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). Textbooks write that as 1 - CDF(x), and that is fine on paper, but it is a poor way to compute it. Once the p-value drops below about 1e-16 the subtraction has no digits left and the answer collapses to zero. Swapping the two arguments of the incomplete beta gives the upper tail directly, which is what this page does, and it is why a p-value here can read 1.9e-13 instead of “0”. 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. Note that df1 and df2 are not interchangeable. F(0.05; 5, 20) is 2.71 but F(0.05; 20, 5) is 4.56, so getting the order backwards is the single most common way to misread an F table.

Worked example. Six fertiliser treatments are compared across 26 plots, giving df1 = 5 (six groups minus one) and df2 = 20. The between-group mean square comes out 3.5 times the within-group mean square, so F = 3.5. Enter 5, 20 and 3.5 and the upper tail is p = 0.0196. That clears the 5% bar (3.5 > 2.71) but not the 1% bar (3.5 < 4.10), so the usual write-up is “significant at 0.05”. Notice how fast the tail falls: raising F from 2.71 to 4.10 cuts the p-value from 0.05 to 0.01.

The F and t distributions are related: if T follows a t distribution with n degrees of freedom, then T squared follows F(1, n). A two-sided two-sample t-test assuming equal variances and the one-way ANOVA F-test on those same two groups return identical p-values, because the F statistic there is literally the square of the t statistic. That equivalence does not carry over to the F-test for equal variances, which uses the same distribution to answer a different question.

The chart plots the upper-tail probability P(F > x) as a function of x, with your own F value or critical value marked, so you can see how much probability mass sits beyond it.