22 •23 .24

0'85

0'8023

1'25

•86 •87 •88 •89

•8051 •8078 •8106 •8133

•26 •27 •28 -29

0'05 •06

0.5199 '5239

0.45 .46

.07

.5279

*47

•08

'5319

'09

'5359

•48 .49

0.6736 .6772 •6808 •6844 .6879

0•10

0'5398

0'50

0'6915

0'90

0'8159

1•30

'II •12

'5438 '5478

•13 '14

.5517 '5557

•51 •52 .53 '54

•6950 •6985 .7019 '7054

'91 •92 •93 .94

•8186 •8212 .8238 .82,64

•3I -32 .33 '34

0.15 •16 •17 •18 •19

0-5596 •5636 .5675 .5714 '5753

0.55 •56 '57 •58 '59

0.7088 •7123 '7157 •7190 '7224

o.95 •96 .97 •98 .99

0.8289 •8315 .8340 •8365 .8389

0'20 '21 '22

0.5793

0.60

0.7257

100

'5832 '5871

'61

'7291

'01

•62

•02 '03 '04

'5910

'63

'5948

'64

'7324 '7357 '7389

0.25 •26

0 '5987

0'65

0'7422

1.05

•6026 •6064 '6103 •6141

'7454 '7486 .7517 '7549

•06

•27 '28

•66 •67 •68

•o8 •09

.23 '24

•29

'69

'07

•8888

'9015 0'9032 *9049

'98077

•98124 '98169 0'98214 '98257 '98300

•9066 .9082 '9099

.71 '72 .73 '74

1.35 •36 '37 •38 '39

0-9115 •9131 '9147 •9162 '9177

r75 •76 .77 •78 '79

0 '9599

2'15

0'98422

•9608 .9616 •9625 '9633

•16 .17 •18 '19

•98461 •98500 .98537 '98574

0'8413

1'40

0'9192

1.80

0.9641

'41

.9207

'8,

'9649

2'20 '21

0.98610

.8438 '8461 '8485 '8508

'42

'9222

'9236 '925 1

-9656 '9664 .9671

•22 .23 .24

•98679

'43 .44

•82 .83 .84

'98745

0.8531 .8554 '8577 •8599 •8621

1'45 '46 '47 •48 '49

0 '9265

1.85 •86

0'98778

•26

'9292

'87

'27

'98809 '98840

•9306 '9319

•88 '89

0.9678 •9686 .9693 •9699 '9706

2'25

'9279

•28 •29

•98899

1•50

0.9332 '9345 '9357 '9370 -9382

1'90

0.9713 '9719

2'30

0'98928

.31

.98956

'9726

'32

'98983

'9732 '9738

'33 '34

'99010 '99036 0.99061 '99086

•II •12 .13 •14

0'30 •31 •32

0.6179

0•70 '71 '72

0.7580

1•10

'6217 •6255

'7611 •7642

•I 1

'33 '34

.6293 '6331

'73 '74

'7673 '7704

•13 '14

0.8643 •8665 •8686 '8708 '8729

0.35 •36 '37 •38 '39

0.6368 •6406 '6443 •6480 '65 1 7

0.75 •76 '77 •78 '79

0'7734

r15 •16 •17 •i8 •19

0.8749 •8770 .8790 •8810 •883o

r55 '56 .57 •58 '59

0 '9394

1'95

0 '9744

2'35

•7764 '7794 •7823 '7852

.9406 '9418 •9429 '9441

'96 '97 •98 '99

'9750 '9756 •9761 '9767

-36 .37 •38 '39

0'40

0.6554

0.80

0.7881

1•20

0'8849

1•60

0'9452

2'00

0'9772

2.40

•12

(I)(x)

'51 '52 '53 .54

34

'91 .92 '93 '94

.98341 .98382

•98645 '98713

•98870

'99111

'99134 '991 58 0'99180

TABLE 4. THE NORMAL DISTRIBUTION FUNCTION x

x

(1.(x)

x

(1)(x)

x

(1)(x)

x

(1)(x)

2'40

0.99180

2'55

0'99461

2'70

'99202 '99224 '99245 '99266

•56

'99477

II

2.85 •86

'57 •58

.72 '73 '74

'99683

•88

•59

'99492 .99506 .99520

0.99653 '99664 '99674 '99693

•89

0.99781 .99788 '99795 .99801 -99807

3.00 0.99865

',II

2'60

0 '99534

2.75

0.99813

3•05

0.99886

-76

0.99702 .9971i

2'90

•61 •62 '63 '64

'99547

•9x

.99819

•o6

.99889

'99560

.77

.99720

'92

'99825

•07

'99893

'49

0'99286 '99305 '99324 '99343 '99361

'93 '94

'9983 1 '99836

•o8 .09

'99900

2'50

0.99379

.52

'99396 '99413

0'99841 - 99846

3.10

. 51

•I2

'53 '54

•13

'99913

'14

'99916

2'55

3'15

0'99918

'42 '43 '44 2'45

'46 '47 '48

'87

'99573

•78

.99728

'99585

'79

'99736

2.65

0'99598

•99609 .99621

2.8o •8x

2'95

'99430 '99446

•66 .67 •68 .69

-99

.99851 .99856 .99861

0'99461

2.70

3'00

0'99865

.99632

•83

'99643

*84

(3'99744 '99752 .9976o '99767 '99774

0.99653

2'85

0'99781

•82

.96

.97

.98

•ox •02 '03 •04

•I

x

(D(x) .99869 '99874 '99878

'99882

.99896

0.99903 .99906 '99910

0(x)

3'15 •x6

0.99918

•x7 •i8 •19

'99924 .99926 '99929

3'20 '2I '22 6 23 •24

0 '99931

3'25

0 '99942

'26 '27

•28 -29

'99944 '99946 '99948 '99950

3.30

0.99952

.99921

'99934 '99936 '99938 '99940

The critical table below gives on the left the range of values of x for which 0(x) takes the value on the right, correct to the last figure given; in critical cases, take the upper of the two values of (1)(x) indicated. 3.075 0.9990 3o5 0.9991 '130 0'9992 3.215 3'174 0.9993 0 '9994

3'263 09994

3-320 09995 3.389 0.9996 3'480 0.9997 3.615 0.9998 0 '9999

When x > 3.3 the formula -0(x) *

xV2IT

0

99990

3 .916 0 '99995 3.976 099996

0 3.826 0.99993 3.867 0.99994

4'055 0.99997 4'173 0.99998 4'417 099999 1•00000

3'73ro .99991 3159 3 .791 99992

0 '99995

I 3 15 1051 is very accurate, with relative error 7+ 78+71x

less than 945/x1°.

TABLE 5. PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION This table gives percentage points x(P) defined by the equation

P

ioo

=

lc° e-It2 dt. x(P)

If X is a variable, normally distributed with zero mean and unit variance, P/Ioo is the probability that X x(P). The lower P per cent points are given by symmetry as - x(P), x(P) is 2PI loo. and the probability that IXI

P

x(P)

P

50

0'0000

45 40 35

0'1257

30 25

0'2533 0 '3853 0'5244 0 '6 745 0.8416

x(P)

P

x(P)

P

x(P)

5'0

P6449

2'0

2'0537

I'0

2'3263

0•10

3'0902

1.6646 1'6849

3'0 2'9

1'8808

4.8 4'6 4'4

P8957

2'8

1 .91 10

2.0749 2•0969

2.7

0'07

P7279

2'6

P9268 1.9431

2.3656 2•089 2.4573

4*2

I'6

2'1444

0.9 o•8 0.7 o•6

0.09 0T8

1.7060

1.9 1.8 r7

2.5121

0'06

3'1214 3.1559 3'1947 3.2389

P7507

2.5

P9600

I.5

2•1701

o•5

1'7744

P9774 1 '9954

1.4

2.1 973

0.4

2.5758 2.6521

1 . 7991

2'4 2'3

1'8250 P8522

2*2 2'1

2'0141

r3 1•2

2'0335

I •I

2•2262 2'2571 2'2904

0'3 0•2 0•1

I0

P2816

4'0 3.8 3•6 3.4

5

P6449

3'2

20 15

1.0364

35

2•i20i

P

x(P)

2/478 2•8782

3.0902

P

x(P)

0•0x 0.005

3.2905 3.7190 3.8906

0.001 0.0005

4.4172

0.05

4.2649

TABLE 6. LOGARITHMS OF FACTORIALS n

log10 n!

n

0

0'0000

i

o•0000

logio n!

n

log10 n!

n

log10 n!

n

loglo n!

100

157'9700

377'2001

250 251

161'9829

267'1177

200 201 202

374.8969

159'9743

15 151 152

26 62 4 '9 7 35. 6 59 9

Ica

379'5054

163.9958

153

269.3024

203

381.8129

252 253

154

271'4899

204

384.1226

254

492.5096 494'9093 497'3107 499'7138 502'1186

227753.'86783043

205 206

255 256

278'0693 280'2679

207 208

282'4693

209

386 '4343 388.7482 39 P 0642 393.3822 395'7024

210 211 212

398'0246 400'3489 402'6752

log10 n!

n

50

64.483 1

5x

102

103

2

0'3010

52

3 4

0.7782

53

66.1906 67.9066 69.6309

I • 3802

54

71'3633

104

166•0128

5 6 7 8 9

2'0792 2'8573

73.1037 74.8519 76.6077 78.3712 80. 1420

105 xo6 107 xo8 109

168.0340 170.0593

3'7024 4'6055 5'5598

55 56 57 58 59

172'0887

156 157

174•1221 176.1595

158 1 59

IO II

6.5598

6o

81•9202 83'7055

178'2009 180•2462

284'6735

61

II0 III

160

7'6012

161

12 13 14

8.6803 9'7943 10'9404

62 63 64

85'4979 87.2972 89'1034

112 1 xx 13 4

182.2955 118846...43045845

262 163

286.8803 289•o898

15

12'1165

65

90'9163

115

188'4661

16

13.3206

92'7359 94'5619 96'3945 98.2333

xx6

440075..734 360

291'3020

258 259

504'5252 506'9334 509'3433 511'7549 514.1682

260 261 262 263

518.9999 52 r 4182 523.8381

264

526•2597 528.6830 531.1078 533'5344 535.9625 538.3922

257

164 293.5168

214 215

409.6664

297.9544

216 217

265 266

4112:3 4 03 07 03 9

ix 8

165 ,66 167 ,68

295'7343

190.5306 192.5988 194.6707

2,8 416.6758

267 268

119

196'7462

169

219

419.0162

269

220 221 222

421'3587 423'7031 426'0494

270 271 272

17

14'5511

66 67

x8

15.8063

68

19

17.0851

69

20

18.3861

70

100'0784

120

198.8254

170

306.8608

21 22 23

19'7083 21'0508 22'4125

71 72

101 .9297

200'9082

171

309'0938

202 '9945

24

23'7927

107. 5196

121 122 123 124

172 173 174

311.3293 313'5674 315'8079

223 224

428'3977 430'7480

273 274

25

25.1906 26.6056 28.0370 29%4841 30'9465

75

109'3946 111. 2754

125

209-2748

175

318.0509

225

433'1002

126

211'3751

176

320'2965

226

322'5444 324"7948 327'0477

227 228 229

435'4543 437.8103 4.40.1682 442.5281

275 276 277

8o 81 83

124.5961

34

38- 4702

84

126.5204

329.3030 331.5607 333.8207 336.0832 338'3480

230 231

82

120'7632 122'6770

33

32.4237 33.9150 35'4202 36.9387

85 86

128.4498 130'3843

X85

340'6152

235

186

342•8847

26 27

28 29 30 31 32

35

40'0142

36 37

41'5705 43.1387

38

44 • 7185

39 46.3096

73 74

76 77

78 79

103'7870 105'6503

117

205'0844 207'1 779

113.1619

127

213.4790

115-0540

128

215.5862

177 178

116.9516

129

217.6967

179

118.8547

130 131 132

219'8107 221'9280 224'0485

181

133 134

226 .1 724 228'2995

135 136 137 138

230.4298 232- 5634 234.7001 236 . 8400

87

132-3238

88 89

134.2683 136-2177

180 182 183 184

300'1771 302'4024 304'6303

189

283 284

572.5753 575' 0287

285

:5569:7 02 99 6:

286

577'4835 579.9399

461'4742 463 .8508

288

349'70 71

239

466.2292

289

468'6094 470'9914 473'375 2 475'7608 478.1482

47'91 16

90

138'1719

140

241'1291

190

351'9859

240

49'5244 51' 1477 52'7811 54'4246

91

140.1310

141

243.2783

354.2669

241

45

56.0778

46 47 48 49

57'7406 59'4127 61'0939 62. 7841

50

64.4831

92

142'0948

142

245'4306

93 94

144. 0632 146'0364

143

144

247'5860 249'7443

191 192 193 194

95 96 97 98

148.0141 1 49'9964 151•9831 1 53'9744

145 146

251'9057 254.0700

147

256'2374

368'0003

99

155.9700

148 149

258.4076 260.5808

197 198 1 99

100

57'9700

150

262'7569

200

For large n, logio n!

565.2246 567-6733

237

40

570'1235

584 2;8 39 577 587.3180

290

589/804

291

592'2443

292 293 294

594'7097 597.1766 599'6449

295 296 297

6o2'1147

356'5502

242

358.8358 361.1236

243 244

195

363'4136

245

196

365.7059

246

480'5374 482.9283

370.2970 372'5959

248 2 47 8 249

487 71 5 5'32140

298

609'5330

490' 1116

299

612.0087

374.8969

250

492.5096

300

614.4858

0.39909 + (n+ logio n - 0.4342945 n.

36

553'0044 555'4453 557.8878 560.3318 562 '7774

280 281 282

238

41 42 43 44

540'8236

543.2566 545.6912 548'1273 550'5651

444.8898 447'2 534 449-6189 451.9862 454'3555

3 3 :=

187

139 238.9830

232

233 234

278

279

516'5832

604.586o 607.0588

TABLE 7. THE x'-DISTRIBUTION FUNCTION F,(x)

The function tabulated is FAx)

-

jo

tiv-le-igdt 0 (The above shape applies for v > 3 only. When v < 3 the mode is at the origin.)

for integer v < 25. Fp(x) is the probability that a random variable X, distributed as X2 with v degrees of freedom, will be less than or equal to x. Note that F,(x) = 20(xi) - (cf. Table 4). For certain values of x and v > 25 use may be made of the following relation between the X2 and Poisson distributions :

with mean v and variance 2v. A better approximation is usually obtained by using the formula

Fv(x) * (1)(V-z; - zy -

Fv(x) = 1 - F(iv - x I ix)

where 4(s) is the normal distribution function (see Table 4)• Omitted entries to the left and right of tabulated values are r and o respectively (to four decimal places).

where F(rlit) is the Poisson distribution function (see Table z). If v > 25, X is approximately normally distributed

= x = 0.0 'I '2

'3 '4

I

V =

2

V =

x = 4.0

0 '9545

x = 0•0

0.0000

x = 4.0

'I '2

'9571

•1

I

V =

o•0000

'2482 '3453 .4161 '4729

2

V =

0•8647 •8713

v=

x = 0.0

o•0000

x = 4-o

:2 1

:0 00 22 84 2

'2

-.4 6

0.7385 '7593 :7 79 78 66 5

. 0598

•8

8130

0.5 -6

o.o811

5.0

0.8282

•1036

-2

.99°04963

•7

'9137

'9

'4 •6 •8

'8423 '8553

'9

•1268 •1505 •1746

0'3935 '4231 '4512

5.o

0.9179 •9219

1•0 •I

0'1987 '2229

6•o

•I '2

'2

'8 977

'3

.2 .3

'2470 '2709

'5034

'4

'4

'2945

'4 '6 '8

•9063

.4780

'9257 '9293 '9328

5'5 '6

0 '9361

1.5

0'3177

6 .7

.3406 .3631

7'0 '2

0'9281

'9392

'8 27 0

'3851 '4066

:4 6

:93 49 508

.8

*9497

0.4276 '4481

8-o

•4681

':4 86

0'9540 '9579 6 199 .. 966647

9-o

0'9707

'2

'4

'9733 '9756

•i

'9596

'2

:0 04 98 58 2

'2

' 8775

.9619

'3 '4

'1393

'8835

'3

-o400

.1813

'3 -4.

.8892

'4

0.9661 •9680 '9698 .9715 '9731

0'5

0'2212

4:5

0 • 8946

•6 '7

•2592

•6

'8997

'2953

'8

'3297

'9

'3624

0'9747 -9761 '9774 '9787

1'0 'I

'9799

'4

•3 '4

'9641

0•5

0'5205

•6 '7 •8 .9

- 5614

'5972 •6289 -6572

4.5 •6 .7 -8 '9

1•0

0.6827

5•0

'I

'7057

'I

'2

'7267

'2

'3 '4

'7458 '7633

'3 '4

I'5 '6 '7 •8 .9

0 '7793

0.9810 •9820

1.5

0.5276

•6

'9830 '9840

.7

•8203 -8319

5'5 '6 '7 '8 '9

'8

-5507 .5726 '5934

'9849

.9

.6133

'9

'9477

2'0

0'8427

6.o

0.9857

2.: 0 3 1

1 O.:68 33 24

6:.!

2 o:.99945520:

•z

' 8527

'I

'65or

'2

•2

-8620

'2

'9865 '9872

'6671

'3

'4875

'4

'6988

'4 •6 .8

'I '2

'9879 '9886

'9550 '9592 •9631 -9666

2'5

0:55422 457

'794 1 '8o77

.2 '3

'2

3

3

:78

•8

'2

•8672

•8782 o•8884

'9142

'9214

'9342

'3

' 8706

*4

'8787

'3 '4

2•5

0.8862

6.5

o'9892

2•5

0'7135

7.o

0'9698

'6

'8931

'6

'9898

'6

'7275

'2

'9727

'7

'8997

'7

'9904

'9057

'8

'9909

'9

'9114

'9

'9914

'9

'7654

'4 .6 '8

'9753 .9776 '9798

'7 .8 '9

'5598

'8

:78

503: :774

'5927

:6 8

:97 977 97

3'0

0.9167 -9217

7•0

3-o •I

0'7769

8:2 o

7 0:9 98314

3• o .1

o • 6o84 •6235

10.0

•1

'2

'7981 •8o8o '8173

'4

'9850

'2

•6

'3

'8

•9864 '9877

'4

'6382 '6524 - 666o

'4 '6 •8

0.9814 •9831 '9845 '9859 •9871

o•8262 '8347 '8428

9'0 '2

o•9889 '9899

'4

'9909

'8577

'6 '8

'9918 '9926

o.679z •6920 '7043 •7161 '7275

I•o

'8504

3.5 -6 '7 -8 '9

o'8647

I0•0

0.9933

4'0

0'7385

'2

'9264

'2

'3 '4

'9307 '9348

'3 '4

0.9918 '9923 '9927 '9931 '9935

3'5 '6 '7 '8 '9

0'9386 '9422 '9456 '9487

7.5 '6

0'9938 '9942

'7

'9945

'8

'9517

'9

'9948 '9951

3.5 '6 '7 '8 '9

4'0

0 '9545

8'o

0.9953

4'0

'I

•3 '4

•7878

37

. 5765

•2

•2

0.9883 •9893

'4

'9903

•6 '8

.9911 .9919

12'0

0'9926

TABLE 7. THE x2-DISTRIBUTION FUNCTION 9

xo

II

.00x 8 •0073 •0190

o'0006 •0029 •oo85

0'0002

0•0001

•0011 •0037

0.0729

0.0383

•1150

'0656

0.0191 .0357

0.0091 •0186

v=

4

5

6

7

8

X = 0'5 1•0

0'0265 •0902

0'0079

0'0022

0'0006

0'000 I

'0374

1.5

.1734

•o869

•0144 •0405

2'0

•2642

•1509

•0803

•0052 •0177 •0402

2'5 3•0

0.3554

0.2235

0.1315

'4422

•3000

•1912

12

13

14

•0004 •0015

0.0001 •0006

0•0002

0.0001

0'0042

0'0018

•0093

•0045

00008 •0021

0.0003 -0009

3'5 4'0 4'5

•5221

'3766

'2560

•1648

•'008

'0589

'0329

'0177

'0091

'0046

'0022

'5940

'4506 '5201

'3233 . 3907

•2202 .2793

•1429 '1906

-o886 •1245

•o527 •0780

•0301 -4471

.ol 66 •0274

•oo88 •0154

'0045 ' 0084

5.0

0.7127

0.4562 •5185 . 5768 '6304 .6792

0.3400 •4008 '4603 •5173 '5711

o'2424 •2970 •3528 •4086 *4634

0.1657 •2113 •26o x •3110 *3629

0.1088 .1446 •1847 •2283 .2746

0.0688 .o954 •1266 •16zo •2009

0'0248

0'0142

•7603 •8009

0.5841 •6421 .6938 '7394 . 7794

0'0420

5.5 6•o 6.5 7.0

•0608 •o839 •1112 .1424

'0375 .0538 - 0978

'0224 '0335 '0477 '0653

0.8140 .8438 .8693 .8909 *9093

0.7229 .7619 '7963 '8264 '8527

0.6213 '6674 *7094 '7473 •7813

0.5162 '5665 .6138 '6577 •6981

0.4148 '4659 •5154 '5627 •6075

0.3225 '3712 '4199 '4679 .5146

0.2427 •2867 '3321 *3781 '4242

0.1771 •2149 *2551 •2971 •3403

0.1254 •1564 '1904 •2271 •2658

0.0863 •1107 -1383 •1689 -zozz

0'9248 *9378 '9486 .9577 •9652

0'8753 .8949 •9116 .9259 •9380

0.8114 -838o •8614 •8818 -8994

0'7350 •7683 '7983 •8251 .8488

0.6495 •6885 '7243 •7570 .7867

0'5595

•6022 '6425 •6801 . 7149

0'4696 •5140 '5567 •5976 '6364

0.3840 *4278 . 4711 .5134 '5543

0•3061 '3474 •3892 .4310 '4724

0.2378 .2752 •3140 .3536 '3937

0.9715 *9766 '9809 '9844 '9873

0'9483

0'9147

'9279 '9392 .9488 '9570

o'8697 •8882 .9042 •9182 '9304

0.8134 * 8374 .8587 •8777 .8944

0.7470 '7763 .8030 .8270 *8486

0.6727 .7067 .7381 7 6 7o '7935

0.5936 .6310 •6662 .6993 '7301

0.5129 . 5522 . 5900 •6262 •6604

0'4338

'9570 '9643 '9704 '9755

0'9896 '9916 '9932 '9944 '9955

0'9797 .9833

•9862 '9887 '9907

0.9640 *9699 '9749 '9791 •9826

0.9409 '9499 '9576 '9642 '9699

0.9091 '9219 '9331 .9429 '9513

0.8679 .8851 •9004 •9138 .9256

0.8175 '8393 •8589 •8764 •8921

0.7586 '7848 •8088 •8306 •8504

o•6926 .7228 •7509 •7768 '8007

o.6218 '6551 •6866 •7162 '7438

0.9924 .9938 '9949 .9958 *9966

0.9856 •9880 '9901 .9918 '9932

0'9747 .9788

'9822 '9851 '9876

0'9586 '9648 '9702 .9748 .9787

0.9360 •9450 '9529 '9597 .9656

0.9061 '9184 *9293 '9389 '9473

0.8683 '8843 *8987 '9115 .9228

0.822.6 '8425 -8606 .8769 •8916

0.7695 .7932

'9992 '9994

0. 9964 '9971 '9976 '9981 .9984

20 21 22 23

0'9995

0'9988

0'8699

'9999

0.9707 '9789 '9849 '9893 '9924

0 '9048

24

0.9821 *9873 '9911 '9938 '9957

0 '9329

'9962 '9975 '9983 '9989

0.9897 *9929 '9951 '9966 '9977

0 '9547

'9992 '9995 '9997 '9998

0.9972 •9982 '9988 '9992 '9995

0 '9944

'9997

'9666 '9756 *9823 '9873

*9496 '9625 '9723 '9797

.9271 '9446 *9583 '9689

*8984 '9214 '9397 '9542

25 26 27

0 '9999

0 '9999

0 '9997

0.9984 '9989 '9993 '9995 '9997

0'9970 '9980 '9986 '9990 '9994

0.9769 .9830

0'9654

'9963 '9974 '9982 '9988

0'9909 '9935 '9954 '9968 '9977

0'9852

'9998 '9999 '9999 '9999

0'9992 '9995 '9997 '9998 '9999

0 '9947

'9999

0 '9999

0'9998

0'9996

0.9991

0'9984

0 '9972

'6575

*8352

-8641

7'5 8•o 8.5 9.0 9'5

o-8883 '9084

10.0 xo.5 x•o

-9251 .9389 '9503

11•5

0.9596 .9672 '9734 .9785

12.0

•9826

12•5

13.0 13.5

0.9860 •9887 '9909

14•0

*9927

14.5

'9941

15.0 15.5 16•o

0 '9953

16.5 17'0

.9976 .9981

17.5

0 '9985

18•o 18'5 19.0 19.5

'9988

28 29 30

'9962 .9970

'9990

'9998 '9999

'9999

38

6

'9893 '9923 '9945 '9961

' 0739

'4735 •5124 •5503 •5868

•8151 .8351 '8533

'9876 '9910 '9935

-9741 -9807 '9858 '9895

0 '9953

0 '9924

TABLE 7. THE f-DISTRIBUTION FUNCTION v= X

15

x6

17

18

19

=3 4

0.0004 •0023

0'0002

0'0001

•001I

'0005

0'0002

0'000 I

5 6 7 8 9

0.0079 •0203 •0424 -0762 •I225

0'0042

0'0022

0'001 I

•0 I 19

•oo68

•o267

•0165

'051 I

•0335

•o866

•0597

•0038 •0099 '0214 •0403

10 II 12

0'1803

0.1334 •1905

0.0964 •1434

•2560

•1999

•3977 '4745

•3272 .4013

'2638 •3329

o•o681 •1056 •1528 •2084 '2709

13

'5 16 17 x8 19

0 '5486

0.4754 '5470 .6144 .6761 .7313

0 '4045

0•3380

•4762 .5456 -6112 .6715

20 21 22 23 24

0'8281

0.7708 '821 5 •8568 •8863 '9105

25 26 27

0'9501

14

'2474 '3210

•6179 •6811 '7373 •7863

•8632 •8922 •9159

'9349

20

21

22

23

24

25

0•0006

0'0003

0.0001

•0021 •0058 •0133 •0265

•00II

'0006

•0033 •oo8 1 .0 r 7 1

•0019 •0049 •0'08

0.0001 •0003 •00I0 •oo28 •oo67

0.0001 •0005 -00'6 •0040

0.0001 •0003 •0009 •0024

0.0001 •0005 •0014

0.0471 •0762 •1144 •1614 •2163

0'0318

0'0211

0'0137

0'0087

0'0055

0'0033

-0538 •0839 •1226 •1695

.0372 •0604 •0914 '1304

•0253 •0426 •0668 •0985

•0168 •0295 •0480 .0731

.oi To •0201 •0339 .0533

'0235 •0383

'4075 '4769 •5443 •6082

0.2774 '3427 •4101 '4776 '5432

0.2236 •2834 •3470 '4126 .4782

0.1770 .2303 •2889 '3510 '4149

0.1378 .1841 •2366 '2940 '3547

0'1054 •1447 •1907 •2425 *2988

0.0792 •x '19 •1513 .1970 .2480

0.0586 '0852 . i '82 '1576 .2029

0.7258 '7737 •8153 •8507 •8806

0.6672 •7206 •7680 •8094 •8450

0.6054 •6632 •7157 •7627 .8038

0.5421 •6029 .6595 -7112 . 7576

0.4787 '5411 '6005

0.4170 '4793 '5401

'6560

* 5983

•7069

.6528

0.3581 '4189 '4797 '5392 . 5962

0.3032 '3613 '4207 .4802 '5384

0.2532 '3074 '3643 •4224 -4806

0.9053 *9255 '9419 '9551 '9655

0.8751 •9002 •9210 '9379 '9516

0.8395 •8698 -8953 .9166 '9340

0'7986 .8342 •8647 •8906 '9122

0.7528 *7936 •8291 •8598 •886o

0.7029

0.6497 '6991 .7440 '7842 '8197

0'5942

'7483 •7888 •8243 •8551

0.5376 '5924 '6441 -6921 '7361

•0071 '0134

28

'9713 '9784

29

'9839

0.9302 '9460 '9585 •9684 '9761

30 31 32 33

0.9881 '9912 .9936 '9953 '9966

0.9820 •9865 .9900 .9926 '9946

0 '9737

0'9626 '9712 .9780 '9833 '9874

0.9482 '9596 '9687 -9760 '98,6

0'9301 '9448 '9567 .9663 '9739

0.9080 '9263 '9414 .9538 '9638

0.8815 •9039 '9226 -9381 '9509

0.8506 •8772 .8999 '9189 '9348

o.8x 52 •8462 •8730 .8959 .9153

0 '7757

•9800 '9850 .9887 '9916

35 36 37 38 39

0'9975

0.9960 '9971 '9979 '9985 '9989

0.9938 '9954 '9966 '9975 .9982

0.9905 '9929 '9948 '9961 '9972

0.9860 .9894 '9921 '9941 -9956

0 '9799

'9846 '9883 .9911 '9933

0'9718 .9781 '9832 .9871 .9902

0.9613 '9696 .9763 '9817 '9859

0'9480 '9587 '9675 '9745 '9802

0.9316 '9451 '9562 '9653 .9727

0'9118 '9284 '9423 '9537 .9632

40 41 42 43 44

0 '9995

0'9992 '9994 '9996 '9997 '9998

0'9987

'9997 '9998 '9998 '9999

'9991 '9993 '9995 '9997

0.9979 •9985 '9989 .9992 '9994

0'9967 '9976 .9982 .9987 '9991

0.9950 '9963 '9972 .9980 '9985

0.9926 .9944 .9958 .9969 '9977

0.9892 '9918 '9937 '9953 '9965

0.9846 '9882 .9909 .9931 '9947

0.9786 .9833 '9871 .9901 '9924

0.9708 .9770 '9820 •986o '9892

45 46 47 48 49

0 '9999

0 '9999

0'9996 '9997 '9998 '9998 '9999

0 '9973

0 '9960

0.9942

0 '9916

'9995 '9996 '9997 '9998

0'9989 '9992 '9994 '9996 '9997

0'9983

'9999 '9999

0'9998 '9998 '9999 '9999 '9999

0 '9993

'9999

'9987 '9991 '9993 '9995

-9980 '9985 '9989 '9992

'9970 '9978 '9983 .9988

'9956 '9967 '9975 '9981

'9936 '9951 '9963 .9972

0 '9999

0 '9999

0 '9998

0 '9996

0 '9994

0.9991

0'9986

0.9979

34

50

•9620

'9982 .9987 '9991 '9994

39

'6468 '6955 .7400 '7799

-8i io •842o •8689 '8921

TABLE 8. PERCENTAGE POINTS OF THE x2-DISTRIBUTION This table gives percentage points equation

g(p)

P/100

defined by the

co -

12 1-1/PN f,p)X1P-1

100 2'

e-i' dx.

If X is a variable distributed as X2 with v degrees of freedom, Phoo is the probability that X 26(P). For v > loo, ✓2X is approximately normally distributed with mean ✓2v-1 and unit variance.

V

=

0 x(P) (The above shape applies for v 3 3 only. When v < 3 the mode is at the origin.)

8o

P

99'95

99'9

99'5

99

97'5

95

90

I

0.06 3927 0.00I000 0 . 01528

0'051571 0'002001 0'02430

0'043927 0.01003 0'07172

0'031571 0'02010 0'1148

0'039821 0.05064 0'2158

0'003932 0'1026 0'3518

0'01579

0;6 04 69 418

0;0 26 10 47 0- 5844

0.4463 roo5

0.06392

0 09080

0.2070

0.2971

0.4844

0.7107

2

3 4 5 6 7 8 9

.

0.1581

0.2102

0.4117

0 '5543

0.8312

P145

1.610

2 '343

0.2994

0'3811

0'6757

0.8721

V237

P635

2'204

3'070

0 '4849

0 '5985 0.8571 1.152

0'9893 1.344 P735

1'239 1.646

1.690 2.180

2.088

2/00

2.167 2133 3'325

2.833 3'490 4'168

3.822 4'594 5.380 6.179 6.989 7.807 8.634 9'467

0/104

0.9717

xo

1'265

1 '479

2.156

2'558

II

1 .587

1 . 834

2.214

3-053 3'571 4'107 4'660

3'247 3.816 4'404 5.009 5.629

3'940 4'575 5.226 5.892 6.571

4'865 5'578 6.304 7.042 7'790

7.261 7.962 8.672 9'390 10•12

8.547 9.312 10.09 10•86 11. 65

12.86 13.72

10-85 11.59 12- 34 13.09 13.85

12.44 13.24 14'04 14'85 15.66

12

1 '934

13 14

2 . 305

2. 617

2.697

3.041

2.603 3'074 3.565 4'075

15 16

3-108 3'536 3.980 4'439 4'912

3'483 3'942 4'416 4.905 5'407

4'601 5. 142 5.697 6.265 6.844

5.229 5.812 6.408 7.015 7633

6.262 6.908 7'564 8'231 8.907

5'398 5.896

5.921 6.447

7'434 8.034 8- 643 9-260 9.886

8.260 8.897

9'591 10•28 :0;9689

17

18 19 20 21 22 23 24

6'404

6.983

6- 924 7'453

7-529 8.085

1902 10.86

12.40

70

6o

0 . 1485 0/133 12 94 5 1 2:4

0'2750 P022 3 76 59 2 1.- 8

3-000 3.828 4'671 5'527 6 '393

3.655 4.570 5'493 6.423 7'357

7.267 8.148 99 9:02 3:

8.295 9'237 io.: Ir3 8

1o•82

12-08

11.72 12.62 13'53 14'44 15.35

13.03 13.98 14'94 15 .89 16.85

14'58 15'44 16. 31 17.19 18.06

16.27 17.18 18.10

17.81 18.77 19.73

19.02

20.69

19'94

21.65

20.87 2P79 22.72

22.62 23'58

10.31 11.15 12'00

25

7'991

8.649

10.52

11.52

13.12

14'61

16'47

18.94

26

8.538

9.222

11'16

12'20

13'84

15'38

17.29

19.82

27 28 29

9.093

18•1i

20.70 21.59 22.48

ir81

1z.88

14'57

12'46 13.12

13'56

15'31 16.05

0.15 16.93

18.94

10.23

10'39 10.99

17.71

19.77

10.80 11.98 13.18

11.59 12.81 14.06

18'49

205,-60

25'51

20'07

22- 27 22 26 53-"93 14 56

-64

38

15'64

16 61

.

.

2:98r:382491 22-88

21.66 23 27 24.88

23.95

15'32

14'95 16.36 17'79 10.23 20 69

16.79

14.40

13.79 15.13 16.50 17 89 19 29

28.73 30'54

27-37 29.24 31 12 32 99 .

27-44 29-38 31.31 33.25 35-19

40 50

16.9z 23-46 30'34 37'47 44'79

17 92

20'71

22 . 16

24'43

26.51

24-67 31'74 39'04 46'52

27-99 35'53 178 4 51 3:2

29-71

32.36 4570-.4185 48.76

3520. 3 64 41'45

34'87 1 81 44 5;3

37'13 66 4 56.-82

63.35 72.92

66.40 76 19

52•28 59.90

54' 16 61.92

59.20 67.33

30

32

34 36

6o 70 8o 90 I00

9'656

9.803

.

.

.

14'26

8 3 47 5:444 53'54 61.75 70.06

65.65 74' 22

40

.

434 37196 51'74 6o 39

27-34 29.05 37-69 5456:3463

23.65 24.58

.

24'54 25.51 26.48

.

64.28

59'90 69.2!

69.13 77'93

73'29 82.36

78.56

82'51

85 . 99

87.95

92-13

95.81

.

TABLE 8. PERCENTAGE POINTS OF THE f-DISTRIBUTION This table gives percentage points x;,(P) defined by the equation

rco 100

2142

rq)

A( p)

xiv-1 e- P dx. 0

If Xis a variable distributed as x2 with v degrees of freedom, Phoo is the probability that X x,2,(P). For v > ioo, VzX is approximately normally distributed with meanzi .s/ and unit variance.

P

50

v=I

40

(The above shape applies for v at the origin.)

30

20

To

5 3.841 5'991 7'815 9.488

2 3 4

1.386 2.366 3'357

0'7083 1.833 2.946 4'045

1.074 2.408 3'665 4'878

1.642 3'219 4.642 5'989

2.706 4'605 6.251 7'779

5 6 7 8 9

4'351 5.348 6.346 7'344 8.343

5.132 6.2i 1 7.283 8'351 9'414

6.064 7.231 8.383 9'524

7.289 8.558

9.236 10.64

II

9'342 10.34

0 '4549

X,2,'(P)

2'5

I

3 only. When v < 3 the mode is

0'5

0•I

0'05

5.024 7'378 9'348 11.14

6.635 9.210 11'34 13.28

7.879 10•6o 12.84 14.86

10'83

12'12

13.82 16'27 18'47

15.2o 17'73 20.00

11.07 12'59

12.83 14'45

15 '09

20'52 22'46 24'32 26'12 27'88

22'II 24'10 26'02 27'87 29'67

12'02

14'07

16'01

18.48

11.03

13.36

15'51

17'53

10•66

12'24

14'68

16'92

19'02

20'09 21'67

16.75 18.55 20.28 21.95 23.59

10'47 1P53

11.78 12'90

13.44 14.63

15'99 17.28

18.31

20'48 21'92

23.21 24'72

25.19

29'59

31'42

19'68

2616

31'26

14'01 15'12

15•81 16'98

18'55 19'81

21.03

23.34

26.22

22'36

24'74

27'69

16.22

18.15

21.06

23.68

26.12

29.14

28.30 29.82 31.32

32.91 34'53 36.12

33'14 34'82 36.48 38.11

19.31 20'47 z1•61 22.76 23.90

22.31 23'54 24'77 25'99 27.20

25.00 26.3o 2759 28.87 30.14

27'49 28'85

32.80 34'27 35'72 3716 38.58

37'70 39'25 40'79 42.31 43'82

39'72 41.31

30.19 31•53 32.85

30.58 32.00 33'41 34' 81 36.19

28'41

27'10

25.04 26.17 27.30 28'43 29.55

29'62 30'81 32'01 33'20

31.41 32.67 33'92 35'17 36.42

34'17 35'48 36.78 38.08 39'36

37'57 38.93 40'29 41'64 42 .98

40.00 4P40 42.80 44' 18 45'56

45'31 46.80 48'27 49'73 51.18

47'50 49'01 50.51 52.00 53'48

29

28.17 29.25 30.32 31.39 32.46

30.68 31•79 32.91 34'03 35'14

34'38 35'56 36.74 37'92 39'09

3765 38.89 40.11 41'34 42.56

40.65 41'92 43'19 44'46 45'72

44'31 45'64 46 .96 48.28 49'59

46'93 48.29 49'64 50'99 52. 34

52.62 54'0 5 55'48

27'34 28'34

26.14 27.18 28.21 29.25 30.28

56-89 58'30

54'95 56.41 57.86 59'30 60.73

30 32 34 36 38

29. 34 31'34 33'34 35'34 37'34

31'32 33'38 35'44 37'50 39'56

33'53 35'66 37'80 39'92 42.05

36.25 38'47 40.68 42.88 45.08

40'26 42'58 44'90 4721 49'51

43'77 46'19 48.60 51'00 53.38

46'98 49'48 51.97 54'44 56.90

50'89 53'49 56.06 58.6z 61•16

53'67 56'33 58.96 61.58 64.18

5910 62.49 65.25 67.99 70'70

62.16 65•oo 67.8o 70'59 73'35

40 50 6o

39.34 49'33 59.33 69.33 79.33

41.62 51'89 62.13 72.36 82.57

44'16 54'72 65.23 75.69 86.,2

47' 27 58.16 68.97 79'71 90'41

51.81 63.17 74'40 85'53 96.58

55'76 67.5o 79.08 90.53 I01.9

59'34 71.42 83.3o 95.02 106.6

63.69 76.15 88.38 100'4 rI2•3

66.77 79'49 91'95 104.2 116.3

73'40 86.66 99.61 112.3 124.8

76'09 89'56 102'7

89.33 99.33

92.76 102.9

96.52 106.9

124.3

118•1 129.6

124'1 135.8

128.3 140.2

149'4

12

11'34

12'58

13 14

12.34 13.34

13'64 14.69

15

14'34 15'34

15'73

1732

16.78 17.82 18.87 19.91

18.42 19'51 20.60

20'95 21.99

22.77 23.86 24'94 26'02

16 17 18 19

16'34 17'34 18'34

20

19.34

21

20'34

22 23

21 '34

24

23'34

25 26

24'34 25'34

27 28

70

8o 90

zoo

22 '34

26 '34

23.03 24'07 25•11

21'69

9'803

111.7

107.6 118.5

41

16.81

1372

45'97

115.6 128.3 140.8 153'2

TABLE 9. THE t-DISTRIBUTION FUNCTION The function tabulated is

F„(t)

ray +1-) f

=

,

VV 7T

- co k I m

ds svoi(v+i) .

F„(t) is the probability that a random variable, distributed as t with v degrees of freedom, will be less than or equal to t. When t < o use F„(t) = i F„( t), the t distribution being symmetric about zero. The limiting distribution of t as v tends to infinity is the normal distribution with zero mean and unit variance (see Table 4). When v is large interpolation in v should be harmonic. -

V

=

I

-

V =

V =

Omitted entries to the right of tabulated values are (to four decimal places).

V =

2

t = o 0 0.5000 •1 '5317

t = 4•0 4.2

0.9220 .9256

'2

'5628

.9289 '9319 '9346

•5700

-2

.5928 •621

4'4 4'6 4.8

'2

.3 '4

•3 '4

-6038 •6361

'3 '4

0.5

0'6476

•6

•6720

.7

'6944

•8 '9

•7148 '7333

5.0 5'5 6.0 6.5 7.0

0.9372 '9428 '9474 '9514 '9548

0.5 -6 '7 -8 '9

0.6667 .6953 •7218 '7462 • 684

I•0

0.7500

I.0

•7651

7.5 8.0

0.9578

I

'2

'7789

8.5

•9627

'2

'3

'7913 •8026

9.0 9.5

.9648 .9666

1.5 •6 •7 •8 '9

0.8128

10. 0 I0•5

-8386 '8458

II-5 I2•0

2•0

0.8524 •,_, R585

12.5

'2

•3 '4 2'5

0.8789

•6 •7 -8 .9

•8831 -8871

t = 0.0

0.5000 '5353

2

v= t = 0.0

3

V

=

3

0•5000

t = 4.0

.5367

.1

0.9860 .9869

'2

'5729

'2

'9877

.9760

•3 '4

•6081 •6420

•3 '4

'9891

4'5 '6 '7 -8 '9

0 '9770

0.5

0. 6743

4'5

'9779

-6

.7046

.6

'9788 '9804

.7 •8 •9

-7328 .7589 -7828

'7 .8 '9

5'0

0'9811

I.0

'2

•9818 .9825

•I '2

0 '8045 - 8242 - 8419

5•0 •I •2

0.9923

I

'3 '4

0.7887 8070 -8235 -8384 •8518

•3 '4

.9831

'9837

•3 '4

.8578 •872o

'3 '4

'9934 '9938

0.9683 •9698 .9711 '9724 '9735

r5 •6 '7 .8 .9

0.8638 .8746 - 8844 .8932 .9011

5'5

0.9842 •9848 '9853 .9858 •9862

/.5 -6 •7 -8 •9

0.8847 •8960

5.5 .6 '7 .8 '9

0'9941

0'9746 .9756

21)

0'9082

6.0

0.9303 '9367

'8642

13.5

'9765

'2

'9206

•I '2

'9875

2'0 'I '2

0 '9954

'9147

0.9867 -9871

6.0

13•0

'9424

.8695

14.0

'9773

14'5

'9781

'9308

'3 '4

'9879 '9882

'3 '4

'9475 .9521

-9960

'8743

'3 '4

•2 •3 '4

. 9961

15

0.9788

2.5

6.5

0'9561

•6 .7 -8 •9

.9598 .9631

- 8943

.6 '7 .8 '9

-9687

6.5 -6 -7 -8 '9

0.9963

•9801 •9813 -9823 .9833

0.9886 -9889

2'5

16 17 i8 19

3•0 •I '2

0. 8976 •9007 '9036

20 21 22

0'9841

3•0

'I

0.9712 '9734

7.0 •I

-3 '4

•9063 -9089

23

•9862

0.9970 .9971 -9972 '9973

24

3.5 •6 .7 -8 •9

0'9114

25

•9138 -916o •918z •9201

30

4'0

0'9220

'4

•8222 -8307

-8908

•9604

t = 4*0 •I

-

•6 '7 •8 •9

'9259

0.9352 -9392

•6 .7 -8

'9429 -9463 '9494

0 '9714

.9727 '9739 '9750

-9796

-9892 .9895

•9062 •9152 .9232

-9661

1

•9884

0.9898 •9903 .9909 . 9914

'9919 -9927 '9931

'9944 '9946 '9949

'9951 -9956 -9958

.9965 -9966

.9967

.9

-9898

7•0 •I '2

.9753

•2

'9867

'3 '4

0.9901 '9904 .9906 .9909 •9911

3•0

'3 '4

0.9523 '9549 '9573 '9596 •9617

'3 '4

'9771 '9788

'3 '4

3'5 •6 .7 -8 '9

0.9636 - 9654 •9670 •9686 •97ot

7.5 -6 -7 •8 •9

0.9913 •9916 •9918 •9920 .9922

3.5 -6 '7 .8 .9

0.9803 •9816 -9829 '9840 •9850

7'5 .6 '7 -8 '9

0 '9975

45

0.9873 '9894 '9909 •9920 '9929

50

0'9936

4 '0 0.9714

8.0

0.9924

4.0

0.9860

8.0

0.9980

35

40

'9849 '9855

'2

42

2

'9969

'9974

.9976 '9977 -9978 '9979

TABLE 9. THE t-DISTRIBUTION FUNCTION 4

5

6

7

8

9

xo

II

12

13

14

t = 0.0

0.5000

0•50o0

0.5000

0.500o

0.5000

0.5000

0.5000

0.500o

0.500o

0.500o

0.5000

'I •2

'5374 '5744

'3 '4

•6104 . 6452

'5379 '5753 •6119 '6472

.5382 .5760 •6129 .6485

.5384 '5764 •6136 . 6495

.5386 .5768 •6141 •6502

'5387 '5770 •6145 •6508

.5388 '5773 •6148 •6512

•5389 '5774 •6151 •6516

'5390 '5776 •6153 •6519

'5391 '5777 •6155 •6522

'5391 '5778 .6157 •6524

0.5 •6 '7 .8

0•6783

0.6809

0.6826

•7096 .7387 .7657

.7127

•7148

'9

•7905

'7424 •7700 '7953

'7449 '7729 .7986

0•6838 •7163 '7467 '7750 •8oio

0•6847 '7174 •7481 .7766 -8028

0•6855 -7183 '7492 '7778 •8042

0.6861 .7191 •7501 .7788 •8054

0•6865 '7197 •7508 '7797 •8063

0•6869 •7202 •7514 •7804 •8o71

0•6873 •7206 .7519 •7810 •8078

0•6876 •7210 •7523 •7815 •8083

ro •x

o•813o •8335 •8518 •8683 •8829

0•8184 •8393 •8581 .8748 .8898

0.8220 - 8433 •8623 •8793 .8945

0•8247 •8461 •8654 •8826 '8979

0.8267 •8483 •8678 •8851 •9005

0•8283 •85o1 •8696 •8870 •9025

0•8296 '8514 •8711 •8886 •9041

0.8306 .8526 •8723 •8899 -9055

0•8315 •8535 •8734 •8910 .9066

0•8322 .8544 '8742 •8919 .9075

0.8329 .8551 •875o .8927 -9084

I.5 •6

0.8960

0.9030

•9076

•9148

0'9079 •9196

'7 .8 '9

•9178 •9269 '9349

'925 1 '9341

0.9140 .9259 •9362 0 5 532 :9 94

0.9161 -9280 .9383 '9473 '9551

0.9177 '9297 •9400 '9490 •9567

0.9191 .9310 '9414 •9503 -958o

0.9203 '9322 156 1 5529 994 .9

9421

'9390 9469

0.9114 •9232 '9335 9426 .9 04

0.9212 '9332 '9435 '9525 •9601

0.9221 '9340 '9444 '9533 •9609

2'0

0'9419

'I '2

'9482 '9537

'3 '4

'9585 •9628

0.9490 '9551 •9605 •9651 '9692

0.9538 '9598 •9649 •9694 '9734

0.9572 -9631 •9681 -9725 •9763

0.9597 •9655 .9705 '9748 •9784

0.9617 •9674 '9723 '9765 •9801

0.9633 '9690 '9738 '9779 •9813

0.9646 .9702 '9750 '9790 •9824

0.9657 .9712 '9759 '9799 •9832

0.9666 '9721 .9768 .9807 •9840

0•9674 .9728 '9774 .9813 -9846

2'5 .6

0.9666

0.9728 '9759

'9730

•9786 •9810 •9831

0•9767 '9797 •9822 '9844 '9863

0'9795

•9700

'9823 •9847 -9867 •9885

0.9815 '9842 -9865 •9884 .9901

0.9831 -9856 •9878 .9896 '9912

0.9843 •9868 •9888 •9906 -9921

0.9852 .9877 •9897 '9914 -9928

0.9860 .9884 •9903 '9920 '9933

0•9867 .9890 -9909 '9925 '9938

0.9873 -9895 '9914 '9929 '9942

0.985o •9866 -9880 '9893 •9904

0.9880 .9894 '9907 .9918 -9928

0.9900 •9913 .9925 '9934 '9943

0.9915 '9927 '9937 '9946 '9953

0.9925 '9936 '9946 '9954 '9961

0.9933 '9944 '9953 -996o '9966

0.9940 '9949 '9958 .9965 '9970

0.9945 '9954 .9962 .9968 '9974

0.9949 '9958 .9965 '9971 '9976

0.9952 -9961 '9968 '9974 '9978

0.9914 '9922 '9930 '9937 '9943

0.9936 '9943 '9950 '9955 •9960

0'9950

0.9960 •9965 '9970 '9974 '9977

0.9966 '9971 '9975 '9979 •9982

0.9971 '9976 '9979 •9983 '9985

0'9975

'9956 .9962 .9966 '9971

'9979 •9982 •9985 -9988

0.9978 '9982 '9985 •9987 -9989

0.998o .9984 .9987 .9989 '9991

0.9982 .9986 '9988 '9990 '9992

0.9948 '9953 '9958 •9961 '9965

0.9964 .9968 '9972 '9975 '9977

0.9974 '9977 .9980 -9982 '9984

0•9980 •9983 '9985 •9987 •9989

0.9984 .9987 -9988 '9990 '9991

0'9987

0'9990

0.9991

0'9992

•9989 '9991 '9992 '9993

'9991 '9993 '9994 '9995

'9993 '9994 '9995 '9996

'9994 '9995 '9996 '9996

0.9993 '9995 '9996 '9996 '9997

0.9979 '9982 '9983 .9985 -9986

0.9986 •9988 '9989 '9990 '9991

0.9990 '9991 '9992 '9993 '9994

0.9993 '9994 '9994 '9995 '9996

0.9994 '9995 '9996 '9996 '9997

0.9995 '9996 '9997 '9997 '9998

0.9996 '9997 '9997 '9998 '9998

0 '9997

0 '9998

'9998 '9998 '9998 '9999

'9998 '9998 '9999 '9999

0.9988

0.9992

0.9995

0 '9996

0'9997

0.9998

0 '9998

0.9999

0 '9999

v=

'2 '3 '4

'7 .8 '9 3'0 'I '2 '3 '4 3.5 •6

'9756 '9779 0.980o •9819

'9835 •9850 '9864 0.9876

'9

•9886 •9896 '9904 •9912

4'0

0 '9919

'I '2 '3 '4

•9926 '9932

'7 .8

'9937 '9942

4.5 .6

0.9946

'7 .8 '9

'9953 '9957 .9960

0.9968 '9971 '9973 '9976 '9978

5'0

0.9963

0 '9979

'9950

.9300

43

TABLE 9. THE t-DISTRIBUTION FUNCTION v=

15

16

17

i8

19

20

24

30

40

6o

c0

t = 0.0 •1 .2 •3 '4

0.5000 .5392 *5779 •6159 '6526

0'5000

0'5000

0'5000

0'5000

0'5000

0'5000

0'5000

0'5000

0.5000

.5392

'5780 •6160 •6528

*5392 '5781 •6161 •6529

'5393 '5781 •6162 .6531

'5393 '5782 •6163 . 6532

'5393 '5782 •6164 *6533

'5394 '5784 •6166 '6537

'5395 '5786 •6169 '6540

'5396 '5788 -6171 '6544

'5397 '5789 •6174 '6547

0.5000 '5398 '5793 '6179 '6554

0.5 •6 '7 •8 '9

0.6878 •7213 '7527 •7819 •8088

0.6881 •7215 '7530 •7823 •8093

0.6883 •7218 '7533 •7826 •8097

0.6884 •722o '7536 •7829 -81oo

0.6886 •7222 '7538 . 7832 -8103

0.6887 •7224 '7540 . 7834 •8xo6

0.6892 •7229 '7547 '7842 •8115

0.6896 '7235 '7553 •7850 •8124

0.6901 '7241 '7560 •7858 •8132

0.6905 '7246 . 7567 •7866 '8141

0.6915 '7257 .7580 •7881 '8159

1.0

0.8339 •8562 '8762 '8940 '9097

0.8343 •8567 '8767 '8945 '9103

0'8347 .8571 -8772 *8950 '9107

0 '8351

0 '8354

0 '8364

•8578 '8779 *8958 •9116

•8589 .8791 '8970 •9128

0.8383 •8610 •8814 '8995 •9154

0 '8413

.8575 '8776 *8954 •9112

0.8373 •8600 •8802 '8982 •9141

0 '8393

'3 '4

0'8334 •8557 •8756 '8934 '9091

•8621 •8826 '9007 •9167

'8643 •8849 '9032 -9192

1.5

0'9228

0'9235

0'9240

0'9245

0'0250

0'9254

0'9267

0'9280

.6 '7 .8 '9

'9348 '9451 '9540 '9616

'9354 '9458 '9546 •9622

'9360 '9463 '9552 •9627

'9365 .9468 '9557 •9632

'9370 '9473 '9561 •9636

'9374 '9477 '9565 •9640

'9387 '9490 '9578 •9652

'9400 '9503 '9590 •9665

0.9293 '9413 .9516 •9603 •9677

0.9306 .9426 '9528 •9616 •9689

0.9332 '9452 '9554 -9641 .9713

2'0

0.9680 '9735

0.9691 '9745 '9790 '9828 '9859

0.9696 '9750 '9794 .9832 '9863

0.9700 '9753 '9798 .9835 •9866

0 '9704

0.9715 '9768 -9812 '9848 •9877

0'9727 '9779 •9822 '9857 •9886

0.9738 '9790 •9832 '9866 '9894

0.9750 •9800 •9842 '9875 '9902

0.9772 •9821 •9861 '9893 •9918

•I •z

'2

'9781

'3 '4

'981 9 '985 1

0.9686 '9740 .9786 '9824 '9855

2.5 '6 '7 .8 '9

0.9877 '9900 '9918 '9933 '9945

0.9882 '9903 '9921 '9936 '9948

0.9885 '9907 '9924 '9938 *9950

0.9888 '9910 '9927 .9941 '9952

0.9891 '9912 '9929 '9943 '9954

0'9894 '9914 '9931 '9945 '9956

0'9902

0'9909

'9921 '9937 '9950 *9961

'9928 '9944 '9956 '9965

0'9917 '9935 '9949 '9961 '9970

o'9924 '9941 '9955 •9966 '9974

0'9938 *9953 '9965 '9974 '9981

3'0

0.9955 .9963 '9970 .9976 '9980

0 '9958

0.9960 .9967 '9974 '9979 '9983

0.9962 -9969 '9975 -9980 '9984

0.9963 .9971 '9976 -9981 '9985

0 '9965

0 '9969

0.9973

0'9977

•9972 '9978 •9982 '9986

.9976 '9981 '9985 *9988

.9979 '9984 .9988 '9990

.9982 '9987 .9990 '9992

0.9980 .9985 *9989 .9992 '9994

0.9987 .9990 '9993 .9995 '9997

3'5 '6 '7 .8 '9

0 '9984

0 '9985

0.9987 '9990 '9992 '9993 '9995

0.9988 '9990 '9992 '9994 '9995

0.9989 '9991 '9993 '9994 '9996

0.9991 '9993 '9994 '9996 '9997

0 '9994

'9988 '9990 .9992 '9994

0.9986 '9989 '9991 '9993 '9994

0 '9993

.9987 '9989 '9991 '9993

'9994 '9996 '9997 '9997

'9996 .9997 '9998 '9998

0.9996 '9997 '9998 '9998 '9999

0.9998 '9998 '9999 '9999

4'0 'I

0 '9994

0 '9995

0 '9995

0.9999

'9998 '9998 '9999 '9999

'9999 '9999 '9999 '9999

'9999 '9999 '9999

'9999

'3 '4

0.9996 '9997 '9998 '9998 '9999

0 '9999

'9996 '9997 '9998 '9998

0.9996 '9997 '9998 '9998 '9998

0 '9998

'9996 '9997 '9997 '9998

0'9996 '9997 *9997 '9998 '9998

0 '9997

'9995 '9996 '9997 '9997

4,5

0 '9998

0'9998

0'9998

0 '9999

0 '9999

0 '9999

0 '9999

•1

•I

'2

•3 '4

'2

.9966 *9972 .9977 '9982

44

'9757 •9801 '9838 •9869

TABLE 10. PERCENTAGE POINTS OF THE t-DISTRIBUTION This table gives percentage points tv(P) defined by the equation

P too

rav + - vw, r(#y)

tv(p)(1

dt t 2I p)i(P +1) •

Let Xi and Xz be independent random variables having a normal distribution with zero mean and unit variance and a X'-distribution with v degrees of freedom respectively; then t = XJVX2/P has Student's t-distribution with v degrees of freedom, and the probability that t t„ (P)is P/Ioo. The lower percentage points are given by symmetry as - t„ (P), and the probability that Iti t,,(P) is 2Phoo.

P (To)

30

40

25

20

V = I

0.3249

0.

7265

I•0000

1 '3764

2 3 4

0'2887

0'6172

0'2767 0'2707

0'5844 0'5686

0.8165 0. 7649 0.7407

-o607 0.9785 09410

5 6 7 8 9

The limiting distribution of t as v tends to infinity is the normal distribution with zero mean and unit variance. When v is large interpolation in v should be harmonic.

15

to

5%

2.5 31.82 6.965 4'541 3'747

r 963 I .386

3.078

I'250 1'190

1.638 1.533

6.314 2.920 2.353 2.132 2.015 1'943 1.895 1.860 1.833

2'571

2'262

3.365 3'143 2.998 2.896 2.821

1.812

2. 228

2.764

3.169

2'201 2'179 2'160 2'145

2'718 2'681 2'650 2'624

3'106

1 . 886

0.2672

0'5594

0/267

0. 2648

0- 5534 0'5491

0.2619 o- 26io

0'5459 0'5435

0.9195 0.9057 0.8960 0.8889 0.8834

1.156 1' 134

0'2632

0.7176 0.711I 0.7064 0.7027

1. 108 1•100

1'476 r 440 1'415 1 '397 1.383

0.2602 0.2596

0.5415 0'5399

0'6998 0'6974

0.8791 0'8755

r 093 1. 088

1.372 F363

0'2590 0. 2586 0'2582

0.5386

13 14

0.5375

0.6955 0. 6938 0'6924

0.8726 0'8702 0. 8681

I'083 I' 079 1'076

1'356 1'350 1'345

1'796 1'782 I'771 1'761

15

0.2579

0 '5357

0. 6912 0.69o1 0'6892 0.6884 0. 6876

0.8662 0.8647 0.8633 0.8620 o• 8610

r 074 .071 1069 1.067 r•o66

I • 341 1.337 1'333 1'330 1.328

1'753 1/46 1' 740 1'734 x729

2.131

0'2576

2. 120 2' I I0 2' IOI 2'093

0.6870 0.6864

o.8600 0.8591

0'6858

0. 8583 0'8575

I'064 1'063 1'06' I .06o

I ' 325 1 . 323 1'321 1'319

I ' 725 I . 721 I'717 1'714

0.8569

r 059

1.318

1.711 1'708 P 706

to II

12

0'5366

0.5

12.71 4.303 3.182 2.776

'I19

2'447 2.365 2.306

0•X

63.66 318.3 636.6 31.6o 9.925 22'33 12.92 5.841 10.21 8.610 7.173 4'604 4'032 3.707 3'499 3'355 3'250

5'893 5.208 4'785 4'501 4'297

2. 947 2.921 2.898

3'733 3.686 3.646

2'878

3'610

2.861

3.579

4'073 4.0 x 5 3.965 3.922 3.883

2.086

2.528

2.845

2. 080 2'074

2'518 2'508

2.o69 2.064

2.500 2.492

2'831 2'819 2'807

3.552 3.527 3- 505 3.485 3'467

3.850 3.819 3-792 3.768 3'745

2. 060 2. 056 2.052 2'048 2-045

2.485 2.479

3'450 3'435 3.421 3'408 3.396

3.725 3.707 3.690 3'674 3.659

2'042 2'037 2.032 2. 028 2. 024

2.457 2.449 2441 2434 2429

2.750 2. 738 2'728 2'719

3'385 3. 365 3'348 3'333 3.319

3.646

3'551 3'496 3. 460 3'373 3.291

0.2571 0.2569

20 21 22 23

0'2567 0'2566 0'2564 0'2563

0'5329 0'5325 0'5321 0.5317

24

0.2562

0.5314

0.6853 0.6848

25 26 27 28

0. 2561 0. 2560 0' 2559 0.2558 0 ' 2557

0.5312 0.5309 0.5306 0.5304 0.5302

0'6844

0'8562

1. 058

1'316

0.6840 0.6837 0'6834 0.6830

0.8557 0' 8551 0.8546 0.8542

r 058 r 057 i• 056

P315 1 '314 1.313

1.055

1.31

1.699

30

0- 2556

0.5300

0.6828

0.8538

r 055

1.310

32

0'2555

0'5297

0'6822

0'8530

0.2553

0.5294

0.68,8

0.8523

36 38

0'2552 0'2551

0'5291 0'5288

0'6814 0'6810

0'8517 0'8512

1.309 1.307 1'306 1'304

I '697 r 694

34

I'054 I'052 1'052 1'051

1.688 1.686

40 50 6o

0.2550 0- 2547 0.2545

0.5286 0.5278 0.5272

0.6807 0.6794 0.6786

2'403 2'390

2/04 2'678 2•660

0. 6765

1.684 1676 P671 1658

2423

0'5258

1. 303 r 299 1.296 1.2.89

2'009 2'000

0'2539

r 050 r 047 1. 045 1-041

2.021

X20

0.8507 0 ' 8489 0.8477 0. 8446

P980

2.358

2.617

3-307 3.261 3.232 3.160

op

0.2533

0.5244

o•6745

o•8416

i•o36

1•282

r645

1.960

2'326

2'576

3.090

45

1-69.

4.587 4'437 4.318

2. 602 2. 583 2. 567 2'552 2'539

0 ' 2573

29

5'041 4'781

3.055 3.012 2 '977

17 18 19

I.701

6.869 5'959 5'408

4'144 4'025 3'930 3.852 3.787

0- 5350 0 '5344 0.5338 0.5333

1'703

0'05

2'473 2'467 2.462

v 797 2.787 2/79 2. 771 2'763 2.756

2'71[2

4'221 4' 140

3.622 3.6ox 3.582 3.566

TABLE 11(a). 2.5 PER CENT POINTS OF BEHRENS' DISTRIBUTION 0 vz

=

o°

17.36 8'344 7.123 6.771 6.636 6.577

15.56 6.34o 4.960 4'469 4.218 4'074

I I•04

9'065

6'546

3'980

2'365

11.03

9.060 9.055 9.052

6.529 6.511 6-5o1

3'917 3.835 3.786

2.306

9•046

6'485

3'685

2'228 2'179 2'064

9.040

6.473

3.615

1.960

4'563 3'645 3.312 3'145 3'045 2 '979 2 '933 2 '873 2 '835 2•750 2.679

4'414 3.36o 2 '978 2.784 2.667 2.589 2 '534 2460 2 '414

4'303 3.182 2.776 2.571 2.447

3.191 2.816 2.626 2.513 2.437

17.36 11.54

3 4 5 6 7 8

12/ 1 12'71 12'71

I2•29 12'28 12•28

1 I•I x

12.71

12.28 12.28

2

12'71 12/1 12'71 12'71 12/1 12'71

= 2

4'303 4'303 4'303 4'303 4.303 4'303

4.303

I0

4.303

12

4.303

24

4.303 4'303

00

v, = 3 4 5 6 7 8

12'28

12.28

11'03

12'28 12•28 12'28

11.03 I I•03 I I•03

4.190

3'882

4'187 4.186 4.184

3'867 3'857 3'846 3'840 3.828 3.818

4'624 3'903 3'653 3'535 3'468 3'427 3'400 3.366 3'346 3.306 3.276

3.225 3.088 3.026

3.244 3.012 2.897

3.225 2.913 2.756

4'414 4'240 4'205 4'194

4.182 4.180 4.178

4'563 4'100 3'964 3'909

0

go° 12.71 4'303 3.182 2.776 2.571 2 '447

2'305 2•206

2'365 2'306 2'228 2'179 2'064 I•060

3.182

3.191

3'182 3'182 3.182 3.182 3.182 3.182 3.182 3.182 3.182

3'149 3.134 3.127 3.122 3'120 3.117 3.115 3.111 3.1o8

2'992 2'972 2'958

2'831 2'787 2'758

2'663 2'600 V556

2.942 2.933 2.913 2.898

2.719 v696 2.644 2.603

2.498 2.462 2.378 2304

2.312 2.267 2.162 2.067

2/76

2.776 2.776

2.772 2.754 2/46

2 '779 2.717 2.682

2.787 2.675 2.610

2.779 2.625 2532

2.772 2.582 2 '468

2/76 2.571 2 '447

2/76 2/76

2'741 2'738

24

2.776 2.776 2.776

2'365 2'306 2'228

CO

2'776

IO 12 24

00

v1 = 4 5 6 7 8

=4

I•o6 11.04 11•04

17.97 10.14 9.303 9.136 9.090 9.073

15.56 12.41

3 4 5 6 7 8

V2

75

1211

00

=3

6o°

12.71

24

V2

45°

2

IO

=

30°

P1 = I

12

V2

150

IO 12

2'384

2'660

2'567

2•471

2'392

2'734 2.732 2.727

2'646 2.628 2.617 2'594

2.428 2'371 2'335 2.252

2.339 2•268

2•723

2'576

2.537 2.498 2.475 2.421 2'377

If t1and t2 are two independent random variables distributed as t with v1, 1/2 degrees of freedom respectively, the random variable d = t1sin 0 t2 cos 0 has Behrens' distribution with parameters VI, V2 and 0. The function tabulated in Table II is dp = dP(vi, v2 0) such that

Pi

2'178

2'223 2'118 2•024

3.182 2.776 2.571 2.447 2'365 V306 2'228 2 '179 2 '064

1.960

2 '179 2 ' 064

1.96o

v2. When v1 < V2 use the result that

dp(vi , 1/2, 0) = dP(P2,

90° - 0)•

-

Behrens' distribution is symmetric about zero, so Pr (1d1 > dP) = zP/Ioo.

,

Pr (d > dp) = Phoo

Notice that in this table 0 is measured in degrees rather than radians.

for P = 2.5 and o•5 and a range of values of v1and v2 with

46

TABLE 11(a). 2.5 PER CENT POINTS OF BEHRENS' DISTRIBUTION 0 V2 =

5

=

7

v2 = 8

V2 = 10

V2=12

1/2 = 24

1/2 = CO

75°

90°

2.564

2'562

2'565

2'562

2'564

2'571

2 '554

2.470 2.410 2.367 2•310 2.274

2.449 2. 374 2.320

24

2.571 2.571 2.571

2.549 2 '546 2.541 2 '539 2.533

2'228 2'179 2'064

2'571

2'529

2.191 2'118

2'248 2'203 2'098

00

2.500 2.458 2.428 2.390 2.366 y312 2.266

2.447

2-571

2.527 2.505 2.490 2 '471 2.460 2.436 2.416

2•004

P960

2.435 2'413 2.398 2.379 2.367

2.436 2-394 2.364 2.325 2.301

2.435 2.375 2.331 2.274 2.239

2 '440 2'364 2'310 2'238 2'193

2 '447 2'365 2'306 2'228 2.179

2'342 2'322

2.247 2•201

2.156

2•088 P993

2.064 1.960

2'352 2'322 2'283

2'352 2'309 2'252

2'358 2.304 2'232 2'187 2'082

2'365 V306 2.228 2'179 2'064

P987

1.960

2. 300 2'228 2.183

2.306 2'228 2'179

2.077 1.982

2.064 P960

2'223 2'178 2'072

2179

1.977

1.960

2.571

2'365

2.306

v1 = 6

2 '447

2.440

7 8

2 '447

2 '434

2.447

xo

2 '447

12 24

2.447 2.447

00

2 '447

2.431 2.426 2.423 2.418 2 '413

7 8

2.365

2.358

2'352

2.365

2. 354

2.337

ICI 12 24 CO

2'365 2'365 2'365 2'365

2•350

2'317

2. 347

2.306

2.259

2.216

2'341 2'336

2'280 2'259

2'205 2•158

2•133 2'060

1,1 = 8 I0 12 24

2'306 2'306 2•306 2'306

2.300 2.295 2'292 2'286

2'294 2'274 2'262 2'236

2'292 2'254 2'230 2.175

2'294 2'237 2'201 2'118

00

2.306

2.281

2.215

2.128

2.044

V1 = 10 12 24 CO

2'228 2'228 2'228 V228

2'223 2.220 2'214 V209

2'217 2'205 2•178 2'157

2'215 2'191 2'136 2'089

2'217 2'181 V098 2'024

= 12

2'179

2'175

2'169

2'167

2'179

y179

2.168

2.142

2.112

2.179

2.163

2'120

2'064

2'169 2'085 2.01 1

2'175

24 CO

2.069 1.973

2.064 1.960

V1 = 24 00

2'064 2'064

2'062 2•056

2•058 2'035

2'056

V058

2'062

2'064

y009

1.983

P966

1•96o

vl = 00

1.960

1•96o

1.960

1'960

1.96o

1.96o

i.96o

V1

=

2'064

Pi

Pi

-

j.

V, v2

2'228

-

0 = tan-1 ( s -- - i s 2 )' 0 being 07

4 measured in degrees. Define r = I - + s2 and d =

2.082

If d > dp the confidence level associated with it 2 is less than P per cent, and if d < dp the confidence level associated with ft 2 is less than P per cent. (See H. Cramer, Mathematical Methods of Statistics, Princeton University Press (1946), Princeton, N.J., pp. 520-523.) Also, the values of Pi -#2 such that 1(Yi - - (#1-P2)1 < rdp provide a Ioo 2P per cent Bayesian credibility interval for #2.

This distribution arises in investigating the difference between the means #1, P2 of two normal distributions without assuming, as does the t-statistic, that the variances are equal. Let o i, p72 be the means and 4., 4 the variances of two independent samples of sizes n1, n2 from normal distributions, let v1= n1-1, v2 = n2 - 1 and

6o°

45°

2.571

5 6 7 8

12

v2

30°

2 '571

v1 =

xo

v2 = 6

15.

o°

I. -072 .

r

47

pi

-

TABLE 11(b). 0-5 PER CENT POINTS OF BEHRENS' DISTRIBUTION v2 = I

vl = I

2

3 4 5 6 7

8 io

12 24 c0 V2 = 2

V1 = 2

3 4 5 6 7 8 10

12

24

V2

=3

P2 = 4

o°

x5

3o°

450

6o°

75.

go°

63.66 63.66 63.66 63'66 63.66 63.66 63.66 63.66 63.66 63.66 63.66 63.66

77.96 61.61 61 '49 61'49 61 '49 61'49 61 '49 61 '49 61 '49 61 '49 61'49 61'49

86.96 55.62 55'15 55'14 55'14 55'14 55'13 55'13 55'13 55'13 55'13 55'13

90.02 46.18 45' 08 45'04 45.03 45.03 45.03 45.03 45'03 45.03 45.02 45.02

86.96 34'18 32.04 31.89 31.87 31.86 31.86 31.86 31.86 31.86 31.85 31.85

7796 21'11 17'28 16'70 16'59 16'57 16.56 16.55 16.55 16.54 16'54 16.53

63.66 9.925 5 .841 4'604 4. 032 3.707 3'499 3'355 3.169 3.055 2.797 2 '576

10'01

10'14 8'905 8.717 8.676 8.663 8.657 8.653 8.649 8. 647 8. 642 8.638

10'19 7'937 7'428 7'270 7'210 7'183 7'169 7155 7.148 7'134 7'124

10'14 6.966 6.082 5.716 5'535 5'434 5'373 5.308 5'275 5.223 5'194

10'0!

9'640 9'609 9'604 9'602 9.601 9'600 9.600 9'599 9'598 9'597

6.187 5 .049 4'5 28 4'235 4'049 3'921 3'759 3.660 3'446 3.276

9.925 5'841 4'604 4'032 3.707 3'499 3'355 3.169 3-055 2.797 2.576

5'841 5'841

5'754 5'694

5'640 5'349

5.841

5.681

5.256

5.841 5'841

5.676 5'673

5.218 5'199

5'640 4'720 4'316 4'095 3'958 3.866 3'753 3.686 3'548 3'449

5'754 4'601 4'076 3.782 3'595, 3'467 3'302 3'201 2 '977 2.789

5'841 4.604 4'032 3.707 3'499 3'355 3.169 3.055 2.797 2.576

4'400 3'983 3'755 3.613 3'517 3'395 3'323 3.167 3'045

4'525

4'604 4'032 3.707 3'499 3'355 3.169 3.055 2.797 2.576

9.925 9'925 9'925 9'925 9.925 9.925 9.925 9.925 9'925 9'925 9'925

.

Pi = 3 4 5 6 7 8 xo

5.841

5.671

5.189

5.841

1205 g.

5 'E1 1

5'669 5.668 5.666 5 .664

5'177 5.171 5.159 5.150

5'598 4'986 4'739 4'617 4'548 4'506 4'459 4'434 4'389 4'361

v1 = 4

4'604 4'604 4'604 4'604

4'525 4:505 4'497 4'493 4'490 4.487 4'486 4'482 4'479

4'400 4.283 4.229 4'201 4'184 4.165 4'155 4'135 4'i2i

4'350 4'084 3'945 3.862 3.809 3'745 3.709 3. 64° 3'592

5 6 7 8 I0

12 24

4. 604 4.604 4.604 4'604

4'604

If t1and t2 are two independent random variables distributed as t with v1, v2 degrees of freedom respectively, the random variable d = t1sin 0- t2 cos 0 has Behrens' distribution with parameters V1, V2 and 0. The function tabulated in Table it is dp = dP(vi, v2, 0) such that Pr (d

Pi

3'993

3-694 3'504 3'373 3.206 3'104 2.876 2•685

P2. When v, < V2 use the result that

0). dp(Vi, v2, 0) = dp(v2, V1, 90° Behrens' distribution is symmetric about zero, so Pr (1d1

> dp) = P/ioo

> dp) = 213 1 ioo.

Notice that in this table 0 is measured in degrees rather than radians.

for P = 2.5 and 0.5 and a range of values of vi and v2 with

48

TABLE 11(b). 0.5 PER CENT POINTS OF BEHRENS' DISTRIBUTION 0 V2 = 5

vl

= 5

6 7 8 10 12 24 00

=6 7 8 10 12

v2 = 6

24

00 V2 =7

v, = 7 8 xo 12

24 c0 v2 = 8

v1 = 8 10 12

24 00

o°

xs°

30°

450

6o°

750

900

4.032 4.032 4'032 4.032 4.032 4.032 4.032 4'032

3'968 3'957 3'952 3'949 3'945 3'943 3.938 3'934

3.856 3'794 3'760 3'739 3'715 3 '702 3'677 3'658

3.809 3.663 3'575 3'518 3'447 3'407 3'325 3'266

3.856 3'622 3'476 3'378 3'253 3. I 78 3'016 2.886

3.968 3.666 3'474 3'342 3.173 3.069 2.840 2.646

4.032 3.707 3'499 3'355 3.169 3.055 2'797 2.576

3'707 3'707 3'707 3.707 3.707 3.707 3'707

3'654 3'648 3'644 3'639 3'637 3'631 3'627

3'556 3'519 3'496 3'468 3'453 3.424 3.402

3'514 3'423 3'363 3.289 3'246 3'158 3.093

3'556 3'408 3'308 3.180 3'104 2.938 2'804

3'654 3-461 3'328 3.158 3'053 2.822 2.627

3'707 3'499 3'355 3.169 3'055 2.797 v576

3'499 3'499 3'499 3'499 3'499 3'499

3'454 3'450 3'445 3'442 3'436 3'431

3'369 3'344 3'314 3'298 3.265 3'241

3'331 3'269 3'193 3'149 3.056 2'987

3'369 3.267 3'138 3'060 2.892 2'755

3'454 3.321 3.149 3'045 2.812 2•616

3'499 3'355 3.169 3'055 2.797 2.576

3'355 3.355 3.355 3'355 3.355

3'316 3.310 3.307 3'301 3.295

3'241 3.210 3.192 3.158 3.132

3.206 3.129 3.083 2.988 2.916

3.241 3.110 3-032 2.862 2.723

3.316 4 13 49 3 :0

3'355

3'169

3'138

3.169 3.169 3.169

3.135 3'127 3.121

3.078 3.059 3'021 2.993

3.049 3'002

00

3.055 3.055 3.055

3'029 3.021 3.015

2'978 2.939 2.909

V2 = 24 VI = 24 00

2'797 2.797

2'784 2.777

P2 = CO V1 = 00

2 '576

2.576

V2 = 10

V1 = 10 12 24

00 V2= 12

vl = 12 24

3.078

3.138 3'033

2'904

2'998 2'825

2.828

2.684

2'798 2.600

3.169 3-055 2'797 2.576

2'954 2'853 2.775

2'978 2.803 2.661

3'029 2.794 2.595

3'055 2'797 2.576

2'759 2726

2 '747 2.664

2'759

2'784

2'797

2'613

2'584

2'576

2.576

2.576

2'576

2'576

2'576

If d > dp the confidence level associated with illt-C. /12 is less than P per cent, and if d < dp the confidence level associated with 121 11 2 is less than P per cent. (See H. Cramer, Mathematical Methods of Statistics, Princeton University Press (2946), Princeton, N.J., pp. 520-523.) Also, the values of -#2 such that I(x1- x2) - (01-#2)1 5 rdp provide a 200-2P per cent Bayesian credibility interval for it, -#2.

This distribution arises in investigating the difference between the means ltil #2 of two normal distributions without assuming, as does the t-statistic, that the variances are equal. Let oil, x2 be the means and 4, 4 the variances of two independent samples of sizes n1, n2 from normal distributions, / S2 •

-

,

let = - V2 = 712 -1 and 0 =

(071 N/V2)

S2 S 2

measured in degrees. Define r = ji- +1and d vi V2

2.806 2.608

3'169

3.055 2.797 2.576

U being -X2 r

49

TABLE 12(a). 10 PER CENT POINTS OF THE F-DISTRIBUTION The function tabulated is F(P) = F(Plv,, v2) defined by the equation

P zoo

rco

ITO “JTI)

Fip.-

r(i-v2) vl v2b2 F(p)(v2 +P,FP( P'+"')c1F'

for P = io, 5, 2.5, I, 0.5 and 0•1. The lower percentage points, that is the values F'(P) = F'(PIP1, P2) such that the probability that F < F'(P) is equal to Pim°, may be found by the formula

F'(P IPi,

VI =

I

P2) = I/F(Plv2,

2

3

(This shape applies only when vl > 3. When v1< 3 the mode is at the origin.)

4

8

xo

12

24

CO

58.91 9'349 5.266 3'979

59.44 9'367 5.252 3'955

60.19 9.392 5.230 3'920

60.71 9.408 5.216 3.896

62.00 9.450 5.176 3.831

63'33 9.491 5'134 3.761

5

6

7

39.86 8.526 5'538 4'545

49.50 9.000 5'462 4'325

53'59 9.162 5'391 4.191

55'83 9.243 5'343 4.107

57.24 9'293 5'309 4.051

58.20 9'326 5'285 4.010

4'060 3.776 3'589 3.458

3.78o 3.463 3'257 3.113

3.619 3.289 3.074

3.520 3.181 2.961

3'453 3.108 2.883

2'924

2'806

2.726

3'006

2.813

•693

2.611

3'368 3'014 2.785 2.624 2.505

3'339 2'983 2.752 2.589 2.469

3'297 2.937 2.703 2.538 2.416

3.268 2.905 2.668 2.5o2 2'379

3.191 2.818 2.575 2.404 2'277

3.105 2.722 2.471 2'293

3'360

3'405 3.055 2.827 2•668 2.551

3.285 3.225 3.177 3.136 3.102

2.924

2728 2.66o

2•414

2'377 2.304

2.323 2.248

2.284

2'245

2560 2.522

2.195

2'188 2'138

2'209 2'147 2'097

2'178 2'100 2'036 1.983

2 '055

2'451 2'394 v347 2.307

2'461 2'389

2.763 2.726

2.605 2•536 2.480 2.434 2.395

2 '154

2.095

2.0 54

P938

P904 1.846 P797

3'073 3.048 3'026 3.007

2'490

V361

2'273

2'208

2'158

2'119

2'059

2'017

2.462 2'437 2.416 2.397

2.333 2.308 2.286

2.244 2.218

2.178 2.152

2.128 2.102

2'990

2 '695 2.668 2.645 2.624 2.606

2'266

2'196 2'176

2'130 2'109

2.058

2.088 2.061 2.038 2.017

2.028 voox 1.977 P956

P985 P958 P933 I•912

P899 P866 P836 P810 1.787

P755 P718 P686 P657 P631

2.589 2.575 2.561 2'549

2.380 v365

2'249

2'158

2'091

2'040

2.233

2'142

2'075

2'351

2'219

2'128

2'060

23

2.975 2.961 2.949 2.937

2'339

2.207

2.115

2.047

24

2'927

2'538

2'327

V195

2'103

2'035

2.023 2.008 P995 P983

P999 P982 1.967 P953 1.941

P937 P920 P904 1.890 1.877

1.892 P875 P859 1.845 P832

P767 1.748 1.731 1.716 P702

P607 P586 P567 P549 1'533

25 26 27 28 29

2.918

2.528

2'317

2'184

2'092

2'024

2'909 2'901

2'519 2'511

2.307

2.174

vo82.

2.014

2'073

2'005

P952

2.503 2.495

2'299 2'291

2'165

v894 2.887

V I 57

2.283

2'149

2'064 2.057

P996 P988

P943 P935

1.929 P919 P909 p900 x.892

P866 P855 1.845 P836 P827

1•820 p809 P799 P790 1.781

P689 P677 1•666 1.656 P647

1.518 P504 1.491 P478 P467

30

2'881

P980 P967

P819

P773

1.638

P913

P870

1'805

P758

P622

34 36 38

2'859 2.850

2'466

2'252

V049 2'036 2'024

1.884

2.263

2.456

2.243

2'142 2'129 2'118 2'108

P927

v869

2.489 2.477

2'276

32

v014

P6o8 P595

2'234

2'099

2'005

1.858 P847 1.838

P745 P734

2'448

P901 P891 1•881

P793 1.781

2'842

P955 P945 P935

P772

P724

P584

1.456 P437 P419 1404 1.390

40 60

2.835 2/91

2'440 V393

2'226

2'347

P722

1'652

p574 1.511 1.447

P377 P291 1 '193

V303

1.873 P819 P767 1.717

P715 P657 I .601

2/06

P927 P875 1.824 P774

P763 1.707

2.748

P997 P946 1.896 1.847

1.829 p775

120 00

2.091 2.041 1.992 1.945

p670

P599

P546

P383

P000

P2 = I 2

3 4 5 6 7 8 9 10 II 12

13 14 15 16 17

18 19 20 21 22

2'86o 2'807

2'606

v177 2' I 3o 2.084

2'522

2.331 2.283 2.243

50

2'342 2'283

2.234 2 ' 193

2'079

P971 P961

2 '159

P972

TABLE 12(b). 5 PER CENT POINTS OF THE F-DISTRIBUTION

--1/-?X

If F = X

112

, where X1and X2 are independent random

variables distributed as X2 with v1and v2 degrees of freedom respectively, then the probabilities that F F(P) and that F F'(P) are both equal to Pilot). Linear interpolation in v1and v2 will generally be sufficiently accurate except when either v1> 1z or v2 > 40, when harmonic interpolation should be used. 0

F(P)

(This shape applies only when v1>. 3. When v, < 3 the mode is at the origin.) I

2

3

4

5

6

7

8

xo

12

24

CO

V2 = I

161 4

199.5

215.7

224.6

230.2

234•0

2

18'51 10'13

19'00

19.16

19.25

19-30

19.33

9.277 6.591

9.117 6.388

9.013 6•256

8'941

24P9 19.40 8.786 5.964

249'1 19.45 8.639

19.50 8.526

6.163

238.9 19'37 8.845 6.041

243'9 19'41

9'552 6.944

236.8 19'35 8.887 6.094

5'774

5.628 4'365 3.669 3.230 2.928 2.707

Ili =

3 4

7'709

8.745 5'912

254'3

5

6.6o8

5.786

5'409

5'192

5.050

4'950

4.876

4.818

4'735

4'678

4'527

6 7 8 9

5.987 5'591 5.318

5'143

4'757

4'534

4'387

4'284

4.207

4'147

4.06o

4.000

3.841

5.117

4'737 4'459 4'25 6

4'347 4.066 3.863

4'120 3.838 3.633

3'972 3.687 3'482

3.866 3.581 3'374

3.787 3.500 3.293

3.726 3'438 3.230

3.637 3'347 3.137

3'575 3.284 3.073

3.410 3.115 2.900

4.965 4.844 4'747 4.667 4.600

4.103 3.982 3.885 3.806 3'739

3.708 3.587 3'490 3.411 3'344

3'478 3'357 3.259 3.179 3.112

3.326 3.204 3.106 3.025 2.958

3.217 3.095 2.996 2.915 2.848

3.135 3.012 2.913 2.832 2.764

3.072 2.948 2.849 2.767 2.699

2.978 2.854 2.753 2.671 2.602

2.913 2.788 2.687 2.604 2.534

2 '737 2.609 2.505 2.42o 2.349

2.538

4'543 4'494 4'451 4'414 4.381

3.682 3.634 3'592 3'555 3.522

3.287 3'239 3.197 3.160 3.127

3.056 3.007 2.965 2.928 2.895

2.901 2.852 2.810 2.773 2.740

2.790 2'741 2.699 2.661 2.628

2.707 2.657 2.614 2.577 2.544

2.641 2.591 2.548 2.510 2'477

2.544

2.475 2.425 2.381

2.288 2.235

2.066

4.351 4'325 4'301 4'279

3'493 3'467 3'443 3.422

2'711 2.685 2.661 2.64o 2.621

2.514 2.488

2.420

3'403

2.866 2.84o 2817 2.796 2/76

2.599 2.573 2.549

4.26o

3.098 3.072 3.049 3.028 3.009

4'242 4.225

2.991 2.975 2.96o 2.947 2.934

2/59 2 '743

2'728

2 '572

2 '459

2.714

4.183

3.385 3.369 3'354 3'340 3.328

2.558 2.545

30

4.171

3.316

2'922

32 34

4'149

3.295

2.901

2.690 2.668

4'130 4.113 4.098

3'276

2'883

2'650

2.534 2.512 2'494

2'399 2.380

3'259 3.245

v866 2.852

2.634 2.619

2463

2.364 2'349

3.232 3.I50 3.072 2.996

2.839 2/58 2.68o 2.605

2.606 2.525

2'449

2'336

2.249

2.18o

2368

2'447

2'290

2. 254 2'175

2'372

2.214

2.099

2'167 2'087 2'010

2'097 2'016 P938

I0

II 12

13 14 15

16 17

18 19 20 21 22 23 24

25 26 27 28 29

36

38

4'210 4.196

40

4•o85

6o

4.001 3.920 3.841

120

00

2/01

2 '494

2.45o 2.412 2-378

2 '404

2.296 2.2o6 2'131

2'010 1.960

2 '342

2'190 2.150

2.308

2.114

2.278 2.250 2.226 2.204

1'843

P917 p878

2'397

2'528

2'464 2 '442

2. 375

2'348 2.321 2'297 2.275

2.508

2.423

2.355

2'255

2'183

2•082 v054 2.028 2.005 P984

2'603

2'490

2'474

2.405 2.388 2.373

2.337 2.321 2.305

2.236

2.587

2•220

2'165 2'148

P964 1•946

1•71 I 1'691

2.204

2.132

1.930

1.672

2'445 2.432

2 '359

2.291

2.346

2.278

2'190 2'177

2'118 2.104

1.915 P901

1.654 1'638

2.421

2.334 2313

2'165 2'142 2'123 2'106 2'091

2'092 2.070 2'050 2.033 2'017

I .887 1.864 P843 P824 1'808

2.077 I '993

2'003

P910 1'831'

P834 P752

P793 x.700 I .608

P509 P389 1.254

1.517

1•000

2 '477

51

2.447

2.266

2. 294

2'244 2'225

2.277

2.209

2'262

2'194

I•917

1 .812

P783 P757 P733

1'622

1.594 P569 1'547 I .527

TABLE 12(c). 2.5 PER CENT POINTS OF THE F-DISTRIBUTION The function tabulated is F(P) = the equation

P

r(iPi+ i-v2)

100

F( v1)r(iv2)

V2iv2

v2) defined by

F(p)(P2+

viflttpx+vo dF,

for P = 1o, 5, 2.5, I, 0.5 and o•I. The lower percentage points, that is the values F'(P) = FTP1v1, v,) such that the probability that F t F'(P) is equal to P/Ioo, may be found by the formula

0

F'(Plvi, v2) = IF(PIP2, PO.

v1=

F(P)

(This shape applies only when v1> 3. When v1 < 3 the mode is at the origin.)

5

6

7

8

ro

12

24

00

921'8

937'1 39'33 14'73 9.197

948.2 39'36 14.62 9'074

956.7 39'37 14'54 8.98o

968.6 39'40 14'42 8.844

976.7 39'41 14'34 8.751

997'2 39'46 14.12 8.511

1018 39'50 13'90 8'257

6.978 5.820

6.757 5.6o0 4'899 4'433

6.619

6'525

5.461

5'366

4'102

4'761 4'295 3'964

4'666 4' 200 3.868

6.278 5.117 4'415 3'947 3'614

6.015 4'849 4'142 3.670 3'333

3.365 3.173 3.019 2.893 2.789

3.080 2.883

2'701 2'625 2'560 2'503 2'452

v395 2.316 2.247 2.187 2-133

2'602 2'570 2'541

2'408 2'368 2'331 2'299 2'269

2'085 2'042 2'003

2.515 2'491 2'469 2'448 2'430

2 '242 2.217 2.195 2'174 2'154

I'906 1.878

2.412 2.38, 2.353 2.329 2.307

2'136 2'103 2'075 2'049 2'027

1.787 1.750 1.717 1.687 1.661

2.288

2'007 1'882 1'760 I'640

1'637 1'482 1'310 1'000

2

3

4

864.2 39.17 15.44 9'979

899.6 39'25 15'10 9'605

39'30 14' 88 9'364

V2 = i 2

647'8 38'51

3 4

12'22

799'5 39.0o 16.04 10•65

5 6 7 8 9

8.813 8.073 7'571 7.209

8'434 7.260 6'542 6.059 5.715

7'764 6.599 5.890 5'416 5.078

7.388 6.227 5.523 5.053 4'718

7.146 5.988 5-285 4'817 4'484

4'652 4'320

6.853 5.695 4'995 4'529 4'197

5'456 5.256 5.096 4'96 5 4'857

4'826 4'630 4'474 4'347 4'242

4%168 4'275 4'121 3'996 3.892

4'236 4'044 3.891 3-767 3.663

4'072 3.881 3.728 3 . 604 3.501

3'950 3'759 3.607 3'483 3.38o

3'855 3.664 3'512 3.388 3.285

3'717 3.526 3'374 3.250 3'147

3'621

13 14

6.937 6.724 6'554 6'414 6.298

Is

6'200

6.115 6'042

5.978 5.922

3.804 3'729 3.665 3.608 3'559

3'576 3.5oz 3'438 3.382 3'333

3'415 36 341 3.277

x8

4'153 4'077 4' 0 II 3'954 3.903

3.293 3.219 3.156 3.100 3.051

3'199 3.125 3.061 3.005 2.956

3.060 2.986

17

4'765 4'687 4'619 4'5 6o 4'508

2'963

z6

2'817

2'769 2'720

4%161 4'420 4'383 4'349 4'319

3'859 3.819 3.783 3'750 3.721

3.515 3'475 3'440 3'408 3'379

3.289

3.128 3.090 3.055 3.023 2.995

3.007 2 '969 2'934

2'913

2 '774 2 '735

2-637

2'902 2'874

2.874 2.839 2.808 2.779

3.129 3.105 3.083 3:063 3'044

2.753 2.729

2.613

2'923 2'903 2'884

2'802 2'782 2/63

2'707 2'687

2.568 2 '547

4'201

3'353 3.329 3'307 3.286 3.267

2'848 2.824

5.588

3'694 3.670 3'647 3.62,6 3.607

2.969 2 '945

5'610

4'291 4.265 4'242 4.221

2.669

2'529

4'182 4'149 4'1zo 4'094 4'071

3.589 3'557 3'529 3.505 3'483

3'250

3.026 2'995 2.968 2'944

2.867 2.836 2.808 2.785 2.763

2/46 2.715

2.651 2.620 2.593 2.569 v548

2'511 2'480

34 36 38

5.568 5.531 5'499 5'471 5.446

2.453

40 6o

5'424 5.286

3'463 3'343

120 co

5'152 5'024

4'05I 3'925 3.805 3.689

2'529 2'412 2'299 2'192

v388 2.27o 2 'I57 2.048

zo zz

12

zo 20 21 22 23

24 25 26 27

28 29 30 32

17'44

5.871 5'827

5.786 5-750 5'717 5.686 5.659 5'633

3'227 3'116

3'218 3'191

3.167 3'145 3.126 3.008 2. 894 v786

3'250 3'215

3.183 3'155

2'923

2. 904 2.786 v674 2.567

5'119

3'221 3'172

2.688 2.664 v643

2 '744

2.624

2.627 2.515 2.408

2.507 2.395 2.288

52

2'922

2.866

2'700

2.668 2•64o

2'590

2%429 2 407

3'430 3.277 3'153 3.050

2.889 2.825

2'676

2'169 2'055

1.945

2'725

2.595 2.487

r968 1'935

1.853 1'829 1'807

TABLE 12(d). 1 PER CENT POINTS OF THE F-DISTRIBUTION --2, where X1and X2 are independent random Xpi11X F=V2

If

variables distributed as X2 with v1and v2 degrees of freedom respectively, then the probabilities that F F(P) and that F < F'(P) are both equal to Nioo. Linear interpolation in vi or v2 will generally be sufficiently accurate except when either v1 > 12 or v2 > 4o, when harmonic interpolation should be used. 0

F(P)

(This shape applies only when v1 at the origin.) vl =

I

v2 = I 2

3 4 5 6 7 8 9 xo II 12

2

3

4

5

6

3. When v1< 3 the mode is

7

8

to

12

24

00

5928 99'36 27.67 14.98

5981 99'37 27.49 14'80

6056 99'40 27.23 14'55

6,o6 99'42 27.05 4'37

6235 99'46 26.6o 13'93

6366 99'50 26.13 13'46 9'020 6.88o

4052 98'50 34- 12 21.20

4999 99'00 30.82 18.00

5403 99'17 29.46 16.69

5625 99'25 28.71 15.98

5764 99'30 28.24 15'52

16.26 13.75

13.27 10.92

12.06 9.780

I1•39 9.148

10.97

10.67

10.46

10•29

8.746

8.466

8.26o

8.102

10.05 7'874

9.888 7718

9.466 7313

12.25 11.26

9'547 8'649

8'451 7.591

7'847 7.006

7'460 6.632

7.191 6.371

6.993 6.178

6.840 6.029

6.620 5'814

6.469 5'667

6.074 5'279

10.56

8'022

6'992

6'422

6'057

5'802

5.613

5 .467

5'257

5'111

4.729

4'859 4.31 1

10'04 9.646

6'552 6.217 5'953 5'739 5 .564

5'994 5 .668 5'412 5'205 5'035

5'636

5.386

5.200

5.316 5.064 4'862 4'695

5.069 4.821 4'620 4'456

4.886 4'640 4'441 4'278

5.057 4'744 4'499 4'302 4'140

4'849 4'539 4'296 4'100 3'939

4'706 4'397 4'155 3 .960 3'800

4'327 4'021 3'780 3'587

3.909 3.602 3.361 3 .165

3'427

3'004

3'434

3'666 3'553 3'455 3.371 3'297

3'294 3'181 3'084 2.999 2'925

2.868 2.753 2 '653 2.566 2.489

5859 99'33 27.91 15.21

5.650

13

9'074

7'559 7.206 6'927 6'701

14

8.862

6.515

15 16 17 18 19

8.683

6'359

5'417

4'893

4'556

4'318

4'142

4'004

3805

8.531 8.400 8.285 8.185

6.226

4'773 4'669 4'579 4.500

4'437 4'336 4'248 4'171

4'202 4'102 3'939

4.026 3'927 3-841 3'765

3.890 3'791 3.705 3'631

3'691 3'593

6.013 5.926

5'292 5'185 5'092 5010

20

8.096 8.017 7'945 7.881 7.823

5'849 5'780 5'719 5.664 5.614

4'938 4'874 4'817 4'765 4.718

4'431 4'369 4'313 4'264 4.218

4.103 4'042 3'988 3'939 3.895

3.871 3.812 3'758 3'710 3'667

3'699 3.640 3'587 3'539 3'496

3.564 3.506 3'453 3'406 3'363

3.368 3'310 3'258 3.211

3.231 3'173 3.121 3.074

2.859 2.8o1 2.749 2.702

2.421 2.360 2.305 2.256

3'168

3.032

2.659

2.211

7'770 7'721 7.677 7.636 7'598

5'568 5'526 5.488 5'453 5'420

4'675 4'637 4'601 4'568 4.538

4'177 4'140 4'106 4'074 4'045

3'855 3.818 3'785 3'754 3'725

3'627 3'591 3'558 3'528 3'499

3'457 3'421 3.388 3'358 3'330

3'324 3.288 3.256 3.226 3.198

3'129 3.094 3062 3.032 3.005

2'993 2.958 2.926 2.896 2-868

2.62o 2.585 2.552 2'522 2.495

2.169 2.131 2 '097 2 ' 064 2 '034

5.390 5'336 5'289 5'248 5'211

4'510

4.018

3'699

4'459 4'416 4'377 4'343

3'969 3'927 3'890 3'858

3'652 3'611 3'574 3'542

3'473 3'427 3 .386 3'351 3'319

3'304 3.258 3.218 3.183 3.152

3'173 3.127 3.087 3'052 3.021

2'979 2.934 2.894 2.859 2.828

2'843 2.798 2'758 2.723 2.692

2.469 2.423 2.383 2'347 2.316

2.006

34 36 38

7.562 7'499 7'444 7.396 7'353

1.911 1.872 1'837

40 6o

7'314 7077

5'179 4'977 4'787 4.605

4'313 4'126 3'949 3.782

3'828 3'649 3'480 3.319

3'514 3'339 3'174 3.017

3.291 3'119 2.956 2.802

3.124 2'953 2.792 2.639

2'993 2.823 2.663 2.511

2.801 2.632 2.472 2.321

2.665 2.496 2.336 2.185

2.288 2.115 1•950 1.791

1.8o5 1.601 1.381 1.000

21 22 23 24 25 26 27

28 29 30 32

9.330

120

6'851

00

6.635

4.015

53

3.508

1'956

TABLE 12(e). 0.5 PER CENT POINTS OF THE F-DISTRIBUTION The function tabulated is F(P) = F(Plv,, v2) defined by the equation

P

ravi+ -PO v

Ioo r(iv,) ra v2) 1

v iv2 _ co z

fF(p) (12+ viF)1(P.+92) dF,

for P = 1o, 5, 2.5, I, 0•5 and o•i. The lower percentage points, that is the values F'(P) = F'(Plvi, v2) such that the probability that F < F'(P) is equal to P/Ioo, may be found by the formula

F'(Plvi, v2) = i/F(Plv2, v1)•

Vi. =

I

2

3

4

1/2 = I

20000 199'0

21615

22500

2

16211 198.5

199'2

199'2

3 4

55'55 31•33

49'80 26.28

47'47 24.26

46'19 23.15

5 6 7 8 9

22.78 16.24 14.69 13.61

18.31 14.54 12.40 11'04 IO II

16.53 12'92 10•88 9'596 8.7i7

12.83 12.23 I r 75 1 P 37 ii•o6

9.427 8.912 8.510 8.186 7.922

15 i6 17 18 19

10•80 1o•58 10•38

20

(This shape applies only when vl at the origin.) 7

8

zo

12

24

(X)

23715 199'4 44'43 2 I .62

24224 43'69 20'97

24426 199'4 43'39 20.70

24940 199'5 42'62 20.03

25464

22'46

23437 199'3 44'84 2 1 •97

23925 199'4 44'13 2r35

15'56 12.03 1o•o5 8.805 7956

14'94 11.46 9'522 8.3oz 7'471

14'51 11.07 9'155 7952 7134

14'20 10.79 8.885 7694 6.885

13.96 /0'57 8.678 7496 6.693

13.62 10.25 8.38o 7.211 6.417

13.38 10'03 8.176 7.015 6.227

12.78 9'474 7-645 6'503 5/29

12 '14 8'879

8.081 7.600 7226 6.926 6.68o

7'343 6.88 6.521 6.233 5'998

6'872 6.422 6.071 5'791 5'562

6'545 6.102 5'757 5'482 5'257

6.3o2 5.865 5'525 5'253 5'031

6.116 5.682 5'345 5'076 4'857

5.847 5.418 5'085 4'820 4'603

5.661 5.236 4'906 4'643 4'428

5.173 4-756 4'431 4'173 3.961

3'904 3'647 3.436

7701 7.514 7354

6 '476 6.303 6'156

5'803 5.638 5'497

10'22

7215

6'028

5'375

10.07

7'093

5'916

5.268

5'372 5.212 5'075 4'956 4'853

5'071 4'913 4'779 4'663 4'561

4'847 4'692 4'559 4'445 4'345

4'674 4'521 4'389 4'276 4'177

4'424 4'272 4'142 4.030 3'933

4'250 4'099 3'971 3.860 3'763

3/86 3.638 3.5 3'402 3'306

3.260 3•112 2 '984 2'873 2.776

6.986 6.891 6.806 6/30 6.661

5.818 5.730 5.652 5'582 5.519

5'174 5.091 5.017 4'950 4.890

4'762 4.681 4.609 4'544 4'486

4'472 4'393 4.322 4'259 4'202

4.257 4'179 4'109 4'047 3.991

4.090 4'013 3'944 3'882 3.826

3.847 3'771 3'703 3'642 3.587

3.678 3.602 3'535 3'475 3.420

3.222 3.147 3'081 3'021 2.967

2'690

23 24

9.944 9.830 9.727 9.635 9'551

2.428

25

9.475

26 27

9.406

5.462 5'409 5.361 5.317

4'835 4/85 4.740 4.698 4-659

4'433 4'384 4.340 4.300 4.262

4'150 4'103 4.059 4.020 3'983

3'939 3'893 3.850 3.811 3'775

3'776 3'730 3.687 3.649 3'613

3'537 3'492 3'450 3'412 3'377

3'370 3'325 3.284 3'246 3.211

2.918 2'873 v832

2.377 2'330 2.287

2 '794

2.247

2.759

2.210

4'228 4'166

3'949 3'889

3'742 3'682

4'112

3'836

3'630

4'065 4'023

3'790 3'749

3'585 3'545

3'580 3.521 3.470 3.425 3'385

3'344 3.286 3.235 3.191 3.152

3'179 3.121 3.071 3.027 2.988

2.727 2.670 2.620 2.576 2.537

2.176 2'114 2.060 2•013 1.970

3'986 3/60 3'548 3'350

3'713 3.492 3'285 3'091

3'509 3.291 3'087 2.897

3'350 3.134 2 '933 2.744

3.117 2 '904 2 '705 2.519

2'953 2'742 2 '544

2.502

1.932 r689 1.431 I•000

io II 12

13 1 4

2/ 22

18.63

z8

9'342 9.284

6'598 6'541 6'489 6•44o

29

9'230

6'396

5'276

30

9.180

6 '355

5'239

32

9'090

6'281

5'171

34 36 38

0.012 8.943 8.882

6.217 6.161 6•

5.113 5.062 5•o16

4'623 4'559 4'504 4'455 4.412

8'828

6.066 5'795 5'539 5.298

4'976 4'729 4'497 4.279

4'374 4'140 3'921 3'715

40 6o 120 00

8'495

8'179 7.879

,5 23056 199'3 45'39

6

3. When v1< 3 the mode is

54

199'4

2'358

2'290

2.089 1.898

199'5

41-83 19.32

7.076 5-951 5.188 4'639 4'226

2.614

v545 2 '484

TABLE 12(f). 0.1 PER CENT POINTS OF THE F-DISTRIBUTION If

IX , F= X v, V2 -2

where X1and .7C2 are independent random

variables distributed as X' with v1and v2 degrees of freedom respectively, then the probabilities that F F(P) and that F < F'(P) are both equal to Piro°. Linear interpolation in v1or v2 will generally be sufficiently accurate except when either vl > 12 or v2 > 4o, when harmonic interpolation should be used. (This shape applies only when V1 at the origin.) vl =

I

v2 = I * 2

3 4 5 6 7 8 9

2

3

4053 998'5 167•0 74'14

5000 999'0 148.5 61.25

5404 999'2 141.1 56.18

47'18 35.51

37'12 27.00

33.2o 23.70

4

5

5625 999'2 137.1 53'44

5764 999'3 134.6 51'71

6 5859 999'3 132.8 50'53

3. When v, < 3 the mode is

7

8

I0

12

24

a)

5929 999'4 131•6 49'66

5981 999'4 130.6 49'00

6056 999'4 129.2 48'05

6107 999'4 128.3 47'41

6235 999'5 125.9 45'77

6366 999'5 123'5 44.05

26.92 18.41

26.42 17.99 13.71 11•19 9'570

25.13 16.90 12.73 10.30 8.724

23'79 15'75 11.70 9'334 7-813

8'445

7638 6.847 6 .249 5'781 5'407

6.762 5'998 5'420 4'967 4'604

5'101

31.09

29.75

28.83

28.16

27.65

20•80 16'21

20'03 15'52

19'46 15'02

19'03

29'25

21'69

18'77

21'92 17'20

14'63

14'08

25.41 22.86

18'49 16.39

15'83 13.90

14'39 12.56

13.48 11.71

12.86 11.13

12.40 10.70

12.05 10.37

11.54 9'894

21.04 19.69 18.64 17.82 17.14

14.91 13.81 12.97 12.31 11.78

12.55 1r56 io•8o 10.21 9.729

11•28 10.35 9.633 9'073 8.622

10•48 9.578 8.892 8'354 7.922

9.926 9'047 8.379 7.856 7'436

9.517 8.655 8•ooi 7'489 7.077

9.204 8-355 7.710 7.206 6.8o2

8.754 7922 7292 6-799

6. 404

7626 7.005 6-519 6.130

11'34 10'97

9'335

10•66 10.39 10•16

8.727 8.487 8.280

8.253 7944 7.683 7'459 7.265

7022

18 19

16-59 16.x2 15.72 15.38 15.08

6.808 6.622

6•081 5.812 5'584 5'390 5'222

5.812 5'547 5'324 5.132 4'967

4'631 4'447 4' 288

4'307 4'059 3.85o 3.670 3'514

20 2x

14'82 14'59

9'953 9'772

22 23 24

14'38

9'612

14'20 14'03

9'469 9'339

8•098 7'938 7'796 7669 7'554

7.096 6'947 6.814 6.696 6.589

6'461 6.318 6.191 6•078 5'977

13.88 13'74 13.61 13.5o 13.39

9.223 9'116 9.019 8.931 8'849

7'451 7'357 7.272 7.193 7.121

6'493 6.406 6.326 6.253 6.186

5.885 5.802 5.726 5.656 5'593

13.29 13.12

8.773 8.639

6.125 6•014 5.919 5.836 5.763 5.698 5.307 4'947 4.617

4'757 4'416 4'103

10 II 12

13 14 15 16 17

25 26 27

28 29

9'006

34 36 38

12'97

8'522

I2•83

8.420 8.331

7.054 6.936 6.833 6.744 6.665

40

6o

12.6i 11.97

120

11 . 38

00

1o•83

8.251 7.768 7321 6.908

6.595 6.171 5.781 5.422

3o 32

1211

7'567

7092

6.741

6.471

7.272

6.805

6.460

6.195

6'562

6'223

6.355 6.175

6.021 5'845

5.962 5.763 5'590

6•019 5.881 5'758 5'649 5'550

5.692 5'557 5'438 5'331 5-235

5.308 5.190 5.085 4'991

5.075 4'946 4'832 4'730 4'638

4'823 4'696 4'583 4'483 4'393

4'149 4'027 3.919 3.822 3'735

3'378 3.257 3.151 3.055 2.969

5'462 5.381 5.308 5'241 5.179

5'148 5.070 4'998 4'933 4'873

4'906 4'829 4'759 4'695 4'636

4'555 4480 4'412 4'349 4'292

4'312 4.2 38 4'171 4'109 4'053

3.657 3.586 3.521 3'462 3'407

2'890 2'819

5'534 5'429 5'339 5.26o 5.190

5.122 5.021 4'934 4'857 4'790

4.817

4'719 4'633 4'559 4'494

4.581 4'485 4'401 4'328 4-264

4'239 4'145 4.063 3'992 3'930

4.001 3.908 3.828 3'758 3.697

3'357 3.268 3.191 3.123 3'064

2.589 2498 2 '419 2'349 2.288

5•128

4'731 4'372 4'044 3'743

4'436 4' 086 3-767 3'475

4'207 3.865 3'552 3.266

3'874 3'541 3'237 2.959

3.642 3.315

3'011 2'694 2'402 2'132

v233

5.44o

* Entries in the row v2 = I must be multiplied by ioo.

55

3'016 2'742

4.846

2'754 2.695 2'640

I .890

1.543 I•000

TABLE 13. PERCENTAGE POINTS OF THE CORRELATION COEFFICIENT r WHEN p = 0 The function tabulated is r(P) = r(P1v) defined by the equation 2 1)

-

\ITT

1

rp)fr(P)

(1 - r2) 2 dr = P

2

Let r be a partial correlation coefficient, after s variables have been eliminated, in a sample of size n from a multivariate normal population with corresponding true partial correlation coefficient p = o, and let v = n - s. This table gives upper P per cent points of r; the corresponding lower P per cent points are given by - r(P), and the tabulated values are also upper 2P per cent points of For s = o we have v = n and r is the ordinary correlation coefficient. When v > 130 use the results that r is approximately normally distributed with

-1

6 7 8 9 xo II 12 13 14

5

2'5

I

0'5

0.9877 '9000

0.9969 •9500

0 '9995

0 '9999

•9800

.9900

Tables of the distribution of r for various values of p are given by, for example, F. N. David, Tables of the Ordinates

and Probability Integral of the Distribution of the Correlation Coefficient in Small Samples, Cambridge University Press (1954), and R. E. Odeh, ' Critical values of the sample product-moment correlation coefficient in the bivariate normal distribution', Commun. Statist. - Simula Computa. II (I) (1982), pp. x-26. The z-transformation may also be used (cf. Tables 16 and 17).

0 '9343

0 '9587

0 '9859

•8822 '8329 •7887 7498

•9172 . 8745 . 8343 '7977

'9633 '9350 '9049 -8751

0'5494 •5214 '4973 '4762 '4575

0.6319 •6021 '5760 '5529 '5324

0.7155 •6851 -6581 '6339 •6120

0.7646 '7348 -7079 '6835 •6614

0.8467

0'4409

0'5140

'4259 '4124 '4000 •3887

'4973 '4821 '4683 '4555

20 21 22 23

0.3783 .3687 •3598 '3515 '3438

0.4438 '4329 '422 7 '4132 '4044

'

0.641 I •6226 •6055 '5897 '5751

0.7301 '7114 '6940 '6777 •6624

0 '5155

0.5614 '5487 •5368 •5256 •5151

0.6481 '6346 •62,19 '6099 .5986

'5034 '4921 '4815 '4716

0 '3365

'3297 •3233 •3172 •3115

0.3961 •3882 •3809 '3739 .3673

0.4622 '4534 '4451 '4372 '4297

30 31 32 33 34

0.3061 •3009 -2960 '2913 •2869

0.3610 '3550 •3494 '3440 •3388

04226

'4158 '4093 '4032 '3972

0.5052 '4958 '4869 '4785 '4705

0-5879 '5776 '5679 '5587 '5499

0.4629 '4556 '4487 '4421 '4357

0 '5415

0.2826

0'3338

0'3916

0 '4296

785

'3862 •3810

'4238

'2709

•3291 •3246 •3202

' 2673

'3160

' 2

'2746

'8199

'7950 -7717 •7501

0.5923 '5742 '5577 '5425 '5285

25 26 27 28 29

35 36 37 38 39

'9980

0.8783 •8114 '7545 .7067 •6664

15 16 17 18

24

0 '999995

o.8054 '7293 •6694 •6215 .5822

'3760 '3712

'4182 '4128 '4076

1

U-shaped.)

zero mean and variance -- (cf. Tables 16 and 17). v-3

P

r(P)

(This shape applies for v > 5 only. When v = 4 the distribution is uniform and when v = 3 the probability density function is

zero mean and variance--, or (more accurately) that v -1 z = tanh-1r is approximately normally distributed with

v --- 3 4 5

0

P v = 40

5

2'5

0'2638

0'3120

0'3665

42

'2573

'3044

44 46 48

'2512

'2455 '2403

-2973 ' 2907 - 2845

'3578 •3496

5c1 52 54 56 58

0'2353 '2306 '2262 '2221 •218I

0'2787

6o 62 64 66 68

0'2144 •2I08 ' 2075 '2042 '2012

0'2542

70 72 74 76 78

0.1982 •1954 1927 .1901 •1876

0.2352 '2319 •2287 '2257

80

0.1852 '1829 '1807 •1786 '1765

0'2199 '2172

90 92 94 96 98

0 '1745

•1726 1707 -1689 •1671

100

105 II0 115 120 125 130

0'1478 '1449

82 84 86 88

'5334 '5257 '5184 -5113 0'5045 '4979 '4916 '4856 '4797

56

0%5

O•I 0 '4741

'3420 '3348 0.3281 •3218 '3158 •3102 •3048

0.4026 '3932 '3843 •3761 •3683 0.3610 •3542 '3477 '3415 '3357

0.4267 '4188 '4114 '4043 '3976

0.2997 '2948

0.3301 •3248

0'3912

'2902

'3198

.2858 •2816

.3150 •3104

•3850 '3792 '3736 •3683

0.2776 '2737 '2700 •2664 •2630

0.3060 •3017 '2977 '2938 •2900

0.3632 •3583 '3536 '3490 '3447

0 '2597

0.2864 •2830 '2796 .2764 '2732

0'3405

•2565 '2535 .2505 '2477

0.2072 •2050 •2028 •2006 .1986

0.2449

0'2 702

0' 3215

'2422 '2396 '2371 ' 2 347

' 2673

- 2645

'3181 '3148

•2617 '2591

•3116 •3085

0. 1 654

0.1966 '1918

1576

. 1874

. 1541

•1832 '1793

0.2324 -2268 -2216 •2167

0.2565 '2504 •2446 '2393

0- 3054

- 1614

•2I22

'2343

0'2079 -2039

0.2296

1509

'2732 •2681 '2632 •2586 '2500

' 2461 '2423 '2387

'2227

' 2146 '2120 '2096

0.1757 •1723

'2252

'4633 '4533 '4439 '4351

'3364 '3325 •3287 '3251

'2983 '2915 '2853 '2794

0.2738 •2686

TABLE 14. PERCENTAGE POINTS OF SPEARMAN'S S TABLE 15. PERCENTAGE POINTS OF KENDALL'S K Spearman's S and Kendall's K are both used to measure the degree of association between two rankings of n objects. Let di (1 5 i n) be the difference in the ranks of the ith object;

-51-02(n+ 1)2(n— I) for S and ,72,1-n(n — t)(2n + 5) for K, and when n > 4o both statistics are approximately normally distributed; more accurately, the distribution function of X = [S n)]I[]n(n +1)' ” .17---1] -- is approximately equal to

Spearman's S is defined as E 4. To define Kendall's K, re-

Y

(130 (x)—

order the pairs of ranks so that the first set is in natural order from left to right, and let mi (1 5 i n) be the number of ranks greater than i in the second ranking which are to the

e

241/27T

SPEARMAN'S S 2'5

5

I

KENDALL'S K

0'5

0•I

5 6 7 8 9

2

0

6 16 3o 48

4

0 2

12 22

6

4

O

14

36

26

IO 20

4 Io

58 84 118 160

42 64 92 128 170

34 54 78 108 146

20

165

zo

34 52 76 104

220

II

286 364 455

222

194

140

616

284 354 436 53o

248 312 388 474

184 236 298 370

72 102

O

2'5

5

n=4

I0

—

P

i(n3—

0

II

—o.c:14(ign2+ 5n-36) *(n3—n)

and 110(x) is the normal distribution function (see Table 4). A test of the null hypothesis of independent rankings is provided by rejecting at the P per cent level if S x(P), or K x(P), when the alternative is contrary rankings. The other points are similarly used when the alternative is similar rankings. To cover both alternatives reject at the 2P per cent level if S, or K, lies in either tail. Spearman's rank correlation coefficient rsis defined as 1 — 6S/(n3 — n), and has upper and lower P per cent points I — 6x(P)/(n3— n) and — [1— 6x(P)/(n3— n)] respectively. Kendall's rank correlation coefficient ric is defined as 4K I[n(n — 1, and has upper and lower P per cent points 4x(P)/[n(n — 1)] r and I} respectively. — {4x(P)I[n(n—

right of rank i. Kendall's K is defined as E mi. i =1 For Table 14 the tabulated value x(P) is the lower percentage point, i.e. the largest value x such that, in independent rankings, Pr(S < x) P/ loo; in Table 15, K replaces S and the upper percentage point is given. A dash indicates that there is no value with the required property. The distributions are symmetric about means (n3— n) for S and in(n-1) for K, with maxima equal to twice the means; hence the upper percentage points of S are -i(n3 — n) — x(P) and the lower percentage points of K are in(n-1)— x(P). The variances are

P

(x3 — 3x), where y =

I0

n=4

20

5 6 7 8 9

I

0'5

9

I0

10

14

14

15

22

19 24 3o

20 25

27

18 23 28

12 13 14

33 39 46 53 62

34 41 48 56 64

56o 68o 816 969 1140

15 16 17 18 19

7o 79 89 99 II0

73 83 93 103 114

1330

121 133 146

126 138 151

159

164

172

178

144 157 171 185 216 232 248 266

120

5

31

26 33

7'5 I0.5 14 18

36 43 51 59 67

37 44 52 61 69

40 47 55 64 73

22.5 27.5 33 39 45.5

77 86 97 1(38

79 89 I00

83 94 105 117 129

6o 68 76-5

142 156 170 184 200

95 zos xxs.s 126.5 138

238 254 272

216 232 249 267 285

150 162.5 175'5 189 203

303

323 342 363 384

217.5 232.5 248 264 280.5 297.5 315 333 351'5 370.s 390

2I

I2

142

13 14

188 244

xs x6 17 18 19

388 478 58o 694

20 21 22 23 24

824 970 1132 1310 1508

736 868 to18 1182 1364

636 756 890 1040 1206

572 684 8o8

452 544 65o

1771

948 1102

768

2024

900

2300

20 21 22 23 24

25 26 27 28 29

1724

1566

1272 1460 1664 1888 2132

1584 1796

260o 2925 3276 3654 4060

25 26 27 28 29

186 201 216 232

2794

1784 2022 2282 2564

1388 1588 1806 2044 2304

It:48

1958

248

193 208 223 239 256

3o 31 32 33 34

3118 3466 3840 4240 4666

2866 3194 3544 3920 4322

2584 2884 3210 3558 3930

2396 2682 2988 3318 3672

2028 2280 2552 2844 316o

4495 496o 5456 5984 6545

3o 31 32 33 34

265 282 300 318 337

273 291 309 328 347

283 301 320 340 359

290 308 328 347 368

35 36 37 38 39

512o 5604 6118 666z 7238

4750 5206 5692 6206 675o

4330 4754 5206 5686 6196

4050 4454 4884 5342 5826

3498 3858 4244 4656

35 36 37 38 39

356 376 397 418 440

367 388 409 430 452

380 401

410

5092

7140 7770 8436 9139 988o

444 467

432 454 477

405 428 450 473 497

40

7846

7326

6736

6342

5556

10660

40

462

475

490

501

522

310

2214 2492

210

268 338 418 512

1210 1388

1540

57

n(n — I) 3

13

35 56 84

0•I

6

119 131

200

422

III 123 135 148

161 176 190 205 221

388

52.5

85.5

TABLE 16. THE z-TRANSFORMATION OF THE CORRELATION COEFFICIENT The function tabulated is

coefficient p, and let v = n-s. Then z is approximately normally distributed with mean tanh-1p+plz(v I) (or, less accurately, tanh-1p) and variance - 3). If s = o we have v = n and r is the ordinary correlation coefficient. For p = o the exact percentage points are given in Table 13.

z = tanh-1 r = loge (I

. If r < o use the negative of the value of z for -r. Let r be a partial correlation coefficient, after s variables have been eliminated, in a sample of size n from a multivariate normal population with the corresponding true partial correlation

-

I

r

z

r

z

r

z

r

z

r

z

0'00 •OI •02 '03

0'0000 '0 I 00 '0200

0.500 .5o5 •po '515 •52o

0.5493 .556o •5627 •5695 •5763

0'750

0.9730 0 '9845 0.9962 z•oo82 1.0203

0•9I0

•912 '9,4 •916 .918

1.5275 '5393 '5513 •5636 .5762

0.9700 •9705 •9710 '9715 •9720

2.0923 •xoo8 •1o95 •1183 .1273

09950 '995x '9952 '9953 '9954

'755 •760 .765 .770

r

z 2 '9945

3.0046 3'0149 3'0255

'04

•0300 •0400

0.05 o6 •07 •o8 •09

0.0500 •o6oi •0701 -o8o2 •0902

0'525

0'5832

0'775

.5901 '5971 •6042 -611 2

•78o .785 '790 '795

1.0327 .0454 .0583 •0714 . o849

0'920 '922 '924 '926 '928

1.5890 -6022 .6157 -6296 '6438

0.9725 .9730 '9735 '9740 '9745

2.1364 '1457 •1552 •1649 '1747

0'9955

•530 '535 '540 '545

0'I0 •II •I2

0'1003 •II04 •1206

0'550

'93, '932 '933 '934

1.6584 -6658 .6734 •681x •6888

0.9750 '9755 '9760 .9765 '9770

2'1847 •I950 - 2054 •2I6o

•1409

-56o .565 .57o

1.0986 •1127 •127o 1417 •1568

0'9960

1 307

o.800 .8o5 •810 .815 •820

0'930

.13 '14

0.6184 •6256 •6328 •64ox .6475

0.15 x6 '17 18 •19

0'1511 '1614 •1717 •1820 '1923

0.575 .58o .585 '590 '595

0. 6550

0'825 '830 '835

1'1723

0'935 .936 '937 .938 '939

1.6967 '7047 -7129 -7211 .7295

0'9775

2.2380 '2494 -2610 .2729

0.9965 •9966 '9967

•285I

•9969

'2027 •218I '2340

0'20 •2I '22 '23 '24

0'2027 •2I32 •2237

o.600 •6os .6 zo .615 •62o

0.6931 •70I0 '7089 •7169 •7250

o. 85o .852 '854 .856 -858

1•2562

0.940 '94x '942 '943 '944

1.738o '7467 '7555 .7645 '7736

0.9800 •9805 •98,o •9815 •9820

2'2976

0.9970 '9971 '9972 '9973 '9974

3.2504 .2674 . 2849 •3031 -3220

0'25 '26 '27

0 '2554

0•625 '630 '635 -640

0.7332 '7414 '7498 -7582 •7667

o 86o •862 '864 •866 •868

1-2933 •3011 •3089 •3169 '3249

0'945

r7828 '7923 •8019 •8117 -8216

0.9825 -983o .9835 •9840 - 9845

2.3650 '3796 '3946 '4101 '4261

0'9975

'946 '947 '948 '949

'9976 '9977 '9978 '9979

3'3417 •3621 '3834 '4057 -4290

0•870 '872

0.950 '95x .952 '953 '954

1.8318 •8421 •8527 -8635 - 8745

0.9850 .9855 •9860 •9865 •9870

2'4427

•876 .878

r333r '3414 '3498 •3583 •3670

'4597 '4774 '4957 '5147

0.9980 •9981 •9982 '9983 '9984

3'4534 '4790 •5061 '5347 •5650

2'5345 '5550 '5764 .5987 •6221

0.9985 -9986 •9987 '9988 .9989

3'5973 -6319 -6689 •7090 .7525

0.9990 '9991 '9992 '9993 '9994

3.8002 3'8529 3.9118 3.9786 4'0557 4'1469 .2585 '4024 •6o51 '9517

•28 .29

'2342 .2448 •2661 •2769 •2877 -2986

'555

'645

•6625 -6700 •6777 .6854

840 '845

•1881 '2044 •2212 •2384

•2634 •2707 •2782 .2857

•9780 .9785 '9790 '9795

-2269

•31o3 -3235 .3369 •3507

0.30 '31 .32 '33 '34

0 '3095

o.65o -655 -66o •665 •67o

0'7753

•3205 -3316 '3428 '3541

0.35 •36 '37 •38 '39

0. 3654 •3769 •3884 '4001 '4118

0.675 •68o .685 •690 .695

0.8199 •8291 -8385 -8480 -8576

o.88o •882 .884 •886 •888

1•3758 •3847 '3938 '4030 '4124

0'955 .956 .957 .958 '959

1.8857 -8972 •9090 .9210 '9333

0.9875 •9880

040

0.4236 '4356 '4477 '4599

0'700

0.890 .892

1.4219 '4316

1'9459

•720

.9588 •9721 '9857 '9996

0.9900 '9905 •9910 '9915 •9920

2'6467 - 6724 '6996 '7283

'4722

0.8673 -8772 •8872 '8973 •9076

0.4847 '4973 •510I •523o -5361

0'725

0.9181

•46 '47 '48 '49

'730 '735 '740 '745

0.50

0'5493

0'750

'41 '42 '43 '44 0'45

'

705

.7 zo

'7,5

'7840 .7928 -8017 -81o7

'874

'9885

-9890 •9895

'9956 '9957 '9958 '9959

'9961 •9962 '9963 '9964

'9968

'894

'441 5

'896 .898

'4516

•4618

0.960 '961 •962 '963 '964

0'900 •902

'9395 .9505 •9616

'904 '906 '908

1.4722 '4828 '4937 '5047 .5 x 6o

0.965 •966 '967 •968 '969

2.0139 -0287 .0439 .0595 .0756

0.9925 '9930 '9935 •9940 '9945

v7911 -8257 -8629 •9031 '9467

0'9995

'9287

0.9730

0'910

1.5275

0.970

2'0923

0.9950

2'9945

I'0000

58

•7587

'9996 '9997 .9998 '9999

3'0363

3.0473 .0585 •0701 -0819 '0939 3.1063 -I190 •132o •1454 .1591 3'1732 •1877

00

TABLE 17. THE INVERSE OF THE z-TRANSFORMATION The function tabulated is r = tanh z =

e22 -

+

. If z < o, use the negative of the value of r for -z.

z

r

z

r

z

r

z

r

z

r

z

r

0'00 '01 •02

0'0000 '0100

0'50

•52 '53 '54

0.7616 •7658 -7699 '7739 '7779

0.905! •9069 •9087 •9104 -9121

2'00 •02 '04

0.9640 .9654 •9667 -9680 .9693

3'00 •02

'0200 •0300 '0400

1'00 •OI '02 •03 '04

x.50 '51

'03 •04

0.4621 '4699 '4777 '4854 '4930

0.9951 '9952 '9954 '9956 '9958

0'05

0'0500

.0599

1.05 -06 -07 -08 .09

0.7818 -7857 •7895 '7932 -7969

x-55 •56 •57

0.9138 '9154

2•I0 •I2

'9170

-14

- 58

•9186

'59

- 9201

•x6 -18

0.9705 -9716 -9727 '9737 '9748

3'10 •I2

.0798 •o898

0.55 -56 '57 •58 '59

0.5005

•o6 •07 •08 •09 0•10

0.0997

*14

x.60 •6x -62 -63 - 64

0.9217 -9232 '9246 -9261 .9275

3'20 '22

'1293 •I39I

0.8005 •8041 -8076 •8110 - 8144

2'20 - 22

'I3

0.5370 '5441 •5511 •5581 . 5649

0.9757

•1096 '1194

o•6o •61 •62 -63 - 64

PIO

•II •I2

- 24 '26 '28

'9776 -9785 '9793

.28

0'15

0'1489

o•65 •66 -67 •68 -69

0.5717 '5784 •585o •5915 •5980

I•I5

I•65

0.9289 '9302

2'30 - 32

'67

.9316 '9329 '9341

-36 •38

0.9801 -9809 -9816 -9823 •983o

3.3o •32 - 34 •36 •38

0 '9973

-66

•17 -x8 •x9

0.8178 -8210 8243 -8275 •8306

0.70 '71 -72 . 73 '74

0.6044 •6107 -6169 •6231 •6291

I'20 •2I '22 •23 '24

0.8337 •8367 -8397 - 8426 - 8455

1.70 '71 '72 '73 '74

0 '9354

2.40 - 42 '44 '46 •48

0.9837 .9843 .9849 .9855 •9861

3'40 - 42 .44 - 46 •48

0.9978 '9979 '9979 -9980 '9981

0.75 •76 '77 •78 '79

0.6351 - 6411 - 6469 .6527 - 6584

1'25

0.8483 •8511 -8538 .8565 -8591

1'75

0 '9414 - 9425

2'50

'9436 '9447 '9458

•52 '54 -56 •58

0.9866 •9871 •9876 -9881 •9886

3-50 '55 -60 -65 •70

0.9982 '9984 -9985 -9986 -9988

0.6640 -6696 •6751 •68o5 -6858

1'30

x.8o

•32 '33 '34

0.8617 - 8643 •8668 •8692 -8717

-82 -83 -84

0.9468 '9478 '9488 '9498 -9508

2.60 -62 - 64 •66 -68

0.9890 -9895 -9899 .9903 •9906

3.75 •8o .85 •90 '95

0.9989 - 9990 '9991 '9992 '9993

4.00 •05 •xo .x5

0 '9993

- 20

'9994 '9995 '9995 '9996

•r7 -x8 •I9 0'20 •2I '22

•o699

•1586 •1684 '1781 '1877 0'1974 '2070

.2165

•23

•2260

'24

'2355

0.25

0 '2449

-26 •27 •28

'2543 '2636

'29

•282I

0'30

0.2913 •3004

'2729

-5080 '5154 .52,27 .5299

•II '12 - 13

-26 •27 -28 •29

'52

'53 '54

-

-68 -69

•76 '77 •78 '79

-9366 '9379 '9391 '9402

-06 •08

- 34

•9767

'04

-06 •08

'14 •x6 •x8

'24 '26

0 '9959

.9961 •9963 '9964 -9965 0.9967 -9968 •9969 '9971 '9972

'9974 '9975 '9976 '9977

'32

•3095

'33 '34

•3185 '3275

o•80 •81 -82 .83 -84

0.35 •36 - 37 •38 '39

0.3364 •3452 '3540 •3627 '3714

0.85 -86 .87 •88 -89

0.6911 -6963 •7014 '7064 •7114

1-35

0'8741

•36 - 37 •38 '39

-8764 -8787 .8810 -8832

1-85 -86 -87 -88 .89

0.9517 '9527 '9536 '9545 '9554

2'70 '72

'74 76 •78

0.9910 '9914 .9917 -9920 '9923

0'40

0.3799 .3885 •3969 '4053 •4136

0.90 '9I .92 .93 '94

0.7163 -7211 '7259 •7306 '7352

1 40

0.8854 •8875 -8896 •8917 .8937

x•90 '9, .92 '93 '94

0.9562 '9571 '9579 -9587 '9595

2.80 •82 '84 -86 88

0.9926 -9929 '9932 '9935 '9937

4'25 .30 '35 . 40 '45

0.9996 '9996 '9997 '9997 '9997

0.45 '46 '47 '48 '49

0'4219

0.95 •96 '97 •98 '99

0.7398 '7443 '7487 '7531 '7574

1'45

1'95 •96 '97 -98 '99

0.9603 -9611 •9618 -9626 -9633

0 '9940

•46 '47 •48 - 49

0.8957 .8977 -8996 •9015 .9033

2'90

'4301 .4382 '4462 '4542

•92 '94 -96 .98

'9942 '9944 '9946 '9949

4'50 - 55 •60 .65 -7o

0.9998 '9998 '9998 '9998 '9998

0'50

0'4621

1'00

0.7616

I•50

0•905I

2'00

0'9640

3'00

0.9951

4'75

0'9999

'31

'41 '42 '43 '44

.31

'42 '43 '44

59

TABLE 18. PERCENTAGE POINTS OF THE DISTRIBUTION OF THE NUMBER OF RUNS with the required property. When n, and n2 are large, R is

Suppose that th A's and n2 B's (n1 n 2) are arranged at random in a row, and let R be the number of runs (that is, sets of one or more consecutive letters all of the same kind immediately preceded and succeeded by the other letter or the beginning or end of the row). The upper P per cent point x(P) of R is the smallest x such that Pr {R x} 5 P/ioo, and the lower P per cent point x'(P) of R is the largest x such that Pr IR P/ioo. A dash indicates that there is no value

approximately normally distributed with mean

zni n2 + and ni+ n2

2n022(2n1n2— — n2)

. Formulae for the calculation (ni +n2)2 (ni +n2 — I) of this distribution are given by M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 2 (3rd edition, 1973), Griffin, London, Exercise 30.8. variance

UPPER PERCENTAGE POINTS

=3 4

112

P

5

=4 4 5 6

7 8 9 9 9

7 5

5 6

7 8 9 5 6

6

7

7

7

8

8

8

xo 6 7 8 9 xo

I

P

o•x

5

n1 = 8 n2 = 17 18 19 20

9 9

9

9

II

I0 II 12 13

II

14

9 I0 I0

I0 II II

II II

12 12

9 12 12 13 13

13

II 12 13

12 13 13 13

14

13

7

12

13

14

8 9

13

14

13

15 15

9 xo

15 16 I7 18 19 20 TO II

12 13

10 II

13 14

14 15 15 15

10

14 15 16 17 i8

10

19

14 15 15

II

20 II

15 15 15

II

18 19

8 9

13 14

14 15

16 16

xo II 12 13

14 15 15 15

17 17

14

16

15 16 16 17 17

15 ,6

16 16

17 17

12

14

13

14

14 15 16 17

II

12

12

16 16 16 17 14

I

17

17 17

18 18

17 18 18

19 19 19

18 16 16 17 17

19 17 18 18 19

17

19 19 20 20 20

19

18 19 zo

16

16 16 17

18 18

18 18 19 19 19

13

17

14

14

18 19 20 14 15 16 17

I2 13

19 19

20 20 21

14 15 16 17 18

18 19 19 19 20

20 20 2I 21 2I

21 22 22 22 23

19 20 12 13 14

20 20 18 18 19

22 22 19 20 21

23 23 21 22 22

15

19

21

23

I

o•x

20 20 21 21 21

22 22 22 23 23

23

19 20 20 21 21

21 21 22 22 23

23 23

21 22 22 20 21

23 24 24 22 23

25 25 26

21 22 22 23 23

23

25

24 24 24 24

24 24 25

24 24

25

24 24 25

26 26 27

21 22 22 23 23

23 24 24 25 25

25 26 26 27 27

19

24 23 23 24 24

26 24 25 26 26

28 z6 27 28 28

16

20

25

26

29

17

17 18

24 24

26 26

19

25

27

28 29

20

25

27

29

18 19 20

25 25 26

27 27 28

29 30 30

18 19 20

15

15 16 17

18 19

20 20

5

24

18 19 20 20 20 2I 21 21

13 14 15

x6

13

19 19 17 17 18

6o

n2 =

n1 =

16 17 17 18 18

15 15

P

0•I

15

20

16

16 17

x8

x8

19

20

28

19

26

28

20

27

29

30 31

20

27

29

31

TABLE 18. PERCENTAGE POINTS OF THE DISTRIBUTION OF THE NUMBER OF RUNS LOWER PERCENTAGE POINTS

P

=

2

2

n2

=

8 9

3

3

n1 =

8

n2 = 19

5

x

8 8 6 6 6

6 6 4 5 5

5 5 3 3 3

7 7 7 8 8

5 6 6 6 6

4 4 4 4 5

8 8 8 9 6

7 7 7 7 5

5 5 5 5 4

7 7 8 8 8

5 6 6 6 7

4 4 4 5 5

8 9 9 9

7 7 7 8 8

5 5 6 6 6 4 5 5 5 5

12 13

12

2

14

5

2

15 x6 17 18

2

19

5 5 5 5 5

4 4 4 4 4

5 3 4 4 4

4 2

3 —

3 3 3

2

I0

3 4 4 4 4

2

10

2

12

12 13 14

5 5 5 5 5

3 3 3

'3

15 16 z7 18 19

6 6 6 6 6

4 4 5 5 5

3 3 3 3 3

xo

20

5 3 3 4 4

4

II

2

14

8

10

6 4 4 5 5

3

15

9

6 6 6 7 7

II 12 13 Ls 15

5 6 6 6 6

4 4 5 5 5

3 3 3 3 3

II

16 17 18 x9

9 9 io xo xo

7 8 8 8 8

6 6 6 6 7

16 17 18 19

6 7 7 7 7

5 5 5 6 6

4 4 4 4 4

12

8 9 9 9 io

7 7 7 8 8

5 5 5 6 6

5 5 6 6 6

4 4 4 5 5

2

12

3 3 3 3

x8 19

lo io zo

6 7 7 7 5

5 5 5 6 6 6

13

2

14

2 2

=5

5

2 2

2

20

2

2

5 6

2 2

7 8 9

2 2

2

10

3 3

2

12

17 x8 19

4

4

2

4

5 6 7 8 9

3 3 3 3

zo

5 6

6

2

3 3 3 3 3

2

6

2 2

2 2 2

6 7

2 2 2 2

7

2 2 2

2 2

14

3 3 3

2 2

15 16

4 4

3 3

2

12 13

20

6 7 8 9 zo II

3 3 4 4 4

II

n2= I° II

18 19

20

5

2

P

2

2

3 3 3 3

4 5

o•x

II

16

4

I

I0

15

4

5

3 3 3 3 3

13 14

3

P

0' I

4 4 4 4

II

3

I

2 2

15 x6 17 2

5

2

7

2

7 8 9

20

17

4

3

2

8 9 xo

18 19

4 4

3 3

2

II

2

12

20

4 3 3 3 3

3

2

5 6 7 8 9

4

2

8

2

x6

2

17

6 7 7 7 7

x8

8

8

13

2

14

2

15

3

2

8

61

20

2 2

9

9

2

I0

2

II

2

9

2 3 3 3

12

13 14 15 x6 9

17 x8 19 20

2

I0

II

15 16 17

x8 19 20 II

12

2 2

20 12 13 14

xs 16

9

7 8 8

20

II

13

13

9

8 8 9 9 7

4 4 4 4 4

13

14

15 x6 17 x8

9 io io Jo ix

8 8 8 9 9

6 6 6 7 7

4

13

19

II

9

7

/7

TABLE 18. PERCENTAGE POINTS OF THE DISTRIBUTION OF THE NUMBER OF RUNS LOWER PERCENTAGE POINTS

P

nI= 13

722 = 20 14 15

16 17 14

18

15

19 20 15

5

I

0•I

II 10 I0 II II

10 8 8 9 9

8 6 7 7 7

II I2 12

9 10 10 9 9

7 8 8 7 7

II

16

II

P

=

15

5

I

o•x

II

I0 I0

8 8 8 8 8

th = /7 18 19 20

I2 12 I2

IO

16

16

II

10

16

17 18

I2 12 13 13 12

o To

19 20 17

17

II

nI = 17

II II

I0

5

19

n2 =

18 18

8 8 9 9 8

P

19 20

I

0•I

13

II

13

II

9 9 9 9 9

20

13

II

18 19

13

II

14

I2

20 19 20 20

14 14 14 15

12 12 12

13

I0 IO 10 II

TABLE 19. UPPER PERCENTAGE POINTS OF THE TWO-SAMPLE KOLMOGOROV-SMIRNOV DISTRIBUTION .. d(P). When rejecting at the P per cent level if nin2 D(ni, n,)?.

This table gives percentage points of

D(ni, n2) = sup I Fi(x) F2(x)I, where F1(x) and F2(x) are the empirical distribution functions of two independent random samples of sizes n, and n2 respectively, nI < n2 < 20 and n, = n2 loo, from the same population with a continuous distribution function; the function tabulated d(P) is the smallest d such that Pr {nin2 D(ni, n2) d} Phoo. A dash indicates that there is no value with the required property. A test of the hypothesis that two random samples of sizes n, and n2 respectively have the same continuous distribution function is provided by

— D(ni, n2) are n1and n2are large, percentage points of 4/n1n2 ni+ n2

—

5

=

2

"2 =

5

6 7 8 9 2

I0 II

12 13 14 2

16 17 x8 19 2

20

3

3 4 5 6

3

3

7 8 9 xo

14 16 18

P

o•x ni.

16 18

24 26

24

26

24 26 28

26 28 3o

z8 3o 32 34 36

3o 32 34 36 38

38

38

40

40

32 32

34 9 12 15 15 18 2I 21

12 13

27 30

14

33 33 36

=3

I0

5

2'5

I

0•I

36

39 42 42 16

42 45 45 48 16

45 48 51 51 —

48 51 54 57 —

—

16 18 21

20 20 24

20 24 28

—

—

24 27

28 28

28 32

32 36

14

28 29 36 35 38

3o 33 36 39 42

36 36 40 44 44

15 16 17 18 19

40 44 44 46 49

44 48 48 5o 53

45 52

20

52

6o

64

5 6 7 8

20

25

25

24 25 27

24 28 3o

3o 35 35 36 40

35 40 39 43 45

n2 = 17 18 19 20

18 20 22 24

II

16

I

10 12

24 27

15

2'5

approximately given by those in Table 23 with n = co. Formulae for the calculation of this table are given by P. J. Kim and R. I. Jennrich, ' Tables of the exact sampling distribution of the two-sample Kolmogorov—Smirnov criterion D,„„, m < n', Selected Tables in Mathematical Statistics, Vol. 1 (1973), American Mathematical Society, Providence, R.I.

20 22

4

4

4

5 6 7 8 9

4

xo II 12 13

15 18

18

21 21 24 27 30

2I 24 27 30 30

27 3o 33

3o 33 36 36 39

33 36 39 39 42

36 39 42 42 45

4

4 5

5

9 to II 12

13 6z

24 28

36 40

44 48 48

— 52 56

52 56 6o 6o 64

6o 64 68 72 76 76

3o 3o 32

68 25 3o 35 35

36 40 44 45 47

40 45 45 5o 52

45 5o 55 6o 65

52

54 57

TABLE 19. UPPER PERCENTAGE POINTS OF THE TWO-SAMPLE KOLMOGOROV-SMIRNOV DISTRIBUTION P nI

=5

5

5

n2 = 1 4 15 x6 17 x8

42

46

51

56

70

50

52

55 54 55 6o

55 59 6o 65

6o 64 68 70

70 75 8o 85

19

56 6o

61 65

66 75

30 28

30

36

30

30 34

35 36

71 8o 36 36 40

85 90 — — 48

39 40 43 48

13

46

52

44 48 54 54

45 48 54 6o 6o

54 6o 66 66 72

10

12

33 36 38 48

14

48

51 54 56 66

54 57 6o 6z 72

58 63 64 67 78

64 69 72 73 84

78 84 84 85 96

10

x5 x6 17 x8

II

20

6

6

6 7 8 9 I0

Ix

6

6 7

7

9 9

n1 =

9

xs 17 9

76

83

96

34

36

40 42

41 45

88 42 48 49

Ioo 49 56 63

xo

40

46

49

53

63

II 12

44 46 50 56

48 53 56 63

52

56 58 70

59 6o 65 77

7o 72 78 84

56 59 61

62 68

68 73 77

75 77 84

90 96 98

65 69

72 76

So 84

87 91

107

93 56 55 6o 64

112

64 64 7o 77

x3

13

64

72

79

8 9

40

48

40 44 48

46 48 53

86 48 48 54 58

5z 54 58 6o

6o 62 64 67

64 65 70 74

68 72 76 81

8o

72

8o

8o

88

104

x7 18 19

68

77

8o

88

72

8o

86

94

III 112

74

8z

90

98

117

20

8o 54

88 54

96 63

104 63

124 72

To

50

II

52 57

53 59

6o 63

63 70

8o 81

63

69

75

87

9

12

72• 76 81 85 90

t•x

78 84 90 94 99

91 98 105 no 117

90

99

io8

126

98 Ioo 70 68

107 III 8o 77

126 133 90 89

6o 64 68 75

66 70 74 8o

72 77 82 90

8o 84 90 Ioo

96 Ioo io6 115

76

84

go

Ioo

x18

18 19

79 8z 85

89 92 94

96 xoo 103

io6 108 113

126 132 133

20

I00

II0

120

130

150

ix

66

77

77

88

99

xo

12

17

12

64

72

13 14 15 16

67

75 82 84

76 84 87 94

86 91 96

99 1 08 115

102

120

89

96

io6

127 132 140

73 76 8o

92

102

III

122

12

20 12

96 72

107 84

116 96

127 96

146 154 120

12

13

71

95

117

78

81 86

84

14

94

104

120

15 16 17

84

93

99

io8

129

88 go

96 100

104

io8

116 119

136 141

x8 x9

96 99

io8 1o8

20

104

116

13 14

91 78

91 89

15 x6

87 91

101

17

iz

88

90 97

85

93

102

II0

88

97

107

118

96

Ito

126

15o

120 124 104

130 140 117

156

Ioo

104

164 130 129

104 III

115 121

137 143

114

127

152

131 138

156 164

18 19

96 99

105 110

104

114

120 126

13

20

108

14

x4 15 x6 17

98

120 112

130 II2

143 126

154

140

14

18 19 20

63

65 70 75 78 82

x

17 x8 x9

Ica

20

12

Ix

59 63 69 69 74

2'5

89 93 70 6o

15 16

78 42

5

81

x3 14

xx

lo

8o 84 6o 57

II

70

x5 16 17 18 x9

x8 x9 20

xo

72 42

7 8 9

n2 = 13 14 16

64

13 14 x5 x6 8

42

O•I

66 35

I0 II

8

50

I

19

14

7 8

48

2'5

20

13

7

P

Jo

169

92

98

no

123

96 Ioo

106 III

116

126

152

122

134

159

104

116

126

140

166

II0 114

121 126

133 138

148 152

176 180

TABLE 19. UPPER PERCENTAGE POINTS OF THE TWO-SAMPLE KOLMOGOROV-SMIRNOV DISTRIBUTION P ni = 15 n2 = 15 16 17 18 19

110

5

2'5

I

0'1

105 POI

120 114

135 133

165 162

105 III

116 123

135 119 129 135

142

165

147

114

127

141

152

P

xo

5

2.5

I

128

140 136 133

156 153 148

168

200

170 164

204 187

141

166

174

n j. = 16 n2 = 20 17 17 18 19

136 118 126

180

20

130

146

151 160

175

200 209

18

18

144

162

162

18o

216

19

133

20

136 152

142 152 171

159 166 190

176 182 190

212 214 228

144

16o

169

187

225

840 854 868 882 896

900

1020

1080

1320

915 992 ioo8 1024

1037 1054 1071 io88

1098 1178 1197 1216

1342 1364 1386 1408

910

1040

1495

1056 1072

1105 1122

1235

990 1005

4206

112273 54

1088 1104

1224 1242

1292 1380

1564 1587

1190 1207 1224

1260 1278 1296

1400 1420 1440

1610 1704 1728

15

20

125

135

150

160

195

x6

16 17 18 19

zi2 109 116

128 124 128

144 136 140

16o 143 154

176 174 186

19

120

133

145

160

190

20

ni = n2 = 20

160 168 198

180 189 198 230

200 210 220 230 264

220 231 242 253 288

260

21 22 23 24

273 286 299 336

ni = n2 = 6o 6z 62 63 64

275 z86

300

350

65

312

364

324 364 377

405 420 435

66 67 68 69

1020

1035

450 496

70 71

1050 1065 1080

207 216

240

25 26 27 28 29

225 234

2 90

308 319

297 336 348

30

300 310

330 341

360 372

390 403

31

243 280

250 260 270

o•x

19

;54 518 1

32

320

352

384

416

512

72

33 34

330 374

396 408

396 442

462 476

528 544

73 74

1095

1241

1314

1460

1752

II I0

1258

1332

1480

1776

1125 1216 1232 1248

1800

35

385

420

455

490

595

75

36

396

432

468

504

612

76

37 38 39

407 418 429

444 456 468

481 494 546

518 570 585

629 646 702

77 78 79

560 574 588 6oz 616

600 615 630 688 704

720 738 756 774 836

8o 8i

1280

1440

1296

1458

82

1312

1476

83 84

1328 1344

1264

1275

1425

1500

1292

1444

1596

1824

1309 1326 1422

1463 1482 1501

1617 1638

1925

1659

1975

1520

1680

2000

1539

1701

2025

1558

1722

2050

1494 1512

1660 1680

1743 1848

2075 2184

1700 1720

1870 1892

1740 1760 1780

1914 1936 1958

2210 2236 2262

1729 1748 1767 1786

1800 1820 1932 1953 1974

1980 2002 2416 2139 2162

2484 2511 2538

1995 2016 2037 2058

2185 2208 2231 2254

2565 2592 2716 2744

40

440

520

41 42 43 44

492 504 516 528

533 546 559 572

45 46 47 48 49

540

585

675

720

855

85

1360

1530

552 564 576 637

644

690 705

736 752

874

893

720

768

912

735

833

980

86 87 88 89

1462 1479 1496 1513

1548 1566 1672 1691

90 91 92

1530 :5 546.4 7

93 94

1581 1598

658 672 686

50

650

700

750

51

663 676 689

714 728 742

765 832 848

850 867 884 901

1020 1040 1060

702

810

864

918

1134

55

715

56 57

728 798

825 84o 855

88o 896 912

990 1008 1026

1155 1176 1197

95 96

1615 1632

52

53 54

1000

1710

1950

2288 2 314 2430 2457

58

812

870

928

1044

1218

1764

59

8z6

885

1003

1062

1298

9 99

1805 1824 1843 1960

1782

1980

2079

2277

2772

6o

84o

900

1020

1080

1320

I00

I 800

2000

2100

2300

2800

64

TABLE 20. PERCENTAGE POINTS OF WILCOXON'S SIGNED-RANK DISTRIBUTION cent level if W+ x(P); a similar test against u > o is provided by rejecting at the P per cent level if W- 5x(P), and, against # o, one rejects at the 2/:' per cent level if W, the smaller of W+ and W-, is less than or equal to x(P). When n > 85, W-F is approximately normally distributed. Formulae for the calculation of this table are given by F. Wilcoxon, S. K. Katti and R. A. Wilcox, ' Critical values and probability levels for the Wilcoxon rank sum test and the Wilcoxon signed rank test', Selected Tables in Mathematical Statistics, Vol. I (1973), American Mathematical Society, Providence, R.I.

This table gives lower percentage points of W+, the sum of the ranks of the positive observations in a ranking in order of increasing absolute magnitude of a random sample of size n from a continuous distribution, symmetric about zero. The function tabulated x(P) is the largest x such that Pr {W+ < x} P/Ioo. A dash indicates that there is no value with the required property. W-, the sum of the ranks of the negative observations, has the same distribution as W+, with mean ln(n+ 1) and variance -Nn(n + 1) (n + ). A test of the hypothesis that a random sample of size n has arisen from a continuous distribution symmetric about p = o against the alternative that j < o is provided by rejecting at the P per P n= 5

6 7 8 9 10 II

12 13 14

zs

5 0 2

3 5 8 10 13 17 2I 25

2'5 0 2

3 5 8 10

17 21

0'5

I

I

o

3

I

5 7 9

3 5 7 9

I

n = 45

371

46 47 48 49

389 407 426 446

343 361 378 396 415

312 328 345 362 379

322 339 355

249 263 277 292 307

so 51 52 53 54

466 486 507 529 550

434 453 473 494 514

397 416 434 454 473

373 390 408 427 445

323 339 355 372 389

12 15

12

573 595 618 642 666

536 557 579 602 625

493 514 535 556 578

465 484 504 525 546

407 425 443 462 482

I

2

4 6

19 23

17 18 19 20 21 22 23 24

6o 67 75 83 91

25

loo

26 27

110

119

107

92

83

64

28

130

116

IoI

91

29

140

126

I TO

100

71 79

120 130 140 151 162

109 118 128 138 148

159 171 182

0'5 291 307

0.1

32 37

15 19 23 27 32

8 I 14 18 21

55 56 57 58 59

52 58 65 73 8,

43 49 55 62 69

37 42 48 54 61

26 30 35 40 45

6o 61 62 63 64

690 715 741 767 793

648 672 697 721 747

600 623 646 669 693

567 589 61i 634 657

501 521 542 563 584

89 98

76 84

68 75

51 58

65 66 67 68 69

8zo 847 875 903 931

772

879

718 742 768 793 819

681 705 729 754 779

6o6 628 651 674 697

86 94 103

70

960 990 1020

907 936

846 873

805 831

964

112 121

73 74

1050

994

901 928

io81

1023

957

884 912

721 745 770 795 821

131

75

1112

141

76

1144

151 162 173

77 78 79

1209

1053 1084 1115 1147 "79

986 1015 1044 1075 1105

940 968 997 '026 1056

847 873 900 927 955

185

1276 1310 1345 1380 1415

1211 1244 1277 1311

1136

Io86 1116 "47

1345

1168 1200 1232 1265

1178 1210

983 MI 1040 1070 1099

1451

1380

1298

1242

1130

27

151

137

163

147

175 187 200

159

35 36 37 38 39

213 227

195 208 221 235 249

173

40

264 279 294

238 252

43 44

286 302 319 336 353

310 327

45

371

343

41 42

2'5

0

25 29 34 40 46

241 256 271

5

O

30 35 41 47 53

3o 31 32 33 34

P

O'I

170 182

185 198 211

224

194 207

71

72

266 281 296

220 233 247 261 276

209 222 235

8o 81 8z 83 84

312

291

249

85

197

65

1176 1242

798 825 852

858

TABLE 21. PERCENTAGE POINTS OF THE MANN-WHITNEY DISTRIBUTION x(P), and a similar test against level P per cent if UB /LA < ,I.tB is provided by rejecting at the P per cent level if UA x(P). For a test against both alternatives one rejects at the 2P per cent level if U, the smaller of UA and UB, is less than or equal to x(P). If n1and n2 are large UA is approximately normally distributed. Note also that UA +UB = n1n2.

Consider two independent random samples of sizes n1and n2 respectively (n1 < n2) from two continuous populations, A and B. Let all n, + n2 observations be ranked in increasing order and let RA and RB denote the sums of the ranks of the observations in samples A and B respectively. This table gives lower percentage points of UA = RA -ini(ni+ 1); the function tabulated x(P) is the largest x such that, on the assumption that populations A and B are identical, Pr {UA x} < P/too. A dash indicates that there is no value with the required property. On the same assumption, UB = RB — in2(n2+ t) has the same distribution as UA, with mean ini n2 and variance 12-n1n2(n1+ n2 + 1). A test of the hypothesis that the two populations are identical, and in particular that their respective means /tit, itB are equal, against the alternative itA > AB is provided by rejecting at

P =

2

2

5

2'5

I

0'5

Formulae for the calculation of this distribution (which is also referred to as the Wilcoxon rank—sum or Wilcoxon/ Mann—Whitney distribution) are given by F. Wilcoxon, S. K. Katti and R. A. Wilcox, ' Critical values and probability levels for the Wilcoxon rank sum test and the Wilcoxon signed rank test', Selected Tables in Mathematical Statistics, Vol. I (1973), American Mathematical Society, Providence, R.I.

0'1

12 13 14

I0

15 16 I7 x8 19

O

I

O I

15 16 17 18 19

3

I

0

3

I

0

3

2

0

2 2

0 I

0

2

I

0

20

4

3 4 5 6

0

7 8 9 To II

12 13 14

15 16 17 18 I9 20 4 5 6 7 8 9

4 5

4 4

2

4

5 6 7 8 9

0

3

4

7 8 9 to II

II

=4

3

3

n2 = ro

O

O

2

3

2.5

6 7 8 9

12

3

5

n2 = 5

II

2

P

5

0

I

o

2

I

—

12

I0

14

II

I

0.5

3

2

O

4

2

O

5 5 6

3 3 4

O

7 7 8 9 9

5 5 6 6 7

15

II

16

12

17

13

20 5 6 7 8

18 4

14 2

to

8

I

O

5 6 8

3 5 6

2 3 4

I 2

9 ro II

9 II

7 8 9

5 6 7 8 9

3 4 5 6 7

12

12 13

15

12

16 18

13

I0

15

14

II

II

2

I

O

2

O

2

I

O

19

15

12

3 3

I

O

20

17

13

7 8 9 to

I

O

22

18

14

II

5 6 7 7 8

4 4 5 5 6

2

I

2

I

2

I

3 3

2

15 i6 3 4 6

9 9

4 4 4 5

2

O

2

O

3 3

0

II

6 7 7 8

I

O

IO

18 5 6

2

6

I

0

2

I

O

3 4 4

I

O

2

I

3

I

6 7 8 9 ro II 12 13

0

6

2

3 4 5 6

19 20

66

14 15 16 17 18

23

19

25 7 8 to

20

12 14

IO

i6 17 19

13 14 16

5 6 8

II

21

17

23

19

25

21

26

22

28

24

I I I 2 2

3 3 3

I

3 4 4 5

5

0•x

I2 13

0 I I 2 2

3 3 4 5 5 6 7 7

2

3 4

O

7 8 9 II

5 6 7 9

12

I0

2 3 4 4 5

15 16 8 19

12

II 13 15 16

I

6 7 8 9 to

TABLE 21. PERCENTAGE POINTS OF THE MANN-WHITNEY DISTRIBUTION P n1 = 6

7

7

2'5

I

0'5

n2 = 19

30

25

20

17

II

20

32 II

27

22

18

I2

8 10

14

15

50

12

4 6 7

2

15

6 7 9

I

13

3

x6

17 19

14 16

17

13

21 24

14

z6

18 20 22

15

28

16

30

17

7 8 9 10 II 12

7

x8 19

7 8

20

8 9 I0 II

8

P

5

II

9

12 14 16

10 12

O•I

n1 = II

5

2'5

I

0'5

38 42 46 54

33 37 40 44 47

28 31 34 37 41

24 27 30 33 36

57 61 65 69 42

51 55 58 62 37

44 47 50 53 31

39 42 45 48 27

29 32 34 37

47 51 55 6o 64

41 45 49 53 57

35 38 42 46 49

31 34 37 41 44

23 25 28 31 34

18 19

68

61

53

47

37

72

65

56

51

40

20

13

77 51

14

56

69 45 50

6o 39 43

54 34 38

42 z6 29

i5

61 65 70 75 8o

54 59 63 67 72

47 51 55 59 63

42 45 49 53 57

32 35 38 42 45

84 61 66 71 77

76 55 59 64 69

67 47 51 56 6o

6o 42 46 50 54

48 32 36 39 43

20

82 87 92

74 78 83

65 69 73

15 16

72

64

56

77

70

61

58 63 67 51 55

46 50 54 40 43

18 19

83 88 94

75 8o 85 90 75

66 70 75 8o 66

6o 64 69 73 6o

47 51 55 59 48

81 86 92 98 87

71 76 8z 87 77

65 70 74 79 70

52 56 6o 65 57

93 99 105 99 io6

82 88 93 88 94

75 81 86 81 87

61 66 70 66 71

112 113 119 127

I00 101

92

76

107

93 99

114

105

77 82 88

n2 = I2

13

II

17

13 15

5 6 7 8 9

12

12

24

19

16

zo

12

13

26

21

18

II

14

33 35 37

28 30 3z

23

19

13

15

24

21 22

14

x6

15

17

39 15 18

34 13 15 17

z8 9 II 13

24

16

19

15

7 9 II 13

4 5 6 8

22

17 20 22

20 23

z6

18 19 20

12

13

15

9 II

16

12

17

24 26

17 18 20 22

14 15

19

34 36 38 41

28 3o 32 34

24 z6 28 30

17 18 20 21

15

21

17

14

II

7

17

24 27

20 23

13 16

8 10

26 28

14

36

31

26

18 20 22

12

13

30 33

16 18 21

15 x6 17 x8 19

39 42 45 48 51

34 37 39 42 45

28 31 33 36 38

24 27 29 31 33

23 25

9

20

54

10

I0

27

40 19

36 16

26 10

II

22 24

18 21

12

29

13

31 34 37

48 23 26 33

27

24

17

17

14

41

30 33 36 38 41

26 29 31 34 37

17

44 48 51 55

36 39 42 45 48

19

15

23 25 27

44 47

39 42

29 32

25

21

15

26 z8 31 33 36

24 26 29 31

20

39 41 44 47

9 xo

12

13 14 15

16 8

17 18 19

9 9

II 12

9

12

xo

16 17

18 10 XX

19

58

52

20

6z 34

55 30

II

23

13

18

13

20

14

14

x6

14

x8 19

15

14

15

15

17 19 21

17

20

I00

i6

16

83

x6

17 z8 19

89 95 101

20

I07

17

96

14

21

18

18

102

19

109

20

115

x8

109 116

19

18

20

19

19

20

67

20

123 123 130

20

138

0•I

17 20 22 24 27

20

TABLE 22A. EXPECTED VALUES OF NORMAL ORDER STATISTICS (NORMAL SCORES) The values E(n, r) are often referred to as normal scores; they have a number of applications in statistics. In carrying out calculations for some of these applications the sums of squares of normal scores are often required: they are provided in Table 22 B.

Suppose that n independent observations, normally distributed with zero mean and unit variance, are arranged in decreasing order, and let the rth value in this ordering be denoted by Z(r). This table gives expected values E(n, r) of Z(r) for r 5 En+ r); when r > En+ I) use

E(n, r) = - E(n, n+ 1- r). n= r= I

I

2

3

0'0000

0.5642

2

6

4

5

0.8463

1. 0294

1.163o

1•2672

•3522

•0000

0'2970

0'4950

0.6418 02015

0 '7574

0.0000

3 4

7

0.3527 0.0000

8

9

xo

1 4136 0'8522 0%4728 0'1525

1 '4850 0'9323 0*5720 0•2745

1.5388 P0014 0'6561 0*3758

0'0000

0'1227

5

n=

II

12

13

14

15

16

17

18

19

20

1'8445

1.8675 P4076 ply's, 0.9210

Y = I

1'5864

1'6292

1.6680

1.7034

P7359

P7660

2

•0619

1•1157

P1641

P2079

P2479

3 4

0.7288

0.7928 0.5368

0.8498 0.6029

0.9011 0•6618

0 '9477

0.4620

0.7149

1.2847 0.9903 0.7632

P7939 1.3,88 1.0295 0.8074

1.8200 1•3504 1.0657 0.8481

P3799 P0995 0.8859

5 6

0'2249 -0000

0'3122 •1026

0'3883 •1905 •0000

0•4556 .2673 •0882

0'5157

0'5700

0'6195

•3962

'4513

'2338 •0773

'2952

0.6648 •5016 •3508 •2077 •o688

0.7066 '5477 .4016 •2637 •1307

0 '7454

'3353 •1653 •0000

0.0000

0.0620

7 8

•1460 •0000

9 I0

n=

•5903 •4483 •3149 •1870

21

22

23

24

25

26

27

28

29

30

1.8892

1.9097 1.4582

P9292

1'9477

•5034

P9653 .5243

P9822 '5442

P9983 .5633

2'0285

.4814

2'0428 1'6156

1.1582

r=x 2

1'4336

3 4

i•16o5

1.1882

•21414

'2392

•2628

'2851

'3064

2'0137 1.5815 1 '3267

0.9538

0.9846

•0136

•0409

•0668

.0914

•1147

1.1370

5 6

0.7815

0.8153

0'8470

7 8

'491 5 •3620 •2384

0.9570 •82o2 •6973 •5841 .4780

P0261

•7012 .5690 .4461

0.9317 '7929 •6679 .5527 '4444

1•0041

•6667 .5316

0.9050 •7641 •6369 '5193 •4086

0'9812

.6298

•8462 .7251 -6138 •5098

0.8708 0.7515 0.6420 0.5398

0.894.4 0.7767 0.6689 0.5683

04110 •3160 •2239 -1336 •0444

0.4430 •3501 •26o2 .1724 •0859

0 '4733

0'0000

0'0415

9 10 II 12

13

'4056 •2858

•3297

0.8768 '7335 •6040 '4839 •3705

0•1184

0.1700

0.2175

0.2616

0.3027

0.3410

0.3771

•0000

'0564

•1081

'1558 •0518

'2001 •0995

•2413

•2798

•1439 •0478

•1852 •0922

'0000

•0000

•0000

14

15

68

1.5989 1'3462

1•3648 1.1786

•3824 •2945 •2088 •1247

TABLE 22A. EXPECTED VALUES OF NORMAL ORDER STATISTICS (NORMAL SCORES) =

31

32

33

34

35

36

37

38

39

40

T = I

2.0565

2.0697

2'0824

2'0947

2'1066

2'1181

2. 1293

2'401

2'1506

2

1'6317

1• 647 I

1.6620

P6764

1.6902

1.7036

1.7166

P7291

P7413

z• 1608 1.7531

3 4

1•3827

1'3998

1 '4323

1 '4476

1'1980

1.2167

1.4164 1.2347

1•2520

1•2686

1.4624 1.2847

1.4768 1.3002

1.4906 1.3151

1'5040 1.3296

1 '3437

5 6 7 8 9

1.0471 0.9169 0.8007

x.o865 0.9590

1.1051 0.9789

p1230

1'1402

0 '9384

0 '9979

1.0162

1.1568 r0339

0.8455 0.7420 0.6460

0.8666 0.7643 0.6695

0.8868 0.7857 0.6921

0.9063

0'9250

1.1883 1.0674 0.9604

1.2033 1.0833 0'9772

0.8063 0.7138

0r8261 0.7346

0.8451

0.8634

0'5955

0.8236 0.7187 0.6213

1.1728 x.0509 0.9430 0'7547

0'7740

0.881 0.7926

10

0.5021

0.5294

0 '5555

0'6271

0.6490

0'6701

0.6904

0'7099

'4129 '3269

*44 I 8

'4694

13 14

'2432 .1613

'3575 '2757 •1957

•3867 •3065 .2283

0.5804 '4957 '4144 '3358 •2592

0.6043

II 12

•5208 •4409 •3637 •2886

'5449 •4662 •3903 •3166

.5679 '4904 '4158 '3434

•5900 •5136 .4401 .3689

•6113 '5359 4635 '3934

-6318 '5574 '4859 '4169

15

0.0804 •0000

0.1169 •0389

0.1515 .0755 •0000

0.1841 •xi0r -0366

0.2151 •1428 -0712 -0000

0'2446

0'2727

0 '2995

•1739

•2034

•2316

0.3252 •2585

0.3498 •2842

•1040 •0346

•1 351 '0674 '0000

'1647 '0985 •0328

'1929 '1282 •0640

•2199

0'0000

0'0312

x6 17 18

0.6944

1.0672

19 20

n=

1•5170

•1564 •0936

41

42

43

44

45

46

47

48

49

50

r=I

2.1707

2'1803

2'1897

2.2077

2.2164

2'2249

2'2331

I .7646 1•5296

1'7757

1.7865

1.8073

p8173

•827,

1.8366

2'2412 1'8458

2'2491

2

2.1988 P7971

1.5419 1•3705

1.5538 1•3833

1.5653

1. 5875

1. 6187

1•4196

1.5982 1.4311

1.6086

P3957

1'5766 I.4078

1'4422

1'4531

1.6286 1. 4637

1.2456 1•1281 1•0245 0.9308 0.8447

1.2588 P1421 p0392 0.9463

1.2717 1.1558 1.0536 0.9614

1.2842 1.1690 1.0675 0.9760

1.2964 p1819 1.0810 0.9902

1.3083 1.1944 1'0942 I.0040

0.8610

0.8767

0.8920

0.9068

0'9213

1.3198 1.2066 1.107o 1.0174 0 '9353

1'3311 1'2185 P1195 I'0304 0.9489

0.7645 •6889 •6171 '5483 •4820

0.7815 •7067 -6356 -5676 -5022

0'7979

0.8139 '7405 -6709 -6044 '5405

0.8294 .7566 .6877 •6219 '5586

0.8444 -7723 '7040 •6388 '5763

0.8590 -7875 .7198 •6552 '5933

0.8732 -8023 .7351 -6712 '6099

0.4389 '3772 •3170

0'4591

0.4787 '4187 -3602 •3029 •2465

0.4976 '4383 •3806 '3241 •2686

0 '5159

0 '5336

'4573 '4003 '3446 •2899

'4757 '4194 '3644 •3105

0.5508 '4935 '4379 •3836 '3304

0. 1910 •1360

0'2140 •1599

0'2361

0'2575

0'2781

•0814

•1064

'1830 '1303

•2051 '1534

•2265 '1756

'0271

'0531

*0781

•1020

'1251

•0000

•0260

-0509

'0749

0.0000

0'0250

3 4

1 '3573

7

1.2178 1•0987 0 '9935

8 9

0.81 06

1.2319 1.1136 1.0092 0.9148 0.8279

10 II 12

0.7287

0 '7469

•6515 -5780

'6705

13 14

•5075 '4394

•5283 •4611

15 16 1.7 18 19

0'3734

•3089 .2457 •1835 •1219

0.3960 •3326 '2704 -2093 '1490

0.4178 '3553 ' 2942 '2341

'2579

•1749

•1997

20 21 22 23 24

0'0608

0.0892

0•1163

O'1422

'0000

'0297

'0580

'085,

'0000

'0283

5 6

0.8982

'5979

.7238 .6535 •5863 •5217

•3983 '3390 •2808 •2236 0.1671 •IIII '0555 '0000

25

69

1 '8549

TABLE 22B. SUMS OF SQUARES OF NORMAL SCORES This table gives values of S(n) = E [E(n, r)]2. r=1

S(n)

n

S(n)

3 4

o•0000 0•6366 1.432 2'296

xo xx 12 13 14

7.914 8.879 9.848 1o•82o 11.795

5 6 7 8 9

3'195 4.117 5'053 5'999 6.954

15 16 17

10

7.9x 4

n

2

n

S(n)

n

S(n)

n

S(n)

43 44

37479 38'473 39-466 40.460 41'454

20

17'678

30

27'558

40

2x

18.663

3x 32 33 34

28'549 29'540 30'531 3P523

41

32'515 33'507 34'500 35'493 36'486

45 46 47 48 49

42.448 43'443 44'437 45'432 46.427

37'479

50

47'422

22

19.649

23 24

20'635 21.623

12.771 I3.750 14'730

25 26

18

15-711

28

22.610 23'599 24'588 25'577

19

16 '694

29

26'567

35 36 37 38 39

20

17.678

30

27'558

40

27

42

TABLE 23. UPPER PERCENTAGE POINTS OF THE ONE-SAMPLE KOLMOGOROV-SMIRNOV DISTRIBUTION n'D(n) d(P). The distribution of n1D(n) tends to a limit as n tends to infinity and the percentage points of this distribution are given under n = co. This table was calculated using formulae given by J. Durbin, Distribution Theory for Tests Based on the Sample Distribution Function (1973), Society for Industrial and Applied Mathematics, Philadelphia, Pa., Section 2.4.

If F„(x) is the empirical distribution function of a random sample of size n from a population with continuous distribution function F(x), the table gives percentage points of D(n) = sup IFn(X)-F(X)1;the function tabulated is d(P) such that the probability that niD(n) exceeds d(P) is P/Ioo. A test of the hypothesis that the sample has arisen from F(x) is provided by rejecting at the P per cent level if 10

5

2'5

I

0•I

P

10

5

2.5

x

o•x

0.950 r098

0.975 1'191

0.9875 P256

0.995 P314

n = 20

P184

3 4

1'102 1' 130

1•226 1'248

1•330 1'348

1436 1468

0.9995 P383 1. 595

21 22 23

•185 •186 •187

24

•188

P315 •316 '317 •318 '319

1.434 '435 '436 '438 '439

1.576 •578 '579 •58o -582

1.882 .884 .887 •889 •890

5

P139

P260

P370 •382 •391 •399 '404

1'495 '510 •523 .532 '540

P747 '775 '797 .813 •825

25 26 27

1•188 •189 •190

P320

28

•190

29

•1 9 1

'322 •323 •323

P440 '440 '441 '442 '443

P583 •584 '585 •586 '587

1.892 *894 *895 •897 .898

1 '546

1 '835

'551 •556 '559 .563

'844 *851 -856 .862

30

P192 .196

P324 •329 •332 .335 •337

1•444 '449 '453 '456 '458

P588 '594 .598 -601 •604

P899 .908 '914 •918 •921

r565 •568 .570 .572 '574

1866 •87o '874 .877 .88o

P338 '339 •340 '346 '358

1'459 '461 '462 '467 '480

x•605 •6o7 •6o8 '614 '628

1.923 •925 •927 '935 '949

P n=I 2

6

•146

-272

7

•154

'279 •285 •290

8

59

• 1

9

•162

xo

1. x 66

I/ 12

•169 •171

13 14

•174 •176

•303 •3 05

P409 '41 3 '417 '420 '423

15

P177 •179 •180 •182 •183

P308 •309 •311 •313 .314

P425 '427 '429 '431 '432

x6 17 18 19

P294 '298 •301

1'701

40 50

•1 99

6o

•20I

70

•203

8o

P205

90 100 200

'206 '207

00 20

1'184

p315

p434

p576

P882

70

'212 •224

•321

TABLE 24. UPPER PERCENTAGE POINTS OF FRIEDMAN'S DISTRIBUTION Consider nk observations, one for each combination of n blocks and k treatments, and set out the observations in an n x k table, the columns relating to treatments and the rows to blocks. Let the observations in each row be ranked from I to k, and let Ri (j = I, 2, k) denote the sum of the ranks in the jth column. This table gives percentage points of Friedman's statistic

M

12

k

3n(k + 1)

nk( + 1) =

on the assumption of no difference between the treatments; the function tabulated x(P) is the smallest value x such that, on this assumption, Pr {M x} < Pima. A dash indicates that there is no value with the required property. A test of the hypothesis of no difference between the treatments is provided by rejecting at the P per cent level if M x(P). The limiting distribution of M as n tends to infinity is the X'-distribution with k - I degrees of freedom (see Table 8) and the percentage points are given under n =

k=3 P

lo

5

n=3

6•000 6•000

6•000 6.50o

8.000

8•000

5.200 5'333 5'429 5.25o 5'556

6.400 7.000 7'143 6.25o 6.222

7.600 8.333 7'714 7'750 8.000

8.400 9.000 8.857 9•000 9'556

10'00 12'00 12'29 12'25 12'67

9

5-000 5.091 5.167 4'769 5'143

6.2oo

7.800 7.818 8.000 7'538 7'429

9.600

12.6o 13.27 12.67

I0 II 12

12'46 13'29

13 14

4 5

6 7 8 9 I0

II 12

13

14

6'545

6.500 6.615 6.143

2.5

6.400 6.5oo 6.118 6.333 6.421

7.600

4'900 4'95 2 4'727 4'957 5.083

6'300 6•095 6'091

7'500 7'524 7'364 7.913

25 z6 27 z8 29

4'88o

6.080 6.077 6•000 6.5oo 6.276

7'440 7'462 7'407 7'71 4 7.517

3o 31 32 33 34

4'867 4'839 4'750 4'788 416 5

6•zoo 6.000 6•063 6•061 6•059

7400 7548 7.563 7.515

4'605

5.991

4'933 17

20 21 22 23 24

4'875 5.059 4'778 5.053

4'846 4'741 4'571 5. 034

6'348 6'250

7.625 7'41 2

7'444 7.684

7'750

I

9'455

9'500 9.385 9'143

o.z

5 n =3 4

6-600 6.3oo

740o 7.800

8.2oo 8.400

9•o00 9.60o

5 6

6.36o 6.400 6.429 6.3oo 6•zoo

7.800 7.600 7.800 7.650

8.76o 8.800 9.000 cr000

7.667

8.867

9'960 10'20 10'54 10'50 10'73

1 3'46 13'80 14'07

6.36o 6.273 6.300 6.138

7.680 7.691 7.700 7.800

10•68

14'52

7'714

10•75 10•80 10'85 10'89

1 4.80

6. 343

9.000 9.000 9.100 9.092 9.086

7.720 7.800 7.800 7'733 7.863

9.160 9.150 9.212 9.200 9-253

10'92 10'95 11'05 10•93 11'02

7.800 7'815

9.240 9'348

I 1'10

15'36

11'34

16-27

7 8

8.933 9'375 9. 294 9.000 9'579

12.93 13.50 13.06 13.00 1 3'37

18 19

6.280 6.300 6.318 6-333 6'347

9.300

13.3o

20

6.240

9.238

13. 24

CO

6'251

9.091 9'391 9.250

13.13 13.08

8.960 9.308

1. 10 12•60 12'80

1 4'56

14'91 15.09 15.08 15.15 15.28 15.27 1 5'44

1 3'45

5 n=3 4

13'52 13'23

9'407

13'41

9'172

13'50 13.52

9. 214

k= 4 2.5

5 6 7

7467 7.600

8.533 8.800

7680

8.96o 9.067 9'143 9'200

k=5 2.5 9.600 9.800

I

43•x

i0•13 11.20

13'20

'0.40 10'51 10'60

11'68 11'87 12'11 12'30

14'40 15.2o 15.66 16•oo

10'24

I I .47

13'40

8

13'42 13.69

9

7733

9'244

10.67

12'44

16.36

1 3'52 13'41

00

7'779

9.488

11.14

I3.28

18.47

7'471

9.267 9.290 9.25o 9.152 9.176

7'733 7'771 7.700

7'378

9.210

13.82

k=6

c1:4

71

P

zo

5

n= 3 4

8.714 9.000

9.857 10.29

io.81 11.43

11.76 i2-71

13.29 15.29

5 6

9•00o 9.048

10.49 1(3.57

11'74

16'43

00

9'236

11'07

13'23 13'62 15'09

2.5

12'00 12'83

I

O•I

1705 20'52

TABLE 25. UPPER PERCENTAGE POINTS OF THE KRUSKAL-WALLIS DISTRIBUTION Consider k random samples of sizes n,, n2, ..., nk respectively, n1 n 2 ... nk, and let N = n1+ n2 + ... nk. Let all the N observations be ranked in increasing order of size, and let R, (j = I, 2, ..., k) denote the sum of the ranks of the observations belonging to the jth sample. This table gives percentage points of the Kruskal-Wallis statistic k

12

H

N(N+ r)j

tinuous population; the function tabulated, x(P), is the smallest x such that, on this assumption, Pr {H P/ioo. A dash indicates that there is no value with the required property. The limiting distribution of H as N tends to infinity and each ratio n;IN tends to a ppsitive number is the x2-distribution with k I degrees of freedom (see Table 8), and the percentk). A test of age points are given under n;= oo (j = 1, 2, the hypothesis that all k samples are from the same continuous population is provided by rejecting at the P per cent level if H x(P). -

R2

E

3(N+ I)

on the assumption that all k samples are from the same con-

k n1, n2, ns 2,

=

re.

=3 5

I

cvx

ni, n2,

123

6, 5, 6, 6,

5

540 4:007

5:7 92 459 4

6, 6, 6 6, 4 3 2

4'438 4'558 4'548

5.410 5.625 5'724

6, 5 6, 6 I,

44 4...2L47 32

7 5:8o65i

2,

I

4'200

2, 2

4'526

4'571 4'556

5'143 5'361

5.556

3, 4, 4, 4, 4,

3, 3

4'622 4'500 4'458 4'056 4'511

5.600

5.956

5'333 5.208 5'444

5.500 5'833 6.000

6.444

6, 6, 7, 7, 7,

4, 4, 4, 4, 4,

3, 4, 4, 4, 4,

3

4'709 4'167 4'555 4'545 4'654

5'791 4.967 5'455 5'598 5.692

6 '155 6.167 6.327 6 '394 6.6i5

6 '745 6. 667 7.036 7'144 7.654

7, 7, 7, 7, 7,

5, 2,

I

4.200 4'373 4'018

6.000 6.044 6-004

6. 533

4'533

5.000 5.160 4.960 5.251 5.648 4'985 5.273 5.656 5.657 5.127

5'858 6.068 6.410 6 '673 6•000

6 '955

6.346 6.549 6•760 6.740 5.600 5'745 5'945 6.136 6.436 5-856

5,

3, I 3, 2

I

2

3 4

2, 2

5, 3, I 3, 2 5, 3, 3

5,

I

4.714

4'65 1

4, 4, 4, 4,

3 4

5, 5,

I

3'987 4'541 4'549 4'668 4'109

5, 5, 5, 5, 6,

5, 5, 5, 5,

2 3 4 5

4'623 4'545 4'523 4'560

2,

I

4'200

5'338 5'705 5.666 5.780 4.822

6, 6, 6, 6, 6,

2,

3, 3, 3, 4,

2 I 2

4'545 3'909 4'682 4'500 4'038

5'345 4'855 5.348 5'615 4'947

6, 6, 6, 6, 6,

4, 4, 4, 5, 5,

4'494 4'604 4'595 4'128 4'596

5.340 5.610 5.681 4'990 5.338

4'535

5.602

4'522

5•66 1

5, 5, 5, 5,

2

3 I 2

3 4 z 2

6, 5, 3 6, 5, 4

I

-

7.200

8.909 9-269

=3 5

3, I 3, 2

7

P=

Ic•

4'571 4 5 6 42080

2, I 2, 2

k

.o. 2'5

2 I 2

3, 3, 3, 3,

2, 2, 2,

P

4.706 5'143

2

3, 3 4, I 4, 2

6.848 6.889 5'727 5.818 5-758 6.201 6.449

I

0.1

8.028 7.121 7.467 7.725 8•000

10.29 9.692 9.752 10.15 10'34

8.124

10'52

8'222

10-89

7.000 7.030

-

6.184

6.839 7.228 6.986 7.321

8. 654 9.262 9.198

5 .'6520 3

6:7 5078 7

7 7:5 14 0

4:0 415

5 :0 39 634

1 :9 2:3

06510 7:4

3, I 3,

2'5 6.788 5-923 6.2to 6.725 6.812

7 4 5'3952

3 4.'5 1 2 4'603 4'121 4'549

4'986

5191

5'376

4:55 6 2 27

5.620

4, 3 4, 4 5, I

8-727

7, 7, 7, 7, 7,

5, 3

4'535

5'607

6.627

7.697

9. 67o 9'841 9.178 9.640 9.874

7.205 7'445 7'760 7.309

8.591 8'795 9.168 -

7, 7, 7, 7, 7,

5, 5, 6, 6, 6,

4 5 I

4'542 4'571 4'033

6'738 6.835 6.067 6'223 6.694

7.931 8.108 7'254 7- 490 7.756

io•16 10'45 9'747 10.06 to.26

7.338 7'578 7.823 8•000 -

8.938 9'284 9.606 9.920 -

7, 7, 7, 7, 7,

6, 6, 6, 7, 7,

2

6.970 7.410 7.106

8.692 -

7, 7, 7, 7, 7,

7, 7, 7, 7, 7,

6.667 5.951 6.196

7.340 7.500 7'795 7.182 7.376

8.827 9.170 9.681 9.189

6.667 6.750

7'590

7.936

9.669 9.961

6'315

6.186 6%538

6-909 7.079

-

6'655 6 '873

5, 2

2

4'500

3

4'550

5'733 5-708 5.067 5'357 5.680

4 5 6

4'562 4'560 4'530 3'986 4'491

5.706 5170 5'730 4.986 5-398

6.787 6 '857 6.897 6.057 6.328

8.039 8'157 8.257 7.157 7.491

10.46 10'75 11.00 9.871 10 '24

3 4 5 6 7

4'613 4'563 4'546

5.688 5.766 5.746 5'793 5.818

6.708 6.788 6.886 6.927 6.954

7.810 8.142 8.257 8-345 8.378

10 '45 10.69 10.92 11.13 11.32

8, x, I

4'418 4.011 4.010 4'451

4.909 5.356 4.881 5.316

5.420 5:0 86 14 7 6 6.195

4...050 31

5(D364 9134 7

6.588 g.•189 835

8,

2,

I

I

8, 2, 2 8, 3, x 8, 3, 2

4'568

4'594

4.587

8, 3, 3

8, 4, 1 8, 4, 2

72

6 6..8 60 64 3

7.022

8.791

6 7...3 3 97 50 3

89%940 921 36

TABLE 25. UPPER PERCENTAGE POINTS OF THE KRUSKAL-WALLIS DISTRIBUTION k=3 P = to

nb n2, n5 8, 8, 8, 8, 8,

k =4 -

5

2.5

I

0• I

ni, n2, n3, n4

9'742 icror 9'579 9.781

4, 4, I, 4, 4, 2, I

P=

3 4 I 2 3

4'529 4'561 3.967 4'466 4'514

5.623 5'779 4'869 5'415 5.614

6.562 6750

5.864 6.260 6. 614

7'585 7853 7.110 7'440 7.706

8, 5, 4 5 8, 6, 8, 6, 2 8, 6, 3

4'549 4'555 4'015 4'463 4'575

5.718 5.769 5.015 5'404 5.678

6.782 6. 843 5'933 6. 294 6658

7'992 8.146 7.256 7'522 7'796

10. 29

8, 8, 8, 8, 8,

6, 4 6, 5 6, 6 7, 7, 2

4.563 4'550 4. 599 4'045 4'451

5'743 5.750 5'770 5'041 5'403

6'795 6.867 6.932 6'047 6 '339

8-045 8.226 8.313 7.308 7. 571

10.63 10.89 11.10 10.03 10.36

8, 8, 8, 8, 8,

7, 7, 7, 7, 7,

3 4 5 6 7

4'556 4'548 4'551 4'553 4'585

5.698 5'759 5.782 5.781 5.802

6.671 6.837 6. 884 6.917 6.98o

7.827

8.1,8 8. 242 8.333 8.363

10•54 10.84 11.03 44.28 11.42

2, 2,

2, 2,

I, 2,

I, I,

I I

2, 2,

2, 2,

2, 2,

2, 2,

I 2

3,

2,

X,

X,

X

8, 8, 8, 8, 8,

8, 8, 2 8, 3 8, 4 8, 5

4'044 4'509 4'555 4'579 4'573

5-039 5.408 5.734 5'743 5761

6'oo5

io•16 10.46 10.69 10.97 1.18

3, 3, 3,

2, 2,

2,

I,

I

2,

I

2,

2,

2, 2,

6.920

7'314 7.654 7889 8.468 8.297

8, 8, 6 8, 8, 7 8, 8, 8

4'572 4'571 4'595 4.582

5'779 5'791 5.805 5.845

6.953 6.98o 6995 7'041

8'3 67 8.41 9 8.465 8.564

11.37 44.55

4, 4, 5, 5, 5,

8, 5,

9, 9, 9

4' 60 5

00, 00, OD

5'991

6.351 6.682 6.886

7'378

9. 210

10'04

10.64

9'840 io.n I0 '37

n4

2,

3,

2, 2, 2,

2, 2, I,

I 2 I

3, 3,

2, 2, 2, 2,

I

2,

3, 3, 3, 3, 3, 3,

2

I,

3, 2, I 3, 2, 2 3, 3, 3, 3, 2

3, 3, 3, 3 4, 2, I, 4,

2,

2,

4, 2, 2, 2 4, 3, I, I 4, 4, 4, 4, 4,

3, 3, 3, 3, 3,

2, I 2,

2

3, I 3, 2 3, 3

P= 5'357 5.667 5.143 5.556 5'644

5

25

I

5'679 6.467

6.667

6.667

5.833 6.333

6.250 6.978

6 '333 6. 244 6 '527 6 .600 6.727

6'333 6.689 7'055 7.036 7.515

7.200 7.636 7.400 8.015

6.026

7'ooO

7.667

8.538

5.250 5'533 5'755 5.067

5.833 6.133 6'545 6.178

6.533 7'064 6.741

7.000 7'391 7.067

5'591 5'750 5.689 5.872 6•016

6-309

6'955

6.621 6.545 6.795 6. 984

7.326 7.326

7'455 7.871 7'758 8.333 8.659

7'564 7'775

2.5

I

0•I

5'945 6.386 6.731 6.635 6'874

6'955 7.159 7.538 7.500 7'747

7.909 7.909 8.346 8.23, 8.621

8.909 9'462 9.327 9'945

8876 8.588 8.874 9'075 9.287

10'47 9.758 10.43 10'93 1I-36

4: 4,3 2: 3, 4, 4, 3,

2 I 2

5.182 5.568 5.808 5.692 5'901

4, 4, 4, 4, 4,

3 I 2 3 4

6.019 5.654 5'914 6.042 6•088

7.038 6.725 6 '957 7.142 7.235

7.929 7.648 7'914 8'079 8.228

oo, 00, oo, oo

6.251

7'815

9'348

4, 4, 4, 4, 4,

3, 4, 4, 4, 4,

n4, n2, n3, n4, n5

3, 3,

41.95

5.786 6.250 6.600

11.34

6'982 6'139

6.511 6.709 6955 6.311 6.6o0

2

I,

I

5

2'5

6.750 7.333 7.964

7.533 8.291

6'583

--

--

6.800 7'309

7.200 7'745

7'600

7.682

8.182

7.1n 7.200

7.467 7.618

8.538 8-06 z 8'449 8.8,3 8.703 9.o38 9'233

2,

I 2

6.788 7.026 6.788

2,

I

6'910

2,

2

7'121

7'591 7'910 7'576 7'769 8.044

3, 3, 3, 3, I 3, 3, 3, 3, 2 3, 3, 3, 3, 3

7.077 7.210 7'333

8.0o0 8.2oo 8.333

CO, 00, 2), CO, 00

7'779

9'488

3, 2, 3, 2, 3, 3, 3, 3, 3, 3,

I

6.750 7.133 7.418

2,

3, 3, 3, 3, 3,

11'70

2,

P = zo

I

I, 3, 3, I,

16.27

0'I

-

8.12,

-

--

8'127 8.682 8.073

9'364 -

8.576 9.115 8'424 9.051 9.505

9.303 10403 9'455 9'974

9'451 9.876

10'59

10-64

13'82

0.1

k= P=

2-5

-

I

7.600 8•348 8'455

7.800 8.345 8.864

8'455

2,

2

8.154

8.846

9.385

I,

I

7'467 7'945 8.348 8'731 9'033

7.667 8.236 8.727 9.248 9.648

7.909 8.303

8'564

I,

2,

I,

I,

2,

2, 2,

2, 2,

I, 2,

2,

2,

2,

2,

I,

2, 2,

2, 2,

2, 2,

I I I

Io

• 17

10'20

11'67

13.28

18'47

6

5

I,

2,

11.14

6.833 7.267 7.527 7.909

/42,

7'133

5'333 5.689 5'745 5.655 5.879

5

k=5

k=4 nb nz, n3,

70

0. I

8.648 9'227 9'846

9'773 I 0'54

8.509 9.136 9.692

9.682 I0'38

9'o30

3, 2, 3, 2, 3, 2, 3, 2, 3, 2,

9.513 -

2,

I, 2,

I I

7.133 7418 7.727 7.987

2,

2,

2

8'198

I,

I,

2, 2,

I, I, I 2,

2, 2,

3, I, I, I, I moo 3, 2, I, I, I 7.697 3, 2, 2, I, I 7'872 3, 2, 2, 2, I 8.077 3, 3,2, 2, 2, 2 8.305

3, 3, 3, 3,

8.909 9.482 9'455

8.667

10'22

11'11

8'564 9'045

8'615

9.128

8'923

9'549

10-15

n -oi

9.190

9.914

10-61

11'68

15.09

20*52

9.628

10.31

10'02 CO, 00, 00 , 00, CO, 00

73

9.236

11.07

12.83

• TABLE 26. HYPERGEOMETRIC PROBABILITIES Suppose that of N objects, R are of type A and N— R of type B, with R N—R. Suppose that n of the objects, n < N—n, are selected at random without replacement and X are found to be of type A. Then X follows a hypergeometric distribution with the probability that X = r given by

Here the rows correspond to types A and B and the columns to ' selected' and not selected' respectively, and the marginal totals are given. Fisher's exact test of no association between rows and columns, or of homogeneity of types A and B, is provided by rejecting the null hypothesis at the P per cent level if the sum of the probabilities for all tables with at least as extreme values of X as that observed is less than or equal to Pj roo. More extreme' means having smaller probability than the observed value r of X, given the same marginal totals. This test may be either one- or two-sided, as shown below.

p(rjN, R, n) =

(':)(Nn — T(Nn) This table gives these probabilities for N < 57 and n < R (if not, use the result that p(rIN, R, n) = p(rIN, n, R)). For N > 57 these probabilities may be calculated by using binomial coefficients (Table 3) or logarithms of factorials (Table 6). When N is large and RI N < c•r, X is approximately binomially distributed with index R and parameter p = n/N (see Table r); similarly, if N is large and n/N < o•1, X is approximately binomially distributed with index n and parameter p = R/N. If N is large and neither R/N nor n/N

Example. I 5 4 4 5 9

(X+.', — nR I N)I[R(N — R) n(N — n)I Nz(N — 1)11 is approximately normally distributed with zero mean and unit variance; a continuity correction of 2 , as with the binomial distribution, has been used. A representation of the data in the form of a 2 x z contingency table is useful:

✓ R— r R n—r N—R—n+r N — R n N—n N N

2 I I = 0 0'5000 I '5000

5 =

I 2

Rn 2 2

I

0'7500

•6000 .r000

I

.1429

I

•2500

0.8333 •1667

O I

O 06667

4 O I

2 0'5000 •5000

4

2 2

O

0•1667

x

•6667

2

'1667

N R n

2 I

= 0

6 2 x I

8

0-8571

4 I I 0'7500 '2500

I I

= 0

6 x O

7

N R n

0'3000

3

O 06667 I '3333

N R n

7

2 I

8

2 2

O I

01143 2857

O I

0.5357

2

-0357

7

2 2

•4286

2 2

I

0.0714 '4286

2

'4286

0

3

8 3

2

I

*4762

O

06250

2

•0476

I

•3750

3 4

O

0'4000

I

'5333

2

•0667

7 3

8 3

03571 '5357

2

'1071

001 43 •2286 '5143 •2286 •0143

028 57

6 3

I

. 571 4

0

O I

2

•1429

I

2

5 x

O

O cr8000 I .2000

•

-6000

2

'2000

0'2000

5 2 I

6 3 3

O o.6000 I •4000

O I

0 0500

2

'4500 *4500

3

•0500

8 3 3

o•1786

2

7 3 3 6 3

9 I z

O 0.8889 I

'III'

2

O

o•5000 -5000

2

O 0.1543 I '5143 2

'3429

3

•o2.86

8

O 0.8750 •125o I

3

'5357 2679 •0179

8 4 O o.s000 I -5000 8 4 2 O 0. 2143 I '5714 2

74

'2 1 43

2

0

0.4167

I

•5000

2

'0833

9

2 I

O 0.7778 I '2222

O

0'2381

I

2

'5357 ' 2143

3

'0119

9 4 I O I

2 2

O I

0.5833 '3889

2

'oz78

0.5556 '4444 2

O I

0'2778 *5556

2

'1667

9 4 3 I

2

3

9 3 I

0

O 06667

I

'3333

I

0

O o 8000 '2000 I I0 2 2 O I

0'6222

2

'0222

'3556

zo 3 I I

'3000

xo 3

2

O 04667 '4667 • 2

'0667

0'1190

. 4762 •3571 '0476

9 4 4

I

I0 I

= 0 0'9000 I ' 100

O 0.7000 9 4

0

9

N R n

I0 2 I

.0714

8 4 4 I

O I

9 3

r=

9 3 3

O 04762

7 3 I O 0. 5714 I •4286

N R n

8 4 3

r=

'3333

6

14

From the tables p(1114, 6, 5) = •2098. A more extreme onesided value is r = o, giving a total probability of •2378, not significant evidence of association or of inhomogeneity. If a two-sided test is required, r = 4 and r = 5 have probabilities •0599 and •003o respectively, less than •2098; the total is now .3007. When N > 17 and nR/N is not too small, a (two-sided) test of the hypothesis of no association, or of homogeneity, is provided by rejecting at the P per cent level approximately if x2 = N[rN — nR] 21[R(N — R) n(N — n)] exceeds A(P) (see Table 8). (Cf. H. Cramer, Mathematical Methods of Statistics (1946), Princeton University Press, Princeton, N.J., Sections 30.5 and 30.6.)

is less than o•1,

N R n

6 8

013397 '3175 2 '4762 •i587 3 •0079 4

xo 3 3 O I

0'2917 •5250

2

-175o •0083

3 I0 0 I

4

o.6000 '4000

TABLE 26. HYPERGEOMETRIC PROBABILITIES N R n 10 4

r=

2

N R n II 3

2

N R n 12 I I

0

0.3333

1' = 0

0.5091

Y= 0

I

'5333

I

x

2

.1333

2

.4364 *0545

0.9167 .0833

12 2

10 4 3 O I

0.1667 - 5000

2

- 3000

3 I0

O

'0333 4 4 0.0714

xi 3 3 0 0.3394 I

•5091

z 3

•1455 •006I

Ix 4 1 I

'1143 •0048

xx

io 5 x O I

0.5000 -5000

0.8333 •1667

12 2 2

o

0.6818

I 2

'3030 '0152

0 0.6364

I '3810 2 -4286

3 4

0 I

I

•3636

4 2

0 0.3818 •5091 x 2

'1091

II 4 3

12

3 i

o I

0'7500 *2500

12 0

3

2

I

0.5455 .4091

2

-0455

o 0'2121 22 3 3 12 - 5091 I 0'2222 2 •2545 0 0.3818

N R n

N R n

N R n

N R n

12

13

13 5 5

14 3

5 4

Y= 0 I 2

0'0707

3 4

' 1414 •oioi

'3535 '4242

I

.5556

3

•0242

2 '2222 10

5 3

ix 4 4 0 0•1061

O

0.0833

I 2

'4167

I 2

•4167 •0833

4

3

3

'4242

•3818 •0848 •0030

1

.4909

2

*1227

3

.0045

12 4

I

0

0.6667

I

'3333

12 4 2 I0

5 4

O

0'0238

I

•2381 .4762 - 2381 '0238

2

3 4

xo 5 5 0

0'0040

x

-0992.

2

•3968

3 4 5

-3968 .0992 .0040

II I I O

I

0'0091 '0909

II 2 I

O 1

0.8182 •1818

II 2 2

O •

0. 6545 .3273

2

'0182

II

3 O 0.7273 I •2727

II 5 I 0 I II

0. 5455 •4545 5 2

0

0'2727

I 2

O I 2

0'4242

•4848 '0909

5 5

12

.5455

O x

0. 2545 •5091

'1818

2

'2182

3

•.0182

II 5 3 O

0.1212

x

-2210

2

3

'441 9 '2652

4

' 044 2

5

•0013

I

'4545

0 0. 1414

*3636

3

.0606

I 2

*3394

3 4

.0646 .0020

II 5 4

•4525

O

0.0455

I

•3030

12

5 I

0 I

0.5833

2

*4545

3

•1818

4

'0152 12

6 x

12 0 I

0.5000 •5000

6

12

Ix

5 5

O I 2

0'0130 -1623 '4329

3 4 5

.3247 -0649 •0022

O I 2

2

0.3182 .5303 .1515

0 I

0 2273 . '5455 ' 2273

6

12

3

0 0'0909

x

.4091

2

409 1

3

.0909

12

6

4

0

0'0303

I 2 3

• 2424 .4545 . 2424 0303

4

0 I 2

6

5

0'0076 •1136

-3788 •3788 .1136 .0076

3 4 5

12 6 6 0 I 2

0'0011 •0390

3 4 5 6

'4329 '2435 .0390 .00 1 1

'2435

13 I I O I

0- 9231 -0769

5 3

O

0.8462

O

0'1591

I

'1538

•

'4773 •3182. .0455

2

3

r = 0 2704 4335 51 I .2.720 3

13 3 I I

'2308

13 3

2

0 0. 5769 '3846 I '0385 2

0 0.4196

x

-47zo

2

'1049

3

.0035

13 4 I 0

0'6923

x

•3077

13 4 0 I 2

2

0 . 4615 •4615 '0769

13 4 3

13 2 I 12

- 0128

2

'4167

5

•2821

2

x3 3 3

12 4 4

2

1

0 0-0265

12 12 4 3

7 = 0 0'7051

4 5

•2176 3 01 51 00

0 0.7692

xo 5 O

2 2

75

13 6 I O I

0'5385

O I 2

2

0'2692 '1923

x3 6

3

O I 2

0'1224 *4406 '3671

3

-0699

13 O I 2

3 4

6 4 0.0490 .2937 .4406 -1958 •0210

x3 6

5

I

- 1632

3

2 '4079

I

-4699

2 '3021

3 4 13 O I

.0503 .00i4 5 i 0.6154

3846

13 5 0

I 2

3 4

13 O I 2

6 6 0'0041 '0734 - 3059

'4079 '1836 •0245 6 •0006

3 4 5

'0714

'3916 '1119 '0070

0. 7143 - 2857

14

O I 2

4 0'4945 .4396 •0659

14 4 3

0 I

0.3297 '4945

2

'1648

3

•0110

14 4 4 O I 2

0'2098 .4795 - 2697

3 •0400 4 •0010 14 5 I x

0.9286

2

0 I

.3571

5 •0047

I

0.0979 .3916

-0907

•0027

0 0.6429

O

O I

2

3

•0816

14

13 5 4

0'4533 '4533

4

5128 '1282

2 '2797 .0350 3

O I

.3263

2

13 5 3 0 0.1958 -4895 I

14 3 3

3

0.3590 •

'0330

.5385

0.0163

0 0- 1762

• 3626

14 4 I 13 6

O

13 4 4

I 2

'4615

0 0'2937 -5035 I •1888 2

•0140

2

r = 0 0. 6044

14 2 I

O I

0.8571 •1429

14 2 2 O I 2

0/253 '2637 '0110

14 3

O 0.7857 '2143 I

14 5 2 O I

0.3956

2

'1099

'4945

14 5 3 O

0'2308

I 2 3

*4945 '2473 -0275

14 5 4

O I 2

0.1259 •4196

3 4

.0899

'3596 '0050

14 5 5

O

0'0629

• 2

'3 147 '4196

3 4

'0225

5

.1798

•0005

TABLE 26. HYPERGEOMETRIC PROBABILITIES N

R n

14

6

1

N 14

R n 7

5

N 15

R n

N

R n

N

R n

N

R n

N

4

15

6

x6

x

16

5

16

3

r= 0

0'5714

7' = 0

0'0105

r= 0

0.3626

I

-4286

x

•1224

I

. 4835

2

'3671

2

3 4 5

•3671 •1224

3

•1451 •0088

•0105

15

14 o

6 2 0.3077

I

•5275

2

.1648

14

6

3

0

14 0

7

6

•1510

I

2

14

7 7

0

0.0003

1

'3357

2

•4196

3 4

•1598

I

•0143

•0150

2

'1285

'2098

2

'4196

3 4 5

'2797 '0599

14

0030

6

6

0

0'0093

I

'1119

IS

•1285 0143 ;0003

I

1

•-0667 2

I

0.8667 '1333

2

3497

3 4

'3730 '1399

I

5

'0160

15

6

*0003

0

0.7429

I

'2476 '0095

14

7

1

0

0'5000

I

•5000

14 0

7

2

2 15

2

3

2

1

0

0.8000

I

•2000

I

0.2308 -5385

15

2

'2308

0

0.6286

I

'3429

2

•0286

14

7

3

2

0.0962, '4038 '4038

3

-0962

0

I

14

7

4

15 0

3

2

3

3

-0022

0'0350

' 2448

15

2

' 4406 ' 2448 ' 0350

0 j

15

1

0.6667

1

'3333

15 0

5

2

0 .4286

I

• 4762

2

'0952

15

5

3

0

0'2637

X

'4945

2

'2198

3

'0220

15

5

4

0

0'1538

I

'4396

2

'3297

3 4

. 0037

15 0

'0733

5

5

0-0839

I

'3497

2

'3996

3 4

'1499 •0167

5

'0003

15 0 I

15 0

6

1

0.6000 ' 4000

6

2

0-3429

I

5143

2

' 1429

0'4835 '4352

I

3 4

3

I 2

0

5

0

x

0.9333

0

15

3569

'3569

0

15

16

'3776 '3357

'4079 . 2448 '0490 •0023

0.0280

6

I

3 4 5 6

I

6

2

'3297 ' 0549

5

'0020

'0007

2

6

0 I

•0322

'0490 '2448

14 o

15

16

3 4

I

3 4 5 6 7

3 4 5

'4835

2

00699

-4196 '2398 '0450

'2418

•4615

4

2

r= 0 I

1

0.1538

6

0'0420 *2517

2

I

14 0

4

0 1

0'0023

0

3

4

0'2418

r=

5

'0791

4

I

0.7333 '2667

4

2

15

6

3

0 1

0.1846

2

*2 967

3 x5

o

0.0168

3 4 5 6 15

'1079 •0108 •0002 7

x

0

0 '5333

I

-4667

15

7

0.2667

2

'2000

7

6¢

0

0'0923

I

.3692

0

0'5238

2

x

•4190

'3956

2

'0571

3 4

•0110

'1319

2

I

0'8750

'1250

3

2

0

0•6500

I

' 3 2 50

'0250

3

3

0

05107

I

. 4179

2

•o696

3

.00'8

.0769

0

0•7500

I

.2500

•2872 4308 '2051

3 4 15

0

•0256

7 5 0•0186

I

1632

2

•3916 •3263

3 4

'0932 '0070

5 x5 0 I

7

0.0056

'0783

-2937 -3916 . 1958

2

3 4 5 6

.0336 •0014 7

7

2

0'5500

-4000

2

'0500

0

4 3 0.3929

1

'4714

2

•1286

3

•0071 4

5

5

•3231 '4154

•0002 6

x

16

I

'3750

3 4 5

'0721

x6

6

2

0

01750

I

•5000

2

'1250

16

6

3

o

0'2143

x 2

'4821 •2679

3

'0357

16

6

4

16

•Ixoi '3304

3 4 5 6

.3671

x6 0 I

.0514

2

* 2313

.1099

3 4

' 2570

3 4 16

•0082

6

5

o

0.0577

x

•2885

2

'4121

3 4 5

.2060 0343 •0014 6

6

$

6 7 16

'3125

16

-1828

16

2

'1964

'0002

3

'01 79

0 1

'0843 .0075 •0001

5

3

0- 2946

49 11

7

8

z

•5000

16 0

8

2

0.2333 '5333 '2333

8

3

0

0'1000

I 2

'4000 '4000

3

•1000

I

0•5625

x6

x

'4375

0

7

'0001

x

0

16

•0664

•0055

0•5000

16

3 4 5 6

3855

0

'3934 '2997

.3807 -3046 .0914 •0087

7

-3709

2

•0833

7

0'0031

•3956

•0005

'4583

0009

I 2

•0264

I

'0236

0- 1154

3

2

'1573

0

4

'0305

6

I

I

0'0012

7

2

2

I 2 3 4 5 6 7

- 0048

0'0105

•1888

2

-2885

0

0•0262

5

.20'9 '4038

x

0.4583

5

0.0288

0

o

7

0 x

'2176

x

'0192

'0126

'4835

5

.1731

2

4

0•6875

4

0'0692

I

0 I

7

0

2

x6

'4500

0.6250

16

x

0.1500

0

0•2720

0

76

x6

0

16 15

4

z

x

x6

6

4

0

x6

z6 16

3 4

3

x

•0625

'3777 ' 1259

i fs

2

3

2

'1875

x6

•0604 •0027

3 4 5

2

4

3 4

'2333

' 4308 '3692

7

'3375

'0083

1

0'0513

2

I

0. 123I

0

'4533

•3022

2

x

3

I

'3777

3

7

2

0•1058

0

15

0.1813

R n

r= 0 I

0

o

2

0'8125

16

r=

4

1

2

0.7583

I

2

3

0 '9375 '0625

0

x6

'5333

'4747 '0 440

x6

2

0 x

15

0

z

2

0

0'3000

z

.525o

2

* 1750

8

4

0'0385

I

•2462

2

'4308

3 4

•2462

. 0385

TABLE 26. HYPERGEOMETRIC PROBABILITIES N R n x6 8 5 r= o o.0128 I

'1282

2

'3590 '3590 • I282 •oiz8

3 4 5

16 8 6 O

0.0035

I

.0559

2

'2448

3 4

5

-3916 '2448 ' 0559

6

'0035

x6 8 7 O

0'0007

x

•0196 .1371 '3427 '3427 ' 1371 .0196 •0007

2

3 4 5 6 7

16 8 8 O I

0'0001 '0050

2

'0609 '2437 '3807 '2437 '0609

3 4 5 6 7 8

'0050 '0001

N R n 17 3 3 r = 0 0'5353 I

O

0'9412

I

•0588

17

I

2

'0618

2

3

'0015

17 4 2 0 0. 5735 I •3824 2

'0441

17

4 3 O 0.4206 1 '4588 2

' 1147

3

'0059

17 4 4 o 0-3004 x -48o7 .1966 2 3 .0218 '0004 4 17

5

1

O

0'7059

I

•2941

x7 5

2

O 04853 I '4412 2 '0735 5 3 O 0.3235 '4853 I

2 3

'1765 •0147

17

5 4 O 0.2080 I

'4622

2 2

2

'2773

O

0'7721

I

'2206

3 4

'0504 '0021

17

2 '0074 17 17

3 O 0.8235 x -1765

17 3

2

O 0.6691 I '3088 2

'0221

2

.4853 '1103

17 6 3 17 4 1 o 0.7647 .2353 I

2 I

O 0.8824 I •1176

17 6

r = 0 04044

'4015

17

17 I I

N R n

5 5

O I

0'1280 '4000

2

'3555 .1o67 '0097

3 4 5

'0002

17 6 I o I

6 0._47r •3529

o 0.2426 i '4853 2

3

N R n

N R n

17 7 6

x7 8 7

r = 0 0'0170

I 2

3 4 5 6

'1425 .3563 '3394 ' 1273 '0170

•0006

'2426

.0294

17 6 4

17 7 7 0 o•oo62 .0756 I

= 0 0'0019 I '0346

2

'1814

3 •3628 4 '3023 •1037 5 '0130 6 7 •0004 17 8 8 0 0'0004 '0118

0 0.1387

2

'2721

I

I

'4160

'3466

'3779 .2160 •0486 .0036 •000i

2 '0968

2

3 4 5 6 7

3 '0924 4 •0063 17 6 5 0

0.0747

I 2

'3200 '4000

3 4 5

' 1 778

-oz67 '0010

17 6 6 O

0'0373

x

'2240 '4000

2

•2666 3 4 •o667 5 '0053 6 •000l

17 8 x 0 I

0'5294 '4706

x7 8 0 I

2

0.2647 • 5294

2 '2059

x7 8 3 0 O'1235

I

*4235

2

'3706

3

•o824

17 8 4 17

O I

7 I

o

0'5882 '4118

x

17 7 2 O

0'3309

I

'5147

2 '1544 17

7 3 O o.1765 •4632 I '3088 2 .0515 3

17

7 4 O 0.0882 I '3529 2

'3971

3 4

.14.71 '0147

2

3 4

0 0.0204 I '1629 2 '3801

3 4 5

3 4 5

'3258 .1o18 '0090

17 8 6 o x

o-oo68 .0814 .2851 2 '3801 3 '2036 4 5 '0407 6 '0023

7 5 O 0.0407 I

'4235 .2118 '0294

x7 8 5

17

2

0'0529 '2824

'2376 '4072

'2545 •0566 '0034

77

3 .2.903 •3628 4 '1935 5 6 '041 5 '0030 7 8 •0000

TABLE 27. RANDOM SAMPLING NUMBERS Each digit is an independent sample from a population in which the digits o to 9 are equally likely, that is, each has a probability of-116. 84 28 64 49 o6 75 09 73 49 64 93 39 89 77 86 95

53 03 65 76 31 97 97 o8 4! 70 44 z6

87 57 37 49 24 65 97 52 46 33 59 68

75 09 93 64 33

12

83

76

z6 29 51

45 70 14

OI

14

04

49 59 50 32 73 8z 98 27 74 00 78 6o

36 88 8o 79 98 77 12 39 04 40 71 92

23 92 z6 69 o8 73 19 16 79 86 16 99

36 17 74 41 05 o8 82 42 72 92 41 6o 25 35 05 44 8o 31 79 46 65 8z 31 96 40 50 44 68

36 78 65 96 81 75 74 o6 96 37 69 17 61 97 01 97 89 13 24 34 8o 31 81 o6 64 6z 77 58 33 57 36 65 92 02 65 63 22

50

4z 87 41 46 56 35 81 69 6o 05 88 34 29 75 98

56 77 39 73 69 96 89 II 8o 49 36 ii 19 03 51 77

22

09

30 77 03 46 65 68 93 61 54 z8 61 68

12

21

76

13

39

73 85

59 68

53 66

04

60

30

10

44 89 55 67 57 21

63 38

8o 13 77 42 76 97 95 85 98 17

46

99 85 50 92 69 51 27 44 40 92

20 40 19 72 89 73 52 12

84 74 69 25 03 59 91

02

30 76 81 6z 49 07

48 46 6z 68 12

16 98 04 61 64 27 86 93 25 00

20

29 96

75 68 48 82

20

14

20

10 73 62 73 19 92

18 85 76 06 52 68

91 86 46 70 67 68 00 76 64 85 26 32 71 57 99 51 81 14 35 II 15 17 24 78 76 49 97 56 11 76 04 47 18 85 z6 04 92 27 28 47 61 o8 89 81 zi

o6 69 53 91

87 22

19 99 97

63 55 38 31 97 56 43 15

4 6o 52 5

2 17 5

09 14 61 09 91 93 19 58 85 24 35

54 14 65 59 28 19 77

12 z8

zi

09 57 74 70

76 46 II 99 85 6o 39 22

8z 72

58 83 54

z6 21

64 40 56 72 0! 90 49 85 88 19 37 37 04 79 64 II 83 4 94 64 67 54 io 88

16

96 27 94 42

16 53 z8 78 36

92

52

12

83 97 II s 15 3 1 -1 28 94 54 6o 39 16 16 33 33 46 6 3 37 15 89 94 15 97 16 54 32 76 86 21 25 i8 6o 48 64 6i 48 63 10 76 58 38 98 R 17 32 2 3 33 23 89 45 08 44 6o 15 84 (D 7II 44 39 98 68 io 66 69 87 95 87 65 70 5 85 83 12 33 43 24 96 56 97 63 97 17 83 00 6o 65 09 44 77 96 43 40 11 36 44 33 05 40 43 42 91 65 62 83 53 05 20 53 70 52 51 62 2 3 74 76 96 88 83 69 o8 24 6z 95 47 58 6z 35 22 35 72 22 73 91 58 76 56 87 00 6z io 22 06 84 03 83 87 00 87 76 6z 31 65 91 30 71 56 08 3 03 74 8 o 77 40 59 16 0 7 72 96 25 59 35 69 71 3! 20

66 86 09 37 07 97

91 18 92 47

2 02 20 24 22 86 73 49 00 42 27 44 23

83

37 35

37 96

25 34

88 84 00 33 35 30 61 34 35

1 61 9 70 84 83 87 67 67 22 03 17

24

14

6o 8o 42 08 57 24 58 44 33 75

83 17 29 54 05 33 89 43 12 15

78

69 78 10 02 43 10

6o 76 06 64

41

91 14 26

98 34 65 28 37 6o 92

8o 30

9z

91 30 17 41 57 41 46 II 68 77

90

69

93 73 95 32 72 64 87

68 52 52 oit 78 77 63

29 21

26

90

55 27 74 48 41 02 o6 42 87

66 09

87

o8 8 92 67 o8 93 19 72 47 89 29 02

62 99 81

21

17 68 03 12 8z 82 o6

67

40

53 19 8z 41 93 94 16 61

65 33 63 48 52 68 66 89 14 94 45 o6

21

04 07 6o

38 88

73

22 21

40

22

6z 64 03 95 45 of 34 76

52

42

63 8 67

72 79

10

36 33 04

21

o6

14 63

04

97 99 94 13 30 95 II 49 90 86 51 55 90 19 39 67 88

9 98

4 67

39 76 53 38 70 56

90 07 47 04 48 83 90 36 97 56

02

74 55 67 43 72 63 40 03 07 47 02 62 20

19

45 23 8o 99

16 89 52 85 91 00 z8

86 39 33 73 48 14 91 z6

75 58 30 54 52 of

68 33 95 70 02

00 20

41 30 44 70

72 49 7

29 33

48 47 52 88 7 46 5 36

47 21 86 6i 0 8: 89

22

85 47 95 Jo

22

76 01 23 44 65 92 15 17 55 09 83 z6 o8 62 78 39 54 55 19 57

02

z6 44 58 40 o6 59 12

04 17 69 65 31 65 58 01 85 90 74

TABLE 28. RANDOM NORMAL DEVIATES Each number in this table is an independent sample from the normal distribution with zero mean and unit variance. 0.7691 -0.5256 0.9614 0.3003 1•1853

1.0861 P5109 0-3639 1.7218 -1.7850

0'2411 0'2614 1'0204 1'0167 PI2I1

-0•2628 0'3413 0.8185 -0-1489 0.4711

0.2836 - P7888 -0. 4654 0.7887 0.9046

-0.9189

-0.2884

0.9222 0.0989 -0.8744

-0'1051 -0'2588 -0'7164 0•1110 -0'3172

-0'7442

P9225 0.8966

-0'2911

- P4II9 - p2664 -0'4255 -0'2201

1'5256

0•6820

0 '4547 0.0213

0.8897 0.4814 -0.4014 -0.9607 P258o

1.8266 0.8452 - p4908 0.2071

-0 ' 40 70

-0'5601

-0'2685

-

-0'3797 -0.9013 1.6169

-0.6995 -0.0617 1.2541 P4531 -0.8250

0.8620 24288 0.0450 0.2532 -0.9363

-0.8698 0.4890 -0.1291 0.1939 0.2668

-0.1144 -0.0339 -0.1236 01285 -0.7122

0.5823 0.7836 0.1600 1.0383 0•1671

0.1137 0.4104 -0.3707 -0.4590

-1.1334 0.2755 0.3674

2'1522

0'2133

0'3379 0. 34-44 -0-3912 -0-5941 -0.0768

- V06 37 -0 ' 0948

0. 4198 -1•8812 0. 0778 0.8105

23696 -0.3968 1.1401 -0.1992 0.7894

-0.5056 -0.5669 0.7913 0- 9914 1.7055

-1.9071 -0-3260 0.4862 -0.8312 -1-9095

0.7738 -I-3150 P2719

I•I404 0'1964 -0'2309 0'9651 - P1403

P 6399 0.7164 1.9058 -0'5440 -1'3378 1'5939

0.7378 p4320 0.5276 -2.1245 P6638 -0.0429 -0'1320 -0'2098 -0'1563 -0'8492 -1'8692

-0'5447

0.6299 -0.1507 - P1798 1.3921 I•1049

P2474 P1397

-

P9211 0 '4393

-0'4597

0'7258

I.4880 -0.4037 -1.1855 -0.0251 -0.4311

PI002

0'1176

-0'6501

1'7248 -1'0621 0'9133

-0.456o 0.8729 0.3646 0.1885

-0-9633 0.6117 1.0033 2.1098 0.8366

P2725 1.0492 -1.1969 0.2146 -0'3594

P2193 0'2024 0'4722 -0'2230 -0'4389

- P5498 -I•5027 -16761 0. 1433 0 '7375

-0.5608 -2'6278

0.7064

-0.5104 -0.6110

0'2239 -0. 6841 0'2177

0•5802 0.0556 2'0196

-18613 0.8646 - V0438 0.2533 -0. 4953

0'8258

0.6571 0.0679 0.9970 0.6705 -0.1985

-0'0917

0•9517 0'7272 -0'2223

0.5697 0.4869 -1.3296 -0.1765 -0.2777

-0.7707 0.3990 0. 9649 -1'5953 -0.0382

-0.2558 - P5290 -0.7695 0.0153 -2.6190

-0.3052 - P2785 -0.3767 •o193 - P6919

-0.1825

0.2461 0.8169 0.0790 0.7225 0.9014

-0'4089

0.3838 0.5219 0 '3779 -1'9919

-0.1679 -0.004I -0.3111 0.7127

0.1428

-0'3749

0.0722 -o-7849

-0'6237 1.5668 0.45 21

0.1263 0.1663 -0.2830 1.2061 -1.4135

P0313 1'8116 - P9062 2'2544 -0'6327

0. 5464 -1.4139

0.1833 -1.6417

-0'4379

-0'3937 -0.0851 -0.2088

0.2889 -0. 1720

0.3946 -0. 0909

0'7673 - P9853

p6459 0.8910 P1387 0.6764

0. 2794

0'8007

-0/296 -0.5887

0.2325

- P0242 - P4929

-0'1122

-

1.2370

0'7125 P1813 -0'8344 -0'8015

0.6259

-1.4433

-0'3211

-1.2630

0•7890 - P2187 -2.6399

-0'7494

-0.9592 - 1.1750 -1.2106

-0-7991 1.0306

-0'6252

0'4124

-

-1.5349 0 '3377

-0'7473

0.4011 -0.2193

-0'5749

-0.4263 -0.1614 0.5114

-0'3337

-0. 5094 0.2414 -0.5231 1.8868 0. 6994

0'2013 P3071 - P2255 -0•6109 -0'7522

0.4899 0.9586 -1' 2 947 0.5067 0.5021

1.3002

0'4787 -11240 - 2'0142 -0/707 0'1095

-0'0994

1'0811 P5762 0/726 -0'8558 1'3333 -0.1512 -0-0416 P1621 -o.2743

v7767 - p8o3I 2.1364 0.1295 0. 2454

1.8961 0.3863 -0.0943 -0.7132 0.248,

V1558

-0-0814

-0'7431

P4721

0.5274

-0'3755

-0-3730 0.1049 -0.1819 I.2704 0.6109

-0.8716 0.36o2

1'3133 0.4986 -12309 0.2453 -i•1675

0•1115 0.6226

-0'2521 -0•5518 -11584

79

-0.6933

-

-1.1761 -0.5156 -0.5671

P2094 1'0500 0'6314 P5742

0.6715

-0'7459

0. 1543 -0.1108 0.5970 0.0842 - P5805

-0. 2542 -0-4762 0.9149 -0.8178 -0.5567

0. 0945

0.7350 - P6734 -1.8237

-1'0210

P6674 -0.8427 0.0398 0.5787

1'5481

0.2236

-0.8393 -0.2523 1.4846 0.8527 0 '5541

- 0 • 7841 -0'9212 -2'4283 0'0046

-0.3490 -0.6525 -10642 -i.o6o6

0 '4431

0 '9379

3'4347 0.7703 -0.6135 P3934

P0013 0'8129

-0.4194 -0.0173 0.5664 -0.7216 -0'3349

0.8688 I.41 I0 -

P40II 0'6272 P2943

P4732 2'1097 0'5529 -0'1047

0.6484 0.9714

0.0982 -0.6845 P6357 P2928 0.1785

-0.2936 -1.1208 0.7254 -1•1351 P4302

0.6226 0.6017 1.1673

-0.0962 -11846

- P4375 po786

0 '5354 0.6161

0.6177 -0.4680 0.5638 0.3650 -0.1867

1.0176 -0.6125

-0.0772 0.9166 p4564 -0.2898 1.0821

1.1889 p5315 -0.7601 -0.2105 -0.6962

0 '4598

- P7724

0.5089 P3761

TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL PARAMETER If r is an observation from a binomial distribution (Table 1) of known index n and unknown parameter p, then, for an assigned probability C per cent, the pair of entries gives a C per cent Bayesian confidence interval for p. That is, there is C per cent probability that p lies between the values given. The intervals are the shortest possible, compatible with the requirement on probability. The tabulation is restricted to r < Zn. If r > Zn replace r by n - r and take 1 minus the tabulated entries, in reverse order. Example 1. r = 7,n = 12. Use n = 12 and r = 5 in the Table, which at a confidence level of 95 per cent gives 0.1856 and 0•6768, yielding the interval 0.3232 to 0-8144. The intervals have been calculated using the reference prior which is uniform over the entire range (0,1) of p. The entries can be used for any beta prior with density proportional to pa(1-p)b , where a and b are non-negative integers, by replacing r with r + a and n with n + a + b. If r + a is outside the tabulated range, replace r + a with n - r + b and n with n + a+ b, and take 1 minus the entries, in reverse order. Example 2. r = 7, n = 12. If the prior has a = 2, b = 1, then r +a = 9, n+ a + b = 15 and n - r + b = 6. Use n = 15

posterior probability density of p

,

p

(This shape applies only when 0 < r < n. When r = 0 or

r = n, the intervals are one-sided.) and r = 6 in the Table, which at a confidence level of 95 per cent gives 0.1909 and 0.6381, yielding the interval 0.3619 to 0.8091. When n exceeds 30, C per cent limits for p are given approximately by

± x(P)[i(1 - P)/n]1 where /5 = r/n, P = 1(100 C) and x(P) is the P percentage point of the normal distribution (Table 5). -

CONFIDENCE LEVEL PER CENT

90 n =1 r=0

95

99

99'9

0.0000

0.6838

0.0000

0.7764

0.0000

0.9000

0.0000

0•9684

0'0000

0'5358

0'0000

0'6316

0. 0000

0'7846

0•0000

0'9000

'1354

.8646

.0943

.9057

-0414

'9586

•0130

•9870

0.0000

0'4377 •7122

0.0000 -0438

0.5271 •7723

0.0000 .0159

0•6838 •8668

0.0000 -0037

0'8222

'0679

r=0

0. 0000

0. 4507

0.0000

0. 6019

0•0000

0. 7488

•0425 •1893

0•3690 •6048

0.0000

I

•0260

•6701

•0083

•7820

•0016

'8788

•8107

•1466

'8534

•0828

•9172

-0375

.9625

0.0000

0. 3187

0.0000

0. 3930

0'0000

•0302

•5253

-0178

•5906

•0052

05358 •7083

0.0000 .0009

0•6838 •8186

•I380

•7

•I 048

•7613

*0567

'8441

•0242

.9133

n

=2

r =0

n

=3 =

n=

0

4 2

n

'9377

=5 =0

2

8o

TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL PARAMETER CONFIDENCE LEVEL PER CENT

90

r=0

99

95

n=6 '4641

0.0000 '0133

•22 53

•6317 '7747

•0805 '1841

•5273 -6846 •8159

r= 0

0•0000

0'2505

0.0000

0'3123

0•0000

0 '4377

I

•0185

•0105 •065o •1488

'4759 •6210 '7459

•0028 .0331 .0934

.5913 •7174 .8227

0.0000 •0004 •0129 .0495

0.5783 •7113 •8115 '8912

0. 0000

I

n=

2

'02 31 '1076

3

0.2803

99'9

0'3482

0'0000

0'4821

0'0000

o'6272

•0037 •0421 •1177

•6452 •7769 •8823

-0006 •0171 •0639

•7625 •8616 •9361

7 2

•0878

3

•1839

'4155 •5677 •7008

r= 0

0'0000

0'2257

0'0000

0'2831

0'0000

0'4005

0.0000

0'5358

I

•0154 -0739 '1549

•3761 '5152 .6388

•0086 '0542 •1245

'4334 •5676 •6854

.0022 -0271 •0769

'5451

'2514

'7486

'2120

'7880

•1461

•6651 •7679 '8539

•0003 -0103 •0400 .0884

•6651 '7645 •8463 •9116

r=0

0.0000

0'2057

0.0000

0.2589

0.0000

0•3690

I

'01 32 •0638 '1337

'3435

'4714 •5863

'0073 •0464 •io68

'3978 •5224 '6332

•0018 •0229 '0652

•5053 •6192 '7184

'2165

•6901

•1816

'7316

'1237

'8039

0.0000 •0003 •0085 '0333 '0739

0.4988 '62 37 •7212 •8032 •8714

0'1889

0•0000

0'2384

0.0000

3 4

'1175 •1899

•0063 •0406 .0934 •1586

•3675 '4837 •5880 •6818

•0016 '0197 •0564 .1071

0'0000 '0002

0'4663

•3160 '4344 '5416 .6393

0•3421 '4706

2

0'0000 '0115 •0560

'5788 -6741 '7578

•0072 •0284 •0632

-5866 •6817 •7627 •8320

5

0'2712

0'7288

0.2338

0'7662

0'1693

0'8307

0'1100

0'8900

r=0

0'0000

•0102

0.1746 •2926

0'0000

I 2

'0499

3 4

•1047 •1691

5

0'2411

=8

n

2

3 4 n

=9 2

3 4

n

= Io r=0 I

n=

II 0'2209

0'0000

01187

0.0000

.4027 .5030 '5951

-0055 '0360 •0829 '1407

'3415 '4502 '5485 '6377

.0013 •0173 '0497 .0943

'4402 '5431 '6344 '7156

•0002 •0062 .0248 '0551

0'4377 '5534 -6456 •7252 '7943

0.6803

0'2070

0'7191

0'1488

0'7878

0'0958

0'8539

See page 8o for explanation of the use of this table.

81

TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL PARAMETER CONFIDENCE LEVEL PER CENT

90

99

95

99'9

rt = 12 r

=0

0•0000

I

•0091

2

'0449

3 4

'0944 •1524

•5564

5 6

0.2169 •2870

I

0.1623 •2724

0.0000 •0049

0.2058 •3188

0.0000 •0012

'3753 '4695

•0323 '0745 •1263

•4210 '5138 •5987

•0154 '0443 •0841

0'6374 '7130

0.1856 .2513

0•6768 '7487

0.0000 •0082

0.1517 •2548

0.0000 •0044

2

-0409

3 4

•1386

'3514 '4400 •5223

•2604

0.0000

0.2983

0.0000

0.4122

'4134 •5113 '5987 •6773

•0002 •0055 .0219 •0488

•5234 •6128 •6905 '7588

0.1326 •1887

0 '7479

•8113

0.0848 •1290

0.8188 •8710

0.1926 •2990

0.0000 . 001 I

0.2803 '3896

0.0000 •0001

0.3895

•0293 •0676 '1146

'3953 •4832 '5639

•0139 •0400 '0759

•4829 •5666 '6424

-0049 •0196 '0437

0'5994 •6717

0-1682

0.6388

0.1195

•7082

•1698

0.7112 '7738

0.0759 '11 54

0'7855

•2274

n = 13 r

=0

5 6 n = 14 r=0 I

.o859

0.1971

'4963 •5828 '6585 '7257

'8384

0'1423

0'0000

0.1810

0.0000

0.2644

0.0000

0.3691

•0075

'2394

.0375 •0788

.3303 '4140

.0040 -0267 •0619

•1271

.4921

'1048

•2814 -3726 '4559 '5329

.0009 -0126 '0364 •0691

•3684 '4574 '5376 •6106

-0001 '0044 •0177 '0396

•4718 '5554 •6290 '6948

o•i8o6

0-1537 '2075 •2659

0.6045 '6715 '7341

0.1087 '1542 '20 51

0.6775 •7388 '7949

0.0687 •1043 '1457

0 '7539

-3000

0•5654 '6346 •7000

r=0

0'0000

0'1340

0.0000 •0009

0.3506

•2257

0.1708 •2658

0.0000

•0069

0.0000 '0037

0.2505

i 2

-0346

-3116

3 4

•0728

'3909

'1173

'4650

•0246 -0570 '0966

'3522 '4315 '5049

'0115 .0334 '0634

'3493 '4344 .5113 •5817

•0001 '0040 •0162 •0361

'4495 '5303 •6017 •6660

0.1666

0'5349 •6012 •6641

0.1415

0'5736 •6381

0'0997 .1413

0.6465

0.0627

. 1909

•7063

.0951

'2442

•6988

•1876

-7615

•1327

0.7242 '7770 •8246

2

3 4 5 6 7

'2383

•8070 '8543

n = 15

5 6 7

•2197 •2762

See page 8o for explanation of the use of this table.

82

TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL PARAMETER CONFIDENCE LEVEL PER CENT 90

99

95

99'9

n = 16 =

0

0.0000

0.1267

0.0000

0.1616

0.0000

0.2373

•oo64

•2135

2

•0321

•2949

3 4

•0676 .1090

.3703

•0034 •0228 •0528

'4408

•o895

•2518 '3340 .4095 '4797

•0008 •oio6 •0308 '0585

•3320 '4135 '4874 '5552

0.0000 •0001 .0037 •0148 .0332

0'3339

i

5 6 7 8

0.1546

0.5075

0.1311

-2037 '2558 •3108

'5710

0'5455 •6076

•6314 •6892

•1767 •2258

•2781

•7219

o•o920 •1303 •1728 -2193

o•618o •6762 .7304 •7807

0.0576 •0873 •1217 •i6o6

0•6964 •7486 •7962 '8394

It = 17 r= 0

0.0000

0.1201

0.0000

i

•006o

-0032 •0492 .0834

0•1533 .2393 .3175 •3897 '4568

o•0000 •0007 '0099 •0286 '0544

0.2257 •3164 '3945 '4656 .5310

o•0000 •0001 .0034 '01 37 .0307

0.3187 •4105 •4860 '5533 '61 44

0.5200 '5798 •6366 •6905

0.0854 •1209 •i6oi •2030

0.5917 •6483 .7013 '7508

0.0533 •0807 •1124 •1481

0.6703 •7217 -7690 -8124

0.2152 •3021

0.0000 •0001

0.3048

-0031 •0128 •0286

.3934 '4665 .5317 '5912

r

2

•0300

3 4

•0631 •1017

•2025 .2799 •3516 '4189

5 6 7 8

0.1442

0.4827

0.1221

'5435

•0213

•6664

'4292 .5073 .5766 .6393

•1899 •2383 •2893

•6o17 '6574

•1644 -2099 -2583

0.1141 •1926 •2663

0.0000 •0030 . 0199

0.1459

0.0000

2

0.0000 •0056 •0281

•2279 '3026

•0007 •0092

3 4

'0592 '0953

'3348 '3991

.0461 •0781

'3716 -4360

-0267 •0508

'3771 *4455 •5086

5 6 7 8 9

0.1351

0.4602

0.1142

0.4967

0 '0797

0'5674

•1778 .2230 -2705 •3201

•5185 '5745

-1537 •1962

•6282

•2412

•6799

•2886

'5543 •6092 -6615 . 7114

•1127 '1492 •1889 •2316

•6224 '6741 •7228 •7684

0.0496 '0750 .1044 '1374 •1738

0.6459 '6964 '7432 -7864 •8262

r= 0

0.0000

0• I087

0'0000

0'1391

I

-0052

•1836

•0028

•2175

2

•0264

'2540

•0187

•2890

'3551 •4170

0•0000 •0006 •0086 '0251 .0476

0.2057 •2891 •3612 •4271 •4880

o.0000 •0001 •0029 -0119 •0267

0.2920 '3776 '4484 •5117 -5696

0'4754

0.0747 .1056 '1397

0'5449

'5984 •6488

0.0463 •0701 '0974

o•6231 •6726 '7187

•6964

•1281

•7615

'7413

-1619

-8013

n = 18 r=0 I

n = 19

3 4

'0557 -0897

'3194 •3810

.0433 .0734

5 6 7 8 9

0•1271 •1672 •2096 •2540 .3004

0.4396 . 4957 '5495 •6014 •6514

0.1073 .1444 '1841

'5309 •5839

'2261

•6346

•1766

•2704

•6832

•2164

See page 8o for explanation of the use of this table.

83

TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL PARAMETER CONFIDENCE LEVEL PER CENT 90

95

99

99'9

n = 20 r=0

0.0000

0. 1039

0.0000

0.1329

0.0000

0'1969

0'0000

0.2803

I

•0049

'1 754

•0026

2

'0249 •0526

•2428 '3055

•0176

'3645

.0409 '0692

•0006 •oo8i •0236 '0448

•2771 •3466 •4101 '4690

•000i •0027 •0112 •0251

-3630 '4315

-o847

•2080 •2766 •3401 '3995

'5494

5 6 7 8 9

0'1200

0•4208

0'1012

0 '4557

•I578 .1 977 •2395 -2828

'4747 •5266 .5767 .6253

.1361 •1734 •2129 '2544

'5093 5606 •6097 •6569

0.0703 '0993 •1312 •1659 '2030

0.5241 .576o •6253 •6717 •7158

0.0435 •0657 .0913 •1199 .1514

0.6016 •6501 '6955 '7379 '7775

io

0'3281

0.6719

0'2978

0•7022

0.2425

0'7575

0.1856

0•8144

0

0.0000

0.0994

0.0000

0.1273

I

-0047

•0025 •0167

'1994 '2652

0.0000 •0006

-3262 '3834

•0223 '0423

0.1889 -2661 '3331 '3944 '4514

0.0000 •0001 •0026 '0105 •0236

0.2695 '3494 . 4159 '4757 •5306

0.0664 '0937 •1238 •1563 '1912

0.5048 '5552 •6030 •6485 '6917

0.0409 •0619 •0859 •1128 .1423

0.5815 -6290 '6735 •7153 '7546

3 4

'4931

n = 21

r=

n=

2

•0236

•1679 '2325

3 4

•0498 •0802

•2926 '3494

•0387 .0655

5 6 7 8 9

0.1136 ' 1493 ' 1870

0.4035 '4554

0.0957 '1287 '1639

I0

•0076

•2265

'5055 '5539

•2675

•6007

•2402

0.4376 '4894 '5389 •5866 •6324

0.3099

0.6461

0.2809

0•6766

0.2281

0.7328

0.1742

0.7914

0

0.0000

•0044

0'0000 '0005

•0224

'0473 •0762

•2809 '3354

•0621

'2547 '3134 -3686

-0072

3 4

0'0000 '0023 •0158 0367

0'1221 '1914

2

0'0953 •1611 '2231

0.1815

I

'2557 •3206 '3798 '4350

0-0000 •0001 •0024 .0099 •0223

0.2594 '3369 •4014 '4594 •5129

5 6 7 8 9

0.1079

0.3875

0.0908

0.4209

0.0628

-1418 •1 775

'4376 '4859

'4709

.5189 •5651 •6096

•0887 •1171 '1478 •1807

0.4868 '5358 -5823 -6267 -6690

0.0387 •0584 •0811 •1064 -1341

0.5626 •6091 -6528 '6939 •7328

I0

0.7094 '7479

0-1641 '1963

0.7693 •8037

•2011

22

r=

ii

'0211

-0401

•2148

'5327

-2536

-5781

-1220 •1554 '1905 -2274

0.2937 '3351

0.6221

0.2659

0'6526

0.2154

'6649

'3059

'6941

'2521

See page 8o for explanation of the use of this table.

84

TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL PARAMETER CONFIDENCE LEVEL PER CENT 90 II =

95

99

99'9

23

r=0

0.0000

I

•0042

2

•0214

0.0915 . 1547 •2144

0.0000 •0022 .0150

0.1173 •1840 •2450

3 4

.0451 -0726

•2700

.0349

3016

0.0000 •0005 •0069 •0200

0.1746 •2465 •3089 •3663

. 3225

.0591

'3548

.0380

*4197

0.0000 •0001 •0023 •0094 •0211

5 6 7 8 9

0.1027

0 '3728

0.0864

0.4053

0.0597

0.4700

0.0367

0'5448

-1349 -1689 •2043

'4211 *4678

•1160 •1475

'4537 •5005

•5131

•1810

.5450

'2411

'5570

•2160

•5883

'0842 -II 11 •1402 •1712

.5176 •5629 •6062 •6476

.0554 •0768 •1007 •1269

•5903 •6331 .6736 .7119

I0

0.6872 •7251

0.1551 •1854

0.7481

0.2791

0.5998

0.2524

0.6302

0.2041

ii

'3183

•6413

•2902

•6707

•2386

/1 = 24 r=0

0'woo

0•0880

0.0000

0.1129

0.0000

I

•0040

•0021 •0143

•1772

•0005

0.1682 '2377

0.0000 •0001

li

0.2505 •3252 •3878

*4442 '4963

•7824

2

•0204

•1489 •2063

•2360

•0065

•2981

•0022

3 4

'0430 •0692

'2599 •3106

.0333

•2906

•0191

'0090

•0564

•3420

•0362

'3537 '4055

0.2414 •3142 '3753 '4300

•0201

•4807

5 6 7 8 9

0.0980

0.3591

0.0824

0'3909 '4377

0.0568 '0799

0.5280 '5725 '6145 '6543 •6920

I() II

•4828

•1057

'5447

'5374

'1724 '2056

.5263 •5684

'1333 •1627

'5869 •6274

0'0348 •0526 '0729 '0956 •1203

0.2659

0.5789

0.2402

12

•3031 '3414

-6193 •6586

•2760 •3131

0.6092 •6487 -6869

0.1938 -2265 •2607

0•6662 '7035 '7393

0.1471 '1756 •2060

0.7279 •7618 '7940

= 25 r=0

0'0000

0'0848

0.0000

0'1088

0'0000

0' I 623

0.0000

0'2333

'1435

•0020

•1708

•0004

'0195

'1988

•2276

•0062

•0411

•2505

•0318

•2804

•0182

-0662

'2995

'0539

'3301

' 0346

•0001 •0021 •oo85 '0191

•3040

•0137

'2295 •2880 '3419 '3921

5 6 7 8 9

0'0937

0'3464

0.0787

0 '3775

0'0542

0'4395

0.0332

•4228

•0764

•4846

•0501

•ioo8 •1271 '1551

•5276 -5688 -6084

•0694 •0910 -1145

0.5122 '5557 '5969 •6360 •6731

0•1846 •2156 •2480

0.6464 •6830 •7182

0.1398 •1668 '1955

0.7085 •7421 '7741

I

2

3 4

•1287

•1610 '1 948 '2297

'0038

•4058 •4510 '4948

•1106 •1407

-1230 '1 539 -1861

'3916

•1056

'4353 '4778

'1344 '1646

•2194

•5191

•1962

'4665 •5088 '5497

I0

0.2538

0.5594

II

.2893

12

'3257

'5986 .6369

0.2291 •2632 •2983

0.5894 •6279 •6654

0'4543

'5011

See page 8o for explanation of the use of this table.

85

•3631

•4166 •4660

TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL PARAMETER CONFIDENCE LEVEL PER CENT

n

90

= 26 r=

99

95

99.9

0

0'0000

0.0817

0.0000

0.1050

o•0000

0.1568

I

-0037

•1384

•0019

•1649

'0004

'2219

•00005

'2944

2

•0187

3 4

.0394 •0634

•1919 •2418 •2892

.0131 •0305 •0516

-2198 •2709 '3190

•0059 .0174 .0331

•2786 •3308 '3796

•0020 •008, •0183

.3519 '4040 •4522

5 6 7 8 9

o•o898 '1179 "474

0'3345

0.0753 •1011 •1286

0 '4973

'1575 '1877

'4696 •5115 .5517

•2100

'4618 '5019

o•o518 '0731 •0963 .1214

0'0317

'1781

0•3649 •4089 •4513 '4924

0'4257

•3783 •4206

'5322

•1481

'5904

'0478 •o663 •0868 •1091

'5399 •5802 -6185 -6551

10

0•2429

0'5411

0'2190

0'5708

0•1762

o'6276

0.1332

II

'2767

12

-3114 '3470

'5793 •6166 '6530

'2515 •2850 •3195

•6084 •6450 •6805

•2057 •2365 -2686

•6635 •6981 •7314

-1589 -1861 •2148

0.6899 •7232 '7549 •7852

o•0000 •oo18 •0126 '0292 '0495

0.1015 '1594 -2125 •2620 •3086

o•0000 •0004 •0057 •0167 •0317

0.1517 •2148 •2698 •3205 '3679

o•0000 •00005 •0019 '0078 '0175

0.2186 '2854 *3414 '3921 '4391

0'3531

0'0496

0'4127

0.0303

0'4832

-0700 '0922 '1162 .1417

'4554 '4963 '5355 '5733

'0457 •0634 •0829 '1043

'5248 '5643 •6019 '6379

13

0.0000

0.2257

I 1 = 27

r=0 2

•0035 •0179

3 4

'0378 •0609

0.0789 '1337 •1854 '2337 .2796

5 6 7 8 9

0.0861

0.3235

'1131 '1414 •1708

'3658 •4069

0'0723 *0970 ' 12 33

-2014

'4469 '4859

'1509 •1798

'3958 •4370 '4770 '5157

10

0.2328

0.5239

0.2098

0'5534

0.1685

0.6098

o.1272

0.6722

•2408

'5900

•1966

'6450

'1516

'7051

-2260 -2565

•6790 •7118

'1775 •2048

'7365 •7665

o•0000 •0004

0•1468 .208 I

0.0000

0.2119 '2769

I

II

0.0000

•2651

•5611

12

'2983

'5974

13

•3323

•6330

•2728 .3057

•6257 •6605

28 r=0

0.0000

0.0763

o•0000

0.0981

I

•0034

'1293 '1793

'0018

'1543 •2057

ii =

2

•0172

3 4

•0364

5 6 7 8 9

0.0828 •1087 -1358 •1641 '1934

'o585

'00004

'0055

'2615

•0018

.0475

'2537 '2989

•o16o '0304

•3107 '3569

•0075 •0168

'3314 •3809 •4268

0'3131

0'0694

0'3421

0'0476

0'4005

0'0290

0'4698

'3542 '3941 '4329 '4708

'0931 •1184 '1449 '1724

.3836 •4236 •4624 '5004

•o672 •o885 •1114 •1358

'4420 •4819 •5203 '5572

'0438 •0607 '0794 '0998

•51o6 '5493 •5862 •6215

.226 I '2705

•0121 •0281

See page 8o for explanation of the use of this table.

86

TABLE 29. BAYESIAN CONFIDENCE LIMITS FOR A BINOMIAL PARAMETER CONFIDENCE LEVEL PER CENT 90

95

99

99.9

n=28 I' = IO

0.2235

0.5078

0'2013

'2 545 •2863 •3188

'5439 '5794 '6140

.2310 '2615 -2930

'3520

'6480

'3253

r=0

0.0000

I

0'0739 •1253

0'0000 '0017

"737

•0116

3 4

•0032 •0166 •0350 •0564

5 6 7 8 9 10

II 12

13 14

0'1615

0'5929

0.1217

0.6553

.5727 •6075 .6415 .6747

-1884 •2164 .2455 •2757

-6274 •6608 •6931 '7243

'1450 •1697 '1957 .2229

.6877 •7188 '7486 '7771

0'0950

0'0000

0.1423

0'0000

0.2057

•1494 •1993

•0004 '0052

•2018 '2 537

0.5369

n=29 2

II

•00004 •0017 -0072 -016i

•2689 •3220 '3703 '4151

'2190

•0270

•2458

'0154

'3016

•2621

-0458

•2898

•0292

'3465

0.0797 •1046 •1307 .1579 •186o

0.3034

0.0668 '0896

0.3317 •3720

0.0458 •0645

0.3890 '4295

0.0279 '0421

•1138

•4110

'0850

'4684

'0582

'5349

'4197 •4566

.1393 .1659

'4488 '4855

•1070 .1304

-5058 '5419

'0762 '0957

'5711 .6058

0'2150

0.4926 '5278

0.1935

0-5213

o-155o

0.5769

0.1167

0.6391

'5562

•1807

'6107

'1390

'6710

'5623

'2219 '2512

'5903

'6435 '6752 •7060

•1626 '1873 •2133

•7017 '7312 '7595

0'0000

0.1997

'3433 '3820

0'4572

'4970

12

'2 447 '2752

13 14

•3063 '3382

•5962 '6293

•2814 '3123

'6236 •6560

•2075 '2 354 •2642

0'0000

0.1380

•0004 -0050

.1958 •2463

•00004 -0017

•0338 '0543

0'0000 '0016 '0112 '0260

'1449

3 4

0'0716 '1213 '1683 •2123

0'0921

2

0.0000 •0031 •0160

•2385

'0148

'2929

•0069

•3602

'2541

'0441

•2812

•0281

-3366

-0155

•4040

5 6 7 8 9

0.0768 -ioo8 •1260

0.2942 '3330

0.3219 •3612 '3991 '4359 '4717

0.0441 •0621 •0818 •1030 '1254

0'3780 •4176 '4555 '4921 '5274

0.0268 -0404 'o56o •0732 '0919

0'4452

n= 3o r=0

r

'1933

•2613 •3131

.1522

'3707 '4073

'1792

'4432

0.0644 •0863 '1097 .1342 '1597

10

0.2071

0'4782

0'1862

0.5066

0'1490

'2 357

12

'2649

13 14

'2949 •3254

'5126 '5462 '5793 -6116

'2136 '2 417 •2706 -3002

'5407 '5740 -6065 '6383

'1737 -1994 •2261 '2536

0.5616 '5948 •6270 •6582 •6885

0.1120 '1334 •156o •1797 '2045

0•6236

II

15

0'3566

0•6434

0.3306

0•6694

0.2821

0.7179

0.2304

0.7696

See page 8o for explanation of the use of this table.

87

•4842 -5213 -5568 '5909

-6551 •6854 •7146 '7426

TABLE 30. BAYESIAN CONFIDENCE LIMITS FOR A POISSON MEAN If xi , x2,... , is a random sample of size n from a Poisson distribution (Table 2) of unknown mean kt, and r = xi, then, for an assigned probability C per cent, the pair of entries when divided by n gives a C per cent Bayesian confidence interval for lt. That is, there is C per cent probability that /2 lies between the values given. The intervals are the shortest possible, compatible with the requirement on probability. Example. r = 30, n = 10. With a confidence level of 95 per cent, the Table at r = 30 gives 19.66 and 40.91. On division by n = 10, the required interval is 1.966 to 4-091. The intervals have been calculated using the reference prior with density proportional to 12-1, and the posterior density is such that nit= 1A, (Table 8). The entries can be used for any gamma prior with density

posterior probability density of p

0 (This shape applies only when r > 2. When r 1, the intervals are one-sided.)

When r exceeds 45, C per cent limits for it are given approximately by

exp(-mµ)µs-i ms /(s - 1)!,

r

n

r2 x(P)-

where m and s are non-negative integers, by replacing n with m + n and r with r + s. No limits are available in the where P = z (100 - C) and x(P) is the P percentage point of the normal distribution (Table 5). extreme case r = 0. CONFIDENCE LEVEL PER CENT 90

99

95

99.9

r =1

0.000

2. 303

0.000

2

0.084 0'441 0.937

3'932 5'479 6.946

0-042 0.304 0.713

2.996 4'765 6.40i 7'948

0.000 0.009 0.132 0.393

4'605 6.638 8 '451 10.15

0.176

I'509

1•207 1.758 2-350 2.974 3.623

9'430 10•86 12.26 13-63 14'98

0.749

2-785 3'46 7 4'171

8.355 9.723 11.06 12.37 13.66

11.77 13.33 14-84 16.32 17.77

0.399 0•691 1.040 1.433 I.862

19.83 21.39

13 14

4'893 5.629 6.378 7.138 7.908

4'94 16.20 1 7. 45 18.69 19.91

4'292 4'979 5-681 6.395 7.122

16.30 17.61 18.91 20.19

19.19 20.60 21.98 23-35 24.71

2.323 2-811 3.321 3.852 4.401

22.93 2 4'44 25.92 27.39 28.84

15 i6 17 i8 19

8.686 9'472 10.26 I1.06 1I.87

21'14

22.35 23.55 2 4'75 25-95

20 21 22

12.68 1 3'49 14.31 15.14 15.96

27.14 28.32 29.5o 30.68 31.85

11.66 12 '44 13.22 14.01 14.81

3 4 5 6 7 8 9 10 II 12

23 24

2.129

P172

1.646 2.158 2.702 3.272

0.000 0.001 0.042

6.908 9.233 1 P24 13• II

14.88 16.58 18.22

2146

3.864 4.476 5.104 5'746

7.858

22.73

6'402

26.05

8.603 9'355 10-12 10.89

23.98 25.23 27.69

7.069 7'747 8.434 9.131

27.38 28.7o 30.01 31.32

4'96 5 5'545 6.137 6'741 7'356

30.27 31.69 33.10 34'50 35-88

28.92 30.14 31•35 32.56 33'77

9.835 10.55 1 P27 11.99 12.72

32.61 33'90 35-18 36 - 45 37'72

7.981 8.616 9-259 9910 10.57

37-25 38.62 39'97 4P32 42.66

26'46

88

TABLE 30. BAYESIAN CONFIDENCE LIMITS FOR A POISSON MEAN CONFIDENCE LEVEL PER CENT

90

95

r = 25

16.8o

33.02

15.61

26 27 28

17-63 18 '47 19.31

29

20.15

34.18 35'35 36.50 37-66

16.41 17.22 18•03 18.84

30

2P00 2P85

31 32

33 34

99 34'97 36.16 37'35 38'54 39'73

99.9 38.98 40'24 41'49 42'74 43'98

13'46 14.20 14'95 15.70 16'46

11 . 24 i1.91 12-59 13.27 13.96

44'00 45'32 46'64 47'96 49'27

38.81

19'66

40.91

17.22

45' 22

14.66

50.57

39'96 41. II 42.26 43'40

20. 48 22'13 22'96

42 .09 43'27 44'44 45' 61

17.98 18'75 1 9'52 20'30

46'45 47' 68 48'91 50.14

15.36 16•06 16.78 1 7'49

51.87 53.16 54'45 55'74

26'13

44'54 45' 68

46'78 47'94 49' 11 50.27 51'43

2P08 2186 22.65 2 3'43 24'23

51.36 52'57 53'79 55.00

18.21 18-93 19.66 20.39

57.02 58.30 59'57 6o'84

56.21

21'12

62•10

52'58 53'74 54'89

25'02 25'82 26.62

57'41 58.62 59.82

21.86

22.60 23.35

63'37 64.63 65.88

22.70 2 3'55 2 4'41

21•31

35 36 37 38 39

26.99

46.82

27.86 28'72

47'95 49'09

23.79 24'63 25'46 26.30 2 7'14

50.22 51.35 52'48

27'98 28.83 29.68

2 5'27

40

29'59

41 42

30.46 31'33

43 44

32'20

53•60

30'52

56'04

33.08

54'73

31-37

57.19

2 7'42 28.22

61•02 62'21

24'09 2 4'84

68.38

45

33'95

55'85

32.23

58'34

29'03

63'41

25.59

69.63

67'13

TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A MULTIPLE CORRELATION COEFFICIENT For a normal distribution of (k + 1) quantities, let p2 be the square of the true multiple correlation coefficient between the first quantity and the remaining k (sometimes called 'explanatory variables'). p2 is the proportion of the variance of the first quantity that is accounted for by the remaining k. If R2 denotes the square of the corresponding sample multiple correlation coefficient from a random sample of size n (n > k + 1), then, for an assigned probability C per cent, the pair of entries gives a C per cent Bayesian confidence interval for p2. That is, there is C per cent probability that p2 lies between the values given. The entries have been calculated using a reference prior which is uniform over the entire range (0, 1) of p2. The intervals are the shortest possible, compatible with the requirement on probability. When R2 = 1, both the upper and lower limits may be taken to be 1. Interpolation in n and R2 will often be needed. When n is large, C per cent limits for p2are given approximately by R2 ± 2x(P)(1 - R2)(R2I n) 1

posterior probability density of p2

(In some cases this shape does not apply, and the intervals are one-sided.)

where P = (100 C) and x(P) is the P percentage point of the normal distribution (Table 5). More accurate upper limits are found by harmonic interpolation (see page 96) in the function f (n) = Vh(U (n) R2), where U (n) is the upper limit for sample size n. For the lower limit, L(n), use the function f (n) = L(n)); in each case f (oo) = 2x(P)(1 - R2) f122.

89

-

-

TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A MULTIPLE CORRELATION COEFFICIENT

k

I

CONFIDENCE LEVEL PER CENT

90

95

n =3 R2 = o•oo

0. 0000

•I 0 '20 •30

'0000 '0000 •0000

'40

99 0.7764 •7882

0.0000 •0000

99'9 0•9000 •9060

0•6838 •6985

0.0000 •0000

0.0000 •0000

'0000

•8004

'0000

'9122

•0000

•0000

•8132

•0000

•0000

•7140 '7305 '7482

'0000

•8269

'0000

•9186 •9253

•0000 •0000

0.9683 -9704 •9725 '9746 •9768

0'50

0.0000

0.7676

0-0000

•6o

-0000

'0000 '0000 •0000

0•8416 •8579

0.0000 •0000

0.9325 -9402

'8764 '8987

•0000 '0000

0.0000 •0000 •0000 •0000 -0000

0'9793 •9817 •9845 -9878 -9919

•70

'0000

•8o

-0391

'7892 '8142 '8810

•90

' 11 55

.9644

0512

•9672

•0000

'9592 •9724

0-95

0.1525

0.9887

0.0767

0.9898

0.0118

0.9905

0.0000

0.9948

R2= woo

0.0000

0'3421

• I0

'0000

0'0000 '0000

0'5671 '6621

0'0000 •0000

0'7152 '7864

•0000

'5938

•0000

•0000

'4481 '5202 '5809

0'4200 '5264

•20 •30

0'0000 '0000 '0000 '0000

'6491

'0000

'40

'0538

.6754

.0170

'7139

•0000

-7163 •7591 •7968

•0000 •0000 •0000

•8238 •8525 -8772

0.50

0.1167

•6o

•1950 '2959 '4328

0•7864 •8466 •8982

•6357

'9413 '9756

0.8347 •8877 •9287 •9608

'90

'3471 '5563

0.0029 •0384 -0931 •1867 .3781

'9847

0.0000 •0000 -0124 -0531 '1720

0.8996 •9206

'70

o•o666 •1310 •2189

'8o

0.7518 -8183 '8770 '9275 '9690

0.95

0.7846

0.9860

0.7259

0.9892

0'5735

0'9936

0'3448

0'9964

n = 25 R2 = woo

0.0000

'9489

n = HI

'9491 '9744 '9909

0-1623

0.0000

0'2058

0'0000

0'2983

0'0000

0'4122

•JO '20 •30

'0000

'3128

'moo

-366o

•4665

•0262 '0817

'4230

'4655

•0000 •0000

'5747 •6512

'40

' 1 544

-5222 •6106

•0090 '0530

•0000 •0000

0'50

0.2439

0.6906

•6o /0

'3507 '4762

•8o

'6232

*90

0.95

'5536

•5622

'0129

'6339

'0000

'7116

•1158

'6464

'0535

'7091

'0090

'7696

'7637

0.1980 •3009

0.7215 -7890

o•1166 •2058

0.7746 -8320

0.0445 •Io81

0.8248 •8719

•8306

•4270

•8500

•3266

'8921

'5807

'9052

'4881

'8824 '9268

'7958

'9485

'7683

'9551

'7046

0'8935

0'9749

0'8778

0.9782

0.8403

'2101

'9121

'9659

•3678 '6116

'9463 '9754

0.9836

0.7821

0.9883

See page 89 for explanation of the use of this table. 90

TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A MULTIPLE CORRELATION COEFFICIENT

k =2 CONFIDENCE LEVEL PER CENT

90

95

n =4

122 = woo

o-0000

o•6o19

0.0000

•io

•0000

•6179

•20

'0000

•6352

•30 '40

•0000 •0000

•6541 •6751

•0000 •0000 •0000 •0000

0'50

o'0000

0.6986

•6o

•0000

'7255

99

99'9 o•8415 •85o3 •8595 -8694 •8799

0.0000 •0000 •0000 •0000 •0000

0.9369

'7434 •7611

o'0000 •0000 •0000 •0000 •0000

0.7807 •8028 •8284 •8597 •9I00

0.0000 •0000 •0000 •0000 •0000

0.8914 •9040 '9184 '9353 '9572

o•0000 •0000 •0000 •0000 •0000

0.9586 •9636 .9695 •9765 •9848

0.6983 •7123 •7272

'9408 '9448 '9491 '9535

• 0

•0000

'7573

•8o

•0000

•7969

•90

•0614

-9055

o'0000 •0000 •0000 •0000 •oo85

0.95

o•1189

0'9690

0'0540

0'9707

0.0000

0'9720

0'0000

0'9905

122 = woo

0'0000

0'0000 '0000

0'0000

0'5671

0.0000

0.7152

'0000

0'3421 '4101

0'4200

•IO

'4904

'20

'4741

•30

•0000 'am

'40

•0000

'5354 '5953

•0000 •0000 •0000

•5527 -6098 -6641

•0000 •0000 •0000 •0000

•6331 •6857 •7313 •7728

•0000 •0000 •0000 •0000

•7665 •8040 •8352 •8626

0'50

0'0363

0'6854

0.0000

0'7170

•6o

•1122

•7800

-2107

-8o

'9656

'8779 '9321 '9728

0.8118 •8493 •9062 •9523 •9825

o'0000 •0000 •0000 •0105

•90

'3487 '5639

'8557 .9171

0.0000 •0000 '0307 •1014 -2688

0•8877 •9112

• 0

'0557 ' 1351 '2564 '4692

•8073

'0774

'9887

0'95

0.7323

0.9848

0.6562

0.9883

0.4649

0.9929

0.2109

0 '9959

0'0000

0'1623 '2822

0'0000 '0000

0'2058 •3364

0'0000 '0000

0'2983 •4400

0'0000 '0000

0.4122

•3783

•0000 '0242 •0849

'4329 '5334 •6273

•0000 •0000 -0272

•5321 -6079 -6907

•0000 •0000 •0000

0.1667 '2708 •4000 '5589

0.7079

0.0866

0'7628

0'0211

0'8134

•8241

•0786 ' 1772

•8653 •9081

'3352 •5866

'9442 '9746

0.9830

0'7664

0 '9879

it = 10

it = 25 R2 = woo '10 •20

'0000 'moo

•30 '40

.0503 •1225

0'50

0'2130 '3221

•6o

'4938 . 5904

0'6758

•8o

'4514 •6040

'7530 •8234 •8878

•90

'7845

'9465

'7550

'9535

'1742 '2958 •4611 •6865

0.95

0.8873

0 '9739

0•8705

0'9774

o'8298

io

'7794 '8436 -9015

'8774 -9240 -9647

See page 89 for explanation of the use of this table.

91

'9338 •9629

'5525 •6341 •6983 '7532

TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A MULTIPLE CORRELATION COEFFICIENT

k =3 CONFIDENCE LEVEL PER CENT

90

99

95

n =5 R2 = woo

o•0000

0•5358

•IO

•0000

'5532

0.0000 •0000

0•6316 •6477

99'9

o•0000 •000o

0.7846 •7962

o-0000 •0000

•20

•0000

•5721

'0000

'6652

'0000

•8085

'0000

'30

•000o

'40

•0000

'5931 •6166

•0000 •0000

•6842 •7053

•0000 •0000

•8217 •836o

•0000 •0000

0'50

0.0000

0.6433

•0000

0•0000 •0000

0.7289

•6o •70

'0000

•0000 '0000

'7874 '8262

0.0000 •0000 •0000

0•8517 -8691 •8890

0.0000 •000o •0000

•0000

'9125

0•900o -9061 •9126 •9196 •9266 0 '9344

•8o

'0000

'6743 •7114 '7582

'0000

•9634

•90

•oun

•8337

•0000

•8787

•0000

-9426

•0000

•9770

0'95

0 '0951

0'9446

0.0362

0•9470

0.0000

0•9630

0.0000

0.9858

n = 15 R2 = woo

0.0000

0.0000 •0000

0.3123 •3892

O•0000 •0000

0 4377 •5186

0.5784

•0000

'0000

•6522

•20

•0000

•30

- 0000

0'2505 •3204 •3930 '4662

o•0000

•I0

'40

-0000

'5395

•0000 -0000 •0000

•4630 '5340 •6026

•0000 -0000 -0000

•5879 •6498 •707o

•0000 •0000 •0000

'7093 '7576 •8002

0'50

0.0648

0.6565

•1562

'7522

'70

'2779

•8o

'4409

'3635 -602!

0.7602 •8230 •8901 '9381

0•0000 -0000 .0045 •0628

•6656

0•6885 -7820 -8548 '9138 •9620

0.0000 •0153 -0823 '2093

•90

•8309 '8976 '9539

0.0231 '0973 •2053

0.8388

•6o

'4537

'9739

'2546

'9830

0'95

0'8139

0.9783

0.7720

0.9823

0.6638

0.9882

0.4879

0.9926

R2 = o•oo •IO

0.0000

0.1623

0.0000

0.2058

o•0000

0.2983

•2580

'0000

•3123

'0000

•20

•0000

•3514

•30

'0190

'40

•0890

'4556 '5667

•0000 -0000 '0533

'4075 '4941 •6038

•000o •0000 •0033

'4175 •5103 '5895 -6631

o•0000 •0000 -0000 •0000 •0000

0.4122

•0000

0'50

0.1797

0'6592

0-1333

'2909

'7412

'2382

.70

'4242

0.7484 '8152 •8719

0.0014 '0491 •1421

'5827

'90

'7719

'9444

'3704 '5349 '7402

0.0562 '1407 -2623

'8o

.8155 '8830

0.6924 •7688 -8367

0'7947

•6o

'8974 '9517

'4314 •6664

'9209 '9634

'2995 '5586

'9037 '9419 '9737

0.95

0.8804

0'9729

0.8621

0.9766

0.8179

0.9824

0'7484

0'9875

'7558

*9430 •9529

'8745 •9117 •9561

n = 25

See page 89 for explanation of the use of this table.

92

'5330 •6166 -6841 .7421

'8572

TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A MULTIPLE CORRELATION COEFFICIENT

k= 4 CONFIDENCE LEVEL PER CENT

90

95

n =6

99

99'9

R2 = woo

o-0000

o. 482o

o•0000

0•5751

0.0000

0.7317

•10

•5928

•0000

-6121

•0000 •0000

'7458 '7609

•30

•0000

-0000

'7771

•0000

6334 •6571

•0000

'40

•5011 •5203 •5427 •5681

- 0000

'20

'0000 •0000

•0000

•7948

•0000 •0000

0•50

0.0000

0.5971

•6312

0.0000 •0000

0.8143 •8360

0.0000 •0000

0.9092

•0000

0.0000 •0000

0•6839

•6o •70

'0000

•0000

.9346

•8o

•0000

'6724 •7249

•90

'0000

0.95

•0000

0.0000 •0000 •0000

o•86io •8696 •8786 •888o •8982

•9212

•0000

•7147 •7512

•8610

•0000

'7993

-0000 -0000

•7964 •8581

•0000 •0000

•8907 •9287

•0000 •0000

'9499 •9689

0.0731

0.9167

o•o186

o•9199

0•0000

0'9543

0-0000

0.9810

R2 = woo

0.0000

0'4377

'0000

•30

•0000

'4343

-000o

•3732 •4377 -5047

'40

•0000

'5061

•0000

'5731

•0000 •0000 •0000 •0000

•5032 -5661 •6264 •6846

o-0000 •0000 •0000 •0000 •0000

0.5784

'20

0.0000 •0000 •0000

0.0000

•0000

0.2505 •3050 '3669

0.3123

•10

0.50 •6o

0-0110

0.5916 .7160

0.0000

0•6424

0.0000

0'7406

0.0000

o•8258

.0959

'0442

'7438

•0000

• 0

'2130

'8117

'1409

*8363

•8o

'3792

•2952

.9055

'0347 •1384

•90

'6200

•8882 .9505

'7947 -8689 •9306

'5463

'9592

'3770

'9719

•0000 •0000 •0218 •1662

-8641 -9001 -9461 •9813

0.95

0.7850

0.9769

0'7345

0.9812

0.6035

0'9874

0'3957

0.9921

R2 = woo

o•0000

0.1623

0.0000

0'2058

0'0000

0.2983

0.0000

0'4122

•IO '20

'0000

•2399 •3263

-2937 •3835

•30

'0000

'4148

•0000

•4709

'40

•0552

•5367

•0229

•5720

•0000 •0000 -0000 •0000

'3993 •4892 •5702 •6438

-0000 •0000 •0000 -0000

•5170

•0000

•0000 •0000

0•50

0.1441

0.0985 •2030

0.0273

0.7289

0.0000

0.7838

•2570

/0

'3942

0.640o •7282 •8068

0.6741

-60

'7570 •8290 •8930

•8049 •8657 .9175 -9620

-0223 -1059 •2606 •5271

'8456 -8985

'9498

-1054 •2259 •3985 '6436

0 '9757

0.8043

0.9818

0.7278

0.9871

n = 15 6394 •6927 •7408

7849

n=25

•8o

•5591

'90

'7578

'8779 '9422

'3379 .5082 .7235

0•95

o'8726

0.9719

o'8527

See page 89 for explanation of the use of this table.

93

'5993 •6690 •7297

'9393 '9727

TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A MULTIPLE CORRELATION COEFFICIENT

k=5 CONFIDENCE LEVEL PER CENT 90

n=7 R2 =o-oo

99

95

o•0000

0 '4377

•20

'0000 '0000

•30

•0000

'40

•0000

'4563 '4771 '5011 •5269

0'50

o-0000

0.5576

-6o

•0000

'70

•0000

-8o

•0000

•90

0.0000 •0000 -0000 '0000 '0000

0.5271

99'9

'5459 •5665 '5895 '6151

0.0000 •0000 •0000 •0000 '0000

0.6838 •6998

•0000

o'6444 -6783 '7188 •7695 •8392

0.0000 •0000 •0000 •0000 •0000

0'7795

'5940 '6384 '6955 '7774

0.0000 •0000 '0000 •000o •0000

0-95

0.0509

0•8858

0.0000

0•8898

n = 20 R2 = woo

0.0000

0.1969

•10

'0000

•20 •30

'0000

'2525 '3190

'40

•0000

'3943 '4755

0.0000 •0000 •0000 •0000 •0000

0'50

0.0445

0'5933

-6o

' 1 433

•70

•2783

-8o

'4557

•7074 -7986 '8758

•90

'6874

0•95

0.0000 -0000 •0000 •0000 •0000

0.8222 •8327

•8052 '8347 •8700 •9154

0.0000 •0000 •0000 •0000 •0000

0•8837 •8991 •9164 -9362 •9606

0.0000

0.9460

0.0000

0.9761

0.2482 •3115 •3824 •4584 •5368

0.0000 '0000 '0000 -0000 •0000

0.3550 '4276 '5011

0.0000 '0000 '0000 -0000 •0000

0.4821

0.6230

'7791 '8620 '9199

0.0000 •0000 •0063 '0897

0.790o -8374 •8867

'8931 '9514

0-0000 •0104 '0896 '2432 '5121

0.7072

'9427

0.0079 .0879 •2101 '3864 •6351

'9647

'3376

'9416 '9757

0.8310

0'9726

0'7986

0.9769

0.7165

0.9835

0.5824

0.9889

R2= woo

0'0000

'20 •30 •4°

•woo

0'1757 '2576 •3486

'0000

•0610

*5157

0'0000 '0000 •0000 •0000 '0291

0.0000 '0000 -0000 •0000 •0000

0•0000 •0000 •0000 •0000 •0000

0.3596

•0000

0'1381 '2092 •2955 '3879

0.257o

-JO

0'50

0'1556

0.1115 .2230 •3627

0'0391 '1288

-8o *90

'7721

'9373

'5339 '7430

'8835 '9448

'2594 '4370 '6758

0'7098 '7876 '8520 '9079 '9570

0'0000 .0399 •1424 '3124 '5785

0.7598

'2735 '4139 '5788

o•6210 .7119

0. 6543

•6o

0.95

0.8812

0.9694

0.8647

0'9731

0.8251

0.9792

0.7642

0.9847

•IO

n=

'0000

'7378 '8236

'7174 •7361 •7567

'5718 •6409

'8443 •8566 •8696

'5557 '6222 •6830 •7387

30

/0

'7939 •8687

'4408 •5503

'7399 .8155

'3543 '4481 '5354 -6154

See page 89 for explanation of the use of this table.

94

'4649 '5540 •6314 '6994

•8310 -8858 .9304 •9681

TABLE 31. BAYESIAN CONFIDENCE LIMITS FOR THE SQUARE OF A MULTIPLE CORRELATION COEFFICIENT

k= 6 CONFIDENCE LEVEL PER CENT

90

99

95

n =8

99'9

R2= woo

0.0000

0'4005

0.0000

0'4861

0'0000

0'6406

0'0000

0'7845

•0

'0000

•20

•0000

•0000 •0000

-5056 •5271

•0000 •0000

•6584 •6776

'0000 '0000

'5511 •5783

'0000

'6986

•7972 •8114 •8258

•0000

•7216

•0000 •0000 •0000 •0000

0.6094 •6457 •6896

0 '7473

o•0000 -0000 -0000 -0000 •0000

0•8586

•7763 •81oo •8504 •9027

•30

'0000

'40

-0000

'4193 '4404 '4642 '4916

0'50

0'0000

0.5234

0.0000

•6o

•0000

• 0

•0000

•8o

-0000

'5614 '6083 '6692

'0000 '0000 '0000

•90

•0000

'7575

•0000

•82I7

o•0000 •0000 •0000 •0000 '0000

0095

0.0278

0'8520

0'0000

o'8778

0'0000

0'9381

0'0000

0'9713

n= 20 R2 = o•oo

0-0000

o.1969

0'0000 •0000

0.0000

0.4821

•0000

0'2482 •3018

03550

•0

0'0000 •0000

'4173

•0000

-0000 -0000 •0000

'4835 •5517 •6206

•0000 •0000 '0000 •0000

•5462 •6o8o •6691 •7238 0.7769 •8269

'7449

'8415

'8774 •8984 •9227 •9524

•20

'0000

•30

'0000

•2436 •30I3 •3698

'40

-0000

'4477

•0000

•3644 '4350 •5114

0•50

0.0034 '0947

0'5358

•676o

0.0000 •0451

0.5914 •7045

o-0000 •0000

0•6887

-6o • 0

•2257

•7821

•1568

•8079

.0469

'8444

•80

'4084

'8674

090

'6556

'9393

'3334 '5967

'8857 '9486

' 1834 '4584

'9137 .9627

0.0000 •0000 •0000 '0446 •2671

0095

0'8123

0'9711

0'7748

0'9757

0.6797

0.9827

0'5243

0.9884

n = 30 R2 = o•oo

o-0000

O.1381

'0000 '0000

0.1757 •2459 •3297

0.0000 -0000 •0000

0.2570 •3420

•20 •30

'1984 '2764 '3658

0'0000 •0000 '0000 '0000

•4200

'0000

'40

•0327

'4839

•0047

•5151

•0000

o•0000 •0000 -0000 •0000 '0000

0.3596

•0

0•50

0.0155 .0972 •2269 •4086

0•0000 •0172 •1094

0.7506 •8183 •8806

'2785 '5525

'9277 -9670

0'7481

0.9843

•0000

•0000

0.1239

0.6022

o•o809

0.6357

•6o

'2430

'6993

/0

'3872

•8o

'5581

'7855 •8637

•90

'7600

'9350

' 1913 '3339 '5107 '7289

'7284 •8080 '8791 .9428

0.95

0 '8747

0.9683

0-8569

0'9722

'7550

'4305 '5175 '5998 0.6881

'6569

'7773 '8459 •9045 '9556

o•8142

0.9786

See page 89 for explanation of the use of this table.

95

'8737 '9341 '9742

'4531 '5389 •6169 •6871

A NOTE ON INTERPOLATION Part of the tabulation of a function f(x) at intervals h of x is in the form given in the first two columns of the figure :

For x = 0.034, p = (0.034-0.03)/0.01 = 0.4 and the linear interpolate is

xo J o x, 4 x, f,

0 '9790 + 0.4

Ai

The additional term for the quadratic interpolate is

A", A;1

—0.25 x 0.4 x o•6 x ( — 0'0090 — 0'0087) = 0'001i

A°

x, f, where ft = f(x) and x,„ = x,H-h. Interpolation of f(x) at values of x other than those tabulated uses the differences in the last three columns, where each entry is the value in the column immediately to the left and below minus the value to the left and above : thus, Ail =f 2 —fi and A; = These are usually written in units of the last place of decimals in f(x). Linear interpolation between x1 and x2 approximates f(x) by +PAil with p = (x—x1)/h. This simple rule uses only the values within the lines of the figure and is often adequate. Quadratic interpolation between x1and x2 approximates f(x) by +pA',4 -1p(i—p) (A';+

and is not negligible, the quadratic interpolate being 0-9709. This is exact, as is expected since A;1 at 3 is well below 6o. The quadratic method uses fo ana f, (needed for A; and A;) and so fails if either is unavailable, for example at the ends of the range of x or when the interval of tabulation h changes. Modified quadratic forms between xl and x2 are

+pevil

20,

Example. The F-distribution, P = to, v1 = i (Table 2(a), page 5o), interpolation in v2, now x.

0'9929

'03

'9790

—

F(P)

0

2'706

120

1/120

2•748

6o 2/12 0

2.791

40 3/120

2.835

43

+3 —8 7

—316 ' 05

'9245

CONSTANTS e = 2.71828

18285

It = 3'141 59 26536

T1E

=

0

44

h--,±0)/(th) =

—90

-9561

I /v2

Notice that the values of v2 chosen for tabulation are such that the intervals of i/v2 are constsnt, here The differences show that linear interpolation will be adequate. For •5 and the linear interv2 = 8o, p = ( polate is 2.748 + o.5 x 0.043 = 2•77o with the possibility of an error of I in the last place.

139

—229 -04

v2

oo

42

r = 2 (Table t,

page 22), interpolation in p, now x.

(f, missing),

Ocassionally, harmonic interpolation is advisable. To do this the argument x is replaced by i/x and then linear (or quadratic) interpolation performed.

2.

Example. The binomial distribution, n =

—p) A;

f1 +0,4-1p(i —p) A; (fo missing).

This is generally adequate provided Ail is less than 6o in units of the last place of decimals in the tabulation. Notice that the quadratic interpolate consists of the addition of an extra term to the linear one, so that a rough assessment of it will indicate whether the linear form is adequate. The maximum possible value of ip(t —p) is when p =

0•02

x ( — 0-0229) = 0-9698.

0-39894 22804

logio e = 043429 44819 loge Io = 2.30258 50930 10g,

2n = 0.91893 85332

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, no Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521484855 © Cambridge University Press 1984 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1984 Eighth printing 1994 Second edition 1995 Thirteenth printing 2009 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-48485-5 paperback