Variance Test

Used to compare the variability (spread) of two populations.

It checks whether the two population variances are equal or significantly different.

Typical use: • To test the assumption of equal variances before applying a pooled t-test. • To check if two machines, processes, or samples have similar consistency.

Hypotheses Setup

Let

$$\sigma_1^2 = \text{variance of population 1} \quad , \quad \sigma_2^2 = \text{variance of population 2}$$

Then:

$$H_0 : \sigma_1^2 = \sigma_2^2$$

Alternative hypothesis depends on the problem:

$$H_1 : \sigma_1^2 \ne \sigma_2^2 \quad \text{(two-tailed)}$$ $$H_1 : \sigma_1^2 > \sigma_2^2 \quad \text{(right-tailed)}$$ $$H_1 : \sigma_1^2 < \sigma_2^2 \quad \text{(left-tailed)}$$

Test Statistic

When $\mu$ is unknown, $\bar{x}$ is used as an estimate.

When sample size is small ($n \lt 30$):

$$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \mu)^2$$

When sample is large:

$$s^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2$$

The test statistic is:

	Small Sample	Large Sample
$\mu$ is known	$\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_n$	$\frac{ns^2}{\sigma^2} \sim \chi^2_n$
$\mu$ is unknown	$\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}$	$\frac{ns^2}{\sigma^2} \sim \chi^2_{n-1}$