Chi-square distribution - Values

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 or Excel). For an introduction to the Chi-square distribution, see the lecture entitled Chi-square distribution.

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.

This lecture discusses the practical problem of computing numerically the values of when is a specific number (e.g. ).

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.

A Chi-square distribution table is a table where some critical values of are tabulated for several values of the degrees of freedom parameter . A Chi-square distribution table looks something like this:

For example, at the intersection of the row corresponding to degrees of freedom and the column corresponding to a value of the distribution function of , we read the value . This means that:In other words, the probability that a Chi-square random variable with degrees of freedom is less than is .

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.