Table 1: Binomial distribution — probability mass function

p

x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n=1 0 .9900 .9500 .9000 .8500 .8000 .7500 .7000 .6500 .6000 .5500 .5000 11 .0100 .0500 .1000 .1500 .2000 .2500 .3000 .3500 .4000 .4500 .5000 0

n=2 0 .9801 .9025 .8100 .7225 .6400 .5625 .4900 .4225 .3600 .3025 .2500 21 .0198 .0950 .1800 .2550 .3200 .3750 .4200 .4550 .4800 .4950 .5000 12 .0001 .0025 .0100 .0225 .0400 .0625 .0900 .1225 .1600 .2025 .2500 0

n=3 0 .9703 .8574 .7290 .6141 .5120 .4219 .3430 .2746 .2160 .1664 .1250 31 .0294 .1354 .2430 .3251 .3840 .4219 .4410 .4436 .4320 .4084 .3750 22 .0003 .0071 .0270 .0574 .0960 .1406 .1890 .2389 .2880 .3341 .3750 13 .0001 .0010 .0034 .0080 .0156 .0270 .0429 .0640 .0911 .1250 0

n=4 0 .9606 .8145 .6561 .5220 .4096 .3164 .2401 .1785 .1296 .0915 .0625 41 .0388 .1715 .2916 .3685 .4096 .4219 .4116 .3845 .3456 .2995 .2500 32 .0006 .0135 .0486 .0975 .1536 .2109 .2646 .3105 .3456 .3675 .3750 23 .0005 .0036 .0115 .0256 .0469 .0756 .1115 .1536 .2005 .2500 14 .0001 .0005 .0016 .0039 .0081 .0150 .0256 .0410 .0625 0

n=5 0 .9510 .7738 .5905 .4437 .3277 .2373 .1681 .1160 .0778 .0503 .0313 51 .0480 .2036 .3281 .3915 .4096 .3955 .3602 .3124 .2592 .2059 .1563 42 .0010 .0214 .0729 .1382 .2048 .2637 .3087 .3364 .3456 .3369 .3125 33 .0011 .0081 .0244 .0512 .0879 .1323 .1811 .2304 .2757 .3125 24 .0005 .0022 .0064 .0146 .0284 .0488 .0768 .1128 .1563 15 .0001 .0003 .0010 .0024 .0053 .0102 .0185 .0313 0

n=6 0 .9415 .7351 .5314 .3771 .2621 .1780 .1176 .0754 .0467 .0277 .0156 61 .0571 .2321 .3543 .3993 .3932 .3560 .3025 .2437 .1866 .1359 .0938 52 .0014 .0305 .0984 .1762 .2458 .2966 .3241 .3280 .3110 .2780 .2344 43 .0021 .0146 .0415 .0819 .1318 .1852 .2355 .2765 .3032 .3125 34 .0001 .0012 .0055 .0154 .0330 .0595 .0951 .1382 .1861 .2344 25 .0001 .0004 .0015 .0044 .0102 .0205 .0369 .0609 .0938 16 .0001 .0002 .0007 .0018 .0041 .0083 .0156 0

n=7 0 .9321 .6983 .4783 .3206 .2097 .1335 .0824 .0490 .0280 .0152 .0078 71 .0659 .2573 .3720 .3960 .3670 .3115 .2471 .1848 .1306 .0872 .0547 62 .0020 .0406 .1240 .2097 .2753 .3115 .3177 .2985 .2613 .2140 .1641 53 .0036 .0230 .0617 .1147 .1730 .2269 .2679 .2903 .2918 .2734 44 .0002 .0026 .0109 .0287 .0577 .0972 .1442 .1935 .2388 .2734 35 .0002 .0012 .0043 .0115 .0250 .0466 .0774 .1172 .1641 26 .0001 .0004 .0013 .0036 .0084 .0172 .0320 .0547 17 .0001 .0002 .0006 .0016 .0037 .0078 0

n=8 0 .9227 .6634 .4305 .2725 .1678 .1001 .0576 .0319 .0168 .0084 .0039 81 .0746 .2793 .3826 .3847 .3355 .2670 .1977 .1373 .0896 .0548 .0313 72 .0026 .0515 .1488 .2376 .2936 .3115 .2965 .2587 .2090 .1569 .1094 63 .0001 .0054 .0331 .0839 .1468 .2076 .2541 .2786 .2787 .2568 .2188 54 .0004 .0046 .0185 .0459 .0865 .1361 .1875 .2322 .2627 .2734 45 .0004 .0026 .0092 .0231 .0467 .0808 .1239 .1719 .2188 36 .0002 .0011 .0038 .0100 .0217 .0413 .0703 .1094 27 .0001 .0004 .0012 .0033 .0079 .0164 .0313 18 .0001 .0002 .0007 .0017 .0039 0

0.99 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 xp

p

x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n=9 0 .9135 .6302 .3874 .2316 .1342 .0751 .0404 .0207 .0101 .0046 .0020 91 .0830 .2985 .3874 .3679 .3020 .2253 .1556 .1004 .0605 .0339 .0176 82 .0034 .0629 .1722 .2597 .3020 .3003 .2668 .2162 .1612 .1110 .0703 73 .0001 .0077 .0446 .1069 .1762 .2336 .2668 .2716 .2508 .2119 .1641 64 .0006 .0074 .0283 .0661 .1168 .1715 .2194 .2508 .2600 .2461 55 .0008 .0050 .0165 .0389 .0735 .1181 .1672 .2128 .2461 46 .0001 .0006 .0028 .0087 .0210 .0424 .0743 .1160 .1641 37 .0003 .0012 .0039 .0098 .0212 .0407 .0703 28 .0001 .0004 .0013 .0035 .0083 .0176 19 .0001 .0003 .0008 .0020 0

n=10 0 .9044 .5987 .3487 .1969 .1074 .0563 .0282 .0135 .0060 .0025 .0010 101 .0914 .3151 .3874 .3474 .2684 .1877 .1211 .0725 .0403 .0207 .0098 92 .0042 .0746 .1937 .2759 .3020 .2816 .2335 .1757 .1209 .0763 .0439 83 .0001 .0105 .0574 .1298 .2013 .2503 .2668 .2522 .2150 .1665 .1172 74 .0010 .0112 .0401 .0881 .1460 .2001 .2377 .2508 .2384 .2051 65 .0001 .0015 .0085 .0264 .0584 .1029 .1536 .2007 .2340 .2461 56 .0001 .0012 .0055 .0162 .0368 .0689 .1115 .1596 .2051 47 .0001 .0008 .0031 .0090 .0212 .0425 .0746 .1172 38 .0001 .0004 .0014 .0043 .0106 .0229 .0439 29 .0001 .0005 .0016 .0042 .0098 1

10 .0001 .0003 .0010 0

n=11 0 .8953 .5688 .3138 .1673 .0859 .0422 .0198 .0088 .0036 .0014 .0005 111 .0995 .3293 .3835 .3248 .2362 .1549 .0932 .0518 .0266 .0125 .0054 102 .0050 .0867 .2131 .2866 .2953 .2581 .1998 .1395 .0887 .0513 .0269 93 .0002 .0137 .0710 .1517 .2215 .2581 .2568 .2254 .1774 .1259 .0806 84 .0014 .0158 .0536 .1107 .1721 .2201 .2428 .2365 .2060 .1611 75 .0001 .0025 .0132 .0388 .0803 .1321 .1830 .2207 .2360 .2256 66 .0003 .0023 .0097 .0268 .0566 .0985 .1471 .1931 .2256 57 .0003 .0017 .0064 .0173 .0379 .0701 .1128 .1611 48 .0002 .0011 .0037 .0102 .0234 .0462 .0806 39 .0001 .0005 .0018 .0052 .0126 .0269 2

10 .0002 .0007 .0021 .0054 111 .0002 .0005 0

n=12 0 .8864 .5404 .2824 .1422 .0687 .0317 .0138 .0057 .0022 .0008 .0002 121 .1074 .3413 .3766 .3012 .2062 .1267 .0712 .0368 .0174 .0075 .0029 112 .0060 .0988 .2301 .2924 .2835 .2323 .1678 .1088 .0639 .0339 .0161 103 .0002 .0173 .0852 .1720 .2362 .2581 .2397 .1954 .1419 .0923 .0537 94 .0021 .0213 .0683 .1329 .1936 .2311 .2367 .2128 .1700 .1208 85 .0002 .0038 .0193 .0532 .1032 .1585 .2039 .2270 .2225 .1934 76 .0005 .0040 .0155 .0401 .0792 .1281 .1766 .2124 .2256 67 .0006 .0033 .0115 .0291 .0591 .1009 .1489 .1934 58 .0001 .0005 .0024 .0078 .0199 .0420 .0762 .1208 49 .0001 .0004 .0015 .0048 .0125 .0277 .0537 3

10 .0002 .0008 .0025 .0068 .0161 211 .0001 .0003 .0010 .0029 112 .0001 .0002 0

n=13 0 .8775 .5133 .2542 .1209 .0550 .0238 .0097 .0037 .0013 .0004 .0001 131 .1152 .3512 .3672 .2774 .1787 .1029 .0540 .0259 .0113 .0045 .0016 122 .0070 .1109 .2448 .2937 .2680 .2059 .1388 .0836 .0453 .0220 .0095 113 .0003 .0214 .0997 .1900 .2457 .2517 .2181 .1651 .1107 .0660 .0349 104 .0028 .0277 .0838 .1535 .2097 .2337 .2222 .1845 .1350 .0873 95 .0003 .0055 .0266 .0691 .1258 .1803 .2154 .2214 .1989 .1571 86 .0008 .0063 .0230 .0559 .1030 .1546 .1968 .2169 .2095 77 .0001 .0011 .0058 .0186 .0442 .0833 .1312 .1775 .2095 68 .0001 .0011 .0047 .0142 .0336 .0656 .1089 .1571 59 .0001 .0009 .0034 .0101 .0243 .0495 .0873 4

10 .0001 .0006 .0022 .0065 .0162 .0349 311 .0001 .0003 .0012 .0036 .0095 212 .0001 .0005 .0016 113 .0001 0

0.99 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 xp

p

x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n=14 0 .8687 .4877 .2288 .1028 .0440 .0178 .0068 .0024 .0008 .0002 .0001 141 .1229 .3593 .3559 .2539 .1539 .0832 .0407 .0181 .0073 .0027 .0009 132 .0081 .1229 .2570 .2912 .2501 .1802 .1134 .0634 .0317 .0141 .0056 123 .0003 .0259 .1142 .2056 .2501 .2402 .1943 .1366 .0845 .0462 .0222 114 .0037 .0349 .0998 .1720 .2202 .2290 .2022 .1549 .1040 .0611 105 .0004 .0078 .0352 .0860 .1468 .1963 .2178 .2066 .1701 .1222 96 .0013 .0093 .0322 .0734 .1262 .1759 .2066 .2088 .1833 87 .0002 .0019 .0092 .0280 .0618 .1082 .1574 .1952 .2095 78 .0003 .0020 .0082 .0232 .0510 .0918 .1398 .1833 69 .0003 .0018 .0066 .0183 .0408 .0762 .1222 5

10 .0003 .0014 .0049 .0136 .0312 .0611 411 .0002 .0010 .0033 .0093 .0222 312 .0001 .0005 .0019 .0056 213 .0001 .0002 .0009 114 .0001 0

n=15 0 .8601 .4633 .2059 .0874 .0352 .0134 .0047 .0016 .0005 .0001 151 .1303 .3658 .3432 .2312 .1319 .0668 .0305 .0126 .0047 .0016 .0005 142 .0092 .1348 .2669 .2856 .2309 .1559 .0916 .0476 .0219 .0090 .0032 133 .0004 .0307 .1285 .2184 .2501 .2252 .1700 .1110 .0634 .0318 .0139 124 .0049 .0428 .1156 .1876 .2252 .2186 .1792 .1268 .0780 .0417 115 .0006 .0105 .0449 .1032 .1651 .2061 .2123 .1859 .1404 .0916 106 .0019 .0132 .0430 .0917 .1472 .1906 .2066 .1914 .1527 97 .0003 .0030 .0138 .0393 .0811 .1319 .1771 .2013 .1964 88 .0005 .0035 .0131 .0348 .0710 .1181 .1647 .1964 79 .0001 .0007 .0034 .0116 .0298 .0612 .1048 .1527 6

10 .0001 .0007 .0030 .0096 .0245 .0515 .0916 511 .0001 .0006 .0024 .0074 .0191 .0417 412 .0001 .0004 .0016 .0052 .0139 313 .0001 .0003 .0010 .0032 214 .0001 .0005 115 0

n=16 0 .8515 .4401 .1853 .0743 .0281 .0100 .0033 .0010 .0003 .0001 161 .1376 .3706 .3294 .2097 .1126 .0535 .0228 .0087 .0030 .0009 .0002 152 .0104 .1463 .2745 .2775 .2111 .1336 .0732 .0353 .0150 .0056 .0018 143 .0005 .0359 .1423 .2285 .2463 .2079 .1465 .0888 .0468 .0215 .0085 134 .0061 .0514 .1311 .2001 .2252 .2040 .1553 .1014 .0572 .0278 125 .0008 .0137 .0555 .1201 .1802 .2099 .2008 .1623 .1123 .0667 116 .0001 .0028 .0180 .0550 .1101 .1649 .1982 .1983 .1684 .1222 107 .0004 .0045 .0197 .0524 .1010 .1524 .1889 .1969 .1746 98 .0001 .0009 .0055 .0197 .0487 .0923 .1417 .1812 .1964 89 .0001 .0012 .0058 .0185 .0442 .0840 .1318 .1746 7

10 .0002 .0014 .0056 .0167 .0392 .0755 .1222 611 .0002 .0013 .0049 .0142 .0337 .0667 512 .0002 .0011 .0040 .0115 .0278 413 .0000 .0002 .0008 .0029 .0085 314 .0001 .0005 .0018 215 .0001 .0002 116 0

n=17 0 .8429 .4181 .1668 .0631 .0225 .0075 .0023 .0007 .0002 171 .1447 .3741 .3150 .1893 .0957 .0426 .0169 .0060 .0019 .0005 .0001 162 .0117 .1575 .2800 .2673 .1914 .1136 .0581 .0260 .0102 .0035 .0010 153 .0006 .0415 .1556 .2359 .2393 .1893 .1245 .0701 .0341 .0144 .0052 144 .0076 .0605 .1457 .2093 .2209 .1868 .1320 .0796 .0411 .0182 135 .0010 .0175 .0668 .1361 .1914 .2081 .1849 .1379 .0875 .0472 126 .0001 .0039 .0236 .0680 .1276 .1784 .1991 .1839 .1432 .0944 117 .0007 .0065 .0267 .0668 .1201 .1685 .1927 .1841 .1484 108 .0001 .0014 .0084 .0279 .0644 .1134 .1606 .1883 .1855 99 .0003 .0021 .0093 .0276 .0611 .1070 .1540 .1855 8

10 .0004 .0025 .0095 .0263 .0571 .1008 .1484 711 .0001 .0005 .0026 .0090 .0242 .0525 .0944 612 .0001 .0006 .0024 .0081 .0215 .0472 513 .0001 .0005 .0021 .0068 .0182 414 .0001 .0004 .0016 .0052 315 .0001 .0003 .0010 216 .0001 117 0

0.99 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 xp

p

x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n=18 0 .8345 .3972 .1501 .0536 .0180 .0056 .0016 .0004 .0001 181 .1517 .3763 .3002 .1704 .0811 .0338 .0126 .0042 .0012 .0003 .0001 172 .0130 .1683 .2835 .2556 .1723 .0958 .0458 .0190 .0069 .0022 .0006 163 .0007 .0473 .1680 .2406 .2297 .1704 .1046 .0547 .0246 .0095 .0031 154 .0093 .0700 .1592 .2153 .2130 .1681 .1104 .0614 .0291 .0117 145 .0014 .0218 .0787 .1507 .1988 .2017 .1664 .1146 .0666 .0327 136 .0002 .0052 .0301 .0816 .1436 .1873 .1941 .1655 .1181 .0708 127 .0010 .0091 .0350 .0820 .1376 .1792 .1892 .1657 .1214 118 .0002 .0022 .0120 .0376 .0811 .1327 .1734 .1864 .1669 109 .0004 .0033 .0139 .0386 .0794 .1284 .1694 .1855 9

10 .0001 .0008 .0042 .0149 .0385 .0771 .1248 .1669 811 .0001 .0010 .0046 .0151 .0374 .0742 .1214 712 .0002 .0012 .0047 .0145 .0354 .0708 613 .0002 .0012 .0045 .0134 .0327 514 .0002 .0011 .0039 .0117 415 .0002 .0009 .0031 316 .0001 .0006 217 .0001 118 0

n=19 0 .8262 .3774 .1351 .0456 .0144 .0042 .0011 .0003 .0001 191 .1586 .3774 .2852 .1529 .0685 .0268 .0093 .0029 .0008 .0002 182 .0144 .1787 .2852 .2428 .1540 .0803 .0358 .0138 .0046 .0013 .0003 173 .0008 .0533 .1796 .2428 .2182 .1517 .0869 .0422 .0175 .0062 .0018 164 .0112 .0798 .1714 .2182 .2023 .1491 .0909 .0467 .0203 .0074 155 .0018 .0266 .0907 .1636 .2023 .1916 .1468 .0933 .0497 .0222 146 .0002 .0069 .0374 .0955 .1574 .1916 .1844 .1451 .0949 .0518 137 .0014 .0122 .0443 .0974 .1525 .1844 .1797 .1443 .0961 128 .0002 .0032 .0166 .0487 .0981 .1489 .1797 .1771 .1442 119 .0007 .0051 .0198 .0514 .0980 .1464 .1771 .1762 10

10 .0001 .0013 .0066 .0220 .0528 .0976 .1449 .1762 911 .0003 .0018 .0077 .0233 .0532 .0970 .1442 812 .0004 .0022 .0083 .0237 .0529 .0961 713 .0001 .0005 .0024 .0085 .0233 .0518 614 .0001 .0006 .0024 .0082 .0222 515 .0001 .0005 .0022 .0074 416 .0001 .0005 .0018 317 .0001 .0003 218 119 0

n=20 0 .8179 .3585 .1216 .0388 .0115 .0032 .0008 .0002 201 .1652 .3774 .2702 .1368 .0576 .0211 .0068 .0020 .0005 .0001 192 .0159 .1887 .2852 .2293 .1369 .0669 .0278 .0100 .0031 .0008 .0002 183 .0010 .0596 .1901 .2428 .2054 .1339 .0716 .0323 .0123 .0040 .0011 174 .0133 .0898 .1821 .2182 .1897 .1304 .0738 .0350 .0139 .0046 165 .0022 .0319 .1028 .1746 .2023 .1789 .1272 .0746 .0365 .0148 156 .0003 .0089 .0454 .1091 .1686 .1916 .1712 .1244 .0746 .0370 147 .0020 .0160 .0545 .1124 .1643 .1844 .1659 .1221 .0739 138 .0004 .0046 .0222 .0609 .1144 .1614 .1797 .1623 .1201 129 .0001 .0011 .0074 .0271 .0654 .1158 .1597 .1771 .1602 11

10 .0002 .0020 .0099 .0308 .0686 .1171 .1593 .1762 1011 .0005 .0030 .0120 .0336 .0710 .1185 .1602 912 .0001 .0008 .0039 .0136 .0355 .0727 .1201 813 .0002 .0010 .0045 .0146 .0366 .0739 714 .0002 .0012 .0049 .0150 .0370 615 .0003 .0013 .0049 .0148 516 .0003 .0013 .0046 417 .0002 .0011 318 .0002 219 120 0

0.99 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 xp

Table 3: Poisson distribution — probability mass function

λx 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x

0 .9048 .8187 .7408 .6703 .6065 .5488 .4966 .4493 .4066 .3679 01 .0905 .1637 .2222 .2681 .3033 .3293 .3476 .3595 .3659 .3679 12 .0045 .0164 .0333 .0536 .0758 .0988 .1217 .1438 .1647 .1839 23 .0002 .0011 .0033 .0072 .0126 .0198 .0284 .0383 .0494 .0613 34 .0001 .0003 .0007 .0016 .0030 .0050 .0077 .0111 .0153 45 .0001 .0002 .0004 .0007 .0012 .0020 .0031 56 .0001 .0002 .0003 .0005 67 .0001 7

λx 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 x

0 .3329 .3012 .2725 .2466 .2231 .2019 .1827 .1653 .1496 .1353 01 .3662 .3614 .3543 .3452 .3347 .3230 .3106 .2975 .2842 .2707 12 .2014 .2169 .2303 .2417 .2510 .2584 .2640 .2678 .2700 .2707 23 .0738 .0867 .0998 .1128 .1255 .1378 .1496 .1607 .1710 .1804 34 .0203 .0260 .0324 .0395 .0471 .0551 .0636 .0723 .0812 .0902 45 .0045 .0062 .0084 .0111 .0141 .0176 .0216 .0260 .0309 .0361 56 .0008 .0012 .0018 .0026 .0035 .0047 .0061 .0078 .0098 .0120 67 .0001 .0002 .0003 .0005 .0008 .0011 .0015 .0020 .0027 .0034 78 .0001 .0001 .0001 .0002 .0003 .0005 .0006 .0009 89 .0001 .0001 .0001 .0002 9

λx 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 x

0 .1225 .1108 .1003 .0907 .0821 .0743 .0672 .0608 .0550 .0498 01 .2572 .2438 .2306 .2177 .2052 .1931 .1815 .1703 .1596 .1494 12 .2700 .2681 .2652 .2613 .2565 .2510 .2450 .2384 .2314 .2240 23 .1890 .1966 .2033 .2090 .2138 .2176 .2205 .2225 .2237 .2240 34 .0992 .1082 .1169 .1254 .1336 .1414 .1488 .1557 .1622 .1680 45 .0417 .0476 .0538 .0602 .0668 .0735 .0804 .0872 .0940 .1008 56 .0146 .0174 .0206 .0241 .0278 .0319 .0362 .0407 .0455 .0504 67 .0044 .0055 .0068 .0083 .0099 .0118 .0139 .0163 .0188 .0216 78 .0011 .0015 .0019 .0025 .0031 .0038 .0047 .0057 .0068 .0081 89 .0003 .0004 .0005 .0007 .0009 .0011 .0014 .0018 .0022 .0027 9

10 .0001 .0001 .0001 .0002 .0002 .0003 .0004 .0005 .0006 .0008 1011 .0001 .0001 .0001 .0002 .0002 1112 .0001 12

λx 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 x

0 .0450 .0408 .0369 .0334 .0302 .0273 .0247 .0224 .0202 .0183 01 .1397 .1304 .1217 .1135 .1057 .0984 .0915 .0850 .0789 .0733 12 .2165 .2087 .2008 .1929 .1850 .1771 .1692 .1615 .1539 .1465 23 .2237 .2226 .2209 .2186 .2158 .2125 .2087 .2046 .2001 .1954 34 .1733 .1781 .1823 .1858 .1888 .1912 .1931 .1944 .1951 .1954 45 .1075 .1140 .1203 .1264 .1322 .1377 .1429 .1477 .1522 .1563 56 .0555 .0608 .0662 .0716 .0771 .0826 .0881 .0936 .0989 .1042 67 .0246 .0278 .0312 .0348 .0385 .0425 .0466 .0508 .0551 .0595 78 .0095 .0111 .0129 .0148 .0169 .0191 .0215 .0241 .0269 .0298 89 .0033 .0040 .0047 .0056 .0066 .0076 .0089 .0102 .0116 .0132 9

10 .0010 .0013 .0016 .0019 .0023 .0028 .0033 .0039 .0045 .0053 1011 .0003 .0004 .0005 .0006 .0007 .0009 .0011 .0013 .0016 .0019 1112 .0001 .0001 .0001 .0002 .0002 .0003 .0003 .0004 .0005 .0006 1213 .0001 .0001 .0001 .0001 .0002 .0002 1314 .0001 14

λx 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 x

0 .0166 .0150 .0136 .0123 .0111 .0101 .0091 .0082 .0074 .0067 01 .0679 .0630 .0583 .0540 .0500 .0462 .0427 .0395 .0365 .0337 12 .1393 .1323 .1254 .1188 .1125 .1063 .1005 .0948 .0894 .0842 23 .1904 .1852 .1798 .1743 .1687 .1631 .1574 .1517 .1460 .1404 34 .1951 .1944 .1933 .1917 .1898 .1875 .1849 .1820 .1789 .1755 45 .1600 .1633 .1662 .1687 .1708 .1725 .1738 .1747 .1753 .1755 56 .1093 .1143 .1191 .1237 .1281 .1323 .1362 .1398 .1432 .1462 67 .0640 .0686 .0732 .0778 .0824 .0869 .0914 .0959 .1002 .1044 78 .0328 .0360 .0393 .0428 .0463 .0500 .0537 .0575 .0614 .0653 89 .0150 .0168 .0188 .0209 .0232 .0255 .0281 .0307 .0334 .0363 9

10 .0061 .0071 .0081 .0092 .0104 .0118 .0132 .0147 .0164 .0181 1011 .0023 .0027 .0032 .0037 .0043 .0049 .0056 .0064 .0073 .0082 1112 .0008 .0009 .0011 .0013 .0016 .0019 .0022 .0026 .0030 .0034 1213 .0002 .0003 .0004 .0005 .0006 .0007 .0008 .0009 .0011 .0013 1314 .0001 .0001 .0001 .0001 .0002 .0002 .0003 .0003 .0004 .0005 1415 .0001 .0001 .0001 .0001 .0001 .0002 15

λx 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 x

0 .0061 .0055 .0050 .0045 .0041 .0037 .0033 .0030 .0027 .0025 01 .0311 .0287 .0265 .0244 .0225 .0207 .0191 .0176 .0162 .0149 12 .0793 .0746 .0701 .0659 .0618 .0580 .0544 .0509 .0477 .0446 23 .1348 .1293 .1239 .1185 .1133 .1082 .1033 .0985 .0938 .0892 34 .1719 .1681 .1641 .1600 .1558 .1515 .1472 .1428 .1383 .1339 45 .1753 .1748 .1740 .1728 .1714 .1697 .1678 .1656 .1632 .1606 56 .1490 .1515 .1537 .1555 .1571 .1584 .1594 .1601 .1605 .1606 67 .1086 .1125 .1163 .1200 .1234 .1267 .1298 .1326 .1353 .1377 78 .0692 .0731 .0771 .0810 .0849 .0887 .0925 .0962 .0998 .1033 89 .0392 .0423 .0454 .0486 .0519 .0552 .0586 .0620 .0654 .0688 9

10 .0200 .0220 .0241 .0262 .0285 .0309 .0334 .0359 .0386 .0413 1011 .0093 .0104 .0116 .0129 .0143 .0157 .0173 .0190 .0207 .0225 1112 .0039 .0045 .0051 .0058 .0065 .0073 .0082 .0092 .0102 .0113 1213 .0015 .0018 .0021 .0024 .0028 .0032 .0036 .0041 .0046 .0052 1314 .0006 .0007 .0008 .0009 .0011 .0013 .0015 .0017 .0019 .0022 1415 .0002 .0002 .0003 .0003 .0004 .0005 .0006 .0007 .0008 .0009 1516 .0001 .0001 .0001 .0001 .0001 .0002 .0002 .0002 .0003 .0003 1617 .0001 .0001 .0001 .0001 .0001 17

λx 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 x

0 .0022 .0020 .0018 .0017 .0015 .0014 .0012 .0011 .0010 .0009 01 .0137 .0126 .0116 .0106 .0098 .0090 .0082 .0076 .0070 .0064 12 .0417 .0390 .0364 .0340 .0318 .0296 .0276 .0258 .0240 .0223 23 .0848 .0806 .0765 .0726 .0688 .0652 .0617 .0584 .0552 .0521 34 .1294 .1249 .1205 .1162 .1118 .1076 .1034 .0992 .0952 .0912 45 .1579 .1549 .1519 .1487 .1454 .1420 .1385 .1349 .1314 .1277 56 .1605 .1601 .1595 .1586 .1575 .1562 .1546 .1529 .1511 .1490 67 .1399 .1418 .1435 .1450 .1462 .1472 .1480 .1486 .1489 .1490 78 .1066 .1099 .1130 .1160 .1188 .1215 .1240 .1263 .1284 .1304 89 .0723 .0757 .0791 .0825 .0858 .0891 .0923 .0954 .0985 .1014 9

10 .0441 .0469 .0498 .0528 .0558 .0588 .0618 .0649 .0679 .0710 1011 .0244 .0265 .0285 .0307 .0330 .0353 .0377 .0401 .0426 .0452 1112 .0124 .0137 .0150 .0164 .0179 .0194 .0210 .0227 .0245 .0263 1213 .0058 .0065 .0073 .0081 .0089 .0099 .0108 .0119 .0130 .0142 1314 .0025 .0029 .0033 .0037 .0041 .0046 .0052 .0058 .0064 .0071 1415 .0010 .0012 .0014 .0016 .0018 .0020 .0023 .0026 .0029 .0033 1516 .0004 .0005 .0005 .0006 .0007 .0008 .0010 .0011 .0013 .0014 1617 .0001 .0002 .0002 .0002 .0003 .0003 .0004 .0004 .0005 .0006 1718 .0001 .0001 .0001 .0001 .0001 .0001 .0002 .0002 .0002 1819 .0001 .0001 .0001 .0001 19

λx 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 x

0 .0008 .0007 .0007 .0006 .0006 .0005 .0005 .0004 .0004 .0003 01 .0059 .0054 .0049 .0045 .0041 .0038 .0035 .0032 .0029 .0027 12 .0208 .0194 .0180 .0167 .0156 .0145 .0134 .0125 .0116 .0107 23 .0492 .0464 .0438 .0413 .0389 .0366 .0345 .0324 .0305 .0286 34 .0874 .0836 .0799 .0764 .0729 .0696 .0663 .0632 .0602 .0573 45 .1241 .1204 .1167 .1130 .1094 .1057 .1021 .0986 .0951 .0916 56 .1468 .1445 .1420 .1394 .1367 .1339 .1311 .1282 .1252 .1221 67 .1489 .1486 .1481 .1474 .1465 .1454 .1442 .1428 .1413 .1396 78 .1321 .1337 .1351 .1363 .1373 .1381 .1388 .1392 .1395 .1396 89 .1042 .1070 .1096 .1121 .1144 .1167 .1187 .1207 .1224 .1241 9

10 .0740 .0770 .0800 .0829 .0858 .0887 .0914 .0941 .0967 .0993 1011 .0478 .0504 .0531 .0558 .0585 .0613 .0640 .0667 .0695 .0722 1112 .0283 .0303 .0323 .0344 .0366 .0388 .0411 .0434 .0457 .0481 1213 .0154 .0168 .0181 .0196 .0211 .0227 .0243 .0260 .0278 .0296 1314 .0078 .0086 .0095 .0104 .0113 .0123 .0134 .0145 .0157 .0169 1415 .0037 .0041 .0046 .0051 .0057 .0062 .0069 .0075 .0083 .0090 1516 .0016 .0019 .0021 .0024 .0026 .0030 .0033 .0037 .0041 .0045 1617 .0007 .0008 .0009 .0010 .0012 .0013 .0015 .0017 .0019 .0021 1718 .0003 .0003 .0004 .0004 .0005 .0006 .0006 .0007 .0008 .0009 1819 .0001 .0001 .0001 .0002 .0002 .0002 .0003 .0003 .0003 .0004 1920 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0002 2021 .0001 .0001 21

x 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 x

0 .0003 .0003 .0002 .0002 .0002 .0002 .0002 .0002 .0001 .0001 01 .0025 .0023 .0021 .0019 .0017 .0016 .0014 .0013 .0012 .0011 12 .0100 .0092 .0086 .0079 .0074 .0068 .0063 .0058 .0054 .0050 23 .0269 .0252 .0237 .0222 .0208 .0195 .0183 .0171 .0160 .0150 34 .0544 .0517 .0491 .0466 .0443 .0420 .0398 .0377 .0357 .0337 45 .0882 .0849 .0816 .0784 .0752 .0722 .0692 .0663 .0635 .0607 56 .1191 .1160 .1128 .1097 .1066 .1034 .1003 .0972 .0941 .0911 67 .1378 .1358 .1338 .1317 .1294 .1271 .1247 .1222 .1197 .1171 78 .1395 .1392 .1388 .1382 .1375 .1366 .1356 .1344 .1332 .1318 89 .1256 .1269 .1280 .1290 .1299 .1306 .1311 .1315 .1317 .1318 9

10 .1017 .1040 .1063 .1084 .1104 .1123 .1140 .1157 .1172 .1186 1011 .0749 .0776 .0802 .0828 .0853 .0878 .0902 .0925 .0948 .0970 1112 .0505 .0530 .0555 .0579 .0604 .0629 .0654 .0679 .0703 .0728 1213 .0315 .0334 .0354 .0374 .0395 .0416 .0438 .0459 .0481 .0504 1314 .0182 .0196 .0210 .0225 .0240 .0256 .0272 .0289 .0306 .0324 1415 .0098 .0107 .0116 .0126 .0136 .0147 .0158 .0169 .0182 .0194 1516 .0050 .0055 .0060 .0066 .0072 .0079 .0086 .0093 .0101 .0109 1617 .0024 .0026 .0029 .0033 .0036 .0040 .0044 .0048 .0053 .0058 1718 .0011 .0012 .0014 .0015 .0017 .0019 .0021 .0024 .0026 .0029 1819 .0005 .0005 .0006 .0007 .0008 .0009 .0010 .0011 .0012 .0014 1920 .0002 .0002 .0002 .0003 .0003 .0004 .0004 .0005 .0005 .0006 2021 .0001 .0001 .0001 .0001 .0001 .0002 .0002 .0002 .0002 .0003 2122 .0001 .0001 .0001 .0001 .0001 .0001

x 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0 x

0 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 01 .0010 .0009 .0009 .0008 .0007 .0007 .0006 .0005 .0005 .0005 12 .0046 .0043 .0040 .0037 .0034 .0031 .0029 .0027 .0025 .0023 23 .0140 .0131 .0123 .0115 .0107 .0100 .0093 .0087 .0081 .0076 34 .0319 .0302 .0285 .0269 .0254 .0240 .0226 .0213 .0201 .0189 45 .0581 .0555 .0530 .0506 .0483 .0460 .0439 .0418 .0398 .0378 56 .0881 .0851 .0822 .0793 .0764 .0736 .0709 .0682 .0656 .0631 67 .1145 .1118 .1091 .1064 .1037 .1010 .0982 .0955 .0928 .0901 78 .1302 .1286 .1269 .1251 .1232 .1212 .1191 .1170 .1148 .1126 89 .1317 .1315 .1311 .1306 .1300 .1293 .1284 .1274 .1263 .1251 9

10 .1198 .1210 .1219 .1228 .1235 .1241 .1245 .1249 .1250 .1251 1011 .0991 .1012 .1031 .1049 .1067 .1083 .1098 .1112 .1125 .1137 1112 .0752 .0776 .0799 .0822 .0844 .0866 .0888 .0908 .0928 .0948 1213 .0526 .0549 .0572 .0594 .0617 .0640 .0662 .0685 .0707 .0729 1314 .0342 .0361 .0380 .0399 .0419 .0439 .0459 .0479 .0500 .0521 1415 .0208 .0221 .0235 .0250 .0265 .0281 .0297 .0313 .0330 .0347 1516 .0118 .0127 .0137 .0147 .0157 .0168 .0180 .0192 .0204 .0217 1617 .0063 .0069 .0075 .0081 .0088 .0095 .0103 .0111 .0119 .0128 1718 .0032 .0035 .0039 .0042 .0046 .0051 .0055 .0060 .0065 .0071 1819 .0015 .0017 .0019 .0021 .0023 .0026 .0028 .0031 .0034 .0037 1920 .0007 .0008 .0009 .0010 .0011 .0012 .0014 .0015 .0017 .0019 2021 .0003 .0003 .0004 .0004 .0005 .0006 .0006 .0007 .0008 .0009 2122 .0001 .0001 .0002 .0002 .0002 .0002 .0003 .0003 .0004 .0004 2223 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0002 .0002 2324 .0001 .0001 .0001 24

Table 4: Normal distribution — cumulative distribution function

x 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 4 8 12 16 20 24 28 32 360.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 4 8 12 16 20 24 28 32 350.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 4 8 12 15 19 23 27 31 350.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 4 8 11 15 19 23 26 30 340.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 4 7 11 14 18 22 25 29 32

0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 3 7 10 14 17 21 24 27 310.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 3 6 10 13 16 19 23 26 290.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 3 6 9 12 15 18 21 24 270.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 3 6 8 11 14 17 19 22 250.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 3 5 8 10 13 15 18 20 23

1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 2 5 7 9 12 14 16 18 211.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 2 4 6 8 10 12 14 16 191.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 2 4 6 7 9 11 13 15 161.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 2 3 5 6 8 10 11 13 141.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1 3 4 6 7 8 10 11 13

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1 2 4 5 6 7 8 10 111.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 1 2 3 4 5 6 7 8 91.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1 2 3 3 4 5 6 7 81.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1 1 2 3 4 4 5 6 61.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 1 1 2 2 3 4 4 5 5

2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 0 1 1 2 2 3 3 4 42.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 0 1 1 2 2 2 3 3 42.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 0 1 1 1 2 2 2 3 32.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916 0 1 1 1 1 2 2 2 22.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936 0 0 1 1 1 1 1 2 2

2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 0 0 0 1 1 1 1 1 12.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964 0 0 0 0 1 1 1 1 12.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 0 0 0 0 0 1 1 1 12.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 0 0 0 0 0 0 0 1 12.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986 0 0 0 0 0 0 0 0 0

3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990 0 0 0 0 0 0 0 0 03.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993 0 0 0 0 0 0 0 0 03.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995 0 0 0 0 0 0 0 0 03.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997 0 0 0 0 0 0 0 0 03.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998 0 0 0 0 0 0 0 0 0

3.5 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 0 0 0 0 0 0 0 0 03.6 .9998 .9998 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 0 0 0 0 0 0 0 0 03.7 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 0 0 0 0 0 0 0 0 03.8 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 0 0 0 0 0 0 0 0 0

Table 5: Normal distribution — inverse cdf

q cq q cq q cq q cq q cq q cq

0.50 0.0000 0.60 0.2533 0.70 0.5244 0.80 0.8416 0.90 1.2816 0.99 2.32630.51 0.0251 0.61 0.2793 0.71 0.5534 0.81 0.8779 0.91 1.3408 0.991 2.36560.52 0.0502 0.62 0.3055 0.72 0.5828 0.82 0.9154 0.92 1.4051 0.992 2.40890.53 0.0753 0.63 0.3319 0.73 0.6128 0.83 0.9542 0.93 1.4758 0.993 2.45730.54 0.1004 0.64 0.3585 0.74 0.6433 0.84 0.9945 0.94 1.5548 0.994 2.51210.55 0.1257 0.65 0.3853 0.75 0.6745 0.85 1.0364 0.95 1.6449 0.995 2.57580.56 0.1510 0.66 0.4125 0.76 0.7063 0.86 1.0803 0.96 1.7507 0.996 2.65210.57 0.1764 0.67 0.4399 0.77 0.7388 0.87 1.1264 0.97 1.8808 0.997 2.74780.58 0.2019 0.68 0.4677 0.78 0.7722 0.88 1.1750 0.975 1.9600 0.998 2.87820.59 0.2275 0.69 0.4958 0.79 0.8064 0.89 1.2265 0.98 2.0537 0.999 3.0902

Table 6: Normal distribution — mean order statistics

n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13

0.564 0.846 1.029 1.163 1.267 1.352 1.424 1.485 1.539 1.586 1.629 1.668—— 0.000 0.297 0.495 0.642 0.757 0.852 0.932 1.001 1.062 1.116 1.1640.637 —— —— 0.000 0.202 0.353 0.473 0.572 0.656 0.729 0.793 0.850

1.432 2.296 —— —— 0.000 0.153 0.275 0.376 0.603 0.537 0.6033.195 4.117 —— —— 0.000 0.123 0.225 0.312 0.388

5.053 5.999 —— —— 0.000 0.103 0.1916.954 7.914 —— —— 0.000

8.879 9.848 ——Note: the entry below the line is

Pni=1 E(X2

(k)). 10.82

n=14 n=15 n=16 n=17 n=18 n=19 n=20 n=21 n=22 n=23 n=24 n=25

1.703 1.736 1.766 1.820 1.820 1.844 1.867 1.889 1.910 1.929 1.948 1.9651.208 1.248 1.285 1.350 1.350 1.380 1.408 1.434 1.458 1.481 1.503 1.5240.901 0.948 0.990 1.066 1.066 1.099 1.131 1.160 1.188 1.214 1.239 1.2630.662 0.715 0.763 0.848 0.848 0.886 0.921 0.954 0.985 1.014 1.041 1.0670.456 0.516 0.570 0.665 0.665 0.707 0.745 0.782 0.815 0.847 0.877 0.9050.267 0.335 0.396 0.502 0.502 0.548 0.590 0.630 0.667 0.701 0.734 0.7640.088 0.165 0.234 0.351 0.351 0.402 0.448 0.491 0.532 0.569 0.604 0.637—— 0.000 0.077 0.208 0.208 0.264 0.315 0.362 0.406 0.446 0.484 0.51911.79 —— —— 0.069 0.069 0.131 0.187 0.238 0.286 0.330 0.370 0.409

12.77 13.75 —— —— 0.000 0.062 0.118 0.170 0.218 0.262 0.30314.73 15.71 —— —— 0.000 0.056 0.108 0.156 0.200

16.69 17.68 —— —— 0.000 0.052 0.10018.66 19.65 —— —— 0.000

20.64 21.62 ——Note: the entry below the line is

Pni=1 E(X2

(k)). 22.61

Table 7: t distribution — inverse cdf

p

df 0.600 0.750 0.800 0.900 0.950 0.975 0.990 0.995 0.999 0.9995

1 0.325 1.000 1.376 3.078 6.314 12.71 31.82 63.66 318.3 636.62 0.289 0.816 1.061 1.886 2.920 4.303 6.965 9.925 22.33 31.603 0.277 0.765 0.978 1.638 2.353 3.182 4.541 5.841 10.21 12.924 0.271 0.741 0.941 1.533 2.132 2.776 3.747 4.604 7.173 8.6105 0.267 0.727 0.920 1.476 2.015 2.571 3.365 4.032 5.894 6.8696 0.265 0.718 0.906 1.440 1.943 2.447 3.143 3.707 5.208 5.9597 0.263 0.711 0.896 1.415 1.895 2.365 2.998 3.499 4.785 5.4088 0.262 0.706 0.889 1.397 1.860 2.306 2.896 3.355 4.501 5.0419 0.261 0.703 0.883 1.383 1.833 2.262 2.821 3.250 4.297 4.781

10 0.260 0.700 0.879 1.372 1.812 2.228 2.764 3.169 4.144 4.587

11 0.260 0.697 0.876 1.363 1.796 2.201 2.718 3.106 4.025 4.43712 0.259 0.695 0.873 1.356 1.782 2.179 2.681 3.055 3.930 4.31813 0.259 0.694 0.870 1.350 1.771 2.160 2.650 3.012 3.852 4.22114 0.258 0.692 0.868 1.345 1.761 2.145 2.624 2.977 3.787 4.14015 0.258 0.691 0.866 1.341 1.753 2.131 2.602 2.947 3.733 4.07316 0.258 0.690 0.865 1.337 1.746 2.120 2.583 2.921 3.686 4.01517 0.257 0.689 0.863 1.333 1.740 2.110 2.567 2.898 3.646 3.96518 0.257 0.688 0.862 1.330 1.734 2.101 2.552 2.878 3.610 3.92219 0.257 0.688 0.861 1.328 1.729 2.093 2.539 2.861 3.579 3.88320 0.257 0.687 0.860 1.325 1.725 2.086 2.528 2.845 3.552 3.850

21 0.257 0.686 0.859 1.323 1.721 2.080 2.518 2.831 3.527 3.81922 0.256 0.686 0.858 1.321 1.717 2.074 2.508 2.819 3.505 3.79223 0.256 0.685 0.858 1.319 1.714 2.069 2.500 2.807 3.485 3.76824 0.256 0.685 0.857 1.318 1.711 2.064 2.492 2.797 3.467 3.74525 0.256 0.684 0.856 1.316 1.708 2.060 2.485 2.787 3.450 3.72526 0.256 0.684 0.856 1.315 1.706 2.056 2.479 2.779 3.435 3.70727 0.256 0.684 0.855 1.314 1.703 2.052 2.473 2.771 3.421 3.68928 0.256 0.683 0.855 1.313 1.701 2.048 2.467 2.763 3.408 3.67429 0.256 0.683 0.854 1.311 1.699 2.045 2.462 2.756 3.396 3.66030 0.256 0.683 0.854 1.310 1.697 2.042 2.457 2.750 3.385 3.646

31 0.256 0.682 0.853 1.309 1.696 2.040 2.453 2.744 3.375 3.63332 0.255 0.682 0.853 1.309 1.694 2.037 2.449 2.738 3.365 3.62233 0.255 0.682 0.853 1.308 1.692 2.035 2.445 2.733 3.356 3.61134 0.255 0.682 0.852 1.307 1.691 2.032 2.441 2.728 3.348 3.60135 0.255 0.682 0.852 1.306 1.690 2.030 2.438 2.724 3.340 3.59136 0.255 0.681 0.852 1.306 1.688 2.028 2.434 2.719 3.333 3.58237 0.255 0.681 0.851 1.305 1.687 2.026 2.431 2.715 3.326 3.57438 0.255 0.681 0.851 1.304 1.686 2.024 2.429 2.712 3.319 3.56639 0.255 0.681 0.851 1.304 1.685 2.023 2.426 2.708 3.313 3.55840 0.255 0.681 0.851 1.303 1.684 2.021 2.423 2.704 3.307 3.551

50 0.255 0.679 0.849 1.299 1.676 2.009 2.403 2.678 3.261 3.49660 0.254 0.679 0.848 1.296 1.671 2.000 2.390 2.660 3.232 3.46070 0.254 0.678 0.847 1.294 1.667 1.994 2.381 2.648 3.211 3.43580 0.254 0.678 0.846 1.292 1.664 1.990 2.374 2.639 3.195 3.41690 0.254 0.677 0.846 1.291 1.662 1.987 2.368 2.632 3.183 3.402

100 0.254 0.677 0.845 1.290 1.660 1.984 2.364 2.626 3.174 3.390

120 0.254 0.677 0.845 1.289 1.658 1.980 2.358 2.617 3.160 3.373160 0.254 0.676 0.844 1.287 1.654 1.975 2.350 2.607 3.142 3.352200 0.254 0.676 0.843 1.286 1.653 1.972 2.345 2.601 3.131 3.340240 0.254 0.676 0.843 1.285 1.651 1.970 2.342 2.596 3.125 3.332300 0.254 0.675 0.843 1.284 1.650 1.968 2.339 2.592 3.118 3.323400 0.254 0.675 0.843 1.284 1.649 1.966 2.336 2.588 3.111 3.315∞ 0.253 0.674 0.842 1.282 1.645 1.960 2.326 2.576 3.090 3.290

Note: Interpolation with respect to df should be linear in 120/df .

Table 8: χ2 distribution — inverse cdf

p

df 0.005 0.010 0.025 0.050 0.100 0.250 0.500 0.750 0.900 0.950 0.975 0.990 0.995 0.999

1 0.000 0.000 0.001 0.004 0.016 0.102 0.455 1.323 2.706 3.841 5.024 6.635 7.879 10.832 0.010 0.020 0.051 0.103 0.211 0.575 1.386 2.773 4.605 5.991 7.378 9.210 10.60 13.823 0.072 0.115 0.216 0.352 0.584 1.213 2.366 4.108 6.251 7.815 9.348 11.34 12.84 16.274 0.207 0.297 0.484 0.711 1.064 1.923 3.357 5.385 7.779 9.488 11.14 13.28 14.86 18.475 0.412 0.554 0.831 1.145 1.610 2.675 4.351 6.626 9.236 11.07 12.83 15.09 16.75 20.516 0.676 0.872 1.237 1.635 2.204 3.455 5.348 7.841 10.64 12.59 14.45 16.81 18.55 22.467 0.989 1.239 1.690 2.167 2.833 4.255 6.346 9.037 12.02 14.07 16.01 18.48 20.28 24.328 1.344 1.647 2.180 2.733 3.490 5.071 7.344 10.22 13.36 15.51 17.53 20.09 21.95 26.129 1.735 2.088 2.700 3.325 4.168 5.899 8.343 11.39 14.68 16.92 19.02 21.67 23.59 27.88

10 2.156 2.558 3.247 3.940 4.865 6.737 9.342 12.55 15.99 18.31 20.48 23.21 25.19 29.59

11 2.603 3.053 3.816 4.575 5.578 7.584 10.34 13.70 17.28 19.68 21.92 24.73 26.76 31.2612 3.074 3.571 4.404 5.226 6.304 8.438 11.34 14.85 18.55 21.03 23.34 26.22 28.30 32.9113 3.565 4.107 5.009 5.892 7.041 9.299 12.34 15.98 19.81 22.36 24.74 27.69 29.82 34.5314 4.075 4.660 5.629 6.571 7.790 10.17 13.34 17.12 21.06 23.68 26.12 29.14 31.32 36.1215 4.601 5.229 6.262 7.261 8.547 11.04 14.34 18.25 22.31 25.00 27.49 30.58 32.80 37.7016 5.142 5.812 6.908 7.962 9.312 11.91 15.34 19.37 23.54 26.30 28.85 32.00 34.27 39.2517 5.697 6.408 7.564 8.672 10.09 12.79 16.34 20.49 24.77 27.59 30.19 33.41 35.72 40.7918 6.265 7.015 8.231 9.390 10.86 13.68 17.34 21.60 25.99 28.87 31.53 34.81 37.16 42.3119 6.844 7.633 8.907 10.12 11.65 14.56 18.34 22.72 27.20 30.14 32.85 36.19 38.58 43.8220 7.434 8.260 9.591 10.85 12.44 15.45 19.34 23.83 28.41 31.41 34.17 37.57 40.00 45.31

21 8.034 8.897 10.28 11.59 13.24 16.34 20.34 24.93 29.62 32.67 35.48 38.93 41.40 46.8022 8.643 9.542 10.98 12.34 14.04 17.24 21.34 26.04 30.81 33.92 36.78 40.29 42.80 48.2723 9.260 10.20 11.69 13.09 14.85 18.14 22.34 27.14 32.01 35.17 38.08 41.64 44.18 49.7324 9.886 10.86 12.40 13.85 15.66 19.04 23.34 28.24 33.20 36.42 39.36 42.98 45.56 51.1825 10.52 11.52 13.12 14.61 16.47 19.94 24.34 29.34 34.38 37.65 40.65 44.31 46.93 52.6226 11.16 12.20 13.84 15.38 17.29 20.84 25.34 30.43 35.56 38.89 41.92 45.64 48.29 54.0527 11.81 12.88 14.57 16.15 18.11 21.75 26.34 31.53 36.74 40.11 43.19 46.96 49.65 55.4828 12.46 13.56 15.31 16.93 18.94 22.66 27.34 32.62 37.92 41.34 44.46 48.28 50.99 56.8929 13.12 14.26 16.05 17.71 19.77 23.57 28.34 33.71 39.09 42.56 45.72 49.59 52.34 58.3030 13.79 14.95 16.79 18.49 20.60 24.48 29.34 34.80 40.26 43.77 46.98 50.89 53.67 59.70

31 14.46 15.66 17.54 19.28 21.43 25.39 30.34 35.89 41.42 44.99 48.23 52.19 55.00 61.1032 15.13 16.36 18.29 20.07 22.27 26.30 31.34 36.97 42.58 46.19 49.48 53.49 56.33 62.4933 15.82 17.07 19.05 20.87 23.11 27.22 32.34 38.06 43.75 47.40 50.73 54.78 57.65 63.8734 16.50 17.79 19.81 21.66 23.95 28.14 33.34 39.14 44.90 48.60 51.97 56.06 58.96 65.2535 17.19 18.51 20.57 22.47 24.80 29.05 34.34 40.22 46.06 49.80 53.20 57.34 60.27 66.6236 17.89 19.23 21.34 23.27 25.64 29.97 35.34 41.30 47.21 51.00 54.44 58.62 61.58 67.9837 18.59 19.96 22.11 24.07 26.49 30.89 36.34 42.38 48.36 52.19 55.67 59.89 62.88 69.3538 19.29 20.69 22.88 24.88 27.34 31.81 37.34 43.46 49.51 53.38 56.90 61.16 64.18 70.7039 20.00 21.43 23.65 25.70 28.20 32.74 38.34 44.54 50.66 54.57 58.12 62.43 65.48 72.0640 20.71 22.16 24.43 26.51 29.05 33.66 39.34 45.62 51.81 55.76 59.34 63.69 66.77 73.40

50 27.99 29.71 32.36 34.76 37.69 42.94 49.33 56.33 63.17 67.50 71.42 76.15 79.49 86.6660 35.53 37.48 40.48 43.19 46.46 52.29 59.33 66.98 74.40 79.08 83.30 88.38 91.95 99.6170 43.28 45.44 48.76 51.74 55.33 61.70 69.33 77.58 85.53 90.53 95.02 100.4 104.2 112.380 51.17 53.54 57.15 60.39 64.28 71.14 79.33 88.13 96.58 101.9 106.6 112.3 116.3 124.890 59.20 61.75 65.65 69.13 73.29 80.62 89.33 98.65 107.6 113.1 118.1 124.1 128.3 137.2

100 67.33 70.06 74.22 77.93 82.36 90.13 99.33 109.1 118.5 124.3 129.6 135.8 140.2 149.4120 83.85 86.92 91.57 95.70 100.6 109.2 119.3 130.1 140.2 146.6 152.2 159.0 163.6 173.6140 100.7 104.0 109.1 113.7 119.0 128.4 139.3 150.9 161.8 168.6 174.6 181.8 186.8 197.4160 117.7 121.3 126.9 131.8 137.5 147.6 159.3 171.7 183.3 190.5 196.9 204.5 209.8 221.0180 134.9 138.8 144.7 150.0 156.2 166.9 179.3 192.4 204.7 212.3 219.0 227.1 232.6 244.4

200 152.2 156.4 162.7 168.3 174.8 186.2 199.3 213.1 226.0 234.0 241.1 249.4 255.3 267.5240 187.3 192.0 199.0 205.1 212.4 224.9 239.3 254.4 268.5 277.1 284.8 293.9 300.2 313.4300 240.7 246.0 253.9 260.9 269.1 283.1 299.3 316.1 331.8 341.4 349.9 359.9 366.8 381.4400 330.9 337.2 346.5 354.6 364.2 380.6 399.3 418.7 436.6 447.6 457.3 468.7 476.6 493.1

Note: Linear interpolation with respect to df should be should be satisfactory for most purposes.

For df > 100, use cq(χ2df ) ≈ 1

2

�cq(N) +

√2 df − 1

�2, where N denotes the standard normal distribution.

Table 9: Standardised range distribution — inverse cdf

k

ν 2 3 4 5 6 7 8 9 10 12 14 16 p

1 18.0 27.0 32.8 37.1 40.4 43.1 45.4 47.4 49.1 52.0 54.3 56.3 0.9590.0 135. 164. 186. 202. 216. 227. 237. 246. 260. 272. 282. 0.99

2 6.09 8.30 9.80 10.9 11.7 12.4 13.0 13.5 14.0 14.7 15.4 15.9 0.9514.0 19.0 22.3 24.7 26.6 28.2 29.5 30.7 31.7 33.4 34.8 36.0 0.99

3 4.50 5.91 6.82 7.50 8.04 8.48 8.85 9.18 9.46 9.95 10.4 10.7 0.958.26 10.6 12.2 13.3 14.2 15.0 15.6 16.2 16.7 17.5 18.2 18.8 0.99

4 3.93 5.04 5.76 6.29 6.71 7.05 7.35 7.60 7.83 8.21 8.52 8.79 0.956.51 8.12 9.17 9.96 10.6 11.1 11.5 11.9 12.3 12.8 13.3 13.7 0.99

5 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99 7.32 7.60 7.83 0.955.70 6.97 7.80 8.42 8.91 9.32 9.67 9.97 10.2 10.7 11.1 11.4 0.99

6 3.46 4.34 4.90 5.31 5.63 5.89 6.12 6.32 6.49 6.79 7.03 7.24 0.955.24 6.33 7.03 7.56 7.97 8.32 8.61 8.87 9.10 9.49 9.81 10.1 0.99

7 3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16 6.43 6.66 6.85 0.954.95 5.92 6.54 7.01 7.37 7.68 7.94 8.17 8.37 8.71 9.00 9.24 0.99

8 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 6.18 6.39 6.57 0.954.74 5.63 6.20 6.63 6.96 7.24 7.47 7.68 7.87 8.18 8.44 8.66 0.99

10 3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60 5.83 6.03 6.20 0.954.48 5.27 5.77 6.14 6.43 6.67 6.87 7.05 7.21 7.48 7.71 7.91 0.99

12 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.40 5.62 5.80 5.95 0.954.32 5.04 5.50 5.84 6.10 6.32 6.51 6.67 6.81 7.06 7.26 7.44 0.99

16 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15 5.35 5.52 5.66 0.954.13 4.78 5.19 5.49 5.72 5.92 6.08 6.22 6.35 6.56 6.74 6.90 0.99

20 2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01 5.20 5.36 5.49 0.954.02 4.64 5.02 5.29 5.51 5.69 5.84 5.97 6.09 6.29 6.45 6.59 0.99

30 2.89 3.49 3.84 4.10 4.30 4.46 4.60 4.72 4.83 5.00 5.15 5.27 0.953.89 4.45 4.80 5.05 5.24 5.40 5.54 5.65 5.76 5.93 6.08 6.20 0.99

40 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.74 4.91 5.05 5.16 0.953.82 4.37 4.70 4.93 5.11 5.27 5.39 5.50 5.60 5.77 5.90 6.02 0.99

60 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65 4.81 4.94 5.06 0.953.76 4.28 4.60 4.82 4.99 5.13 5.25 5.36 5.45 5.60 5.73 5.84 0.99

120 2.80 3.36 3.69 3.92 4.10 4.24 4.36 4.48 4.56 4.72 4.84 4.95 0.953.70 4.20 4.50 4.71 4.87 5.01 5.12 5.21 5.30 5.44 5.56 5.66 0.99

∞ 2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47 4.62 4.74 4.85 0.953.64 4.12 4.40 4.60 4.76 4.88 4.99 5.08 5.16 5.29 5.40 5.49 0.99

Table 10: F distribution — inverse cdf

df1

df2 1 2 3 4 5 6 7 8 10 12 24 60 120 8 p

1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 241.9 243.9 249.1 252.2 253.3 254.3 0.951 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.6 968.6 976.7 997.3 1010 1014 1018.3 0.9751 4052 4999 5404 5624 5764 5859 5928 5981 6056 6107 6234 6313 6340 6366.0 0.991 405K 500K 540K 563K 576K 586K 593K 598K 606K 610K 624K 631K 634K 637K 0.999

2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.40 19.41 19.45 19.48 19.49 19.50 0.952 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.40 39.41 39.46 39.48 39.49 39.50 0.9752 98.50 99.00 99.16 99.25 99.30 99.33 99.36 99.38 99.40 99.42 99.46 99.48 99.49 99.50 0.992 998.4 998.8 999.3 999.3 999.3 999.3 999.3 999.3 999.3 999.3 999.3 999.3 999.3 999.3 0.999

3 10.13 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.785 8.745 8.638 8.572 8.549 8.526 0.953 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.42 14.34 14.12 13.99 13.95 13.90 0.9753 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.23 27.05 26.60 26.32 26.22 26.13 0.993 167.1 148.5 141.1 137.1 134.6 132.8 131.6 130.6 129.2 128.3 125.9 124.4 124.0 123.5 0.999

4 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.964 5.912 5.774 5.688 5.658 5.628 0.954 12.218 10.649 9.979 9.604 9.364 9.197 9.074 8.980 8.844 8.751 8.511 8.360 8.309 8.257 0.9754 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.55 14.37 13.93 13.65 13.56 13.46 0.994 74.13 61.25 56.17 53.43 51.72 50.52 49.65 49.00 48.05 47.41 45.77 44.75 44.40 44.05 0.999

5 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.735 4.678 4.527 4.431 4.398 4.365 0.955 10.01 8.434 7.764 7.388 7.146 6.978 6.853 6.757 6.619 6.525 6.278 6.123 6.069 6.015 0.9755 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.05 9.888 9.466 9.202 9.112 9.020 0.995 47.18 37.12 33.20 31.08 29.75 28.83 28.17 27.65 26.91 26.42 25.13 24.33 24.06 23.78 0.999

6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.060 4.000 3.841 3.740 3.705 3.669 0.956 8.813 7.260 6.599 6.227 5.988 5.820 5.695 5.600 5.461 5.366 5.117 4.959 4.904 4.849 0.9756 13.75 10.92 9.780 9.148 8.746 8.466 8.260 8.102 7.874 7.718 7.313 7.057 6.969 6.880 0.996 35.51 27.00 23.71 21.92 20.80 20.03 19.46 19.03 18.41 17.99 16.90 16.21 15.98 15.75 0.999

7 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.637 3.575 3.410 3.304 3.267 3.230 0.957 8.073 6.542 5.890 5.523 5.285 5.119 4.995 4.899 4.761 4.666 4.415 4.254 4.199 4.142 0.9757 12.25 9.547 8.451 7.847 7.460 7.191 6.993 6.840 6.620 6.469 6.074 5.824 5.737 5.650 0.997 29.25 21.69 18.77 17.20 16.21 15.52 15.02 14.63 14.08 13.71 12.73 12.12 11.91 11.70 0.999

8 5.318 4.459 4.066 3.838 3.688 3.581 3.500 3.438 3.347 3.284 3.115 3.005 2.967 2.928 0.958 7.571 6.059 5.416 5.053 4.817 4.652 4.529 4.433 4.295 4.200 3.947 3.784 3.728 3.670 0.9758 11.26 8.649 7.591 7.006 6.632 6.371 6.178 6.029 5.814 5.667 5.279 5.032 4.946 4.859 0.998 25.41 18.49 15.83 14.39 13.48 12.86 12.40 12.05 11.54 11.19 10.30 9.728 9.532 9.333 0.999

9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.137 3.073 2.900 2.787 2.748 2.707 0.959 7.209 5.715 5.078 4.718 4.484 4.320 4.197 4.102 3.964 3.868 3.614 3.449 3.392 3.333 0.9759 10.56 8.022 6.992 6.422 6.057 5.802 5.613 5.467 5.257 5.111 4.729 4.483 4.398 4.311 0.999 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 9.894 9.570 8.724 8.186 8.002 7.813 0.999

10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 2.978 2.913 2.737 2.621 2.580 2.538 0.9510 6.937 5.456 4.826 4.468 4.236 4.072 3.950 3.855 3.717 3.621 3.365 3.198 3.140 3.080 0.97510 10.04 7.559 6.552 5.994 5.636 5.386 5.200 5.057 4.849 4.706 4.327 4.082 3.996 3.909 0.9910 21.04 14.90 12.55 11.28 10.48 9.926 9.517 9.204 8.754 8.446 7.638 7.122 6.944 6.763 0.999

11 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.854 2.788 2.609 2.490 2.448 2.404 0.9511 6.724 5.256 4.630 4.275 4.044 3.881 3.759 3.664 3.526 3.430 3.173 3.004 2.944 2.883 0.97511 9.646 7.206 6.217 5.668 5.316 5.069 4.886 4.744 4.539 4.397 4.021 3.776 3.690 3.602 0.9911 19.69 13.81 11.56 10.35 9.579 9.047 8.655 8.355 7.923 7.625 6.848 6.348 6.175 5.999 0.999

12 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.753 2.687 2.505 2.384 2.341 2.296 0.9512 6.554 5.096 4.474 4.121 3.891 3.728 3.607 3.512 3.374 3.277 3.019 2.848 2.787 2.725 0.97512 9.330 6.927 5.953 5.412 5.064 4.821 4.640 4.499 4.296 4.155 3.780 3.535 3.449 3.361 0.9912 18.64 12.97 10.80 9.633 8.892 8.378 8.001 7.711 7.292 7.005 6.249 5.763 5.593 5.420 0.999

14 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.602 2.534 2.349 2.223 2.178 2.131 0.9514 6.298 4.857 4.242 3.892 3.663 3.501 3.380 3.285 3.147 3.050 2.789 2.614 2.552 2.487 0.97514 8.862 6.515 5.564 5.035 4.695 4.456 4.278 4.140 3.939 3.800 3.427 3.181 3.094 3.004 0.9914 17.14 11.78 9.730 8.622 7.922 7.436 7.078 6.802 6.404 6.130 5.407 4.938 4.773 4.604 0.999

df1

df2 1 2 3 4 5 6 7 8 10 12 24 60 120 8 p

16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.494 2.425 2.235 2.106 2.059 2.010 0.9516 6.115 4.687 4.077 3.729 3.502 3.341 3.219 3.125 2.986 2.889 2.625 2.447 2.383 2.316 0.97516 8.531 6.226 5.292 4.773 4.437 4.202 4.026 3.890 3.691 3.553 3.181 2.933 2.845 2.753 0.9916 16.12 10.97 9.006 7.944 7.272 6.805 6.460 6.195 5.812 5.547 4.846 4.388 4.226 4.059 0.999

18 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.412 2.342 2.150 2.017 1.968 1.917 0.9518 5.978 4.560 3.954 3.608 3.382 3.221 3.100 3.005 2.866 2.769 2.503 2.321 2.256 2.187 0.97518 8.285 6.013 5.092 4.579 4.248 4.015 3.841 3.705 3.508 3.371 2.999 2.749 2.660 2.566 0.9918 15.38 10.39 8.487 7.460 6.808 6.355 6.021 5.763 5.390 5.132 4.447 3.996 3.836 3.670 0.999

20 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.348 2.278 2.082 1.946 1.896 1.843 0.9520 5.871 4.461 3.859 3.515 3.289 3.128 3.007 2.913 2.774 2.676 2.408 2.223 2.156 2.085 0.97520 8.096 5.849 4.938 4.431 4.103 3.871 3.699 3.564 3.368 3.231 2.859 2.608 2.517 2.421 0.9920 14.82 9.953 8.098 7.096 6.461 6.019 5.692 5.440 5.075 4.823 4.149 3.703 3.544 3.378 0.999

24 4.260 3.403 3.009 2.776 2.621 2.508 2.423 2.355 2.255 2.183 1.984 1.842 1.790 1.733 0.9524 5.717 4.319 3.721 3.379 3.155 2.995 2.874 2.779 2.640 2.541 2.269 2.080 2.010 1.935 0.97524 7.823 5.614 4.718 4.218 3.895 3.667 3.496 3.363 3.168 3.032 2.659 2.403 2.310 2.211 0.9924 14.03 9.340 7.554 6.589 5.977 5.551 5.235 4.991 4.638 4.393 3.735 3.295 3.136 2.969 0.999

30 4.171 3.316 2.922 2.690 2.534 2.421 2.334 2.266 2.165 2.092 1.887 1.740 1.683 1.622 0.9530 5.568 4.182 3.589 3.250 3.026 2.867 2.746 2.651 2.511 2.412 2.136 1.940 1.866 1.787 0.97530 7.562 5.390 4.510 4.018 3.699 3.473 3.305 3.173 2.979 2.843 2.469 2.208 2.111 2.006 0.9930 13.29 8.773 7.054 6.125 5.534 5.122 4.817 4.582 4.239 4.001 3.357 2.920 2.760 2.589 0.999

40 4.085 3.232 2.839 2.606 2.449 2.336 2.249 2.180 2.077 2.003 1.793 1.637 1.577 1.509 0.9540 5.424 4.051 3.463 3.126 2.904 2.744 2.624 2.529 2.388 2.288 2.007 1.803 1.724 1.637 0.97540 7.314 5.178 4.313 3.828 3.514 3.291 3.124 2.993 2.801 2.665 2.288 2.019 1.917 1.805 0.9940 12.61 8.251 6.595 5.698 5.128 4.731 4.436 4.207 3.874 3.643 3.011 2.574 2.410 2.233 0.999

50 4.034 3.183 2.790 2.557 2.400 2.286 2.199 2.130 2.026 1.952 1.737 1.576 1.511 1.438 0.9550 5.340 3.975 3.390 3.054 2.833 2.674 2.553 2.458 2.317 2.216 1.931 1.721 1.639 1.545 0.97550 7.171 5.057 4.199 3.720 3.408 3.186 3.020 2.890 2.698 2.563 2.183 1.909 1.803 1.683 0.9950 12.22 7.956 6.336 5.459 4.901 4.512 4.222 3.998 3.671 3.443 2.817 2.378 2.211 2.026 0.999

60 4.001 3.150 2.758 2.525 2.368 2.254 2.167 2.097 1.993 1.917 1.700 1.534 1.467 1.389 0.9560 5.286 3.925 3.343 3.008 2.786 2.627 2.507 2.412 2.270 2.169 1.882 1.667 1.581 1.482 0.97560 7.077 4.977 4.126 3.649 3.339 3.119 2.953 2.823 2.632 2.496 2.115 1.836 1.726 1.601 0.9960 11.97 7.768 6.171 5.307 4.757 4.372 4.086 3.865 3.542 3.315 2.694 2.252 2.082 1.890 0.999

80 3.960 3.111 2.719 2.486 2.329 2.214 2.126 2.056 1.951 1.875 1.654 1.482 1.411 1.325 0.9580 5.218 3.864 3.284 2.950 2.730 2.571 2.450 2.355 2.213 2.111 1.820 1.599 1.508 1.400 0.97580 6.963 4.881 4.036 3.563 3.255 3.036 2.871 2.742 2.551 2.415 2.032 1.746 1.630 1.494 0.9980 11.67 7.540 5.972 5.123 4.582 4.204 3.923 3.705 3.386 3.162 2.545 2.099 1.924 1.720 0.999

100 3.936 3.087 2.696 2.463 2.305 2.191 2.103 2.032 1.927 1.850 1.627 1.450 1.376 1.283 0.95100 5.179 3.828 3.250 2.917 2.696 2.537 2.417 2.321 2.179 2.077 1.784 1.558 1.463 1.347 0.975100 6.895 4.824 3.984 3.513 3.206 2.988 2.823 2.694 2.503 2.368 1.983 1.692 1.572 1.427 0.99100 11.50 7.408 5.857 5.017 4.482 4.107 3.829 3.612 3.296 3.074 2.458 2.009 1.829 1.615 0.999

120 3.920 3.072 2.680 2.447 2.290 2.175 2.087 2.016 1.910 1.834 1.608 1.429 1.352 1.254 0.95120 5.152 3.805 3.227 2.894 2.674 2.515 2.395 2.299 2.157 2.055 1.760 1.530 1.433 1.310 0.975120 6.851 4.787 3.949 3.480 3.174 2.956 2.792 2.663 2.472 2.336 1.950 1.656 1.533 1.381 0.99120 11.38 7.321 5.781 4.947 4.416 4.044 3.767 3.552 3.237 3.016 2.402 1.950 1.767 1.543 0.999

∞ 3.841 2.996 2.605 2.372 2.214 2.099 2.010 1.938 1.831 1.752 1.517 1.318 1.221 1.000 0.95∞ 5.024 3.689 3.116 2.786 2.566 2.408 2.288 2.192 2.048 1.945 1.640 1.388 1.268 1.000 0.975∞ 6.635 4.605 3.782 3.319 3.017 2.802 2.639 2.511 2.321 2.185 1.791 1.473 1.325 1.000 0.99∞ 10.83 6.908 5.422 4.617 4.103 3.743 3.474 3.265 2.959 2.742 2.132 1.660 1.447 1.000 0.999

Note: The entry 405K denotes 405 000.

Interpolation with respect to df1, df2 should be linear in 120/df1, 120/df2.

Lower tail quantiles are given by cq(Fdf1,df2 ) = 1/c1−q(Fdf2,df1 ).

Table 11: Random digits

14159 26535 89793 23846 26433 83279 50288 41971 69399 3751058209 74944 59230 78164 06286 20899 86280 34825 34211 7067982148 08651 32823 06647 09384 46095 50582 23172 53594 0812848111 74502 84102 70193 85211 05559 64462 29489 54930 3819644288 10975 66593 34461 28475 64823 37867 83165 27120 19091

45648 56692 34603 48610 45432 66482 13393 60726 02491 4127372458 70066 06315 58817 48815 20920 96282 92540 91715 3643678925 90360 01133 05305 48820 46652 13841 46951 94151 1609433057 27036 57595 91953 09218 61173 81932 61179 31051 1854807446 23799 62749 56735 18857 52724 89122 79381 83011 94912

98336 73362 44065 66430 86021 39494 63952 24737 19070 2179860943 70277 05392 17176 29317 67523 84674 81846 76694 0513200056 81271 45263 56082 77857 71342 75778 96091 73637 1787214684 40901 22495 34301 46549 58537 10507 92279 68925 8923542019 95611 21290 21960 86403 44181 59813 62977 47713 09960

51870 72113 49999 99837 29780 49951 05973 17328 16096 3185950244 59455 34690 83026 42522 30825 33446 85035 26193 1188171010 00313 78387 52886 58753 32083 81420 61717 76691 4730359825 34904 28755 46873 11595 62863 88235 37875 93751 9577818577 80532 17122 68066 13001 92787 66111 95909 21642 01989

38095 25720 10654 85863 27886 59361 53381 82796 82303 0195203530 18529 68995 77362 25994 13891 24972 17752 83479 1315155748 57242 45415 06959 50829 53311 68617 27855 88907 5098381754 63746 49393 19255 06040 09277 01671 13900 98488 2401285836 16035 63707 66010 47101 81942 95559 61989 46767 83744

94482 55379 77472 68471 04047 53464 62080 46684 25906 9491293313 67702 89891 52104 75216 20569 66024 05083 81501 9351125338 24300 35587 64024 74964 73263 91419 92726 04269 9227967823 54781 63600 93417 21641 21992 45863 15030 28618 2974555706 74983 85054 94588 58692 69956 90927 21079 75093 02955

32116 53449 87202 75596 02364 80665 49911 98818 34797 7535663698 07426 54252 78625 51818 41757 46728 90977 77279 3800081647 06001 61452 49192 17321 72147 72350 14144 19735 6854816136 11573 52552 13347 57418 49468 43852 33239 07394 1433345477 62416 86251 98935 69485 56209 92192 22184 27255 02542

56887 67179 04946 01653 46680 49886 27232 79178 60857 8438382796 79766 81454 10095 38837 86360 95068 00642 25125 2051173929 84896 08412 84886 26945 60424 19652 85022 21066 1186306744 27862 20391 94945 04712 37137 86960 95636 43719 1728746776 46575 73962 41389 08658 32645 99581 33904 78027 59009

71828 18284 59045 23536 02874 71352 62249 77572 47093 6999595749 66967 62772 40766 30353 54759 45713 82178 52516 6427427466 39193 20030 59921 81741 35966 29043 57290 03342 9526059563 07381 32328 62794 34907 63233 82988 07531 95251 0190115738 34187 93070 21540 89149 93488 41675 09244 76146 06680

82264 80016 84774 11853 74234 54424 37107 53907 77449 9206955170 27618 38606 26133 13845 83000 75204 49338 26560 2976067371 13200 70932 87091 27443 74704 72306 96977 20931 0141692836 81902 55151 08657 46377 21112 52389 78442 50569 5369677078 54499 69967 94686 44549 05987 93163 68892 30098 79312

77361 78215 42499 92295 76351 48220 82698 95193 66803 3182528869 39849 64651 05820 93923 98924 88793 32036 25094 4311730123 81970 68416 14039 70198 37679 32068 32823 76464 8042953118 02328 78250 98194 55815 30175 67173 61332 06981 1250996181 88159 30416 90351 59888 85193 45807 27386 67385 89422

87922 84998 92086 80582 57492 79610 48419 84443 63463 2449684875 60233 62482 70419 78623 20900 21609 90235 30436 9941849146 31409 34317 38143 64052 62531 52096 18369 08887 0701676839 64243 78140 59271 45635 40961 30310 72085 10383 7505101157 47704 17189 86106 87396 96552 12671 54688 95703 50354

02123 40784 98193 34321 06817 01210 05627 88023 51930 3322474501 58539 04730 41995 77770 93503 66041 69973 29725 0886876966 40355 57071 62268 44716 25607 98826 51787 13419 5124665201 03059 21236 67719 43252 78675 39855 89448 96970 9640975459 18569 56380 23637 01621 12047 74272 28364 89613 42251

64450 78182 44235 29486 36372 14174 02388 93441 24796 3574370263 75529 44483 37998 01612 54922 78509 25778 25620 9262264832 62779 33386 56648 16277 25164 01910 59004 91644 9982893150 56604 72580 27786 31864 15519 56532 44258 69829 4695930801 91529 87211 72556 34754 63964 47910 14590 40905 86298

49679 12874 06870 50489 58586 71747 98546 67757 57320 5681288459 20541 33405 39220 00113 78630 09455 60688 16674 0016984205 58040 33637 95376 45203 04024 32256 61352 78369 5117788386 38744 39662 53224 98506 54995 88623 42818 99707 7332761717 83928 03494 65014 34558 89707 19425 86398 77275 47109

62953 74152 11151 36835 06275 26023 26484 72870 39207 6431005958 41166 12054 52970 30236 47254 92966 69381 15137 3227536450 98889 03136 02057 24817 65851 18063 03644 28123 1496550704 75102 54465 01172 72115 55194 86685 08003 68532 2818315219 60037 35625 27944 95158 28418 82947 87610 85263 98139

Table 12: Critical values of the Wilcoxon rank-sum statistic

p

n1 n2 0.005 0.01 0.025 0.05 0.1 0.9 0.95 0.975 0.99 0.995

3 2 3 93 3 6 7 14 15

4 2 3 114 3 6 7 17 184 4 10 11 13 23 25 26

5 2 3 4 12 135 3 6 7 8 19 20 215 4 10 11 12 14 26 28 29 305 5 15 16 17 19 20 35 36 38 39 40

6 2 3 4 14 156 3 7 8 9 21 22 236 4 10 11 12 13 15 29 31 32 33 346 5 16 17 18 20 22 38 40 42 43 446 6 23 24 26 28 30 48 50 52 54 55

7 2 3 4 16 177 3 6 7 8 10 23 25 26 277 4 10 11 13 14 16 32 34 35 37 387 5 16 18 20 21 23 42 44 45 47 497 6 24 25 27 29 32 52 55 57 59 607 7 32 34 36 39 41 64 66 69 71 73

8 2 3 4 5 17 18 198 3 6 8 9 11 25 27 28 308 4 11 12 14 15 17 35 37 38 40 418 5 17 19 21 23 25 45 47 49 51 538 6 25 27 29 31 34 56 59 61 63 658 7 34 35 38 41 44 68 71 74 77 788 8 43 45 49 51 55 81 85 87 91 93

9 1 1 109 2 3 4 5 19 20 219 3 6 7 8 10 11 28 29 31 32 339 4 11 13 14 16 19 37 40 42 43 459 5 18 20 22 24 27 48 51 53 55 579 6 26 28 31 33 36 60 63 65 68 709 7 35 37 40 43 46 73 76 79 82 849 8 45 47 51 54 58 86 90 93 97 999 9 56 59 62 66 70 101 105 109 112 115

10 1 1 1110 2 3 4 6 20 22 2310 3 6 7 9 10 12 30 32 33 35 3610 4 12 13 15 17 20 40 43 45 47 4810 5 19 21 23 26 28 52 54 57 59 6110 6 27 29 32 35 38 64 67 70 73 7510 7 37 39 42 45 49 77 81 84 87 8910 8 47 49 53 56 60 92 96 99 103 10510 9 58 61 65 69 73 107 111 115 119 12210 10 71 74 78 82 87 123 128 132 136 139

Note: Two samples from the same population containing n1 and n2 observations respectively, where n1 ≤ n2, areranked, and W denotes the sum of the ranks in the smaller sample.The table entry is, for p < 0.5, the largest value of w such that Pr(W ≤ w) ≤ p; and for p > 0.5, the smallestvalue of w such that Pr(W ≥ w) ≤ 1− p. No entry means that so such value exists.

For larger values of n1 and n2, it can be assumed that Wd≈ N( 1

2n1(n1 + n2 + 1), 1

12n1n2(n1 + n2 + 1)).

Table 13: Critical values of the signed-rank-sum statistic

p

n 0.005 0.01 0.025 0.05 0.1 0.9 0.95 0.975 0.99 0.995

5 0 1 14 156 0 2 3 18 19 217 0 2 3 5 23 25 26 288 0 1 3 5 8 28 31 33 35 369 1 3 5 8 10 35 37 40 42 4410 3 5 8 10 14 41 45 47 50 52

11 5 7 10 13 17 49 53 56 59 6112 7 9 13 17 21 57 61 65 69 7113 9 12 17 21 26 65 70 74 79 8214 12 15 21 25 31 74 80 84 90 9315 15 19 25 30 36 84 90 95 101 105

16 19 23 29 35 42 94 101 107 113 11717 23 27 34 41 48 105 112 119 126 13018 27 32 40 47 55 116 124 131 139 14419 32 37 46 53 62 128 137 144 153 15820 37 43 52 60 69 141 150 158 167 173

Note: If a sample of n observations is obtained from a population symmetrically distributed about zero, and Wdenotes the sum of like-signed ranks, then the table entry is, for p < 0.5, the largest value of w such thatPr(W ≤ w) ≤ p; and for p > 0.5, the smallest value of w such that Pr(W ≥ w) ≤ 1− p.No entry means that so such value exists.

For larger values of n, it can be assumed that Wd≈ N

�14n(n + 1), 1

24n(n + 1)(2n + 1)

�.

Table 14: Critical values for the Kruskal-Wallis statistic

p

n1 n2 n3 0.90 0.95 0.99

3 2 1 4.283 2 2 4.50 4.713 3 1 4.57 5.143 3 2 4.55 5.363 3 3 4.62 5.60 7.20

4 2 1 4.504 2 2 4.45 5.334 3 1 4.05 5.204 3 2 4.51 5.44 6.444 3 3 4.70 5.72 6.744 4 1 4.16 4.96 6.664 4 2 4.55 5.45 7.034 4 3 4.54 5.59 7.144 4 4 4.65 5.69 7.65

5 2 1 4.20 5.005 2 2 4.37 5.16 6.535 3 1 4.01 4.965 3 2 4.65 5.25 6.825 3 3 4.53 5.65 7.075 4 1 3.98 4.98 6.955 4 2 4.54 5.26 7.115 4 3 4.54 5.63 7.445 4 4 4.61 5.61 7.765 5 1 4.10 5.12 7.305 5 2 4.62 5.33 7.265 5 3 4.54 5.70 7.545 5 4 4.52 5.64 7.795 5 5 4.56 5.78 7.98

∞ ∞ ∞ 4.60 5.99 9.21

Note: The value in the table is the smallest value of h such that Pr(H ≥ h |H0) ≤ 1 − p, where H denotes theKruskal-Wallis statistic for comparison of the location of three populations based on samples of n1, n2 andn3:

H =12

N(N + 1)

3X

i=1

R2i·

ni− 3(N + 1) =

12

N(N + 1)

3X

i=1

ni(R̄i· − R̄··)2 where N = n1 + n2 + n3

and H0 denotes the hypothesis that the populations are identical.

For larger values of n1, n2 and n3, it can be assumed that Hd≈ χ2

2.

Table 15: Critical values for the Friedman statistic

c = 3 c = 4 c = 5 c = 6 p

r = 3 6.00 7.40 8.58 9.89 0.959.00 10.08 11.69 0.99

r = 4 6.50 7.80 8.84 10.23 0.958.00 9.60 10.93 12.58 0.99

r = 5 6.40 7.80 8.98 10.42 0.958.40 9.96 11.42 13.10 0.99

r = 6 7.00 7.60 9.07 10.54 0.959.00 10.20 11.74 13.44 0.99

r = 7 7.14 7.80 9.13 10.62 0.958.85 10.37 11.98 13.68 0.99

r = 8 6.25 7.65 9.18 10.67 0.959.00 10.35 12.13 13.86 0.99

r = 10 6.20 7.66 9.24 10.76 0.959.60 10.51 12.36 14.11 0.99

r = ∞ 5.99 7.82 9.49 11.07 0.959.21 11.34 13.28 15.09 0.99

Note: The value in the table is the smallest value of u such that Pr(U ≥ u |H0) ≤ 1 − p, where U denotes theFriedman statistic for an r × c classification:

U =12

rc(c + 1)

cX

j=1

T 2j − 3r(c + 1) =

12

rc(c + 1)

cX

j=1

�Tj − 1

2r(c + 1)

�2

where Tj denotes the sum of the ranks in the jth column; and and H0 denotes the hypothesis that the columneffects are identical.

For larger values of r, it can be assumed that Ud≈ χ2

c−1.

Table 16: Critical values for the Kolmogorov-Smirnov statistic

p

n 0.90 0.95 0.99

1 0.950 0.975 0.9952 0.776 0.842 0.9293 0.636 0.708 0.8294 0.565 0.624 0.7345 0.509 0.563 0.669

6 0.468 0.519 0.6177 0.436 0.483 0.5768 0.410 0.454 0.5429 0.387 0.430 0.51310 0.369 0.409 0.486

11 0.352 0.391 0.46812 0.338 0.375 0.44913 0.325 0.361 0.43214 0.314 0.349 0.41815 0.304 0.338 0.404

16 0.295 0.327 0.39217 0.286 0.318 0.38118 0.279 0.309 0.37119 0.271 0.301 0.36120 0.265 0.294 0.352

n > 20 1.224an 1.358an 1.628an

where a−1n = n1/2 + 0.12 + 0.11n−1/2

Test of 0.819bn 0.895bn 1.035bn

N(µ, σ2) where b−1n = n1/2 − 0.01 + 0.85n−1/2

Test of 0.2n−1 + 0.990cn 0.2n−1 + 1.094cn 0.2n−1 + 1.308cn

exp(α) where c−1n = n1/2 + 0.26 + 0.50n−1/2

Note: The value tabulated is the smallest value of c such that Pr(Dn ≥ c |H0) ≤ 1 − p, where Dn denotes theKolmogorov-Smirnov statistic: supx |F̂n(x)− F (x)|.

Table 17: Critical values for the Smirnov two-sample statistic

p

n m 0.90 0.95 0.99

4 3 1.0004 4 1.000 1.000

5 2 1.0005 3 1.000 1.0005 4 0.800 1.0005 5 0.800 1.000 1.000

6 2 1.0006 3 0.833 1.0006 4 0.750 0.833 1.0006 5 0.800 0.800 1.0006 6 0.833 0.833 1.000

7 2 1.0007 3 0.857 1.0007 4 0.750 0.857 1.0007 5 0.714 0.800 1.0007 6 0.667 0.714 0.8577 7 0.714 0.857 0.857

8 2 1.000 1.0008 3 0.875 0.8758 4 0.750 0.875 1.0008 5 0.675 0.750 0.8758 6 0.625 0.708 0.8338 7 0.607 0.714 0.8578 8 0.625 0.750 0.875

9 2 1.000 1.0009 3 0.778 0.889 1.0009 4 0.750 0.778 1.0009 5 0.667 0.778 0.8899 6 0.611 0.722 0.8339 7 0.571 0.667 0.7789 8 0.556 0.639 0.7649 9 0.667 0.667 0.778

10 2 0.900 1.00010 3 0.800 0.900 1.00010 4 0.700 0.750 0.90010 5 0.700 0.800 0.90010 6 0.600 0.667 0.80010 7 0.571 0.657 0.75710 8 0.550 0.600 0.75010 9 0.556 0.589 0.70010 10 0.600 0.700 0.800

m, n > 10 1.22cmn 1.35cmn 1.63cmn

where cmn =q

m+nmn

Table 18: Critical values for the Spearman rank correlation

p

n 0.90 0.95 0.975 0.99 0.995

4 1.000 1.0005 0.800 0.900 1.000 1.0006 0.657 0.829 0.886 0.943 1.0007 0.571 0.714 0.786 0.893 0.9298 0.524 0.643 0.738 0.833 0.8819 0.483 0.600 0.683 0.783 0.83310 0.455 0.564 0.648 0.733 0.794

11 0.427 0.536 0.618 0.709 0.75512 0.392 0.503 0.594 0.692 0.73413 0.374 0.478 0.566 0.670 0.70314 0.358 0.459 0.547 0.644 0.67915 0.346 0.443 0.525 0.625 0.657

16 0.335 0.426 0.509 0.603 0.63517 0.324 0.412 0.490 0.583 0.61818 0.313 0.399 0.444 0.564 0.60019 0.304 0.388 0.463 0.549 0.58420 0.296 0.377 0.451 0.534 0.570

21 0.292 0.370 0.436 0.509 0.55622 0.284 0.361 0.425 0.497 0.54423 0.278 0.353 0.416 0.486 0.53224 0.271 0.344 0.407 0.476 0.52125 0.265 0.337 0.399 0.466 0.511

26 0.260 0.331 0.390 0.457 0.50127 0.255 0.324 0.383 0.449 0.49228 0.250 0.318 0.376 0.441 0.48329 0.245 0.312 0.369 0.433 0.47530 0.240 0.306 0.362 0.426 0.467

Note: The value in the table is the smallest value of r such that Pr(R ≥ r |H0) ≤ 1 − p, where R denotes theSpearman rank correlation; and H0 denotes the hypothesis that the variables are independent. The lowercritical value is −r. No entry means that no such value exists.

For values of n > 30, it can be assumed that, under H0, Rd≈ N(0, 1

n−1).

Table 19: Critical values for the Kendall concordance

p

n 0.90 0.95 0.975 0.99 0.995

4 1.000 1.0005 0.800 0.800 1.000 1.0006 0.600 0.733 0.867 0.867 1.0007 0.524 0.619 0.714 0.810 0.9058 0.429 0.571 0.643 0.714 0.7869 0.389 0.500 0.556 0.667 0.72210 0.378 0.467 0.511 0.600 0.644

11 0.345 0.418 0.491 0.564 0.60012 0.303 0.394 0.455 0.545 0.57613 0.308 0.359 0.436 0.513 0.56414 0.275 0.363 0.407 0.473 0.51615 0.276 0.333 0.390 0.467 0.505

16 0.250 0.317 0.383 0.433 0.48317 0.250 0.309 0.368 0.426 0.47118 0.242 0.294 0.346 0.412 0.45119 0.228 0.287 0.333 0.392 0.43920 0.221 0.274 0.326 0.379 0.421

21 0.210 0.267 0.314 0.371 0.41022 0.203 0.264 0.307 0.359 0.39423 0.202 0.257 0.296 0.352 0.39124 0.196 0.246 0.290 0.341 0.37725 0.193 0.240 0.287 0.333 0.367

26 0.188 0.237 0.280 0.329 0.36027 0.179 0.231 0.271 0.322 0.35628 0.180 0.228 0.265 0.312 0.34429 0.172 0.222 0.261 0.310 0.34030 0.172 0.218 0.255 0.301 0.333

Note: The value in the table is the smallest value of k such that Pr(K ≥ k |H0) ≤ 1 − p, where K denotes theKendall concordance; and H0 denotes the hypothesis that the variables are independent. The lower criticalvalue is −k. No entry means that no such value exists.

For values of n > 30, it can be assumed that, under H0, Kd≈ N(0,

2(2n+5)9n(n−1)

).