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. Then we will look at the **chi-square independence test**, a hypothesis test that is applied to decide whether an association exists between two variables of a population.
The biggest **advantage of chi-square statistics** is that we do not need any parameters to compare frequency distributions. Thus, in case of doubt about the underlying metric of our variables, chi-square statistics provide a **non-metric** alternative to moment-based tests.

**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.

**Citation**

The E-Learning project SOGA-Py was developed at the Department of Earth Sciences by Annette Rudolph, Joachim Krois and Kai Hartmann. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: *Rudolph, A., Krois, J., Hartmann, K. (2023): Statistics and Geodata Analysis
using Python (SOGA-Py). Department of Earth Sciences, Freie Universitaet Berlin.*