Values of the Chi-square distribution

by Marco Taboga, PhD

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 be a Chi-square random variable with degrees of freedom and denote its distribution function by

As we have discussed in the lecture entitled Chi-square distribution, there is no simple analytical expression for 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 in Chi-square distribution tables, where some critical values of were tabulated for several values of the degrees of freedom parameter .

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 thatIn 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 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 , we can use the built-in Excel function CHISQ.DIST(). For example, if we need to compute and the value 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 , we can use the Matlab function chi2cdf(), which takes the value as its first argument and the number of degrees of freedom as its second argument. For example, if we need to compute , 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.

How to cite

Please cite as:

Taboga, Marco (2021). "Values of the Chi-square distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/chi-square-distribution-values.