Appendix H — Lower-tail critical values of the \(\chi^2\)-distribution with \(\nu\) degrees of freedom

The following table displays the lower-tail critical values \(c\) for the \(\chi^2\)-distribution with \(\nu\) degrees of freedom and a significance level \(\alpha\): P\((0 \leq X \leq c) = \alpha\).

Probability less than the critical value (source: NIST/SEMATECH (n.d.), URL: web page) Values rounded to 3 decimal places; 0.000 denotes a value less than 0.0005.
\(\nu\) 0.10 0.05 0.025 0.01 0.001
1 0.016 0.004 0.001 0.000 0.000
2 0.211 0.103 0.051 0.020 0.002
3 0.584 0.352 0.216 0.115 0.024
4 1.064 0.711 0.484 0.297 0.091
5 1.610 1.145 0.831 0.554 0.210
6 2.204 1.635 1.237 0.872 0.381
7 2.833 2.167 1.690 1.239 0.598
8 3.490 2.733 2.180 1.646 0.857
9 4.168 3.325 2.700 2.088 1.152
10 4.865 3.940 3.247 2.558 1.479
11 5.578 4.575 3.816 3.053 1.834
12 6.304 5.226 4.404 3.571 2.214
13 7.042 5.892 5.009 4.107 2.617
14 7.790 6.571 5.629 4.660 3.041
15 8.547 7.261 6.262 5.229 3.483
16 9.312 7.962 6.908 5.812 3.942
17 10.085 8.672 7.564 6.408 4.416
18 10.865 9.390 8.231 7.015 4.905
19 11.651 10.117 8.907 7.633 5.407
20 12.443 10.851 9.591 8.260 5.921
21 13.240 11.591 10.283 8.897 6.447
22 14.041 12.338 10.982 9.542 6.983
23 14.848 13.091 11.689 10.196 7.529
24 15.659 13.848 12.401 10.856 8.085
25 16.473 14.611 13.120 11.524 8.649
26 17.292 15.379 13.844 12.198 9.222
27 18.114 16.151 14.573 12.879 9.803
28 18.939 16.928 15.308 13.565 10.391
29 19.768 17.708 16.047 14.256 10.986
30 20.599 18.493 16.791 14.953 11.588
31 21.434 19.281 17.539 15.655 12.196
32 22.271 20.072 18.291 16.362 12.811
33 23.110 20.867 19.047 17.074 13.431
34 23.952 21.664 19.806 17.789 14.057
35 24.797 22.465 20.569 18.509 14.688
36 25.643 23.269 21.336 19.233 15.324
37 26.492 24.075 22.106 19.960 15.965
38 27.343 24.884 22.878 20.691 16.611
39 28.196 25.695 23.654 21.426 17.262
40 29.051 26.509 24.433 22.164 17.916
41 29.907 27.326 25.215 22.906 18.575
42 30.765 28.144 25.999 23.650 19.239
43 31.625 28.965 26.785 24.398 19.906
44 32.487 29.787 27.575 25.148 20.576
45 33.350 30.612 28.366 25.901 21.251
46 34.215 31.439 29.160 26.657 21.929
47 35.081 32.268 29.956 27.416 22.610
48 35.949 33.098 30.755 28.177 23.295
49 36.818 33.930 31.555 28.941 23.983
50 37.689 34.764 32.357 29.707 24.674
51 38.560 35.600 33.162 30.475 25.368
52 39.433 36.437 33.968 31.246 26.065
53 40.308 37.276 34.776 32.018 26.765
54 41.183 38.116 35.586 32.793 27.468
55 42.060 38.958 36.398 33.570 28.173
56 42.937 39.801 37.212 34.350 28.881
57 43.816 40.646 38.027 35.131 29.592
58 44.696 41.492 38.844 35.913 30.305
59 45.577 42.339 39.662 36.698 31.020
60 46.459 43.188 40.482 37.485 31.738
61 47.342 44.038 41.303 38.273 32.459
62 48.226 44.889 42.126 39.063 33.181
63 49.111 45.741 42.950 39.855 33.906
64 49.996 46.595 43.776 40.649 34.633
65 50.883 47.450 44.603 41.444 35.362
66 51.770 48.305 45.431 42.240 36.093
67 52.659 49.162 46.261 43.038 36.826
68 53.548 50.020 47.092 43.838 37.561
69 54.438 50.879 47.924 44.639 38.298
70 55.329 51.739 48.758 45.442 39.036
71 56.221 52.600 49.592 46.246 39.777
72 57.113 53.462 50.428 47.051 40.519
73 58.006 54.325 51.265 47.858 41.264
74 58.900 55.189 52.103 48.666 42.010
75 59.795 56.054 52.942 49.475 42.757
76 60.690 56.920 53.782 50.286 43.507
77 61.586 57.786 54.623 51.097 44.258
78 62.483 58.654 55.466 51.910 45.010
79 63.380 59.522 56.309 52.725 45.764
80 64.278 60.391 57.153 53.540 46.520
81 65.176 61.261 57.998 54.357 47.277
82 66.076 62.132 58.845 55.174 48.036
83 66.976 63.004 59.692 55.993 48.796
84 67.876 63.876 60.540 56.813 49.557
85 68.777 64.749 61.389 57.634 50.320
86 69.679 65.623 62.239 58.456 51.085
87 70.581 66.498 63.089 59.279 51.850
88 71.484 67.373 63.941 60.103 52.617
89 72.387 68.249 64.793 60.928 53.386
90 73.291 69.126 65.647 61.754 54.155
91 74.196 70.003 66.501 62.581 54.926
92 75.100 70.882 67.356 63.409 55.698
93 76.006 71.760 68.211 64.238 56.472
94 76.912 72.640 69.068 65.068 57.246
95 77.818 73.520 69.925 65.898 58.022
96 78.725 74.401 70.783 66.730 58.799
97 79.633 75.282 71.642 67.562 59.577
98 80.541 76.164 72.501 68.396 60.356
99 81.449 77.046 73.361 69.230 61.137
100 82.358 77.929 74.222 70.065 61.918