STATISTICAL TABLES

ST1 Cumulative Binomial Probabilities

ST2 Tail Probability Under Standard Normal Distribution

ST3 Critical Values Under Chi-Square Distribution

ST4 Student's t-Distribution

ST5 F-Distribution: 5% and 1% Points for the Distribution of F

ST6 Random Normal Numbers, μ = 0 and σ = 1

ST7 Critical Values of the Kolmogorov-Smirnov One-Sample Test Statistic

ST8 Critical Values of the Kolmogorov-Smirnov Test Statistics for Two Samples of Equal Size

ST9 Critical Values of the Kolmogorov-Smirnov Test Statistics for Two Samples of Unequal Size

ST10 Critical Values of the Wilcoxon Signed-Rank Test Statistic

ST11 Critical Values of the Mann-Whitney-Wilcoxon Test Statistic

ST12 Critical Points of Kendall's Tau Statistics

ST13 Critical Values of Spearman's Rank Correlation Statistic

Table ST1. Cumulative Binomial Probabilities, images, r = 0,1,2,…,n − 1

Source: For n = 2 through 10, adapted with permission from E. Parzen, Modern Probability Theory and Its Applications, John Wiley, New York, 1962. For n = 11 through 15, adapted with permission from Tables of Cumulative Binomial Probability Distribution, Harvard University Press, Cambridge, M.A., 1955.

n r p
0.01 0.05 0.10 0.20 0.25 0.30 0.333 0.40 0.50
2 0 0.9801 0.9025 0.8100 0.6400 0.5625 0.4900 0.4444 0.3600 0.2500
1 0.9999 0.9975 0.9900 0.9600 0.9375 0.9100 0.8888 0.8400 0.7500
3 0 0.9703 0.8574 0.7290 0.5120 0.4219 0.3430 0.2963 0.2160 0.1250
1 0.9997 0.9928 0.9720 0.8960 0.8438 0.7840 0.7407 0.6480 0.5000
2 1.0000 0.9999 0.9990 0.9920 0.9844 0.9730 0.9629 0.9360 0.8750
4 0 0.9606 0.8145 0.6561 0.4096 0.3164 0.2401 0.1975 0.1296 0.0625
1 0.9994 0.9860 0.9477 0.8192 0.7383 0.6517 0.5926 0.4742 0.3125
2 1.0000 0.9995 0.9963 0.9728 0.9492 0.9163 0.8889 0.8198 0.6875
3 1.0000 0.9999 0.9984 0.9961 0.9919 0.9877 0.9734 0.9375
5 0 0.9510 0.7738 0.5905 0.3277 0.2373 0.1681 0.1317 0.0778 0.0312
1 0.9990 0.9774 0.9185 0.7373 0.6328 0.5283 0.4609 0.3370 0.1874
2 1.0000 0.9988 0.9914 0.9421 0.8965 0.8370 0.7901 0.6826 0.4999
3 0.9999 0.9995 0.9933 0.9844 0.9693 0.9547 0.9130 0.8124
4 1.0000 1.0000 0.9997 0.9990 0.9977 0.9959 0.9898 0.9686
6 0 0.9415 0.7351 0.5314 0.2621 0.1780 0.1176 0.0878 0.0467 0.0156
1 0.9986 0.9672 0.8857 0.6553 0.5340 0.4201 0.3512 0.2333 0.1094
2 1.0000 0.9977 0.9841 0.9011 0.8306 0.7442 0.6804 0.5443 0.3438
3 0.9998 0.9987 0.9830 0.9624 0.9294 0.8999 0.8208 0.6563
4 0.9999 0.9999 0.9984 0.9954 0.9889 0.9822 0.9590 0.8907
5 1.0000 1.0000 0.9999 0.9998 0.9991 0.9987 0.9959 0.9845
7 0 0.9321 0.6983 0.4783 0.2097 0.1335 0.0824 0.0585 0.0280 0.0078
1 0.9980 0.9556 0.6554 0.5767 0.4450 0.3294 0.2633 0.1586 0.0625
2 1.0000 0.9962 0.8503 0.8520 0.7565 0.6471 0.5706 0.4199 0.2266
3 0.9998 0.9743 0.9667 0.9295 0.8740 0.8267 0.7102 0.5000
4 1.0000 0.9973 0.9953 0.9872 0.9712 0.9547 0.9037 0.7734
5 0.9998 0.9996 0.9987 0.9962 0.9931 0.9812 0.9375
6 1.0000 1.0000 0.9999 0.9998 0.9995 0.9984 0.9922
8 0 0.9227 0.6634 0.4305 0.1678 0.1001 0.0576 0.0390 0.0168 0.0039
1 0.9973 0.9427 0.8131 0.5033 0.3671 0.2553 0.1951 0.1064 0.0352
2 0.9999 0.9942 0.9619 0.7969 0.6786 0.5518 0.4682 0.3154 0.1445
3 1.0000 0.9996 0.9950 0.9437 0.8862 0.8059 0.7413 0.5941 0.3633
4 1.0000 0.9996 0.9896 0.9727 0.9420 0.9120 0.8263 0.6367
5 1.0000 0.9988 0.9958 0.9887 0.9803 0.9502 0.8555
6 1.0000 0.9996 0.9987 0.9974 0.9915 0.9648
7 1.0000 0.9999 0.9998 0.9993 0.9961
9 0 0.9135 0.6302 0.3874 0.1342 0.0751 0.0404 0.0260 0.0101 0.0020
1 0.9965 0.9287 0.7748 0.4362 0.3004 0.1960 0.1431 0.0706 0.0196
2 0.9999 0.9916 0.9470 0.7382 0.6007 0.4628 0.3772 0.2318 0.0899
3 1.0000 0.9993 0.9916 0.9144 0.8343 0.7296 0.6503 0.4826 0.2540
4 0.9999 0.9990 0.9805 0.9511 0.9011 0.8551 0.7334 0.5001
5 1.0000 0.9998 0.9970 0.9900 0.9746 0.9575 0.9006 0.7462
6 0.9999 0.9998 0.9987 0.9956 0.9916 0.9749 0.9103
7 1.0000 1.0000 0.9999 0.9995 0.9989 0.9961 0.9806
8 1.0000 0.9999 0.9998 0.9996 0.9982
10 0 0.9044 0.5987 0.3487 0.1074 0.0563 0.0282 0.0173 0.0060 0.0010
1 0.9958 0.9138 0.7361 0.3758 0.2440 0.1493 0.1040 0.0463 0.0108
2 1.0000 0.9884 0.9298 0.6778 0.5256 0.3828 0.2991 0.1672 0.0547
3 0.9989 0.9872 0.8791 0.7759 0.6496 0.5592 0.3812 0.1719
4 0.9999 0.9984 0.9672 0.9219 0.8497 0.7868 0.6320 0.3770
5 1.0000 0.9999 0.9936 0.9803 0.9526 0.9234 0.8327 0.6231
6 1.0000 0.9991 0.9965 0.9894 0.9803 0.9442 0.8282
7 0.9999 0.9996 0.9984 0.9966 0.9867 0.9454
8 1.0000 1.0000 0.9998 0.9996 0.9973 0.9893
9 1.0000 0.9999 0.9999 0.9991
11 0 0.8954 0.5688 0.3138 0.0859 0.0422 0.0198 0.0116 0.0036 0.0005
1 0.9948 0.8981 0.6974 0.3221 0.1971 0.1130 0.0752 0.0320 0.0059
2 0.9998 0.9848 0.9104 0.6174 0.4552 0.3128 0.2341 0.1189 0.0327
3 1.0000 0.9984 0.9815 0.8389 0.7133 0.5696 0.4726 0.2963 0.1133
4 0.9999 0.9972 0.9496 0.8854 0.7897 0.7110 0.5328 0.2744
5 1.0000 0.9997 0.9884 0.9657 0.9218 0.8779 0.7535 0.5000
6 1.0000 0.9981 0.9924 0.9784 0.9614 0.9007 0.7256
7 0.9998 0.9988 0.9947 0.9912 0.9707 0.8867
8 1.0000 0.9999 0.9994 0.9986 0.9941 0.9673
9 1.0000 0.9999 0.9999 0.9993 0.9941
10 1.0000 1.0000 1.0000 0.9995
12 0 0.8864 0.5404 0.2824 0.0687 0.0317 0.0139 0.0077 0.0022 0.0002
1 0.9938 0.8816 0.6590 0.2749 0.1584 0.0850 0.0540 0.0196 0.0032
2 0.9998 0.9804 0.8892 0.5584 0.3907 0.2528 0.1811 0.0835 0.0193
3 1.0000 0.9978 0.9744 0.7946 0.6488 0.4925 0.3931 0.2254 0.0730
4 1.0000 0.9998 0.9957 0.9806 0.8424 0.7237 0.6315 0.4382 0.1939
5 1.0000 1.0000 0.9995 0.9961 0.9456 0.8822 0.8223 0.6652 0.3872
6 1.0000 0.9994 0.9858 0.9614 0.9336 0.8418 0.6128
7 0.9999 0.9972 0.9905 0.9812 0.9427 0.8062
8 1.0000 0.9996 0.9983 0.9962 0.9848 0.9270
9 10000 0.9998 0.9995 0.9972 0.9807
10 1.0000 0.9999 0.9997 0.9968
11 1.0000 1.0000 0.9998
13 0 0.8775 0.5134 0.2542 0.0550 0.0238 0.0097 0.0052 0.0013 0.0000
1 0.9928 0.8746 0.6214 0.2337 0.1267 0.0637 0.0386 0.0126 0.0017
2 0.9997 0.9755 0.8661 0.5017 0.3326 0.2025 0.1388 0.0579 0.0112
3 1.0000 0.9969 0.9659 0.7473 0.5843 0.4206 0.3224 0.1686 0.0462
4 0.9997 0.9936 0.9009 0.7940 0.6543 0.5521 0.3531 0.1334
5 1.0000 0.9991 0.9700 0.9198 0.8346 0.7587 0.5744 0.2905
6 0.9999 0.9930 0.9757 0.9376 0.8965 0.7712 0.5000
7 1.0000 0.9988 0.9944 0.9818 0.9654 0.9024 0.7095
8 0.9998 0.9990 0.9960 0.9912 0.9679 0.8666
9 1.0000 0.9999 0.9994 0.9984 0.9922 0.9539
10 1.0000 0.9999 0.9998 0.9987 0.9888
11 1.0000 1.0000 0.9999 0.9983
12 1.0000 0.9999
14 0 0.8687 0.4877 0.2288 0.0440 0.0178 0.0068 0.0034 0.0008 0.0000
1 0.9916 0.8470 0.5847 0.1979 0.1010 0.0475 0.0274 0.0081 0.0009
2 0.9997 0.9700 0.8416 0.4480 0.2812 0.1608 0.1054 0.0398 0.0065
3 1.0000 0.9958 0.9559 0.6982 0.5214 0.3552 0.2612 0.1243 0.0287
4 0.9996 0.9908 0.8702 0.7416 0.5842 0.4755 0.2793 0.0898
5 1.0000 0.9986 0.9562 0.8884 0.7805 0.6898 0.4859 0.2120
6 0.9998 0.9884 0.9618 0.9067 0.8506 0.6925 0.3953
7 1.0000 0.9976 0.9897 0.9686 0.9424 0.8499 0.6048
8 0.9996 0.9979 0.9917 0.9826 0.9417 0.7880
9 1.0000 0.9997 0.9984 0.9960 0.9825 0.9102
10 1.0000 0.9998 0.9993 0.9961 0.9713
11 1.0000 0.9999 0.9994 0.9936
12 1.0000 0.9999 0.9991
13 0.9999
15 0 0.8601 0.4633 0.2059 0.0352 0.0134 0.0048 0.0023 0.0005 0.0000
1 0.9904 0.8291 0.5491 0.1672 0.0802 0.0353 0.0194 0.0052 0.0005
2 0.9996 0.9638 0.8160 0.3980 0.2361 0.1268 0.0794 0.0271 0.0037
3 1.0000 0.9946 0.9444 0.6482 0.4613 0.2969 0.2092 0.0905 0.0176
4 0.9994 0.9873 0.8358 0.6865 0.5255 0.4041 0.2173 0.0592
5 1.0000 0.9978 0.9390 0.8516 0.7216 0.6184 0.4032 0.1509
6 0.9997 0.9820 0.9434 0.8689 0.7970 0.6098 0.3036
7 1.0000 0.9958 0.9827 0.9500 0.9118 0.7869 0.5000
8 0.9992 0.9958 0.9848 0.9692 0.9050 0.6964
9 0.9999 0.9992 0.9964 0.9915 0.9662 0.8491
10 1.0000 0.9999 0.9993 0.9982 0.9907 0.9408
11 1.0000 0.9999 0.9997 0.9981 0.9824
12 1.0000 1.0000 0.9997 0.9963
13 1.0000 0.9995
14 1.0000

Table ST2. Tail Probability Under Standard Normal Distributiona

Source: Adapted with permission from P. G. Hoel, Introduction to Mathematical Statistics, 4th ed., Wiley, New York, 1971, p. 391.

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
0.7 0.2420 0.2389 0.2358 0.2327 0.2297 0.2266 0.2231 0.2206 0.2177 0.2148
0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1984 0.1867
0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0017 0.0069 0.0068 0.0066 0.0064
2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010

aThis table gives the probability that the standard normal variable Z will exceed a given positive value z, that is, images. The probabilities for negative values of z are obtained by symmetry.

Table ST3. Critical Values Under Chi-Square Distributiona

Source: Reproduced from Statistical Methods for Research Workers, 14th ed., 1972, with the permission of the Estate of R. A. Fisher, and Hafner Press.

image

aFor degrees of freedom greater than 30, the expression images may be used as a normal deviate with unit variance, where n is the number of degrees of freedom.

Table ST4. Student's t-Distributiona

Source: P. G. Hoel, Introduction to Mathematical Statistics, 4th ed., Wiley, New York, 1971, p. 393. Reprinted by permission of John Wiley & Sons, Inc.

n α
0.10 0.05 0.025 0.01 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
images 1.282 1.645 1.960 2.326 2.576

aThe first column lists the number of degrees of freedom (n). The headings of the other columns give probabilities (α) for t to exceed the entry value. Use symmetry for negative t values.

Table ST5. F-Distribution: 5% (Lightface Type) and 1% (Boldface Type) Points for the Distribution of F

Source: Reprinted by permission from George W. Snedecor and William G. Cochran, Statistical Methods, 6th ed., © 1967 by Iowa State University Press, Ames, I.A.

image
image
image
image

Table ST6. Random Normal Numbers, μ = 0 and σ = 1

Source: From tables of the RAND Corporation, by permission.

1 2 3 4 5 6 7 8 9 10
0.464 0.137 2.455 −0.323 −0.068 0.290 −0.288 1.298 0.241 −0.957
0.060 −2.526 −0.531 −0.194 0.543 −1.558 0.187 −1.190 0.022 0.525
1.486 −0.354 −0.634 0.697 0.926 1.375 0.785 −0.963 −0.853 −1.865
1.022 −0.472 1.279 3.521 0.571 −1.851 0.194 1.192 −0.501 −0.273
1.394 −0.555 0.046 0.321 2.945 1.974 −0.258 0.412 0.439 −0.035
0.906 −0.513 −0.525 0.595 0.881 −0.934 1.579 0.161 −1.885 0.371
1.179 −1.055 0.007 0.769 0.971 0.712 1.090 −0.631 −0.255 −0.702
−1.501 −0.488 −0.162 −0.136 1.033 0.203 0.448 0.748 −0.423 −0.432
−0.690 0.756 −1.618 −0.345 −0.511 −2.051 −0.457 −0.218 0.857 −0.465
1.372 0.225 0.378 0.761 0.181 −0.736 0.960 −1.530 −0.260 0.120
−0.482 1.678 −0.057 −1.229 −0.486 0.856 −0.491 −1.983 −2.830 −0.238
−1.376 −0.150 1.356 −0.561 −0.256 −0.212 0.219 0.779 0.953 −0.869
−1.010 0.598 −0.918 1.598 0.065 0.415 −0.169 0.313 −0.973 −1.016
−0.005 −0.899 0.012 −0.725 1.147 −0.121 1.096 0.481 −1.691 0.417
1.393 1.163 −0.911 1.231 −0.199 −0.246 1.239 −2.574 −0.558 0.056
−1.787 −0.261 1.237 1.046 −0.508 −1.630 −0.146 −0.392 −0.627 0.561
−0.105 −0.357 −1.384 0.360 −0.992 −0.116 −1.698 −2.832 −1.108 −2.357
−1.339 1.827 −0.959 0.424 0.969 −1.141 −1.041 0.362 −1.726 1.956
1.041 0.535 0.731 1.377 0.983 −1.330 1.620 −1.040 0.524 −0.281
0.279 −2.056 0.717 −0.873 −1.096 −1.396 1.047 0.089 −0.573 0.932
−1.805 −2.008 −1.633 0.542 0.250 −0.166 0.032 0.079 0.471 −1.029
−1.186 1.180 1.114 0.882 1.265 −0.202 0.151 −0.376 −0.310 0.479
0.658 −1.141 1.151 −1.210 0.927 0.425 0.290 −0.902 0.610 2.709
−0.439 0.358 −1.939 0.891 −0.227 0.602 0.873 −0.437 −0.220 −0.057
−1.399 −0.230 0.385 −0.649 −0.577 0.237 −0.289 0.513 0.738 −0.300
0.199 0.208 −1.083 −0.219 −0.291 1.221 1.119 0.004 −2.015 −0.594
0.159 0.272 −0.313 0.084 −2.828 −0.430 −0.792 −1.275 −0.623 −1.047
2.273 0.606 0.606 −0.747 0.247 1.291 0.063 −1.793 −0.699 −1.347
0.041 −0.307 0.121 0.790 −0.584 0.541 0.484 −0.986 0.481 0.996
−1.132 −2.098 0.921 0.145 0.446 −1.661 1.045 −1.363 −0.586 −1.023
0.768 0.079 −1.473 0.034 −2.127 0.665 0.084 −0.880 −0.579 0.551
0.375 −1.658 −0.851 0.234 −0.656 0.340 −0.086 −0.158 −0.120 0.418
−0.513 −0.344 0.210 −0.736 1.041 0.008 0.427 −0.831 0.191 0.074
0.292 −0.521 1.266 −1.206 −0.899 0.110 −0.528 −0.813 0.071 0.524
1.026 2.990 −0.574 −0.491 −1.114 1.297 −1.433 −1.345 −3.001 0.479
−1.334 1.278 −0.568 −0.109 −0.515 −0.566 2.923 0.500 0.359 0.326
−0.287 −0.144 −0.254 0.574 −0.451 −1.181 −1.190 −0.318 −0.094 1.114
0.161 −0.886 −0.921 −0.509 1.410 −0.518 0.192 −0.432 1.501 1.068
−1.346 0.193 −1.202 0.394 −1.045 0.843 0.942 1.045 0.031 0.772
1.250 −0.199 −0.288 1.810 1.378 0.584 1.216 0.733 0.402 0.226
0.630 −0.537 0.782 0.060 0.499 −0.431 1.705 1.164 0.884 −0.298
0.375 −1.941 0.247 −0.491 0.665 −0.135 −0.145 −0.498 0.457 1.064
−1.420 0.489 −1.711 −1.186 0.754 −0.732 −0.066 1.006 −0.798 0.162
−0.151 −0.243 −0.430 −0.762 0.298 1.049 1.810 2.885 −0.768 −0.129
−0.309 0.531 0.416 −1.541 1.456 2.040 −0.124 0.196 0.023 −1.204
0.424 −0.444 0.593 0.993 −0.106 0.116 0.484 −1.272 1.066 1.097
0.593 0.658 −1.127 −1.407 −1.579 −1.616 1.458 1.262 0.736 −0.916
0.862 −0.885 −0.142 −0.504 0.532 1.381 0.022 −0.281 −0.342 1.222
0.235 −0.628 −0.023 −0.463 −0.899 −0.394 −0.538 1.707 −0.188 −1.153
−0.853 0.402 0.777 0.833 0.410 −0.349 −1.094 0.580 1.395 1.298

Table ST7. Critical Values of the Kolmogorov-Smirnov One-Sample Test Statistica

Source: Adapted by permission from Table 1 of Leslie H. Miller, Table of Percentage points of Kolmogrov statistics, J.Am. Stat. Assoc. 51 (1956), 111-121.

One-Sided Test:
α = 0.10 0.05 0.025 0.01 0.005 α = 0.10 0.05 0.025 0.01 0.005
Two-Sided Test:
α = 0.20 0.10 0.05 0.02 0.01 α = 0.20 0.10 0.05 0.02 0.01
n = 1 0.900 0.950 0.975 0.990 0.995 n = 21 0.226 0.259 0.287 0.321 0.344
2 0.684 0.776 0.842 0.900 0.929 22 0.221 0.253 0.281 0.314 0.337
3 0.565 0.636 0.708 0.785 0.829 23 0.216 0.247 0.275 0.307 0.330
4 0.493 0.565 0.624 0.689 0.734 24 0.212 0.242 0.269 0.301 0.323
5 0.447 0.509 0.563 0.627 0.669 25 0.208 0.238 0.264 0.295 0.317
6 0.410 0.468 0.519 0.577 0.617 26 0.204 0.233 0.259 0.290 0.311
7 0.381 0.436 0.483 0.538 0.576 27 0.200 0.229 0.254 0.284 0.305
8 0.358 0.410 0.454 0.507 0.542 28 0.197 0.225 0.250 0.279 0.300
9 0.339 0.387 0.430 0.480 0.513 29 0.193 0.221 0.246 0.275 0.295
10 0.323 0.369 0.409 0.457 0.489 30 0.190 0.218 0.242 0.270 0.290
11 0.308 0.352 0.391 0.437 0.468 31 0.187 0.214 0.238 0.266 0.285
12 0.296 0.338 0.375 0.419 0.449 32 0.184 0.211 0.234 0.262 0.281
13 0.285 0.325 0.361 0.404 0.432 33 0.182 0.208 0.231 0.258 0.277
14 0.275 0.314 0.349 0.390 0.418 34 0.179 0.205 0.227 0.254 0.273
15 0.266 0.304 0.338 0.377 0.404 35 0.177 0.202 0.224 0.251 0.269
16 0.258 0.295 0.327 0.366 0.392 36 0.174 0.199 0.221 0.247 0.265
17 0.250 0.286 0.318 0.355 0.381 37 0.172 0.196 0.218 0.244 0.262
18 0.244 0.279 0.309 0.346 0.371 38 0.170 0.194 0.215 0.241 0.258
19 0.237 0.271 0.301 0.337 0.361 39 0.168 0.191 0.213 0.238 0.255
20 0.232 0.265 0.294 0.329 0.352 40 0.165 0.189 0.210 0.235 0.252
Approximation for n > 40 images images images images images

aThis table gives the values of images and Dn,α for which images and images for some selected values of n and α.

Table ST8. Critical Values of the Kolmogorov-Smirnov Test Statistic for Two Samples of Equal Sizea

Source: Adapted by permission from Tables 2 and 3 of Z. W. Birnbaum and R. A. Hall, Small sample distributions for multisample statistics of the Smirnov type, Ann. Math. Stat. 31 (1960), 710–720.

One-Sided Test:
α = 0.10 0.05 0.025 0.01 0.005 α = 0.10 0.05 0.025 0.01 0.005
Two-Sided Test:
α = 0.20 0.10 0.05 0.02 0.01 α = 0.20 0.10 0.05 0.02 0.01
n = 3 2/3 2/3 n = 20 6/20 7/20 8/20 9/20 10/20
4 3/4 3/4 3/4 21 6/21 7/21 8/21 9/21 10/21
5 3/5 3/5 4/5 4/5 4/5 22 7/22 8/22 8/22 10/22 10/22
6 3/6 4/6 4/6 5/6 5/6 23 7/23 8/23 9/23 10/23 10/23
7 4/7 4/7 5/7 5/7 5/7 24 7/24 8/24 9/24 10/24 11/24
8 4/8 4/8 5/8 5/8 6/8 25 7/25 8/25 9/25 10/25 11/25
9 4/9 5/9 5/9 6/9 6/9 26 7/26 8/26 9/26 10/26 11/26
10 4/10 5/10 6/10 6/10 7/10 27 7/27 8/27 9/27 11/27 11/27
11 5/11 5/11 6/11 7/11 7/11 28 8/28 9/28 10/28 11/28 12/28
12 5/12 5/12 6/12 7/12 7/12 29 8/29 9/29 10/29 11/29 12/29
13 5/13 6/13 6/13 7/13 8/13 30 8/30 9/30 10/30 11/30 12/30
14 5/14 6/14 7/14 7/14 8/14 31 8/31 9/31 10/31 11/31 12/31
15 5/15 6/15 7/15 8/15 8/15 32 8/32 9/32 10/32 12/32 12/32
16 6/16 6/16 7/16 8/16 9/16 34 8/34 10/34 11/34 12/34 13/34
17 6/17 7/17 7/17 8/17 9/17 36 9/36 10/36 11/36 12/36 13/36
18 6/18 7/18 8/18 9/18 9/18 38 9/38 10/38 11/38 13/38 14/38
19 6/19 7/19 8/19 9/19 9/19 40 9/40 10/40 12/40 13/40 14/40
Approximation for n > 40: images images images images images

aThis table gives the values of images and Dn,n,α for which images and images for some selected values of n and α.

Table ST9. Critical Values of the Kolmogorov-Smirnov Test Statistic for Two Samples of Unequal Sizea

Source: Adapted by permission from F. J. Massey, Distribution table for the deviation between two sample cumulatives, Ann. Math. Stat. 23 (1952), 435–441.

One-Sided Test: α = 0.10 0.05 0.025 0.01 0.005
Two-Sided Test: α = 0.20 0.10 0.05 0.02 0.01
N1 = 1 N2 = 9 17/18
10 9/10
N1 = 2 N2 = 3 5/6
4 3/4
5 4/5 4/5
6 5/6 5/6
7 5/7 6/7
8 3/4 7/8 7/8
9 7/9 8/9 8/9
10 7/10 4/5 9/10
N1 = 3 N2 = 4 3/4 3/4
5 2/3 4/5 4/5
6 2/3 2/3 5/6
7 2/3 5/7 6/7 6/7
8 5/8 3/4 3/4 7/8
9 2/3 2/3 7/9 8/9 8/9
10 3/5 7/10 4/5 9/10 9/10
12 7/12 2/3 3/4 5/6 11/12
N1 = 4 N2 = 5 3/5 3/4 4/5 4/5
6 7/12 2/3 3/4 5/6 5/6
7 17/28 5/7 3/4 6/7 6/7
8 5/8 5/8 3/4 7/8 7/8
9 5/9 2/3 3/4 7/9 8/9
10 11/20 13/20 7/10 4/5 4/5
12 7/12 2/3 2/3 3/4 5/6
16 9/16 5/8 11/16 3/4 13/16
N1 = 5 N2 = 6 3/5 2/3 2/3 5/6 5/6
7 4/7 23/35 5/7 29/35 6/7
8 11/20 5/8 27/40 4/5 4/5
9 5/9 3/5 31/45 7/9 4/5
10 1/2 3/5 7/10 7/10 4/5
15 8/15 3/5 2/3 11/15 11/15
20 1/2 11/20 3/5 7/10 3/4
N1 = 6 N2 = 7 23/42 4/7 29/42 5/7 5/6
8 1/2 7/12 2/3 3/4 3/4
9 1/2 5/9 2/3 13/18 7/9
10 1/2 17/30 19/30 7/10 11/15
12 1/2 7/12 7/12 2/3 3/4
18 4/9 5/9 11/18 2/3 13/18
24 11/24 1/2 7/12 5/8 2/3
N1 = 7 N2 = 8 27/56 33/56 5/8 41/56 3/4
9 31/63 5/9 40/63 5/7 47/63
10 33/70 39/70 43/70 7/10 5/7
14 3/7 1/2 4/7 9/14 5/7
28 3/7 13/28 15/28 17/28 9/14
N1 = 8 N2 = 9 4/9 13/24 5/8 2/3 3/4
10 19/40 21/40 23/40 27/40 7/10
12 11/24 1/2 7/12 5/8 2/3
16 7/16 1/2 9/16 5/8 5/8
32 13/32 7/16 1/2 9/16 19/32
N1 = 9 N2 = 10 7/15 1/2 26/45 2/3 31/45
12 4/9 1/2 5/9 11/18 2/3
15 19/45 22/45 8/15 3/5 29/45
18 7/18 4/9 1/2 5/9 11/18
36 13/36 5/12 17/36 19/36 5/9
N1 = 10 N2 = 15 2/5 7/15 1/2 17/30 19/30
20 2/5 9/20 1/2 11/20 3/5
40 7/20 2/5 9/20 1/2
N1 = 12 N2 = 15 23/60 9/20 1/2 11/20 7/12
16 3/8 7/16 23/48 13/24 7/12
18 13/36 5/12 17/36 19/36 5/9
20 11/30 5/12 7/15 31/60 17/30
N1 = 15 N2 = 20 7/20 2/5 13/30 29/60 31/60
N1 = 16 N2 = 20 27/80 31/80 17/40 19/40 41/80
Large-sample approximation images images images images images

aThis table gives the values of images and Dm,n,α for which images and images for some selected values of N1 = smaller sample size, N2 = larger sample size, and α.

Table ST10. Critical Values of the Wilcoxon Signed-Ranks Test Statistica

Source: Adapted by permission from Table 1 of R. L. McCornack, Extended tables of the Wilcoxon matched pairs signed-rank statistics, J. Am. Stat. Assoc. 60 (1965), 864–871.

n α
0.01 0.025 0.05 0.10
3 6 6 6 6
4 10 10 10 9
5 15 15 14 12
6 21 20 18 17
7 27 25 24 22
8 34 32 30 27
9 41 39 36 34
10 49 46 44 40
11 58 55 52 48
12 67 64 60 56
13 78 73 69 64
14 89 84 79 73
15 100 94 89 83
16 112 106 100 93
17 125 118 111 104
18 138 130 123 115
19 152 143 136 127
20 166 157 149 140

aThis table gives values of tα for which images for selected values of n and α. Critical values in the lower tail may be obtained by symmetry from the equation images.

Table ST11. Critical Values of the Mann-Whitney-Wilcoxon Test Statistica

Source: Adapted by permission from Table 1 of L. R. Verdooren, Extended tables of critical values for Wilcoxon's test statistic, Biometrika 50 (1963), 177-186, with the kind permission of Professor E. S. Pearson, the author, and the Biometrika Trustees.

m α n
2 3 4 5 6 7 8 9 10
2 0.01 4 6 8 10 12 14 16 18 20
0.025 4 6 8 10 12 14 15 17 19
0.05 4 6 8 9 11 13 14 16 18
0.10 4 5 7 8 10 12 13 15 16
3 0.01 9 12 15 18 20 20 25 28
0.025 9 12 14 16 19 21 24 26
0.05 8 11 13 15 18 20 22 25
0.10 7 10 12 14 16 18 21 23
4 0.01 16 19 22 26 29 32 36
0.025 15 18 21 24 27 31 34
0.05 14 17 20 23 26 29 32
0.10 12 15 18 21 24 26 29
5 0.01 23 27 31 35 39 43
0.025 22 26 29 33 37 41
0.05 20 24 28 31 35 38
0.10 19 22 26 29 32 36
6 0.01 32 37 41 46 51
0.025 30 35 39 43 48
0.05 28 33 37 41 45
0.10 26 30 34 38 42
7 0.01 42 48 53 58
0.025 40 45 50 55
0.05 37 42 47 52
0.10 35 39 44 48
8 0.01 54 60 66
0.025 50 56 62
0.05 48 53 59
0.10 44 49 55
9 0.01 66 73
0.025 63 69
0.05 59 65
0.10 55 61
10 0.01 80
0.025 76
0.05 72
0.10 67

aThis table gives values of uα for which images or some selected values of m, n, and α. Critical values in the lower tail may be obtained by symmetry from the equation images.

Table ST12. Critical Points of Kendall's Tau Test Statistica

Source: Adapted by permission from Table 1, p. 173, of M. G. Kendall, Rank Correlation Methods, 3rd ed., Griffin, London, 1962. For values of n > 11, see W. J. Conover, Practical Nonparametric Statistics, John Wiley, New York, 1971, p. 390.

n α
0.100 0.050 0.025 0.01
3 3 3 3 3
4 4 4 6 6
5 6 6 8 8
6 7 9 11 11
7 9 11 13 15
8 10 14 16 18
9 12 16 18 22
10 15 19 21 25

aThis table gives the values of Sα for which images, where images, for some selected values of α and n. Values in the lower tail may be obtained by symmetry, S1_α = –Sα.

Table ST13. Critical Values of Spearman's Rank Correlation Statistica

Source: Adapted by permission from Table 2, pp. 174-175, of M. G. Kendall, Rank Correlation Methods, 3rd ed., Griffin, London, 1962. For values of images, see W. J. Conover, Practical Nonparametric Statistics, John Wiley, New York, 1971, p. 391.

n α
0.01 0.025 0.05 0.10
3 1.000 1.000 1.000 1.000
4 1.000 1.000 0.800 0.800
5 0.900 0.900 0.800 0.700
6 0.886 0.829 0.771 0.600
7 0.857 0.750 0.679 0.536
8 0.810 0.714 0.619 0.500
9 0.767 0.667 0.583 0.467
10 0.721 0.636 0.552 0.442

aThis table gives the values of Rα for which images for some selected values of n and α. Critical values in the lower tail may be obtained by symmetry, images.

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