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Values of the Chi-square distribution

In this lecture we discuss how to compute the values of the Chi-square distribution function, using Chi-square distribution tables or computer programs (in particular Matlab and Excel).

Let X_n be a Chi-square random variable with n degrees of freedom and denote its distribution function by [eq1]

As we have discussed in the lecture entitled Chi-square distribution, there is no simple analytical expression for [eq2] and its values are usually looked up in a table or computed with a computer algorithm. The next sections discuss these alternatives in detail.

Chi-square distribution tables

In the past, when computers were not widely available, people used to look up the values of [eq3] in Chi-square distribution tables, where some critical values of [eq4] were tabulated for several values of the degrees of freedom parameter n.

A Chi-square distribution table looks something like this:

Degrees of freedom / Probability 0.01 0.05 0.10 0.90 0.95 0.99
1 0.00 0.00 0.02 2.71 3.84 6.63
2 0.02 0.10 0.21 4.61 5.99 9.21
3 0.11 0.35 0.58 6.25 7.81 11.34
4 0.30 0.71 1.06 7.78 9.49 13.28
5 0.55 1.15 1.61 9.24 11.07 15.09
6 0.87 1.64 2.20 10.64 12.59 16.81
7 1.24 2.17 2.83 12.02 14.07 18.48
8 1.65 2.73 3.49 13.36 15.51 20.09
9 2.09 3.33 4.17 14.68 16.92 21.67
10 2.56 3.94 4.87 15.99 18.31 23.21
11 3.05 4.57 5.58 17.28 19.68 24.72
12 3.57 5.23 6.30 18.55 21.03 26.22
13 4.11 5.89 7.04 19.81 22.36 27.69
14 4.66 6.57 7.79 21.06 23.68 29.14
15 5.23 7.26 8.55 22.31 25.00 30.58
16 5.81 7.96 9.31 23.54 26.30 32.00
17 6.41 8.67 10.09 24.77 27.59 33.41
18 7.01 9.39 10.86 25.99 28.87 34.81
19 7.63 10.12 11.65 27.20 30.14 36.19
20 8.26 10.85 12.44 28.41 31.41 37.57
21 8.90 11.59 13.24 29.62 32.67 38.93
22 9.54 12.34 14.04 30.81 33.92 40.29
23 10.20 13.09 14.85 32.01 35.17 41.64
24 10.86 13.85 15.66 33.20 36.42 42.98
25 11.52 14.61 16.47 34.38 37.65 44.31
26 12.20 15.38 17.29 35.56 38.89 45.64
27 12.88 16.15 18.11 36.74 40.11 46.96
28 13.56 16.93 18.94 37.92 41.34 48.28
29 14.26 17.71 19.77 39.09 42.56 49.59
30 14.95 18.49 20.60 40.26 43.77 50.89
31 15.66 19.28 21.43 41.42 44.99 52.19
32 16.36 20.07 22.27 42.58 46.19 53.49
33 17.07 20.87 23.11 43.75 47.40 54.78
34 17.79 21.66 23.95 44.90 48.60 56.06
35 18.51 22.47 24.80 46.06 49.80 57.34
36 19.23 23.27 25.64 47.21 51.00 58.62
37 19.96 24.07 26.49 48.36 52.19 59.89
38 20.69 24.88 27.34 49.51 53.38 61.16
39 21.43 25.70 28.20 50.66 54.57 62.43
40 22.16 26.51 29.05 51.81 55.76 63.69
41 22.91 27.33 29.91 52.95 56.94 64.95
42 23.65 28.14 30.77 54.09 58.12 66.21
43 24.40 28.96 31.63 55.23 59.30 67.46
44 25.15 29.79 32.49 56.37 60.48 68.71
45 25.90 30.61 33.35 57.51 61.66 69.96
46 26.66 31.44 34.22 58.64 62.83 71.20
47 27.42 32.27 35.08 59.77 64.00 72.44
48 28.18 33.10 35.95 60.91 65.17 73.68
49 28.94 33.93 36.82 62.04 66.34 74.92
50 29.71 34.76 37.69 63.17 67.50 76.15

For example, at the intersection of the row corresponding to 5 degrees of freedom and the column corresponding to a value of the distribution function of 0.95, we read the value 11.07. This means that:[eq5]In other words, the realization of a Chi-square random variable with 5 degrees of freedom will be less than 11.07 with probability 0.95.

If we are searching for a value of x that does not correspond to one of the critical values in the first row, then a Chi-square distribution table is not of any help. In this case, we need to use a computer algorithm (see below).

Chi-square distribution values in Excel

To compute the values of the Chi-square distribution function [eq6], we can use the built-in Excel function CHISQ.DIST(). For example, if we need to compute [eq7] and the value 1 is stored in cell A1, we can type in another cell:

=CHISQ.DIST(A1,5)

Chi-square distribution values in Matlab

To compute the values of the Chi-square distribution function [eq8], we can use the Matlab function chi2cdf(), which takes the value x as its first argument and the number of degrees of freedom n as its second argument. For example, if we need to compute [eq9], we can input the following command:

chi2cdf(1,5)

Solved exercises

At the end of the lecture entitled Chi-square distribution, you can find some solved exercises that also require the computation of Chi-square distribution values.

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