The Chi-Square Distribution

What is a Chi-square distribution?

The Chi-square distribution is a theoretical distribution of values for a population.

How is the Chi-square distribution used?

It is used for statistical tests where the test statistic follows a Chi-squared distribution. Two common tests that rely on the Chi-square distribution are the Chi-square goodness of fit test and the Chi-square test of independence.


Introducing the Chi-square distribution

The Chi-square distribution is a family of distributions. Each distribution is defined by the degrees of freedom. (Degrees of freedom are discussed in greater detail on the pages for the goodness of fit test and the test of independence.) The figure below shows three different Chi-square distributions with different degrees of freedom.

Figure 1: Chi-Square distribution with different degrees of freedom

You can see that the blue curve with 8 degrees of freedom is somewhat similar to a normal curve (the familiar bell curve). But, it has a longer tail to the right than a normal distribution and is not symmetric. Compare the blue curve to the orange curve with 4 degrees of freedom. The orange curve is very different from a normal curve. The purple curve has 3 degrees of freedom and looks even less like a normal curve than the other two curves.

The higher the degrees of freedom for a Chi-square distribution, the more it looks like a normal distribution.

Using published Chi-square distribution tables

Most people use software to do Chi-square tests. But many statistics books show Chi-square tables, so understanding how to use a table might be helpful. The steps below describe how to use a typical Chi-square table.

  1. Identify your alpha level. Each column in the table lists values for different alpha levels. If you set α = 0.05 for your test, then find the column for α = 0.05.
  2.  Identify the degrees of freedom for the test you are doing and for your data. The rows in a Chi-square table correspond to different degrees of freedom. Most tables go up to 30 degrees of freedom. 
  3. Find the cell in the table corresponding to your alpha level and degrees of freedom. This is the Chi-square distribution value. Compare your test statistic to the distribution value and make the appropriate conclusion.