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 for comparison of 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.

#### $$\chi^2$$-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.

Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. 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: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.