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.