# 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.

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.

- 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.
- 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.
- 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.