F-Distribution Calculator

The two identities a table assumed you knew

Both are exact, both are checkable in the tool above, and both explain something about how F tables were printed for most of the twentieth century.

F(a,b;p) = 1/F(b,a;1−p). The lower tail of an F with a and b degrees of freedom is the reciprocal of the upper tail of an F with b and a. Tables carried only upper tails because of it, halving the paper — and a reader who had not been taught the identity had no way to get a lower critical value.

The tool computes both sides. The lower critical value directly, and 1/F(b,a;1−p) beside it. They agree to machine precision, across 108 combinations of degrees of freedom and level in the verification suite.

F(1, v) = t(v)², exactly. Set the numerator degrees of freedom to 1 and the F critical value is the square of the two-sided t critical value on the same denominator df. At v = 20: t = 2.085963, squared 4.351244, and F(1, 20; 0.95) = 4.351244.

Which is a statement about tests, not about arithmetic. A one-way ANOVA on two groups and a two-sided t test are the same test written two ways. They cannot disagree. If software reports different p-values for the two on the same data, one of them is one-tailed.

It also explains why ANOVA is one-tailed. Squaring loses the sign, so the two-sided t test’s two rejection regions become one upper region in F. The F test is not one-tailed by convention; it is one-tailed because the statistic already folded both directions together.

Numerator first, and it is not symmetric

F takes two degrees of freedom and they are not interchangeable. Swapping them gives a different distribution and a different critical value, and the mistake is easy to make from a table.

At 95%, F(2, 27) is 3.354 and F(27, 2) is 19.459. Nearly six times larger. There is no symmetry to fall back on.

The numerator is the term being tested. Between-group variation in ANOVA, the model sum of squares in regression, the larger variance in a variance-ratio test. The denominator is the error or residual term.

So the numerator df counts what the model is spending. Groups minus one in a one-way ANOVA, the number of predictors in a regression F, the difference in parameters between two nested models. The denominator counts what is left over.

The distribution is right-skewed, and always positive. It is a ratio of two non-negative quantities, so it cannot go below zero, and its long right tail is why a large F is evidence while a small one is not. An F below 1 means the groups differ less than chance alone would produce — which is not a finding, though a suspiciously small F across many analyses has occasionally been evidence of something else.

The mean is d₂/(d₂ − 2), and it needs d₂ > 2. It is slightly above 1 rather than exactly 1, and it approaches 1 from above as the denominator df grows. The variance needs d₂ > 4. The quantiles, by contrast, always exist — a critical value does not require a moment.

What the F in an ANOVA table actually compares

The statistic is a ratio of two estimates of the same quantity, and the logic only makes sense once that is said out loud.

Both mean squares estimate the error variance, if the null is true. The within-group mean square estimates it always. The between-group mean square estimates it too — but only when the group means are equal.

So their ratio should be about 1 under the null, and larger when the groups genuinely differ, because then the numerator is picking up real separation on top of the noise. That is the whole test.

The degrees of freedom come from the sum-of-squares split. Total splits into between and within, and so do the degrees of freedom: k − 1 and N − k. The sum of squares page covers that decomposition and the condition it depends on.

A significant F names no group. It pools every group into one number, so it cannot be read backwards to say which differs. That is the limitation the ANOVA calculator handles with a post-hoc, and it is inherited by the rank-based Kruskal-Wallis test as well.

Equal variances is the assumption the F test actually needs. It is a ratio of variances, so unequal spreads distort it directly — more than non-normality does. Welch’s ANOVA is the robust alternative, and it is underused for the same reason Welch’s t test was for decades.

The variance ratio test, and why it is a bad idea

The F distribution’s original use was testing whether two variances are equal. It is the one application where the distribution is right and the test is usually wrong to run.

The test is extremely sensitive to non-normality. Far more than the t test it is often used to justify. On heavy-tailed data it rejects far more often than its nominal level, so “the variances differ” frequently means “the data is not normal”.

Which makes it a poor gatekeeper. Running an F test to decide between a pooled t test and Welch’s is a two-stage procedure whose combined error rate is not what either stage advertises, and the first stage is unreliable exactly when the second stage’s choice matters.

Use Welch unconditionally instead. It costs almost nothing when the variances are equal and is correct when they are not. The pre-test buys nothing and adds a failure mode.

If you must test variances, use Levene or Brown-Forsythe. Both work on absolute deviations from the centre rather than on squared ones, and both are far less sensitive to non-normality. Brown-Forsythe uses the median and is the more robust of the two.

The distribution is not the problem. The F distribution is exactly right for a ratio of variances of normal samples. It is the normality assumption that fails, and it fails harder here than almost anywhere else in the standard toolkit.

Comparing two models with an F

The most common modern use is not one-way ANOVA at all: it is asking whether adding predictors to a regression bought anything.

The statistic is the improvement per parameter, over the residual per degree of freedom. F = [(SSE₁ − SSE₂)/(p₂ − p₁)] ÷ [SSE₂/(n − p₂ − 1)], where model 2 nests model 1.

The numerator df is the number of parameters added. Adding four predictors to a model fitted on 100 observations gives F(4, 95), and the critical value at 95% is 2.467.

The models must be nested, and fitted on the same rows. Dropping observations with missing values on a predictor that only one model contains silently breaks the comparison, and the F is then meaningless rather than merely wrong.

R² always rises when a predictor is added, which is why the F test exists: it asks whether the rise exceeds what a useless predictor would have produced on average. The R² calculator covers the same point through adjusted R², which penalises rather than tests.

And a significant overall F does not validate the model. It says the predictors collectively beat nothing at all, which is a low bar. The residuals are what say whether the form is right.

What the printed tables could not do

An F table is a page per significance level, with numerator degrees of freedom across the top and denominator down the side. Several things could not fit, and the workarounds are still taught.

Only a handful of levels. Usually 0.05 and 0.01, sometimes 0.10 and 0.001. A p-value could not be read off at all — only bracketed, which is why older papers report “p < 0.05” rather than a number.

Only integer degrees of freedom, in a sparse grid. The denominator column typically jumps 30, 40, 60, 120, then infinity. Interpolating between 60 and 120 is not linear in the critical value, and the usual advice — interpolate in 1/df — is a workaround for a constraint that no longer exists.

No lower tail. Only the reciprocal identity gave you one, and only if you knew it.

No fractional degrees of freedom. Welch’s ANOVA and mixed models both produce them, and this tool takes any positive real value because the incomplete beta function does not care whether its parameters are integers.

The consequence outlasted the tables. “Significant at the 5% level” is a habit from an era when 5% was one of two columns available. Reporting the exact p-value costs nothing now, and says considerably more.

Sources and methodology

References for the distribution and the tests built on it.

• NIST/SEMATECH e-Handbook — the F distribution, its parameters and its critical values
• NIST/SEMATECH e-Handbook — one-way ANOVA and the mean-square ratio
• Brown & Forsythe, “Robust Tests for the Equality of Variances”, JASA 69(346), 1974 — why the variance-ratio F is a poor gatekeeper
• ANOVA Calculator — the test this distribution is the reference for
• t-Distribution Calculator — the other half of the F(1, v) = t(v)² identity
• How we calculate — methodology

Method. Probabilities come from the regularised incomplete beta function, so any positive real degrees of freedom is exact — including the fractional values Welch’s ANOVA and mixed models produce. Critical values are obtained by inverting the same function. The lower tail is computed directly rather than through the reciprocal identity, and the identity is then shown alongside as a check rather than used as the method. The suite verifies F(1, v) = t(v)² across eight degrees of freedom and four levels, and the reciprocal relation F(a,b;p) = 1/F(b,a;1−p) across 108 combinations, both to a relative gap below 10⁻⁸. That engine is verified on every change against 59 assertions. The count and the per-case breakdown are published on the formula verification page.

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