# Dispersion Tests

## Chi-Square Test for Variance#

##### Description#

Determines if the variance of a population is equal to a specified value.

TailAlt. Hypothesis
$left\hspace{1cm}$$H_{1}: \sigma^{2} \lt \sigma_{0}^{2} right\hspace{1cm}$$ H_{1}: \sigma^{2} \gt \sigma_{0}^{2}$
$both\hspace{1cm}$$H_{1}: \sigma^{2} \neq \sigma_{0}^{2} ##### Returns# • p-value • decision • Chi-Sq statistic ## F-Test for Two Variances# ##### Description# Determines if the variances of two populations are equal. TailAlt. Hypothesis left\hspace{1cm}$$ H_{1}: \sigma_{1}^{2} \lt \sigma_{2}^{2}$
$right\hspace{1cm}$$H_{1}: \sigma_{1}^{2} \gt \sigma_{2}^{2} both\hspace{1cm}$$ H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2}$
• p-value
• decision
• F statistic

## Levene Test for Equal Variances#

##### Description#

Null hypothesis is that the population variances are equal.

The Levene test statistic is defined as: $$W = \frac{(N-k)} {(k-1)} \frac{\sum_{i=1}^{k}N_{i}(\bar{Z}_{i.}-\bar{Z}_{..})^{2} } {\sum_{i=1}^{k}\sum_{j=1}^{N_i}(Z_{ij}-\bar{Z}_{i.})^{2} }$$

where $Z_{ij} = |Y_{ij} - \bar{Y}_{i.}|$ and $\bar{Y}_{i.}$ is the mean of the ith subgroup.

##### Returns#
• p-value
• decision
• F statistic
• degrees of freedom