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Chi-Square Table

Chi Square Table

A Chi-Square table, also known as a chi-squared distribution table, is a valuable tool in statistical analysis. It provides critical values for the chi-square distribution, which is commonly used in hypothesis testing and goodness-of-fit tests.

Understanding the Chi-Square Distribution

The chi-square distribution is a probability distribution that arises in various statistical analyses, particularly when dealing with categorical or count data. It is used to assess the difference between observed and expected frequencies and determine if there is a significant association between categorical variables.

The distribution is characterized by a single parameter called degrees of freedom (df), which affects its shape. The degrees of freedom depend on the specific application and are determined by the number of categories or groups involved in the analysis.

Interpreting the Chi-Square Table

The Chi-Square table provides critical values at different levels of significance and degrees of freedom. These critical values are used to determine whether the observed chi-square statistic exceeds the expected range under the null hypothesis.

To interpret the table, follow these steps:

  1. Identify the degrees of freedom: Determine the degrees of freedom based on the number of categories or groups in your analysis. Locate the corresponding row in the table.

  2. Determine the desired level of significance: Choose the level of significance (commonly denoted as alpha, α) that represents your desired threshold for rejecting the null hypothesis. Typically, alpha is set at 0.05 or 0.01.

  3. Find the critical value: Locate the column in the table that corresponds to your chosen level of significance. The intersection of the row and column will provide the critical value.

  4. Compare the observed chi-square statistic: Calculate the chi-square statistic for your specific analysis based on the observed frequencies. Compare this value to the critical value obtained from the table.

    • If the observed chi-square statistic exceeds the critical value, it suggests that the association between the variables is statistically significant, and the null hypothesis can be rejected.
    • If the observed chi-square statistic falls below the critical value, there is insufficient evidence to reject the null hypothesis, indicating that the association is not statistically significant.

The probability density function (PDF) of the Chi Square distribution is:

Chi Square PDF Formula

Where:

  • f(x) is the probability density function
  • x is the random variable
  • ν is the degrees of freedom
  • Γ(z) is the Gamma function
  • e is the base of the natural logarithm

The shaded area is equal to α for 𝛘2 = 𝛘2α.

df 𝛘2 0.995 𝛘2 0.99 𝛘2 0.975 𝛘2 0.95 𝛘2 0.9 𝛘2 0.1 𝛘2 0.05 𝛘2 0.025 𝛘2 0.01 𝛘2 0.005
1 --- --- 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
2 0.01 0.02 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 4.075 4.66 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20 7.434 8.26 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
23 9.26 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
25 10.52 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
26 11.16 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
80 51.172 53.54 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169
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