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