Statistics Tables: For Mathematicians, Engineers, Economists And The Behavioural Management Sciences

STATISTICS TABLES for mathematicians, engineers, economists and the behavioural and management sciences

H.R.Neave

London and New York

First published 1978 by George Allen & Unwin Routledge is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2009. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. © 1978 H.R.Neave All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-203-01167-8 Master e-book ISBN

ISBN 0-203-15532-7 (Adobe ebook Reader Format) ISBN 0-415-10485-8 (Print Edition)

Preface For several years I have been teaching a first-year undergraduate Statistics course to students from many disciplines including, amongst others, mathematicians, economists and psychologists. It is a broad-based course, covering not only probability, distributions, estimation, hypothesistesting, regression, correlation and analysis of variance, but also non-parametric methods, quality control and some simple operations research, especially simulation. No suitable book of tables seemed to exist for use with this course, and so I collected together a set of tables covering all the topics I needed, and this, after various improvements and extensions, has developed into the current volume. I hope it will now also aid other teachers to extend the objectives of their own applied Statistics courses and to include topics which cannot otherwise be covered very meaningfully except with practice and use of some such convenient set of tables. In preparing this book, I have recomputed the majority of the tables, and have thus been able to extend several beyond what has normally been previously available and have also attempted some kind of consistency in such things as the choice of quantiles (or percentage points) given for the various distributions. Complete consistency throughout has unfortunately seemed unattainable because of the differing uses to which the various tables can be put. Two particular conventions should be mentioned. If, in addition to providing critical values, a table can be used for forming confidence intervals (such as tables of normal, t, χ2 and F distributions), then quantiles q are indicated, i.e. the solutions of F(x)=q, where F ( ) represents the cumulative distribution function of the statistic being tabulated, and x the tabulated values. If a table is normally used only for finding critical values (such as tables for non-parametric tests, correlation coefficients, and the von Neumann and Durbin-Watson statistics), then significance levels are quoted which apply to two-sided or general alternatives. To make clear which of the two conventions is relevant, quantiles are referred to in decimal format (0·025, 0·99, etc.) whereas significance levels are given as percentages (5%, 1%, etc.). I would like to express my gratitude to a number of people who have helped in the production of this book: to Dr D.S.Houghton, Dr G.J.Janacek, Cliff Litton, Arthur Morley and Peter Worthington, who have ‘vetted’ the work at various stages of its progress, to the staff of the Cripps Computing Centre at Nottingham University who have helped in very many ways, to Tonie-Carol Brown for help with preparing data for some of the computer programs, and to Betty Page for typing the text. Responsibility for any errors is mine alone—there should not be many, as the tables have been subjected to many hours of checking and cross-checking, but I would be grateful to anyone who points out to me any possible mistakes, whether they be substantiated or merely suspected. I would also greatly appreciate any suggestions for improvements which might be incorporated in subsequent editions. HENRY NEAVE Nottingham University, June 1977.

iv  Preface

Preface to the second impression The only errors of any substance found in the first impression were in Charts 1.2(a, b), which were drawn a little inaccurately because of a tracking error on an incremental graphplotter; these charts have now been redrawn. Improved versions of Charts 6.1(a, b) and of Tables 6.4 and 6.5 have also been included and a few minor amendments made to the text and the layout. Those who purchased the first impression are welcome to write to the author at the Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, England for correct copies of Charts 1.2(a, b) and details of any other alterations.

Acknowledgements Most of the tables have been newly computed for this publication; the exceptions are cited below. But I would like to give a general note of acknowledgement to all those who have previously published sets of tables for their ideas concerning content and layout which clearly have indirectly contributed to this volume. I am indebted to the publishers for permission to include material in full or in part from the following journals: the Annals of Mathematical Statistics (Tables 2.5(a), (b)), Biometrics (Table 4.1), The Statistician (Table 4.4), the Journal of the American Statistical Association (Table 5.4) and Statistica Neerlandica (Table 6.4). Tables 4.2 and 5.2(c) respectively have been derived from Selected Tables in Mathematical Statistics Vol. 3 (1975), pp. 329–84 and Vol. 1 (1970), pp. 130–70 with permission of the publisher American Mathematical Society. Part of Table 2.3(a) has been derived from Biometrika Tables for Statisticians Vol. 2, Table 1 with permission of the Biometrika Trustees. Tables 6.7 and 6.8 have been taken from On the theory and application of the general linear model by J.Koerts and A.P.J.Abrahamse (Rotterdam University Press) by permission of the publisher and authors. The redrawn Poisson probability chart (Chart 1.3(6)) has been included by permission of H.W.Peel & Co. Table 8.1(a) has been abridged from Table 13 of Attribute Sampling by H.Burstein (McGraw-Hill) by permission of the publisher and author.

Contents     page 1     10     12

                     2.2 ordinates of the standard normal density function     2.3(a) quantiles (percentage points) of the standard   normal distribution                     2.5(a) moments of the range distribution          

        SECTION 4:    4.1  factors for Duncan’s multiple-range test                      

19 21 25 26 28 28 29 29 31 31 33

35 36 38 42 43 43 44

vi  Contents SECTION 5:  NON-PARAMETRIC TESTS

     

   

5.1 

 

5.2(b)   Kolmogorov-Smirnov asymptotic critical   values 5.2(c)   Kolmogorov-Smirnov two-sample test

 

   

5.3  

 

   

5.4  

Mann-Whitney/Wilcoxon/rank-sum twosample test improved Tukey quick test  

5.5  

Wald-Wolfowitz number-of-runs test  

   

  Wilcoxon signed-rank test

5.2(a)     Kolmogorov-Smirnov one-sample test

SECTION 6:  CORRELATION

     

6.1(a), (b)   charts giving 95% and 99% confidence intervals for p 6.2   critical values rn, α of the linear correlation   coefficient r 6.3(a)   the Fisher z-transformation

        6.3(b)   the inverse z-transformation  

6.4

 

6.5

     

       

47 48 49

50 52 52 53 54 55

 

56 57

6.8  

 

57

   

59

 

61

 

62

   

63

 

64

 

7.1 

bounds for critical values of the DurbinWatson statistic RANDOM NUMBERS   random digits

7.2

 

    8.1(a)   the construction of single-sampling plans    

 

45 46

 

  random numbers from the standard normal distribution 7.3   random numbers from the exponential     distribution with unit mean SECTION 8:   QUALITY CONTROL

 

     

45

  critical values of the Kendall rank correlation coefficient 6.6 critical   values of the multiple correlation   coefficient 6.7    critical values of the von Neumann ratio

 

SECTION 7: 

 

  critical values of the Spearman rank

  correlation coefficient

 

45

8.1(b)   the operating characteristic of singlesampling plans

 

Contents

                           

                   

65 66 67 67 68

69 70 71 72 75 77

79 9.6(a) common logarithms: log10 (x)   81       83     Inside back cover  

vii

Descriptions of the Tables (References to appropriate books or articles are given for the less familiar tables) Section 1: Discrete probability distributions The quantity tabulated in Table 1.1 (pp. 20–26) is  

which is the probability of obtaining x or less ‘successes’ in n independent trials of an experiment, where the probability of a success at each trial is p. Individual probabilities are easily obtained using P(0)=F(0) and P(x)=F(x)−F(x−1) for x>0. The table covers all n≤20 and p=0·01(0·01)0·10(0·05)0·50. For values of p>½, probabilities may be calculated by reversing the roles of ‘success’ and ‘failure’. Charts 1.2 (pp. 27–28) give (a) 95% and (b) 99% confidence intervals for p on the basis of a binomial sample of size n in which X ‘successes’ occur. If f=X/n≤½, locate the value of f on the bottom horizontal axis, trace up to the two curves labelled with the appropriate value of n, and read off the confidence limits on the left-hand vertical axis; if f>½, use the top horizontal axis and the right-hand vertical axis. For each value of n, the appropriate points have been plotted for all possible values of f and these points joined by straight lines to aid legibility. The charts may also be used ‘in reverse’ to provide (a) 5% and (b) 1% two-sided critical regions for a hypothesis test of H0: p=p0 against H1:p≠p0, or equivalently (a) 2½% and (b) ½% one-sided critical regions. Results for values of n not included may be obtained by interpolation. The quantity tabulated in Table 1.3(a) (pp. 29–32) is  

this being the cumulative distribution function (c.d.f.) of the Poisson distribution with mean µ. Individual probabilities may be found as in Table 1.1. For µ>2·0, the c.d.f. occupies two or more rows of the table, the first row giving F(0) to F(9), the second row giving F(10) to F(19), etc. The Poisson probability chart, Chart 1.3(b) (p. 33), gives Prob (X≥c)=1−F(c−1) where X has the Poisson distribution with mean µ. The value of µ, ranging from 0·1 to 100, is found on the horizontal axis and the probabilities are read on the left-hand vertical. There is a curve for each of the following values of c: 1(1)25(5)100(10)150. The horizontal axis has a logarithmic scale and the vertical axis a normal probability scale.

2

Statistics Tables

Section 2: The normal distribution Table 2.1(a) (pp. 34−5) gives values of the standard normal c.d.f. Φ(z) for z=−4·00(0·01)3·00, expressed to four decimal places (4 d.p.) and with proportional parts for the third decimal place of z, and also for z=3·00(0·01)5·00 to 6 d.p. Note that proportional parts are subtracted if z0, ρU. Table 7.2 (p. 66) gives 500 random numbers from the standard normal distribution N(0, 1). To convert them to random numbers from the normal distribution with mean µ and variance σ2, N(µ, σ2), multiply by σ and add µ. Table 7.3 (p. 67) gives 500 random numbers from the exponential distribution with mean 1, having a probability density function f(x)=e−x(x≥0). They may be converted to random numbers from the exponential distribution with mean µ by multiplying by µ. Further reading: Freund (1973, §§8.1, 8.3).

Section 8: Quality control A large batch (population) has a proportion p of defective items. A random sample of size n is drawn from the batch and if the number of defectives found in the sample exceeds the acceptance number c the batch is rejected, otherwise it is accepted. It is desired to choose n and c such that if p≤p1 the probability is at most α that the batch is rejected, and if p≥p2 the probability is at most β that the batch is accepted. Table 8.1(a) (p. 68) enables the construction of sampling plans approximately satisfying these conditions for α, β=10%, 5%,1%. First and in the column corresponding to the calculate required values of α and β find the entry nearest to S. Read off the corresponding

8

Statistics Tables

values of c and m, and calculate n as the nearest integer to (m/p2)−½(m−c). E.g. if α=5%, β=10%, p1=0·01, p2=0·04, then S=4·061, c=4, m=7·9936 and n=198. Thus a sample of size 198 would be drawn, the batch being accepted if no more than four defectives are found. (If a lower level of accuracy is sufficient, take S=p2/p1 and n=m/p2.) Table 8.1(b) (p. 69) enables the operating characteristic of such a single-sampling plan to be drawn, i.e. a graph of L(p), the probability of a batch being accepted, against p. For each of 13 values of L(p), the corresponding values of p are found by dividing the numbers in the row corresponding to the appropriate acceptance number c by the sample size n. Thus for the plan constructed above, L(p)=0·900 corresponds to p=2·433/198=0·0123. (Use of this table depends on the Poisson approximation to hyper-geometric or binomial distributions, and is thus only accurate for small p. A tolerance interval (tL, tU) is an interval within which one may assert with a degree of confidence γ that a proportion of at least P of a population lies; i.e. if the c.d.f. of the population distribution is F( ), then there is a probability of at least γ that In Tables 8.2(a) and 8.2(b) (pp. 70–1), the population distribution is presumed to be normal, and suppose a sample of size n has mean and adjusted standard deviation s (see p. 88). Denoting a typical value from Table 8.2(a) by k, the form of the appropriate tolerance interval is either or . In Table 8.2(b), again with k denoting a typical value, the interval is of the form .(The values in Table 8.2(b) are ‘strong’ tolerance limits in the sense that, e.g. with γ=95% and P=0·98, there is a probability of at least 95% that no more than 1% of the population lies to the left of and also that no more than 1% lies to the right of .) In Table 8.3(a) (p. 72), the tolerance interval is again taken in the form (∞, tU) or (tL, ∞) but now tL=Xmin and tU=Xmax, where Xmin and Xmax are simply the minimum and maximum values in a sample of size n. The table gives minimum values of n satisfying the conditions for a variety of values of γ and P. There is no assumption necessary concerning the population distribution. Table 8.3(b) (p. 72) gives the sample sizes necessary for tolerance intervals of the form (tL, tU) where again tL=Xmin and tU =Xmax and tL and tU are ‘strong’ tolerance limits in the above sense. Table 8.4 (p. 73) enables the construction of various types of control charts—for sample means , sample ranges R, and (unadjusted) sample standard deviations S (see p. 88). If the population distribution is normal, warning limits correspond to the 0·025 and 0·975 quantiles, and action limits to the 0·001 and 0·999 quantiles. Variability of the observations is characterised either by the population standard deviation σ or by the average range or standard deviation in pilot samples of the same size n as those to be plotted. For -charts, the central line C is placed at the population mean µ or at the overall mean of pilot observations. Warning limits are placed at and action limits at according to the measure of variability being used. The values r and s give E[R/σ] and E[S/σ] respectively and their reciprocals are also listed. These columns facilitate changes between the three variability characteristics, e.g. if σ has been specified, then and ; on the other hand, and provide unbiased estimators for σ. , Given the R-chart has central line warning limits and and action limits and (L and U indicating Lower and Upper respectively). Given σ, the central line should be placed at rσ and the warning and action limits at

Descriptions of the Tables 9

For S-charts, replace R and r by S and s throughout the last paragraph. Further reading: Table 8.1(a), Burstein (1971, ch. 3); Table 8.2(a), Owen (1965); Table 8.2(b), Owen (1964); Table 8.3, Wilks (1942); Table 8.4, Moroney (1965, ch. 11).

Section 9: Miscellaneous These tables need little explanation. Table 9.1 (p. 74) presents reciprocals, squares, square roots, cubes and cube roots of integers n up to 100. In Table 9.2 (p. 75) the binomial coefficients  

are given for n≤30, n being read in the left-hand column and r along the horizontal (r≤15). For r>15, use  

The factorials n! in Table 9.3(a) (p. 76) are given exactly for n≤30 and to six significant figures for 31≤n≤150. Table 9.3(b) (pp. 77–9) gives logarithms (to base 10) of factorials for n≤850. Table 9.4(a) (p. 80) gives ex for x=0·00(0·01)5·00((0·1)10·9. Table 9.4(b) (p. 81) gives −x e for x=0·00(0·01)3·99 with proportional parts (which are subtracted) for the third d.p. of x, and for x=4·0(0·1)10·9 to at least four significant figures. Table 9.5 (pp. 82–3) gives natural logarithms loge (x)≡ln (x). Tables 9.6(a) (pp. 84–5) and 9.6(b) (pp. 86–7) are standard tables of common logarithms and antilogarithms.

References (An asterisk indicates a reference of above-average difficulty, intended more for the teacher than the student.) Bradley, J.V. 1968. Distribution-free statistical tests. Englewood Cliffs, N.J.: Prentice-Hall. Burstein, H. 1971. Attribute sampling. New York: McGraw-Hill. Freund, J.E. 1973. Modern elementary statistics, 3rd edn. Englewood Cliffs, N.J.: Prentice-Hall. *Koerts, J. and A.P.J.Abrahamse 1969. On the theory and application of the general linear model. Rotterdam: Rotterdam University Press. *Lindgren, B.W. 1976. Statistical theory, 3rd edn. London: Macmillan. Miller, I. and J.E.Freund 1965. Probability and statistics for engineers. Englewood Cliffs, N.J.: Prentice-Hall. Moroney, M.J. 1965. Facts from figures, 2nd edn. Penguin Books Ltd. Neave, H.R. 1966. A development of Tukey’s quick test of location. J. Am. Statist. Assoc. 61, 949–64. Neave, H.R. 1972. Some quick tests for slippage. The Statistician 21, 197–208. *Owen, D.B. 1964. Control of percentages in both tails of the normal distribution. Technometrics 6, 377–87. *Owen, D.B. 1965. A special case of a bivariate non-central t-distribution. Biometrika 52, 437–46. Siegel, S. 1956. Nonparametric statistics for the behavioral sciences. New York: McGraw-Hill. *Wilks, S.S. 1942. Statistical prediction with special reference to the problem of tolerance limits. Ann. Math. Statist. 13, 400–9.

THE TABLES

1·1 the binomial c.d.f.  

Discrete probability distributions 13

14

Statistics Tables

Discrete probability distributions 15

16

Statistics Tables

Discrete probability distributions 17

18

Statistics Tables

Discrete probability distributions 19 1·2(a) chart giving 95% confidence intervals for p

20

Statistics Tables 1·2(b) chart giving 99% confidence intervals for p

Discrete probability distributions 21 1·3(a) the Poisson c.d.f.  

22

Statistics Tables

Discrete probability distributions 23

24

Statistics Tables

Discrete probability distributions 25 1·3(b) Poisson probability chart Prob (X≥c)

2·1(a) the c.d.f. of the standard normal distribution  

The normal distribution  27

28

Statistics Tables 2·1(b) extreme values of the standard normal c.d.f.

2.2 ordinates of the standard normal density function

The normal distribution  29 2.3(a) quantiles (percentage points) of the standard normal distribution

2·3(b) the inverse normal function

30

Statistics Tables

The normal distribution  31 2·4(a) normal scores (expected values of normal order statistics)

2.4(b) sums of squares of normal scores

32

Statistics Tables

The normal distribution  33 2·5(a) moments of the range distribution

34

Statistics Tables 2·5(b) quantiles of the range distribution

3·1 the Student t distribution

36

Statistics Tables 3·2 distribution the χ2 (chi-squared)

Continuous probability distributions  37

38

Statistics Tables 3·3 the F distribution

Continuous probability distributions  39

40

Statistics Tables

Continuous probability distributions  41

4·1 factors for Duncan’s multiple-range test

Analysis of variance  43 4·2 the Kruskal-Wallis test

4·3 the Friedman test critical region: S≥tabulated value

44

Statistics Tables 4·4 quick multi-sample tests

5·1 Wilcoxon signed-rank test critical region: W≤tabulated value

5·2(a) Kolmogorov-Smirnov one-sample test critical region: Dn≥tabulated value

5·2(b) Kolmogorov-Smirnov asymptotic critical values

46

Statistics Tables 5·2(c) Kolmogorov-Smirnov two-sample test

Non-parametric tests  47 5·3 Mann-Whitney/Wilcoxon/rank-sum

48

Statistics Tables 5·4 improved Tukey quick test

Non-parametric tests  49 5·5 Wald-Wolfowitz number-of-runs test

6·1(a) chart giving 95% confidence intervals for ρ

Correlation  51 6·1(b) chart giving 99% confidence intervals for ρ

52

Statistics Tables 6·2 critical values rn, x of the linear correlation coefficient r

6·3(a) the Fisher z-transformation

Correlation  53 6·3(b) the inverse z-transformation r=r(z)=tanh z

 

54

Statistics Tables 6·4 critical values of the Spearman rank correlation coefficient

Correlation  55 6·5 critical values of the Kendall rank correlation coefficient

56

Statistics Tables 6·6 critical values of the multiple correlation coefficient

Correlation  57 6·7 critical values of the von Neumann ratio critical region: V≤tabulated value

6·8 bounds for critical values of the Durbin-Watson statistic critical region: d≤tabulated value

58

Statistics Tables

7·1 random digits

60

Statistics Tables

Random numbers  61 7·2 random numbers from the standard normal distribution

62

Statistics Tables 7·3 random numbers from the exponential distribution with unit mean

8·1(a) the construction of single-sampling plans

64

Statistics Tables 8·1(b) the operating characteristic of single-sampling plans

Quality control 8·2(a) one-sided tolerance factors for normal distributions for tolerance intervals (−∞, +ks) or ( −ks, ∞).

65

Statistics Tables

66

8·2(b) two-sided tolerance factors for normal distributions for tolerance intervals (

)

Quality control 8·3(a) sample sizes for one-sided non-parametric tolerance limits for tolerance intervals (−∞, Xmax) or (Xmin, ∞)

8·3(b) sample sizes for two-sided non-parametric tolerance limits for tolerance intervals (Xmin, Xmax)

67

68

Statistics Tables 8·4 control chart constants

9·1 reciprocals, squares, square roots, cubes, cube roots

70

Statistics Tables 9·2 binomial coefficients

Miscellaneous  71 9·3(a) fa ctorials n!=n(n−1)…2 . 1

 

72

Statistics Tables 9·3(b) logarithms of factorials

Miscellaneous  73

74

Statistics Tables

Miscellaneous  75 9·4(a) the exponential function: ex

76

Statistics Tables 9·4(b) the negative exponential function: e−x

Miscellaneous  77 9·5 natural logarithms: logc (x)

78

Statistics Tables

Miscellaneous 79 9·6(a) common logarithms: log10 (x)

80

Statistics Tables

Miscellaneous 81 9·6(b) antilogarithms: 10x

82

Statistics Tables

Useful constants  

Sample standard deviations If X1, X2,…, Xn is a random sample from a distribution having variance σ2 then

 

is an unbiased estimator for σ2, being the sample mean. s is called the adjusted standard deviation of the sample. The square root S of  

is called the unadjusted standard deviation of the sample; its only specific use in these tables is in the construction of some control charts—see Table 8.4.

Asymptotic (large-sample) distributions A number of statistics included in these tables are asymptotically normally distributed; their means and variances are listed below. In the case of a single sample, n denotes the sample size; in the case of two samples, n1 and n2 denote the samples sizes, and N=n1+n2.