F Distribution Calculator

The F distribution arises naturally as the ratio of two independent chi-squared variables, each divided by its degrees of freedom. If U ~ chi-squared(d1) and V ~ chi-squared(d2), then F = (U/d1) / (V/d2) follows an F distribution with d1 and d2 degrees of freedom.

The distribution is right-skewed and defined only for positive values. With small degrees of freedom it has a heavy right tail; as d1 and d2 both grow large, it becomes increasingly symmetric and concentrated near 1.

The F distribution shows up in three major contexts. First, in ANOVA: the test statistic is the ratio of between-group variance to within-group variance. Second, in linear regression: the overall F-test compares the full model to the null (intercept-only) model. Third, in the F-test for equality of variances, comparing two sample variances.

The CDF of the F distribution is computed via the regularized incomplete beta function: P(F <= x) = I(d1x / (d1x + d2), d1/2, d2/2). This is why F-table values can be computed exactly rather than approximated with polynomial fits.

The p-value for an observed F statistic is the area to the right: P(F > x). For a one-sided test this is 1 - CDF(x). For the two-sided F-test for equal variances, you double the upper-tail probability.

Critical values are the F values that cut off exactly alpha probability in the upper tail. For example, the 0.05 critical value with df1 = 5 and df2 = 20 is about 2.71, meaning 5% of the F distribution lies above that value.

The F and t distributions are related: if T ~ t(n), then T^2 ~ F(1, n). A two-sample t-test and the corresponding F-test always give the same p-value.

The chart plots the upper-tail probability P(F > x) as a function of x, so you can read off p-values visually for any observed F statistic.