F distribution

Take two chi-squared variables, each divided by its own degrees of freedom. Take their ratio. That ratio is F.

F = \frac{U _{1} / d _{1}}{U _{2} / d _{2}}, U_{i} \sim χ_{d_{i}}^{2}

Mode: 0.50 Mean: 1.25 F = 1 (null reference)

d₁ (numerator df) 5 d₂ (denominator df) 10 n (samples) 1500

PDF

f (x; d_{1}, d_{2}) = \frac{\frac{( d _{1} x ) ^{d_{1}} d _{2}^{d_{2}}}{( d _{1} x + d _{2} ) ^{d_{1} + d_{2}}}}{x B ( d _{1} /2 , d _{2} /2 )}

. Equivalently the ratio of two independent chi-squareds, each divided by its degrees of freedom:

F = \frac{U _{1} / d _{1}}{U _{2} / d _{2}}, U_{i} \sim χ_{d_{i}}^{2}

What to notice

Always non-negative, right-skewed. Ratios of positive quantities can only be positive, and the denominator’s lower tail drags the upper tail of the ratio.
d₂ controls the right tail. Small denominator degrees of freedom give heavy tails — a small denominator sometimes blows up, inflating F. Once $d_{2}$ exceeds ~10 the tail behaves.
Under the null, F hovers around 1. The dashed line at F = 1 marks where the two variances are equal. ANOVA rejects when the observed F lands deep in the right tail.
Reciprocal symmetry. Swapping $d_{1}$ and $d_{2}$ is equivalent to inverting the statistic:

$F_{d_{1}, d_{2}} (x) = 1/ F_{d_{2}, d_{1}} (1/ x)$

The F distribution is the reference distribution for every test that compares two variances:

ANOVA. The ratio “between-group mean square” over “within-group mean square” is F under the null of equal group means.
Nested regression models. The F statistic compares the sums of squared residuals of a restricted vs. unrestricted model.
Variance ratio tests. Directly testing $σ_{1}^{2} = σ_{2}^{2}$ .

Visually: whenever a null distribution starts at zero, peaks near 1, and has a long right tail, it’s probably an F.

E [F] = \frac{d _{2}}{d _{2} - 2} (d_{2} > 2)