In addition to inferential methods for hypothesis tests for population parameters, such as the mean, \(\mu\), and the standard deviation, \(\sigma\), there are statistical methods to make inferences about the distribution of a variable. These inferential procedures rely on the chi-square (\(\chi^2\)) distribution, and thus are called **\(\chi^2\)-tests**.

In the following section we discuss the **chi-square goodness-of-fit test**, a hypothesis test that is applied to make inferences about the distribution of a variable and the **chi-square independence test**, a hypothesis test that is applied to decide whether an association exists between two variables of a population.

#### \(\chi^2\)-Distribution

**Basic Properties of \(\chi^2\)-Curves (Weiss 2010)**

- The total area under a \(\chi^2\)-curve equals 1.
- A \(\chi^2\)-curve starts at 0 on the horizontal axis and extends indefinitely to the right, approaching, but never touching, the horizontal axis as it does so.
- A \(\chi^2\)-curve is right skewed.

- As the number of degrees of freedom becomes larger, \(\chi^2\)- curves look increasingly like normal curves.