The chi-square ($$\chi^2$$) distribution is one of the most important continuous probability distributions with many uses in statistical theory and inference (Lovric 2011).

Let $$n > 0$$ be a positive integer. For a random variable that has a $$\chi^2$$-distribution with $$n$$ degrees of freedom $$(df)$$ the probability density function is

$f(x) = \begin{cases} 0 & \text{if x \le 0} \\ \frac{x^{(n/2-1)}e^{-x/2}}{2^{n/2}\Gamma \left(\frac{k}{2}\right)} & \text{if x > 0} \end{cases}$

where $$\Gamma$$ denotes the Gamma function.

The $$\chi^2$$-distribution (with $$n$$ degrees of freedom) is equal to the $$\Gamma$$-distribution with the parameters ($$n/2,2$$), that is, with a mean and variance of $$n$$ and $$2n$$, respectively.

##### 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.

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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.