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, , 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, . The probabilities for negative values of z are obtained by symmetry.
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 |
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 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 |
aThis table gives the values of and Dn,α for which and 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: |
aThis table gives the values of and Dn,n,α for which and 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 |
aThis table gives the values of and Dm,n,α for which and 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 for selected values of n and α. Critical values in the lower tail may be obtained by symmetry from the equation .
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 or some selected values of m, n, and α. Critical values in the lower tail may be obtained by symmetry from the equation .
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 , where , 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 , 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 for some selected values of n and α. Critical values in the lower tail may be obtained by symmetry, .