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Distribution

F-Distribution Calculator — Critical F Table

Free F-distribution calculator. Compute p-values from an F-ratio with numerator and denominator degrees of freedom, plus critical F values — for ANOVA and regression F-tests.

📊 F = 3.885, df(2,12) → p = .0500

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The F-distribution is the distribution of the ratio of two independent chi-square variables, each divided by its own degrees of freedom. It is the reference distribution for comparing variances directly (Levene's test, the F-test for equal variances) and — far more commonly — for the omnibus F-test in ANOVA and multiple regression, where it tests whether the variance explained by a model is large relative to the unexplained (residual) variance. Because it compares two variances, the F-distribution needs TWO degrees-of-freedom parameters: df1 (numerator, associated with the effect being tested) and df2 (denominator, associated with the error/residual).

One-way ANOVA F-ratio

F = \frac{{MS}_{between}}{{MS}_{within}} = \frac{{SS}_{between} / {df}_{1}}{{SS}_{within} / {df}_{2}}

Critical F Values (α = .05, upper tail)

df1 ↓ / df2 →	5	10	20	30	60
1	6.608	4.965	4.351	4.171	4.001
2	5.786	4.103	3.493	3.316	3.150
3	5.409	3.708	3.098	2.922	2.758
4	5.192	3.478	2.866	2.690	2.525
5	5.050	3.326	2.711	2.534	2.368
6	4.950	3.217	2.599	2.421	2.254
8	4.818	3.072	2.447	2.266	2.097
10	4.735	2.978	2.348	2.165	1.993
12	4.678	2.913	2.278	2.092	1.917
15	4.619	2.845	2.203	2.015	1.836
20	4.558	2.774	2.124	1.932	1.748

Generated directly from src/core/distributions.js (fInv) — never hand-typed.

Frequently Asked Questions

Why does the F-test need two degrees of freedom?

Because F is a ratio of two independent variance estimates, each with its own sampling variability that depends on its own sample size. df1 (numerator) reflects the number of groups/predictors being compared; df2 (denominator) reflects the sample size left over for estimating error variance. Both affect how much the ratio would vary by chance alone, so both are needed to find the right critical value.

Is F the square of t?

Yes, in the specific case of df1 = 1: F(1, df2) = t(df2)². This is exactly why a two-group one-way ANOVA gives the identical p-value as an independent-samples pooled-variance t-test — they are testing the same hypothesis via mathematically equivalent statistics.

What if my F-ratio is less than 1?

An F-ratio below 1 means the between-group variance is smaller than the within-group (error) variance — the opposite of what a real effect would produce. It is never statistically significant in a standard one-tailed ANOVA F-test (P(F ≥ x) will always exceed .5), so there is no need to look it up against a critical value at all.

Why does the mini-table only show α = .05?

A full F-table needs a separate grid of df1 × df2 combinations for every α level, which quickly becomes too large for a compact reference table — traditional printed F-tables devote a full page per α. The live calculator above computes the exact critical F for any α, df1, and df2 combination instantly, making a second printed table for other α levels unnecessary.

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F-Distribution Calculator — Critical F Table

Critical F Values (α = .05, upper tail)

Frequently Asked Questions

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