Statistical Tables

Fourth Edition

J. Murdoch and J.A. Barnes

For students of Science, Engineering, Psychology, Business, Management, Finance

Preface to the Fourth Edition

Much has changed in statistical analysis since these tables were first published and indeed since they were last revised. The biggest change has been the development and wide availability of personal computers together with comprehensive software which has become steadily easier to use. Between them, they have automated much that once had to be calculated manually as well as making possible previously impracticable methods of analysis.

However, those learning the subject should still find value in a set of tables such as these. The understanding of statistical concepts and the calculations which support them comes from working through practice problems, ideally with a capable teacher to provide assistance. Part of this learning process is finding out how to use tables — knowing what is tabulated and why and thus how to access the relevant table and how to interpret the result when it has been found. An important feature of some of the tables is the encouragement to consider the use of approximations — something which is basic to the application of statistical models to the real world. The tables should also be useful to the practitioner on those occasions where it is not convenient to have access to a computer.

Tables have been added, principally for distribution-free methods and for control chart applications. Some others — the basic mathematical tables of such as logarithms, squares and square roots — have been left out as their function is duplicated on readily available electronic calculators. As before, examples of the use of some of the tables have been given.

Permissions from other copyright holders are acknowledged with thanks at the foot of the relevant tables. Every effort has been made to trace all the copyright holders but if any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.

Comments on these tables and suggestions for their amendment will be welcome. Please either write to John Barnes care of the publishers or contact him by e-mail at J.A.Barnes@cranfield.ac.uk

Cranfield 1998

J. Murdoch
J. A. Barnes

© J. Murdoch and J.A. Barnes 1968, 1970, 1986, 1998

All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission.

No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1P 9HE.

Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.

The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988.

Editions:

  • First edition 1968
  • Second edition 1970
  • Third edition 1986
  • Fourth edition 1998

Published by MACMILLAN PRESS LTD
Houndmills, Basingstoke, Hampshire RG21 6XS and London

ISBN 0–333–55859–6

Contents

Basic Distribution Tables

Table Title Page
1 Cumulative Binomial Probabilities 4
2 Cumulative Poisson Probabilities 8
3 Areas in Upper Tail of the Normal Distribution 13
4 Percentage Points of the Normal Distribution 14
5 Ordinates of the Normal Distribution 14
6 Exponential Function $e^{-x}$ 15
7 Percentage Points of the $t$ Distribution 17
8 Percentage Points of the $\chi^2$ Distribution 18
9 Percentage Points of the $F$ Distribution 20
10 Percentage Points of the Correlation Coefficient 22
11 Tukey's Wholly Significant Difference (5% Level) 23

Distribution-free (Non-parametric) Tables

Table Title Page
12 Percentage Points of Spearman's Rank Correlation Coefficient 24
13 Percentage Points of Kendall's Rank Correlation Coefficient 25
14 Percentage Points of Nair's 'Studentised' Extreme Deviate from the Mean 26
15 Upper Percentage Points of Dixon's Rank Difference Ratio 27
16 Percentage Points of $D$ in the One-sample Kolmogorov–Smirnov Distribution 28
17 Lower Percentage Points of the Wilcoxon Signed-rank Distribution 29
18 Percentage Points of $D$ in the Two-sample Kolmogorov–Smirnov Distribution 30
19 Percentage Points of the Mann–Whitney $U$-Distribution 36
20 Percentage Points of Friedman's Distribution 38
21 Upper Tails of the Kruskal–Wallis Distribution 40

Statistical Process Control Tables

Table Title Page
22 Control Chart Factors for Sample Mean ($\bar{X}$) 42
23 Control Chart Factors for Sample Range (using $\bar{R}$) 43
24 Control Chart Factors for Sample Range (using $\sigma$) 43
25 Control Chart Factors for Mean and Range (American usage) 44
26 Control Chart Factors for Standard Deviation (American usage) 45
27 Tolerance Factors for the Normal Distribution 46
28 Sample Size for Two-sided Distribution-free Tolerance Limits 47
29 Sample Size for One-sided Distribution-free Tolerance Limits 47

Critical Values for Runs

Table Title Page
30 Number of Runs on Either Side of the Mean: 5% Point 49
31 Number of Runs on Either Side of the Mean: 0.5% Point 49
32 Number of Runs Above and Below the Median 50
33 Lengths of Runs on Either Side of the Median: 5%, 1% and 0.1% Points 51
34 Critical Values of Lengths of Runs Up and Down 51

Attribute Single Sampling Tables

Table Title Page
35 Derivation of Single Sampling Plans 52
36 Construction of O.C. Curves for Single Sampling Plans 53

Random Number Tables

Table Title Page
37 Random Numbers 54
38 Random Standardised Normal Deviates ($Z$ Values) 59

Financial Tables

Table Title Page
39 Present Value Factors 60
40 Cumulative Present Value Factors 64
41 Capital Recovery Factors 68

Examples of the Use of Tables 11 to 16: 72
Some Useful Formulae: 77

Table 1 Cumulative Binomial Probabilities

$p$ = probability of success in a single trial; $n$ = number of trials. The table gives the probability of obtaining $r$ or more successes in $n$ independent trials. That is:

$$\sum_{x=r}^{n} \binom{n}{x} p^x (1-p)^{n-x}$$

When there is no entry for a particular pair of values of $r$ and $p$, this indicates that the appropriate probability is less than 0.00005. Similarly, except for the case $r = 0$, when the entry is exact, a tabulated value of 1.0000 represents a probability greater than 0.99995.

Part 1: $p$ = 0.01 to 0.09

$p=$ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
$n=2$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .0199 .0396 .0591 .0784 .0975 .1164 .1351 .1536 .1719
2 .0001 .0004 .0009 .0016 .0025 .0036 .0049 .0064 .0081
$n=5$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .0490 .0961 .1413 .1846 .2262 .2661 .3043 .3409 .3760
2 .0010 .0038 .0085 .0148 .0226 .0319 .0425 .0544 .0674
3 — .0001 .0003 .0006 .0012 .0020 .0031 .0045 .0063
4 — — — — .0001 .0001 .0002 .0003 .0003
$n=10$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .0956 .1829 .2626 .3352 .4013 .4614 .5160 .5656 .6106
2 .0043 .0162 .0345 .0582 .0861 .1176 .1517 .1879 .2254
3 .0001 .0009 .0028 .0062 .0115 .0188 .0283 .0401 .0540
4 — — .0001 .0004 .0010 .0020 .0036 .0058 .0088
5 — — — — .0001 .0002 .0003 .0006 .0010
6 — — — — — — — .0001 .0001
$n=20$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .1821 .3324 .4562 .5580 .6415 .7099 .7658 .8113 .8484
2 .0169 .0599 .1198 .1897 .2642 .3395 .4131 .4831 .5484
3 .0010 .0071 .0210 .0439 .0755 .1150 .1610 .2121 .2666
4 — .0006 .0027 .0074 .0159 .0290 .0471 .0706 .0993
5 — — .0003 .0010 .0026 .0056 .0107 .0183 .0290
6 — — — .0001 .0003 .0009 .0019 .0038 .0068
7 — — — — — .0001 .0003 .0006 .0013
8 — — — — — — .0001 .0001 .0002
$n=50$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .3950 .6358 .7819 .8701 .9231 .9547 .9734 .9845 .9910
2 .0894 .2642 .4447 .5995 .7206 .8100 .8735 .9173 .9468
3 .0138 .0784 .1892 .3233 .4595 .5838 .6892 .7740 .8395
4 .0016 .0178 .0628 .1391 .2396 .3527 .4673 .5747 .6697
5 .0001 .0032 .0168 .0490 .1036 .1794 .2710 .3710 .4723
6 — .0005 .0037 .0144 .0378 .0776 .1350 .2081 .2928
7 — .0001 .0007 .0036 .0118 .0289 .0583 .1019 .1596
8 — — .0001 .0008 .0032 .0094 .0220 .0438 .0768
9 — — — .0001 .0008 .0027 .0073 .0167 .0328
10 — — — — .0002 .0007 .0022 .0056 .0125
11 — — — — — .0002 .0006 .0017 .0043
12 — — — — — — .0001 .0005 .0013
13 — — — — — — — .0001 .0004
14 — — — — — — — — .0001

Part 2: $p$ = 0.10 to 0.50

$p=$ 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
$n=2$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .1900 .2775 .3600 .4375 .5100 .5775 .6400 .6975 .7500
2 .0100 .0225 .0400 .0625 .0900 .1225 .1600 .2025 .2500
$n=5$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .4095 .5563 .6723 .7627 .8319 .8840 .9222 .9497 .9688
2 .0815 .1648 .2627 .3672 .4718 .5716 .6630 .7438 .8125
3 .0086 .0266 .0579 .1035 .1631 .2352 .3174 .4069 .5000
4 .0005 .0022 .0067 .0156 .0308 .0540 .0870 .1312 .1875
5 — .0001 .0003 .0010 .0024 .0053 .0102 .0185 .0313
$n=10$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .6513 .8031 .8926 .9437 .9718 .9865 .9940 .9975 .9990
2 .2639 .4557 .6242 .7560 .8507 .9140 .9536 .9767 .9893
3 .0702 .1798 .3222 .4744 .6172 .7384 .8327 .9004 .9453
4 .0128 .0500 .1209 .2241 .3504 .4862 .6177 .7430 .8281
5 .0016 .0099 .0328 .0781 .1503 .2485 .3669 .4956 .6230
6 .0001 .0014 .0064 .0197 .0473 .0949 .1662 .2616 .3770
7 — .0001 .0009 .0035 .0106 .0260 .0548 .1020 .1719
8 — — .0001 .0004 .0016 .0048 .0123 .0274 .0547
9 — — — — .0001 .0005 .0017 .0045 .0107
10 — — — — — — .0001 .0003 .0010
$n=20$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .8784 .9612 .9885 .9968 .9992 .9998 1.0000 1.0000 1.0000
2 .6083 .8244 .9308 .9757 .9924 .9979 .9995 .9999 1.0000
3 .3231 .5951 .7939 .9087 .9645 .9879 .9964 .9991 .9998
4 .1330 .3523 .5886 .7748 .8929 .9556 .9840 .9951 .9987
5 .0432 .1702 .3704 .5852 .7625 .8818 .9490 .9811 .9941
6 .0113 .0673 .1958 .3828 .5836 .7546 .8744 .9447 .9793
7 .0024 .0219 .0867 .2142 .3920 .5834 .7500 .8701 .9423
8 .0004 .0059 .0321 .1018 .2277 .3990 .5841 .7480 .8684
9 .0001 .0013 .0100 .0409 .1133 .2376 .4044 .5857 .7483
10 — .0002 .0026 .0139 .0480 .1218 .2447 .4086 .5881
11 — — .0006 .0039 .0171 .0532 .1275 .2493 .4119
12 — — .0001 .0009 .0051 .0196 .0565 .1308 .2517
13 — — — .0002 .0013 .0060 .0210 .0580 .1316
14 — — — — .0003 .0015 .0065 .0214 .0577
15 — — — — — .0003 .0016 .0064 .0207
16 — — — — — — .0003 .0015 .0059
17 — — — — — — — .0003 .0013
18 — — — — — — — — .0002
$n=50$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .9948 .9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 .9662 .9971 .9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3 .8883 .9858 .9987 .9999 1.0000 1.0000 1.0000 1.0000 1.0000
4 .7497 .9540 .9943 .9995 1.0000 1.0000 1.0000 1.0000 1.0000
5 .5688 .8879 .9815 .9979 .9998 1.0000 1.0000 1.0000 1.0000
6 .3839 .7806 .9520 .9930 .9993 .9999 1.0000 1.0000 1.0000
7 .2298 .6387 .8966 .9806 .9975 .9998 1.0000 1.0000 1.0000
8 .1221 .4812 .8096 .9547 .9927 .9992 .9999 1.0000 1.0000
9 .0579 .3319 .6927 .9084 .9817 .9975 .9998 1.0000 1.0000
10 .0245 .2089 .5563 .8363 .9598 .9933 .9992 .9999 1.0000
11 .0094 .1199 .4164 .7378 .9211 .9840 .9978 .9998 1.0000
12 .0032 .0628 .2893 .6184 .8610 .9658 .9943 .9994 1.0000
13 .0010 .0301 .1861 .4890 .7771 .9339 .9867 .9982 .9998
14 .0003 .0132 .1106 .3630 .6721 .8837 .9720 .9955 .9995
15 .0001 .0053 .0607 .2519 .5532 .8122 .9460 .9896 .9987
16 — .0019 .0308 .1631 .4308 .7199 .9045 .9780 .9967
17 — .0007 .0144 .0983 .3161 .6111 .8439 .9573 .9923
18 — .0002 .0063 .0551 .2178 .4940 .7631 .9235 .9836
19 — .0001 .0025 .0287 .1406 .3784 .6644 .8727 .9675
20 — — .0009 .0139 .0848 .2736 .5535 .8026 .9405
21 — — .0003 .0063 .0478 .1861 .4390 .7138 .8987
22 — — .0001 .0026 .0251 .1187 .3299 .6100 .8389
23 — — — .0010 .0123 .0710 .2340 .4981 .7601
24 — — — .0004 .0056 .0396 .1562 .3866 .6641
25 — — — .0001 .0024 .0207 .0978 .2840 .5561
26 — — — — .0009 .0100 .0573 .1966 .4439
27 — — — — .0003 .0045 .0314 .1279 .3359
28 — — — — .0001 .0019 .0160 .0780 .2399
29 — — — — — .0007 .0076 .0444 .1611
30 — — — — — .0003 .0034 .0235 .1013
31 — — — — — .0001 .0014 .0116 .0595
32 — — — — — — .0005 .0053 .0325
33 — — — — — — .0002 .0022 .0164
34 — — — — — — .0001 .0009 .0077
35 — — — — — — — .0003 .0033
36 — — — — — — — .0001 .0013
37 — — — — — — — — .0005
38 — — — — — — — — .0002

Part 3: $n = 100$

$p=$ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
$n=100$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .6340 .8674 .9524 .9831 .9941 .9979 .9993 .9998 .9999
2 .2642 .5967 .8054 .9128 .9629 .9848 .9940 .9977 .9991
3 .0794 .3233 .5802 .7679 .8817 .9434 .9742 .9887 .9952
4 .0184 .1410 .3528 .5705 .7422 .8570 .9256 .9633 .9827
5 .0034 .0508 .1821 .3711 .5640 .7232 .8368 .9097 .9526
6 .0005 .0155 .0808 .2116 .3840 .5593 .7086 .8201 .8955
7 .0001 .0041 .0312 .1064 .2340 .3936 .5557 .6968 .8060
8 — .0009 .0106 .0475 .1280 .2517 .4012 .5529 .6872
9 — .0002 .0032 .0190 .0631 .1463 .2660 .4074 .5506
10 — — .0009 .0068 .0282 .0775 .1620 .2780 .4125
11 — — .0002 .0022 .0115 .0376 .0908 .1757 .2882
12 — — — .0007 .0043 .0169 .0469 .1028 .1876
13 — — — .0002 .0015 .0069 .0224 .0559 .1138
14 — — — — .0005 .0026 .0099 .0282 .0645
15 — — — — .0001 .0009 .0041 .0133 .0341
16 — — — — — .0003 .0016 .0058 .0169
17 — — — — — .0001 .0006 .0024 .0078
18 — — — — — — .0002 .0009 .0034
19 — — — — — — .0001 .0003 .0014
20 — — — — — — — .0001 .0005
21 — — — — — — — — .0002
22 — — — — — — — — .0001
$p=$ 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
$n=100$ $r=0$ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 .9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3 .9981 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4 .9922 .9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
5 .9763 .9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
6 .9424 .9984 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
7 .8828 .9953 .9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
8 .7939 .9878 .9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
9 .6791 .9725 .9991 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
10 .5487 .9449 .9977 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
11 .4168 .9006 .9943 .9999 1.0000 1.0000 1.0000 1.0000 1.0000
12 .2970 .8365 .9874 .9996 1.0000 1.0000 1.0000 1.0000 1.0000
13 .1982 .7527 .9747 .9990 1.0000 1.0000 1.0000 1.0000 1.0000
14 .1239 .6526 .9531 .9975 .9999 1.0000 1.0000 1.0000 1.0000
15 .0726 .5428 .9196 .9946 .9998 1.0000 1.0000 1.0000 1.0000
16 .0399 .4317 .8715 .9889 .9989 1.0000 1.0000 1.0000 1.0000
17 .0206 .3275 .8077 .9789 .9978 1.0000 1.0000 1.0000 1.0000
18 .0100 .2367 .7288 .9624 .9955 .9999 1.0000 1.0000 1.0000
19 .0046 .1628 .6379 .9370 .9915 .9995 1.0000 1.0000 1.0000
20 .0020 .1065 .5398 .9005 .9846 .9987 1.0000 1.0000 1.0000
21 .0008 .0663 .4405 .8512 .9735 .9972 1.0000 1.0000 1.0000
22 .0003 .0393 .3460 .7886 .9560 .9944 .9999 1.0000 1.0000
23 .0001 .0221 .2611 .7136 .9299 .9893 .9993 1.0000 1.0000
24 — .0119 .1891 .6289 .8934 .9806 .9981 1.0000 1.0000
25 — .0061 .1314 .5383 .8439 .9666 .9955 1.0000 1.0000
26 — .0030 .0875 .4465 .7803 .9453 .9904 .9997 1.0000
27 — .0014 .0558 .3583 .7032 .9148 .9816 .9986 1.0000
28 — .0006 .0342 .2776 .6151 .8737 .9668 .9958 1.0000
29 — .0003 .0200 .2075 .5199 .8210 .9435 .9896 .9992
30 — .0001 .0112 .1495 .4232 .7560 .9090 .9777 .9977
31 — — .0061 .1038 .3299 .6791 .8612 .9585 .9940
32 — — .0031 .0693 .2452 .5913 .7989 .9296 .9868
33 — — .0016 .0446 .1736 .4950 .7220 .8881 .9739
34 — — .0007 .0276 .1174 .3942 .6315 .8324 .9530
35 — — .0003 .0164 .0756 .2978 .5300 .7616 .9216
36 — — .0001 .0094 .0464 .2128 .4219 .6766 .8767
37 — — — .0052 .0271 .1444 .3139 .5805 .8156
38 — — — .0027 .0150 .0932 .2210 .4774 .7379
39 — — — .0014 .0079 .0570 .1465 .3738 .6445
40 — — — .0007 .0040 .0329 .0914 .2789 .5394
41 — — — .0003 .0019 .0180 .0536 .1973 .4288
42 — — — .0001 .0009 .0093 .0297 .1326 .3221
43 — — — — .0004 .0045 .0154 .0843 .2318
44 — — — — .0002 .0021 .0075 .0506 .1562
45 — — — — .0001 .0009 .0034 .0286 .0984
46 — — — — — .0004 .0014 .0151 .0578
47 — — — — — .0001 .0006 .0075 .0319
48 — — — — — — .0002 .0035 .0164
49 — — — — — — .0001 .0015 .0078
50 — — — — — — — .0006 .0034
51 — — — — — — — .0002 .0013
52 — — — — — — — .0001 .0005
53 — — — — — — — — .0002
54 — — — — — — — — .0001

Note on Approximations:

Table 1 gives binomial probabilities only for a limited range of values of $n$ and $p$ since, in practice, either the more compact tabulation of the Poisson distribution (Table 2) or that of the Normal distribution (Table 3) can usually be used to give an adequate approximation.

As a reasonable working rule:

(i) use the Poisson approximation if $p < 0.1$, putting $m = np$

(ii) use the Normal approximation if $0.1 \leq p \leq 0.9$ and $np > 5$, putting $\mu = np$ and $\sigma = \sqrt{np(1-p)}$

(iii) use the Poisson approximation if $p > 0.9$, putting $m = n(1-p)$ and working in terms of 'failures'.

Note: For values of $p > 0.5$, work in terms of 'failures' which will have probability $q (= 1 - p)$.

Example: What is the probability that 40 or more seeds will germinate out of 50 if the germination rate is 70%? Since the probability of 'success' is greater than 0.5, the table can not be used directly; however, 40 or more successes is the same as 10 or fewer 'failures'. The probability of 10 or fewer 'failures' = 1 – probability of 11 or more 'failures' = 1 – 0.9211 = 0.0789.

Table 2 Cumulative Poisson Probabilities

$\mu$ = mean of distribution. The table gives the probability of obtaining $r$ or more successes. That is:

$$\sum_{x=r}^{\infty} \frac{e^{-\mu} \mu^x}{x!}$$

When there is no entry for a particular pair of values of $r$ and $\mu$, this indicates that the appropriate probability is less than 0.00005. Similarly, a tabulated value of 1.0000 represents a probability greater than 0.99995.

$r$ $\mu=0.5$ $\mu=1.0$ $\mu=1.5$ $\mu=2.0$ $\mu=2.5$ $\mu=3.0$ $\mu=3.5$ $\mu=4.0$ $\mu=4.5$ $\mu=5.0$
0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .3935 .6321 .7769 .8647 .9179 .9502 .9698 .9817 .9889 .9933
2 .0902 .2642 .4422 .5940 .7127 .8009 .8641 .9084 .9389 .9596
3 .0144 .0803 .1912 .3233 .4562 .5768 .6792 .7619 .8264 .8753
4 .0018 .0190 .0656 .1429 .2424 .3528 .4634 .5665 .6577 .7350
5 .0002 .0037 .0186 .0527 .1088 .1847 .2746 .3712 .4679 .5595
6 — .0006 .0045 .0166 .0420 .0839 .1424 .2149 .2971 .3840
7 — .0001 .0009 .0045 .0142 .0335 .0653 .1107 .1689 .2378
8 — — .0002 .0011 .0042 .0119 .0267 .0511 .0866 .1334
9 — — — .0002 .0011 .0038 .0099 .0214 .0403 .0681
10 — — — — .0003 .0011 .0033 .0081 .0171 .0318
11 — — — — .0001 .0003 .0010 .0028 .0067 .0137
12 — — — — — .0001 .0003 .0009 .0024 .0055
13 — — — — — — .0001 .0003 .0008 .0020
14 — — — — — — — .0001 .0002 .0007
15 — — — — — — — — .0001 .0002
$r$ $\mu=6$ $\mu=7$ $\mu=8$ $\mu=9$ $\mu=10$ $\mu=11$ $\mu=12$ $\mu=13$ $\mu=14$ $\mu=15$
0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1 .9975 .9991 .9997 .9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 .9826 .9927 .9970 .9988 .9995 .9998 .9999 1.0000 1.0000 1.0000
3 .9380 .9704 .9862 .9938 .9972 .9988 .9995 .9998 .9999 1.0000
4 .8488 .9182 .9576 .9788 .9897 .9951 .9977 .9990 .9996 .9998
5 .7149 .8270 .9004 .9450 .9707 .9849 .9924 .9963 .9984 .9993
6 .5543 .6993 .8088 .8843 .9329 .9625 .9797 .9893 .9945 .9972
7 .3937 .5503 .6866 .7932 .8699 .9214 .9542 .9741 .9858 .9924
8 .2560 .4013 .5470 .6761 .7798 .8568 .9105 .9463 .9693 .9830
9 .1528 .2709 .4075 .5443 .6530 .7513 .8311 .8888 .9281 .9558
10 .0839 .1695 .2834 .4126 .5231 .6321 .7261 .8030 .8626 .9074
11 .0426 .0985 .1841 .2940 .3971 .5076 .6142 .7111 .7931 .8614
12 .0201 .0533 .1119 .1969 .2834 .3840 .4895 .5944 .6926 .7780
13 .0088 .0270 .0638 .1242 .1906 .2775 .3751 .4803 .5880 .6815
14 .0036 .0128 .0342 .0739 .1211 .1919 .2740 .3705 .4760 .5852
15 .0014 .0057 .0173 .0415 .0730 .1261 .1914 .2725 .3675 .4811
16 .0005 .0024 .0082 .0220 .0415 .0786 .1284 .1930 .2725 .3684
17 .0002 .0010 .0037 .0111 .0225 .0469 .0822 .1313 .1941 .2699
18 — .0004 .0016 .0053 .0116 .0267 .0503 .0861 .1327 .1892
19 — .0001 .0006 .0024 .0057 .0145 .0293 .0546 .0871 .1272
20 — — .0003 .0011 .0026 .0076 .0163 .0332 .0551 .0821
21 — — .0001 .0005 .0012 .0038 .0087 .0195 .0334 .0508
22 — — — .0002 .0005 .0019 .0044 .0110 .0195 .0302
23 — — — .0001 .0002 .0009 .0022 .0060 .0110 .0173
24 — — — — .0001 .0004 .0010 .0031 .0060 .0095
25 — — — — — .0002 .0005 .0016 .0031 .0050
26 — — — — — .0001 .0002 .0008 .0016 .0026
27 — — — — — — .0001 .0004 .0008 .0013
28 — — — — — — — .0002 .0004 .0006
29 — — — — — — — .0001 .0002 .0003
30 — — — — — — — — .0001 .0002
31 — — — — — — — — — .0001
$r$ $\mu=20$ $\mu=25$ $\mu=30$ $r$ $\mu=20$ $\mu=25$ $\mu=30$
0 1.0000 1.0000 1.0000 21 .0002 .1855 .6472
1 1.0000 1.0000 1.0000 22 — .1444 .5763
2 1.0000 1.0000 1.0000 23 — .1094 .5030
3 1.0000 1.0000 1.0000 24 — .0804 .4289
4 1.0000 1.0000 1.0000 25 — .0574 .3563
5 1.0000 1.0000 1.0000 26 — .0397 .2879
6 1.0000 1.0000 1.0000 27 — .0267 .2270
7 1.0000 1.0000 1.0000 28 — .0174 .1746
8 .9999 1.0000 1.0000 29 — .0110 .1311
9 .9995 1.0000 1.0000 30 — .0068 .0960
10 .9980 1.0000 1.0000 31 — .0041 .0688
11 .9945 1.0000 1.0000 32 — .0024 .0481
12 .9863 .9999 1.0000 33 — .0014 .0328
13 .9689 .9995 1.0000 34 — .0008 .0219
14 .9367 .9986 1.0000 35 — .0004 .0142
15 .8818 .9963 1.0000 36 — .0002 .0090
16 .7982 .9906 1.0000 37 — .0001 .0056
17 .6860 .9789 1.0000 38 — — .0034
18 .5591 .9573 .9999 39 — — .0020
19 .4274 .9213 .9997 40 — — .0012
20 .3037 .8666 .9992 41 — — .0007

Table 3 Areas in Upper Tail of the Normal Distribution

$z$ = standardised normal variable. The table gives the probability that $z$ is greater than the tabulated value. That is:

$$P(Z > z) = \int_{z}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz$$

$z$ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641
0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247
0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859
0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483
0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121
0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .2776
0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451
0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148
0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867
0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611
1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379
1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170
1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985
1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823
1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681
1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559
1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455
1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367
1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294
1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233
2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183
2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143
2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110
2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084
2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064
2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048
2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036
2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026
2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019
2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014
3.0 .00135 .00131 .00126 .00122 .00118 .00114 .00111 .00107 .00104 .00100
3.1 .00097 .00094 .00090 .00087 .00084 .00082 .00079 .00076 .00074 .00071
3.2 .00069 .00066 .00064 .00062 .00060 .00058 .00056 .00054 .00052 .00050
3.3 .00048 .00047 .00045 .00043 .00042 .00040 .00039 .00038 .00036 .00035
3.4 .00034 .00032 .00031 .00030 .00029 .00028 .00027 .00026 .00025 .00024
3.5 .00023 .00022 .00022 .00021 .00020 .00019 .00019 .00018 .00017 .00017
3.6 .00016 .00015 .00015 .00014 .00014 .00013 .00013 .00012 .00012 .00011
3.7 .00011 .00010 .00010 .00010 .00009 .00009 .00008 .00008 .00008 .00008
3.8 .00007 .00007 .00007 .00006 .00006 .00006 .00006 .00005 .00005 .00005
3.9 .00005 .00005 .00004 .00004 .00004 .00004 .00004 .00004 .00003 .00003
4.0 .00003

Table 4 Percentage Points of the Normal Distribution

$P(Z > z_p) = p$ $z_p$
0.5000 0.0000
0.4000 0.2533
0.3000 0.5244
0.2000 0.8416
0.1500 1.0364
0.1000 1.2816
0.0500 1.6449
0.0250 1.9600
0.0100 2.3263
0.0050 2.5758
0.0010 3.0902
0.0005 3.2905
0.0001 3.7190

Table 5 Ordinates of the Normal Distribution

The table gives values of $\phi(z)$ where:

$$\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$$

$z$ 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 .3989 .3989 .3989 .3988 .3986 .3984 .3982 .3980 .3977 .3973
0.1 .3970 .3965 .3961 .3956 .3951 .3945 .3939 .3932 .3925 .3918
0.2 .3910 .3902 .3894 .3885 .3876 .3867 .3857 .3847 .3836 .3825
0.3 .3814 .3802 .3790 .3778 .3765 .3752 .3739 .3725 .3712 .3697
0.4 .3683 .3668 .3653 .3637 .3621 .3605 .3589 .3572 .3555 .3538
0.5 .3521 .3503 .3485 .3467 .3448 .3429 .3410 .3391 .3372 .3352
0.6 .3332 .3312 .3292 .3271 .3251 .3230 .3209 .3187 .3166 .3144
0.7 .3123 .3101 .3079 .3056 .3034 .3011 .2989 .2966 .2943 .2920
0.8 .2897 .2874 .2850 .2827 .2803 .2780 .2756 .2732 .2709 .2685
0.9 .2661 .2637 .2613 .2589 .2565 .2541 .2516 .2492 .2468 .2444
1.0 .2420 .2396 .2371 .2347 .2323 .2299 .2275 .2251 .2227 .2203
1.1 .2179 .2155 .2131 .2107 .2083 .2059 .2036 .2012 .1989 .1965
1.2 .1942 .1919 .1895 .1872 .1849 .1826 .1804 .1781 .1758 .1736
1.3 .1714 .1691 .1669 .1647 .1626 .1604 .1582 .1561 .1539 .1518
1.4 .1497 .1476 .1456 .1435 .1415 .1394 .1374 .1354 .1334 .1315
1.5 .1295 .1276 .1257 .1238 .1219 .1200 .1182 .1163 .1145 .1127
1.6 .1109 .1092 .1074 .1057 .1040 .1023 .1006 .0989 .0973 .0957
1.7 .0940 .0925 .0909 .0893 .0878 .0863 .0848 .0833 .0818 .0804
1.8 .0790 .0775 .0761 .0748 .0734 .0721 .0707 .0694 .0681 .0669
1.9 .0656 .0644 .0632 .0620 .0608 .0596 .0584 .0573 .0562 .0551
2.0 .0540 .0529 .0519 .0508 .0498 .0488 .0478 .0468 .0459 .0449
2.1 .0440 .0431 .0422 .0413 .0404 .0396 .0387 .0379 .0371 .0363
2.2 .0355 .0347 .0339 .0332 .0325 .0317 .0310 .0303 .0297 .0290
2.3 .0283 .0277 .0270 .0264 .0258 .0252 .0246 .0241 .0235 .0229
2.4 .0224 .0219 .0213 .0208 .0203 .0198 .0194 .0189 .0184 .0180
2.5 .0175 .0171 .0167 .0163 .0158 .0154 .0151 .0147 .0143 .0139
2.6 .0136 .0132 .0129 .0126 .0122 .0119 .0116 .0113 .0110 .0107
2.7 .0104 .0101 .0099 .0096 .0093 .0091 .0088 .0086 .0084 .0081
2.8 .0079 .0077 .0075 .0073 .0071 .0069 .0067 .0065 .0063 .0061
2.9 .0060 .0058 .0056 .0055 .0053 .0051 .0050 .0048 .0047 .0046
3.0 .0044

Table 6 Exponential Function $e^{-x}$

$x$ $e^{-x}$ $x$ $e^{-x}$ $x$ $e^{-x}$ $x$ $e^{-x}$
0.0 1.0000 2.5 .0821 5.0 .0067 7.5 .0006
0.1 .9048 2.6 .0743 5.1 .0061 7.6 .0005
0.2 .8187 2.7 .0672 5.2 .0055 7.7 .0005
0.3 .7408 2.8 .0608 5.3 .0050 7.8 .0004
0.4 .6703 2.9 .0550 5.4 .0045 7.9 .0004
0.5 .6065 3.0 .0498 5.5 .0041 8.0 .0003
0.6 .5488 3.1 .0450 5.6 .0037 8.1 .0003
0.7 .4966 3.2 .0408 5.7 .0033 8.2 .0003
0.8 .4493 3.3 .0369 5.8 .0030 8.3 .0002
0.9 .4066 3.4 .0334 5.9 .0027 8.4 .0002
1.0 .3679 3.5 .0302 6.0 .0025 8.5 .0002
1.1 .3329 3.6 .0273 6.1 .0022 8.6 .0002
1.2 .3012 3.7 .0247 6.2 .0020 8.7 .0002
1.3 .2725 3.8 .0224 6.3 .0018 8.8 .0002
1.4 .2466 3.9 .0202 6.4 .0017 8.9 .0001
1.5 .2231 4.0 .0183 6.5 .0015 9.0 .0001
1.6 .2019 4.1 .0166 6.6 .0014 9.5 .0001
1.7 .1827 4.2 .0150 6.7 .0012 10.0 .0000
1.8 .1653 4.3 .0136 6.8 .0011
1.9 .1496 4.4 .0123 6.9 .0010
2.0 .1353 4.5 .0111 7.0 .0009
2.1 .1225 4.6 .0101 7.1 .0008
2.2 .1108 4.7 .0091 7.2 .0007
2.3 .1003 4.8 .0082 7.3 .0007
2.4 .0907 4.9 .0074 7.4 .0006

Table 7 Percentage Points of the $t$ Distribution

The table gives the values of $t_{\nu,\alpha}$ where:

$$P(t_\nu > t_{\nu,\alpha}) = \alpha$$

and $t_\nu$ has a $t$-distribution with $\nu$ degrees of freedom.

$\nu$ $t_{0.100}$ $t_{0.050}$ $t_{0.025}$ $t_{0.010}$ $t_{0.005}$
1 3.078 6.314 12.706 31.821 63.657
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
8 1.397 1.860 2.306 2.896 3.355
9 1.383 1.833 2.262 2.821 3.250
10 1.372 1.812 2.228 2.764 3.169
11 1.363 1.796 2.201 2.718 3.106
12 1.356 1.782 2.179 2.681 3.055
13 1.350 1.771 2.160 2.650 3.012
14 1.345 1.761 2.145 2.624 2.977
15 1.341 1.753 2.131 2.602 2.947
16 1.337 1.746 2.120 2.583 2.921
17 1.333 1.740 2.110 2.567 2.898
18 1.330 1.734 2.101 2.552 2.878
19 1.328 1.729 2.093 2.539 2.861
20 1.325 1.725 2.086 2.528 2.845
21 1.323 1.721 2.080 2.518 2.831
22 1.321 1.717 2.074 2.508 2.819
23 1.319 1.714 2.069 2.500 2.807
24 1.318 1.711 2.064 2.492 2.797
25 1.316 1.708 2.060 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779
27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756
30 1.310 1.697 2.042 2.457 2.750
40 1.303 1.684 2.021 2.423 2.704
60 1.296 1.671 2.000 2.390 2.660
120 1.289 1.658 1.980 2.358 2.617
$\infty$ 1.282 1.645 1.960 2.326 2.576

Table 8 Percentage Points of the $\chi^2$ Distribution

The table gives the values of $\chi^2_{\nu,\alpha}$ where:

$$P(\chi^2_\nu > \chi^2_{\nu,\alpha}) = \alpha$$

and $\chi^2_\nu$ has a $\chi^2$-distribution with $\nu$ degrees of freedom.

$\nu$ $\chi^2_{0.995}$ $\chi^2_{0.990}$ $\chi^2_{0.975}$ $\chi^2_{0.950}$ $\chi^2_{0.900}$ $\chi^2_{0.100}$ $\chi^2_{0.050}$ $\chi^2_{0.025}$ $\chi^2_{0.010}$ $\chi^2_{0.005}$
1 — — 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672

Table 9 Percentage Points of the $F$ Distribution — Upper 5% Points

The table gives the values of $F_{\nu_1,\nu_2,0.05}$ where:

$$P(F_{\nu_1,\nu_2} > F_{\nu_1,\nu_2,0.05}) = 0.05$$

$\nu_2$ $\nu_1=1$ $\nu_1=2$ $\nu_1=3$ $\nu_1=4$ $\nu_1=5$ $\nu_1=6$ $\nu_1=7$ $\nu_1=8$ $\nu_1=9$ $\nu_1=10$ $\nu_1=12$ $\nu_1=15$ $\nu_1=20$ $\nu_1=30$ $\nu_1=\infty$
1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 250.1 254.3
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.46 19.50
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.62 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.75 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.50 4.36
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.81 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.38 3.23
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.08 2.93
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.86 2.71
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.70 2.54
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.65 2.57 2.40
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.47 2.30
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.25 2.07
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.04 1.84
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.84 1.62
$\infty$ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.46 1.00

Table 9 — Upper 1% Points

$\nu_2$ $\nu_1=1$ $\nu_1=2$ $\nu_1=3$ $\nu_1=4$ $\nu_1=5$ $\nu_1=6$ $\nu_1=7$ $\nu_1=8$ $\nu_1=9$ $\nu_1=10$ $\nu_1=12$ $\nu_1=15$ $\nu_1=20$ $\nu_1=30$ $\nu_1=\infty$
1 4052 4999 5403 5625 5764 5859 5928 5981 6022 6056 6106 6157 6209 6261 6366
2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.45 99.47 99.50
3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.05 26.87 26.69 26.50 26.13
4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.37 14.20 14.02 13.84 13.46
5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 9.72 9.55 9.38 9.02
6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 7.40 7.23 6.88
7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 5.99 5.65
8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.20 4.86
9 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.65 4.31
10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 4.41 4.25 3.91
12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 3.86 3.70 3.36
15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 3.37 3.21 2.87
20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.78 2.42
30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.84 2.70 2.55 2.39 2.01
$\infty$ 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.18 2.04 1.88 1.70 1.00

Table 10 Percentage Points of the Correlation Coefficient

The table gives the values of $r_\alpha$ where:

$$P(|r| > r_\alpha) = \alpha$$

$n$ $r_{0.100}$ $r_{0.050}$ $r_{0.020}$ $r_{0.010}$ $r_{0.005}$
3 .988 .997 .9995 .9999 1.0000
4 .900 .950 .980 .990 .9950
5 .805 .878 .934 .959 .9740
6 .729 .811 .882 .917 .9410
7 .669 .754 .833 .874 .9050
8 .621 .707 .789 .834 .8700
9 .582 .666 .750 .798 .8380
10 .549 .632 .715 .765 .8050
11 .521 .602 .685 .735 .7750
12 .497 .576 .658 .708 .7470
13 .476 .553 .634 .684 .7230
14 .458 .532 .612 .661 .6970
15 .441 .514 .592 .641 .6740
16 .426 .497 .574 .623 .6530
17 .412 .482 .558 .606 .6330
18 .400 .468 .543 .590 .6150
19 .389 .456 .529 .575 .5990
20 .378 .444 .516 .561 .5840
22 .359 .423 .492 .537 .5570
24 .343 .404 .470 .515 .5330
26 .329 .388 .451 .495 .5110
28 .317 .374 .434 .478 .4920
30 .306 .361 .418 .462 .4740

Source: Murdoch, J. and Barnes, J.A. (1998). Statistical Tables, 4th Edition. Macmillan Press.

Complete transcription for FAD1015 — Mathematics III