A test statistic with degrees of freedom is computed from the data. For upper one-sided tests, the test statistic is compared with a value from the table of upper critical values. For two-sided tests, the test statistic is compared with values from both the table for the upper critical value and the table for the lower critical value.

The significance level, , 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 = 0.05. If the test statistic is greater than the upper critical value or less than the lower critical value, we reject the null hypothesis. Specific instructions are given below.

Given a specified value for :

1. For a two-sided test, find the column corresponding to /2 in the table for upper critical values and reject the null hypothesis if the test statistic is greater than the tabled value. Similarly, find the column corresponding to 1 - /2 in the table for lower critical values and reject the null hypothesis if the test statistic is less than the tabled value.
2. For an upper one-sided test, find the column corresponding to in the upper critical values table and reject the null hypothesis if the test statistic is greater than the tabled value.
3. For a lower one-sided test, find the column corresponding to 1 - in the lower critical values table and reject the null hypothesis if the computed test statistic is less than the tabled value.

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

             Probability of exceeding the critical value
             0.10      0.05     0.025      0.01     0.001

  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 degrees of 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.        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