F Distribution

Suppose 2 independent populations (of sizes $N_1$ and $N_2$) are given. And samples of sizes $n_1$ and $n_2$ are taken from each population.

F statistic can be computed using:

$$F = \cfrac{ \chi_1^2 / \nu_1 }{ \chi_2^2 / \nu_2 } = \frac{ s_1^2 }{ s_2^2 }\cdot \frac{\sigma_1 ^2}{\sigma_2 ^2}$$

Uses

• To compare variance of 2 different independent samples

Properties

Degrees of Freedom

If sample size is $n$, then the degrees of freedom for the F distribution are $n_1 - 1$ and $n_2 - 1$.

F

$$F_{\nu_1, \nu_2, \alpha} = \frac{1}{F_{\nu_2, \nu_1, 1 - \alpha}}$$