The F Distribution

The F distribution describes the ratio of two independent chi-squared variables, each divided by its own degrees of freedom: $F = \frac{U_1/d_1}{U_2/d_2}$. It shows up wherever you're comparing two estimates of variance, because sums of squared, normally-distributed residuals follow chi-squared distributions, and comparing two such sums naturally produces a ratio of that form. The distribution has two degrees-of-freedom parameters, $d_1$ and $d_2$ — one for the numerator, one for the denominator — and is right-skewed, always positive, and centered near 1 when the two variances being compared are actually equal.

Two common quant uses: testing whether two portfolios or strategies have equal return variance (an F-test on the ratio of sample variances), and testing whether a whole block of regression coefficients is jointly zero — the standard "F-test" reported alongside a regression's R². In the second case, you compare the extra variance explained by adding a group of predictors against the leftover residual variance; a large F-ratio means those predictors explain meaningfully more than noise would.

If strategy A's daily return variance is 0.04 and strategy B's is 0.01 over comparable sample sizes, the ratio $F = 0.04/0.01 = 4$ is compared against the F distribution's critical value for the relevant degrees of freedom; if 4 exceeds that threshold, you'd reject the hypothesis that the two strategies have equal return variance.