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.*