Critical Values of Chi-square
Upper critical values of chi-square distribution with x degrees of freedom
Probability of exceeding the critical value:
df
90%
95%
97.5%
99%
99.9%
⇐ Confidence Level
⇓
0.10
0.05
0.025
0.01
0.001
⇐ Significance Level
1
2.706
3.841
5.024
6.635
10.828
2
4.605
5.991
7.378
9.210
13.816
3
6.251
7.815
9.348
11.345
16.266
4
7.779
9.488
11.143
13.277
18.467
5
9.236
11.070
12.833
15.086
20.515
6
10.645
12.592
14.449
16.812
22.458
7
12.017
14.067
16.013
18.475
24.322
8
13.362
15.507
17.535
20.090
26.125
9
14.684
16.919
19.023
21.666
27.877
10
15.987
18.307
20.483
23.209
29.588
11
17.275
19.675
21.920
24.725
31.264
12
18.549
21.026
23.337
26.217
32.910
13
19.812
22.362
24.736
27.688
34.528
14
21.064
23.685
26.119
29.141
36.123
15
22.307
24.996
27.488
30.578
37.697
16
23.542
26.296
28.845
32.000
39.252
17
24.769
27.587
30.191
33.409
40.790
18
25.989
28.869
31.526
34.805
42.312
19
27.204
30.144
32.852
36.191
43.820
20
28.412
31.410
34.170
37.566
45.315
21
29.615
32.671
35.479
38.932
46.797
22
30.813
33.924
36.781
40.289
48.268
23
32.007
35.172
38.076
41.638
49.728
24
33.196
36.415
39.364
42.980
51.179
25
34.382
37.652
40.646
44.314
52.620
26
35.563
38.885
41.923
45.642
54.052
27
36.741
40.113
43.195
46.963
55.476
28
37.916
41.337
44.461
48.278
56.892
29
39.087
42.557
45.722
49.588
58.301
30
40.256
43.773
46.979
50.892
59.703
31
41.422
44.985
48.232
52.191
61.098
32
42.585
46.194
49.480
53.486
62.487
33
43.745
47.400
50.725
54.776
63.870
34
44.903
48.602
51.966
56.061
65.247
35
46.059
49.802
53.203
57.342
66.619
36
47.212
50.998
54.437
58.619
67.985
37
48.363
52.192
55.668
59.893
69.347
38
49.513
53.384
56.896
61.162
70.703
39
50.660
54.572
58.120
62.428
72.055
40
51.805
55.758
59.342
63.691
73.402
41
52.949
56.942
60.561
64.950
74.745
42
54.090
58.124
61.777
66.206
76.084
43
55.230
59.304
62.990
67.459
77.419
44
56.369
60.481
64.201
68.710
78.750
45
57.505
61.656
65.410
69.957
80.077
46
58.641
62.830
66.617
71.201
81.400
47
59.774
64.001
67.821
72.443
82.720
48
60.907
65.171
69.023
73.683
84.037
49
62.038
66.339
70.222
74.919
85.351
50
63.167
67.505
71.420
76.154
86.661
51
64.295
68.669
72.616
77.386
87.968
52
65.422
69.832
73.810
78.616
89.272
53
66.548
70.993
75.002
79.843
90.573
54
67.673
72.153
76.192
81.069
91.872
55
68.796
73.311
77.380
82.292
93.168
56
69.919
74.468
78.567
83.513
94.461
57
71.040
75.624
79.752
84.733
95.751
58
72.160
76.778
80.936
85.950
97.039
59
73.279
77.931
82.117
87.166
98.324
60
74.397
79.082
83.298
88.379
99.607
61
75.514
80.232
84.476
89.591
100.888
62
76.630
81.381
85.654
90.802
102.166
63
77.745
82.529
86.830
92.010
103.442
64
78.860
83.675
88.004
93.217
104.716
65
79.973
84.821
89.177
94.422
105.988
66
81.085
85.965
90.349
95.626
107.258
67
82.197
87.108
91.519
96.828
108.526
68
83.308
88.250
92.689
98.028
109.791
69
84.418
89.391
93.856
99.228
111.055
70
85.527
90.531
95.023
100.425
112.317
71
86.635
91.670
96.189
101.621
113.577
72
87.743
92.808
97.353
102.816
114.835
73
88.850
93.945
98.516
104.010
116.092
74
89.956
95.081
99.678
105.202
117.346
75
91.061
96.217
100.839
106.393
118.599
76
92.166
97.351
101.999
107.583
119.850
77
93.270
98.484
103.158
108.771
121.100
78
94.374
99.617
104.316
109.958
122.348
79
95.476
100.749
105.473
111.144
123.594
80
96.578
101.879
106.629
112.329
124.839
81
97.680
103.010
107.783
113.512
126.083
82
98.780
104.139
108.937
114.695
127.324
83
99.880
105.267
110.090
115.876
128.565
84
100.980
106.395
111.242
117.057
129.804
85
102.079
107.522
112.393
118.236
131.041
86
103.177
108.648
113.544
119.414
132.277
87
104.275
109.773
114.693
120.591
133.512
88
105.372
110.898
115.841
121.767
134.746
89
106.469
112.022
116.989
122.942
135.978
90
107.565
113.145
118.136
124.116
137.208
91
108.661
114.268
119.282
125.289
138.438
92
109.756
115.390
120.427
126.462
139.666
93
110.850
116.511
121.571
127.633
140.893
94
111.944
117.632
122.715
128.803
142.119
95
113.038
118.752
123.858
129.973
143.344
96
114.131
119.871
125.000
131.141
144.567
97
115.223
120.990
126.141
132.309
145.789
98
116.315
122.108
127.282
133.476
147.010
99
117.407
123.225
128.422
134.642
148.230
100
118.498
124.342
129.561
135.807
149.449
100
118.498
124.342
129.561
135.807
149.449
Lower critical values of chi-square distribution with X degrees freedom
Probability of exceeding the critical value:
| 0.90 | 0.95 | 0.975 | 0.99 | 0.999 | |
| 1. . | 016 . | 004 . | 001 . | 000 . | 000 |
| 2. . | 211 . | 103 . | 051 . | 020 . | 002 |
| 3. . | 584 . | 352 . | 216 . | 115 . | 024 |
| 4. 1 | .064 . | 711 . | 484 . | 297 . | 091 |
| 5. 1 | .610 1 | .145 . | 831 . | 554 . | 210 |
| 6. 2 | .204 1 | .635 1 | .237 . | 872 . | 381 |
| 7. 2 | .833 2 | .167 1 | .690 1 | .239 . | 598 |
| 8. 3 | .490 2 | .733 2 | .180 1 | .646 . | 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. 1 | 0.085 8 | .672 7 | .564 6 | .408 4 | .416 |
| 18. 1 | 0.865 9 | .390 8 | .231 7 | .015 4 | .905 |
| 19. 1 | 1.651 1 | 0.117 8 | .907 7 | .633 5 | .407 |
| 20. 1 | 2.443 1 | 0.851 9 | .591 8 | .260 5 | .921 |
| 21. 1 | 3.240 1 | 1.591 1 | 0.283 8 | .897 6 | .447 |
| 22. 1 | 4.041 1 | 2.338 1 | 0.982 9 | .542 6 | .983 |
| 23. 1 | 4.848 1 | 3.091 1 | 1.689 1 | 0.196 7 | .529 |
| 24. 1 | 5.659 1 | 3.848 1 | 2.401 1 | 0.856 8 | .085 |
| 25. 1 | 6.473 1 | 4.611 1 | 3.120 1 | 1.524 8 | .649 |
| 26. 1 | 7.292 1 | 5.379 1 | 3.844 1 | 2.198 9 | .222 |
| 27. 1 | 8.114 1 | 6.151 1 | 4.573 1 | 2.879 9 | .803 |
| 28. 1 | 8.939 1 | 6.928 1 | 5.308 1 | 3.565 1 | 0.391 |
| 29. 1 | 9.768 1 | 7.708 1 | 6.047 1 | 4.256 1 | 0.986 |
| 30. 2 | 0.599 1 | 8.493 1 | 6.791 1 | 4.953 1 | 1.588 |
| 31. 2 | 1.434 1 | 9.281 1 | 7.539 1 | 5.655 1 | 2.196 |
| 32. 2 | 2.271 2 | 0.072 1 | 8.291 1 | 6.362 1 | 2.811 |
| 33. 2 | 3.110 2 | 0.867 1 | 9.047 1 | 7.074 1 | 3.431 |
| 34. 2 | 3.952 2 | 1.664 1 | 9.806 1 | 7.789 1 | 4.057 |
| 35. 2 | 4.797 2 | 2.465 2 | 0.569 1 | 8.509 1 | 4.688 |
| 36. 2 | 5.643 2 | 3.269 2 | 1.336 1 | 9.233 1 | 5.324 |
| 37. 2 | 6.492 2 | 4.075 2 | 2.106 1 | 9.960 1 | 5.965 |
| 38. 2 | 7.343 2 | 4.884 2 | 2.878 2 | 0.691 1 | 6.611 |
| 39. 2 | 8.196 2 | 5.695 2 | 3.654 2 | 1.426 1 | 7.262 |
| 40. 2 | 9.051 2 | 6.509 2 | 4.433 2 | 2.164 1 | 7.916 |
| 41. 2 | 9.907 2 | 7.326 2 | 5.215 2 | 2.906 1 | 8.575 |
| 42. 3 | 0.765 2 | 8.144 2 | 5.999 2 | 3.650 1 | 9.239 |
| 43. 3 | 1.625 2 | 8.965 2 | 6.785 2 | 4.398 1 | 9.906 |
| 44. 3 | 2.487 2 | 9.787 2 | 7.575 2 | 5.148 2 | 0.576 |
| 45. 3 | 3.350 3 | 0.612 2 | 8.366 2 | 5.901 2 | 1.251 |
| 46. 3 | 4.215 3 | 1.439 2 | 9.160 2 | 6.657 2 | 1.929 |
| 47. 3 | 5.081 3 | 2.268 2 | 9.956 2 | 7.416 2 | 2.610 |
| 48. 3 | 5.949 3 | 3.098 3 | 0.755 2 | 8.177 2 | 3.295 |
| 49. 3 | 6.818 3 | 3.930 3 | 1.555 2 | 8.941 2 | 3.983 |
| 50. 3 | 7.689 3 | 4.764 3 | 2.357 2 | 9.707 2 | 4.674 |
| 51. 3 | 8.560 3 | 5.600 3 | 3.162 3 | 0.475 2 | 5.368 |
| 52. 3 | 9.433 3 | 6.437 3 | 3.968 3 | 1.246 2 | 6.065 |
| 53. 4 | 0.308 3 | 7.276 3 | 4.776 3 | 2.018 2 | 6.765 |
| 54. 4 | 1.183 3 | 8.116 3 | 5.586 3 | 2.793 2 | 7.468 |
| 55. 4 | 2.060 3 | 8.958 3 | 6.398 3 | 3.570 2 | 8.173 |
| 56. 4 | 2.937 3 | 9.801 3 | 7.212 3 | 4.350 2 | 8.881 |
| 57. 4 | 3.816 4 | 0.646 3 | 8.027 3 | 5.131 2 | 9.592 |
| 58. 4 | 4.696 4 | 1.492 3 | 8.844 3 | 5.913 3 | 0.305 |
| 59. 4 | 5.577 4 | 2.339 3 | 9.662 3 | 6.698 3 | 1.020 |
| 60. 4 | 6.459 4 | 3.188 4 | 0.482 3 | 7.485 3 | 1.738 |
| 61. 4 | 7.342 4 | 4.038 4 | 1.303 3 | 8.273 3 | 2.459 |
| 62. 4 | 8.226 4 | 4.889 4 | 2.126 3 | 9.063 3 | 3.181 |
| 63. 4 | 9.111 4 | 5.741 4 | 2.950 3 | 9.855 3 | 3.906 |
| 64. 4 | 9.996 4 | 6.595 4 | 3.776 4 | 0.649 3 | 4.633 |
| 65. 5 | 0.883 4 | 7.450 4 | 4.603 4 | 1.444 3 | 5.362 |
| 66. 5 | 1.770 4 | 8.305 4 | 5.431 4 | 2.240 3 | 6.093 |
| 67. 5 | 2.659 4 | 9.162 4 | 6.261 4 | 3.038 3 | 6.826 |
| 68. 5 | 3.548 5 | 0.020 4 | 7.092 4 | 3.838 3 | 7.561 |
| 69. 5 | 4.438 5 | 0.879 4 | 7.924 4 | 4.639 3 | 8.298 |
| 70. 5 | 5.329 5 | 1.739 4 | 8.758 4 | 5.442 3 | 9.036 |
| 71. 5 | 6.221 5 | 2.600 4 | 9.592 4 | 6.246 3 | 9.777 |
| 72. 5 | 7.113 5 | 3.462 5 | 0.428 4 | 7.051 4 | 0.519 |
| 73. 5 | 8.006 5 | 4.325 5 | 1.265 4 | 7.858 4 | 1.264 |
| 74. 5 | 8.900 5 | 5.189 5 | 2.103 4 | 8.666 4 | 2.010 |
| 75. 5 | 9.795 5 | 6.054 5 | 2.942 4 | 9.475 4 | 2.757 |
| 76. 6 | 0.690 5 | 6.920 5 | 3.782 5 | 0.286 4 | 3.507 |
| 77. 6 | 1.586 5 | 7.786 5 | 4.623 5 | 1.097 4 | 4.258 |
| 78. 6 | 2.483 5 | 8.654 5 | 5.466 5 | 1.910 4 | 5.010 |
| 79. 6 | 3.380 5 | 9.522 5 | 6.309 5 | 2.725 4 | 5.764 |
| 80. 6 | 4.278 6 | 0.391 5 | 7.153 5 | 3.540 4 | 6.520 |
| 81. 6 | 5.176 6 | 1.261 5 | 7.998 5 | 4.357 4 | 7.277 |
| 82. 6 | 6.076 6 | 2.132 5 | 8.845 5 | 5.174 4 | 8.036 |
| 83. 6 | 6.976 6 | 3.004 5 | 9.692 5 | 5.993 4 | 8.796 |
| 84. 6 | 7.876 6 | 3.876 6 | 0.540 5 | 6.813 4 | 9.557 |
| 85. 6 | 8.777 6 | 4.749 6 | 1.389 5 | 7.634 5 | 0.320 |
| 86. 6 | 9.679 6 | 5.623 6 | 2.239 5 | 8.456 5 | 1.085 |
| 87. 7 | 0.581 6 | 6.498 6 | 3.089 5 | 9.279 5 | 1.850 |
| 88. 7 | 1.484 6 | 7.373 6 | 3.941 6 | 0.103 5 | 2.617 |
| 89. 7 | 2.387 6 | 8.249 6 | 4.793 6 | 0.928 5 | 3.386 |
| 90. 7 | 3.291 6 | 9.126 6 | 5.647 6 | 1.754 5 | 4.155 |
| 91. 7 | 4.196 7 | 0.003 6 | 6.501 6 | 2.581 5 | 4.926 |
| 92. 7 | 5.100 7 | 0.882 6 | 7.356 6 | 3.409 5 | 5.698 |
| 93. 7 | 6.006 7 | 1.760 6 | 8.211 6 | 4.238 5 | 6.472 |
| 94. 7 | 6.912 7 | 2.640 6 | 9.068 6 | 5.068 5 | 7.246 |
| 95. 7 | 7.818 7 | 3.520 6 | 9.925 6 | 5.898 5 | 8.022 |
| 96. 7 | 8.725 7 | 4.401 7 | 0.783 6 | 6.730 5 | 8.799 |
| 97. 7 | 9.633 7 | 5.282 7 | 1.642 6 | 7.562 5 | 9.577 |
| 98. 8 | 0.541 7 | 6.164 7 | 2.501 6 | 8.396 6 | 0.356 |
| 99. 8 | 1.449 7 | 7.046 7 | 3.361 6 | 9.230 6 | 1.137 |
| 100. 8 | 2.358 7 | 7.929 7 | 4.222 7 | 0.065 6 | 1.918 |