This table contains the basic estimations of the chi-square circulation. On account of the absence of balance of the chi-square appropriation, separate tables are accommodated the upper and lower tails of the dispersion.

A test measurement with ν degrees of opportunity is processed from the information. For upper-tail uneven tests, the test measurement is contrasted and an incentive from the table of upper-tail basic qualities. For two-sided tests, the test measurement is contrasted and values from both the table for the upper-tail basic qualities and the table for the lower-tail basic qualities.

The significance level, α, is demonstrated with the graph below which shows a chi-square distribution with 3 degrees of freedom for a two-sided test at significance level α = 0.05. If the test statistic is greater than the upper-tail critical value or less than the lower-tail critical value, we reject the null hypothesis. Specific instructions are given below

Given a specified value of α:

For a two-sided test, discover the segment comparing to 1-α/2 in the table for upper-tail basic qualities and reject the invalid theory if the test measurement is more prominent than the postponed worth. Essentially, discover the section comparing to α/2 in the table for lower-tail basic qualities and reject the invalid theory if the test measurement is not exactly the postponed worth.

For an upper-tail uneven test, discover the section comparing to 1-α in the table containing upper-tail basic and reject the invalid speculation if the test measurement is more prominent than the postponed worth.

For a lower-tail uneven test, discover the section comparing to α in the lower-tail basic qualities table and reject the invalid theory if the registered test measurement is not exactly the postponed worth.

Upper-tail critical values of chi-square distribution with ν degrees of freedom

Probability less than the critical value

ν           0.90 0.95     0.975 0.99 0.999

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-tail critical values of chi-square distribution with ν degrees of freedom

Probability less than the critical value

ν           0.10 0.05     0.025 0.01   0.001

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.        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