Appendix 3. Chi-Square Distribution
Upper tail probabilities for the Chi-square distribution are
shown. For larger degrees of freedom, n, the distribution
may be approximated as a normal distribution with

χ2 | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |
0.5 | 0.47950 | 0.77880 | 0.91889 | 0.97350 | 0.99212 |
1.0 | 0.31731 | 0.60653 | 0.80125 | 0.90980 | 0.96257 |
1.5 | 0.22067 | 0.47237 | 0.68227 | 0.82664 | 0.91307 |
2.0 | 0.15730 | 0.36788 | 0.57241 | 0.73576 | 0.84915 |
2.5 | 0.11385 | 0.28651 | 0.47529 | 0.64464 | 0.77650 |
3.0 | 0.08326 | 0.22313 | 0.39163 | 0.55783 | 0.69999 |
3.5 | 0.06137 | 0.17377 | 0.32076 | 0.47788 | 0.62339 |
4.0 | 0.04550 | 0.13534 | 0.26146 | 0.40601 | 0.54942 |
4.5 | 0.03390 | 0.10540 | 0.21229 | 0.34255 | 0.47988 |
5.0 | 0.02535 | 0.08209 | 0.17180 | 0.28730 | 0.41588 |
5.5 | 0.01902 | 0.06393 | 0.13864 | 0.23973 | 0.35795 |
6.0 | 0.01431 | 0.04979 | 0.11161 | 0.19915 | 0.30622 |
6.5 | 0.01079 | 0.03877 | 0.08966 | 0.16479 | 0.26056 |
7.0 | 0.00815 | 0.03020 | 0.07190 | 0.13589 | 0.22064 |
7.5 | 0.00617 | 0.02352 | 0.05756 | 0.11171 | 0.18603 |
8.0 | 0.00468 | 0.01832 | 0.04601 | 0.09158 | 0.15624 |
8.5 | 0.00355 | 0.01426 | 0.03673 | 0.07489 | 0.13075 |
χ2 | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |
9.0 | 0.00270 | 0.01111 | 0.02929 | 0.06110 | 0.10906 |
9.5 | 0.00206 | 0.00865 | 0.02333 | 0.04975 | 0.09071 |
10.0 | 0.00157 | 0.00674 | 0.01857 | 0.04043 | 0.07524 |
10.5 | 0.00119 | 0.00525 | 0.01476 | 0.03280 | 0.06225 |
11.0 | 0.00091 | 0.00409 | 0.01173 | 0.02656 | 0.05138 |
11.5 | 0.00070 | 0.00318 | 0.00931 | 0.02148 | 0.04232 |
12.0 | 0.00053 | 0.00248 | 0.00738 | 0.01735 | 0.03479 |
12.5 | 0.00041 | 0.00193 | 0.00585 | 0.01400 | 0.02854 |
13.0 | 0.00031 | 0.00150 | 0.00464 | 0.01128 | 0.02338 |
13.5 | 0.00024 | 0.00117 | 0.00367 | 0.00907 | 0.01912 |
14.0 | 0.00018 | 0.00091 | 0.00291 | 0.00730 | 0.01561 |
14.5 | 0.00014 | 0.00071 | 0.00230 | 0.00586 | 0.01273 |
15.0 | 0.00011 | 0.00055 | 0.00182 | 0.00470 | 0.01036 |
15.5 | 0.00008 | 0.00043 | 0.00144 | 0.00377 | 0.00843 |
16.0 | 0.00006 | 0.00034 | 0.00113 | 0.00302 | 0.00684 |
16.5 | 0.00005 | 0.00026 | 0.00090 | 0.00242 | 0.00555 |
17.0 | 0.00004 | 0.00020 | 0.00071 | 0.00193 | 0.00450 |
17.5 | 0.00003 | 0.00016 | 0.00056 | 0.00155 | 0.00364 |
χ2 | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 |
18.0 | 0.00002 | 0.00012 | 0.00044 | 0.00123 | 0.00295 |
18.5 | 0.00002 | 0.00010 | 0.00035 | 0.00099 | 0.00238 |
19.0 | 0.00001 | 0.00008 | 0.00027 | 0.00079 | 0.00192 |
19.5 | | 0.00006 | 0.00022 | 0.00063 | 0.00155 |
20.0 | | 0.00005 | 0.00017 | 0.00050 | 0.00125 |
20.5 | | 0.00004 | 0.00013 | 0.00040 | 0.00101 |
21.0 | | 0.00003 | 0.00011 | 0.00032 | 0.00081 |
21.5 | | 0.00002 | 0.00008 | 0.00025 | 0.00065 |
22.0 | | 0.00002 | 0.00007 | 0.00020 | 0.00052 |
22.5 | | 0.00001 | 0.00005 | 0.00016 | 0.00042 |
23.0 | | | 0.00004 | 0.00013 | 0.00034 |
23.5 | | | 0.00003 | 0.00010 | 0.00027 |
24.0 | | | 0.00003 | 0.00008 | 0.00022 |
24.5 | | | 0.00002 | 0.00006 | 0.00017 |
25.0 | | | 0.00002 | 0.00005 | 0.00014 |
χ2 | n = 6 | n = 7 | n = 8 | n = 9 | n = 10 |
5.0 | 0.543813 |
|
|
|
|
6.0 | 0.423190 | 0.539749 |
|
|
|
7.0 | 0.320847 | 0.428880 | 0.536633 |
|
|
8.0 | 0.238103 | 0.332594 | 0.433470 | 0.534146 |
|
9.0 | 0.173578 | 0.252656 | 0.342296 | 0.437274 | 0.532104 |
10.0 | 0.124652 | 0.188574 | 0.265026 | 0.350485 | 0.440493 |
11.0 | 0.088376 | 0.138619 | 0.201699 | 0.275709 | 0.357518 |
12.0 | 0.061969 | 0.100559 | 0.151204 | 0.213309 | 0.285057 |
13.0 | 0.043036 | 0.072108 | 0.111850 | 0.162606 | 0.223672 |
14.0 | 0.029636 | 0.051181 | 0.081765 | 0.122325 | 0.172992 |
15.0 | 0.020257 | 0.035999 | 0.059145 | 0.090936 | 0.132062 |
16.0 | 0.013754 | 0.025116 | 0.042380 | 0.066882 | 0.099632 |
17.0 | 0.009283 | 0.017396 | 0.030109 | 0.048716 | 0.074364 |
18.0 | 0.006232 | 0.011970 | 0.021226 | 0.035174 | 0.054964 |
19.0 | 0.004164 | 0.008187 | 0.014860 | 0.025193 | 0.040263 |
20.0 | 0.002769 | 0.005570 | 0.010336 | 0.017912 | 0.029253 |
21.0 | 0.001835 | 0.003770 | 0.007147 | 0.012650 | 0.021094 |
22.0 | 0.001211 | 0.002540 | 0.004916 | 0.008879 | 0.015105 |
23.0 | 0.000796 | 0.001705 | 0.003364 | 0.006196 | 0.010747 |
24.0 | 0.000522 | 0.001139 | 0.002292 | 0.004301 | 0.007600 |
χ2 | n = 6 | n = 7 | n = 8 | n = 9 | n = 10 |
25.0 | 0.000341 | 0.000759 | 0.001555 | 0.002971 | 0.005346 |
26.0 | 0.000223 | 0.000504 | 0.001050 | 0.002043 | 0.003740 |
27.0 | 0.000145 | 0.000333 | 0.000707 | 0.001399 | 0.002604 |
28.0 | 0.000094 | 0.000220 | 0.000474 | 0.000954 | 0.001805 |
29.0 | 0.000061 | 0.000145 | 0.000317 | 0.000648 | 0.001246 |
30.0 | 0.000039 | 0.000095 | 0.000211 | 0.000439 | 0.000857 |
31.0 | 0.000025 | 0.000062 | 0.000141 | 0.000296 | 0.000587 |
32.0 | 0.000016 | 0.000041 | 0.000093 | 0.000199 | 0.000400 |
33.0 | 0.000010 | 0.000026 | 0.000062 | 0.000134 | 0.000272 |
34.0 | 0.000007 | 0.000017 | 0.000041 | 0.000089 | 0.000185 |
35.0 | 0.000004 | 0.000011 | 0.000027 | 0.000060 | 0.000125 |
36.0 | 0.000003 | 0.000007 | 0.000018 | 0.000040 | 0.000084 |
37.0 | 0.000002 | 0.000005 | 0.000012 | 0.000026 | 0.000057 |
38.0 | 0.000001 | 0.000003 | 0.000008 | 0.000017 | 0.000038 |
39.0 |
| 0.000002 | 0.000005 | 0.000012 | 0.000025 |
40.0 |
| 0.000001 | 0.000003 | 0.000008 | 0.000017 |
χ2 | n = 11 | n = 12 | n = 13 | n = 14 | n = 15 |
10.0 | 0.530387 | 0.615961 | 0.693935 | 0.762183 | 0.819740 |
11.0 | 0.443263 | 0.528919 | 0.610818 | 0.686036 | 0.752594 |
12.0 | 0.363643 | 0.445680 | 0.527644 | 0.606303 | 0.679029 |
13.0 | 0.293325 | 0.369041 | 0.447812 | 0.526524 | 0.602298 |
14.0 | 0.232994 | 0.300708 | 0.373844 | 0.449711 | 0.525529 |
15.0 | 0.182497 | 0.241436 | 0.307353 | 0.378155 | 0.451417 |
16.0 | 0.141131 | 0.191236 | 0.249130 | 0.313374 | 0.382052 |
17.0 | 0.107876 | 0.149597 | 0.199304 | 0.256178 | 0.318864 |
18.0 | 0.081581 | 0.115691 | 0.157519 | 0.206781 | 0.262666 |
19.0 | 0.061094 | 0.088528 | 0.123104 | 0.164949 | 0.213734 |
20.0 | 0.045341 | 0.067086 | 0.095210 | 0.130141 | 0.171933 |
21.0 | 0.033371 | 0.050380 | 0.072929 | 0.101633 | 0.136829 |
22.0 | 0.024373 | 0.037520 | 0.055362 | 0.078614 | 0.107804 |
23.0 | 0.017675 | 0.027726 | 0.041676 | 0.060270 | 0.084140 |
24.0 | 0.012733 | 0.020341 | 0.031130 | 0.045822 | 0.065093 |
25.0 | 0.009117 | 0.014823 | 0.023084 | 0.034567 | 0.049943 |
26.0 | 0.006490 | 0.010734 | 0.017001 | 0.025887 | 0.038023 |
27.0 | 0.004595 | 0.007727 | 0.012441 | 0.019254 | 0.028736 |
28.0 | 0.003237 | 0.005532 | 0.009050 | 0.014228 | 0.021569 |
29.0 | 0.002270 | 0.003940 | 0.006546 | 0.010450 | 0.016085 |
χ2 | n = 11 | n = 12 | n = 13 | n = 14 | n = 15 |
30.0 | 0.001585 | 0.002792 | 0.004710 | 0.007632 | 0.011921 |
31.0 | 0.001102 | 0.001970 | 0.003372 | 0.005544 | 0.008785 |
32.0 | 0.000763 | 0.001384 | 0.002402 | 0.004006 | 0.006438 |
33.0 | 0.000526 | 0.000968 | 0.001704 | 0.002881 | 0.004694 |
34.0 | 0.000362 | 0.000675 | 0.001204 | 0.002062 | 0.003405 |
35.0 | 0.000248 | 0.000468 | 0.000847 | 0.001470 | 0.002459 |
36.0 | 0.000169 | 0.000324 | 0.000593 | 0.001043 | 0.001768 |
37.0 | 0.000115 | 0.000223 | 0.000414 | 0.000738 | 0.001266 |
38.0 | 0.000078 | 0.000154 | 0.000288 | 0.000520 | 0.000902 |
39.0 | 0.000053 | 0.000105 | 0.000200 | 0.000365 | 0.000641 |
40.0 | 0.000036 | 0.000072 | 0.000138 | 0.000255 | 0.000453 |
41.0 | 0.000024 | 0.000049 | 0.000095 | 0.000178 | 0.000320 |
42.0 | 0.000016 | 0.000033 | 0.000065 | 0.000124 | 0.000225 |
43.0 | 0.000011 | 0.000023 | 0.000045 | 0.000086 | 0.000157 |
44.0 | 0.000007 | 0.000015 | 0.000031 | 0.000059 | 0.000110 |
45.0 | 0.000005 | 0.000010 | 0.000021 | 0.000041 | 0.000077 |
46.0 | 0.000003 | 0.000007 | 0.000014 | 0.000028 | 0.000053 |
47.0 | 0.000002 | 0.000005 | 0.000010 | 0.000019 | 0.000037 |
48.0 | 0.000001 | 0.000003 | 0.000007 | 0.000013 | 0.000025 |
χ2 | n = 16 | n = 17 | n = 18 | n = 19 | n =20 |
15.0 | 0.524639 | 0.595482 | 0.661967 | 0.722597 | 0.776408 |
16.0 | 0.452961 | 0.523835 | 0.592547 | 0.657278 | 0.716624 |
17.0 | 0.385597 | 0.454366 | 0.523105 | 0.589868 | 0.652974 |
18.0 | 0.323897 | 0.388841 | 0.455653 | 0.522438 | 0.587408 |
19.0 | 0.268663 | 0.328532 | 0.391823 | 0.456836 | 0.521826 |
20.0 | 0.220221 | 0.274229 | 0.332820 | 0.394578 | 0.457930 |
21.0 | 0.178511 | 0.226290 | 0.279413 | 0.336801 | 0.397133 |
22.0 | 0.143192 | 0.184719 | 0.231985 | 0.284256 | 0.340511 |
23.0 | 0.113735 | 0.149251 | 0.190590 | 0.237342 | 0.288795 |
24.0 | 0.089504 | 0.119435 | 0.155028 | 0.196152 | 0.242392 |
25.0 | 0.069825 | 0.094710 | 0.124916 | 0.160542 | 0.201431 |
26.0 | 0.054028 | 0.074461 | 0.099758 | 0.130189 | 0.165812 |
27.0 | 0.041483 | 0.058068 | 0.078995 | 0.104653 | 0.135264 |
28.0 | 0.031620 | 0.044938 | 0.062055 | 0.083429 | 0.109399 |
29.0 | 0.023936 | 0.034526 | 0.048379 | 0.065985 | 0.087759 |
30.0 | 0.018002 | 0.026345 | 0.037446 | 0.051798 | 0.069854 |
31.0 | 0.013456 | 0.019972 | 0.028787 | 0.040373 | 0.055190 |
32.0 | 0.010000 | 0.015048 | 0.021987 | 0.031255 | 0.043298 |
33.0 | 0.007390 | 0.011272 | 0.016690 | 0.024040 | 0.033741 |
34.0 | 0.005433 | 0.008396 | 0.012596 | 0.018378 | 0.026125 |
35.0 | 0.003974 | 0.006221 | 0.009452 | 0.013967 | 0.020104 |
χ2 | n = 16 | n = 17 | n = 18 | n = 19 | n =20 |
36.0 | 0.002893 | 0.004587 | 0.007056 | 0.010556 | 0.015381 |
37.0 | 0.002097 | 0.003365 | 0.005241 | 0.007935 | 0.011702 |
38.0 | 0.001513 | 0.002458 | 0.003873 | 0.005935 | 0.008856 |
39.0 | 0.001088 | 0.001787 | 0.002850 | 0.004417 | 0.006667 |
40.0 | 0.000779 | 0.001294 | 0.002087 | 0.003272 | 0.004995 |
41.0 | 0.000555 | 0.000933 | 0.001522 | 0.002413 | 0.003725 |
42.0 | 0.000395 | 0.000671 | 0.001106 | 0.001772 | 0.002766 |
43.0 | 0.000279 | 0.000480 | 0.000800 | 0.001296 | 0.002044 |
44.0 | 0.000197 | 0.000343 | 0.000577 | 0.000944 | 0.001505 |
45.0 | 0.000139 | 0.000244 | 0.000414 | 0.000685 | 0.001103 |
46.0 | 0.000097 | 0.000173 | 0.000297 | 0.000496 | 0.000806 |
47.0 | 0.000068 | 0.000122 | 0.000212 | 0.000357 | 0.000587 |
48.0 | 0.000047 | 0.000086 | 0.000151 | 0.000257 | 0.000425 |
49.0 | 0.000033 | 0.000060 | 0.000107 | 0.000184 | 0.000307 |
50.0 | 0.000023 | 0.000042 | 0.000075 | 0.000131 | 0.000221 |
51.0 | 0.000016 | 0.000030 | 0.000053 | 0.000093 | 0.000159 |
52.0 | 0.000011 | 0.000021 | 0.000037 | 0.000066 | 0.000114 |
53.0 | 0.000008 | 0.000014 | 0.000026 | 0.000047 | 0.000081 |
54.0 | 0.000005 | 0.000010 | 0.000018 | 0.000033 | 0.000058 |
55.0 | 0.000004 | 0.000007 | 0.000013 | 0.000023 | 0.000041 |