F distribution

The F distribution is the distribution of a ratio of two independent chi-square variables, each divided by its degrees of freedom; it is used for comparing variances and in ANOVA and regression.

F = \frac{\chi^2_{d_1}/d_1}{\chi^2_{d_2}/d_2} \sim F_{d_1, d_2}

Definition

$d_1$: numerator degrees of freedom (e.g. $k - 1$ groups in ANOVA)
$d_2$: denominator degrees of freedom (e.g. $N - k$ in ANOVA)

\E(F) = \frac{d_2}{d_2 - 2} \quad (d_2 > 2)

Mean (close to 1 for large d₂)

$F$: a ratio of two variance estimates

When to use

One-way ANOVA (F = MS between / MS within), the overall F test and nested-model tests in regression, the F test for two variances. ANOVA and regression F tests use the upper tail.

How to interpret

Under the null hypothesis both numerator and denominator estimate the same variance, so F is near 1. Large F means the numerator variance is bigger than chance would explain.
Order of df matters: $F_{2,12}$ and $F_{12,2}$ are different distributions.

Common mistakes

Swapping numerator and denominator df.
Using the F test for two variances as a preliminary check before a t-test: it is very sensitive to non-normality (prefer Levene/Brown–Forsythe, or simply use Welch's test).

Example

With df $(2, 12)$ the 5% critical value is $F_{0.95; 2, 12} = 3.8853$ ; an observed $F = 4.1$ has p-value $0.0440$ . For df $(3, 20)$ the critical value is $3.0984$ .