F Distribution Calculator

StatisticsLast updated: August 22, 2026

The F distribution is the distribution of the ratio of two independent chi-square variables, each divided by its degrees of freedom. It starts at zero, is right-skewed, and is defined by two degrees of freedom (numerator df₁ and denominator df₂). The F statistic of analysis of variance (ANOVA), the overall significance test of a regression model, the test for the equality of two variances, and the comparison of nested models all rest on this distribution.

This calculator works in both directions: given an F statistic and two degrees of freedom it returns the exact upper-tail p-value; given a significance level it produces the critical F without any table lookup. Because printed F tables carry only the upper tail, it also verifies the frequently needed reciprocal identity F(1−α; d₂, d₁) = 1 / F(α; d₁, d₂), both as a lower critical value and numerically in the table.

Use the first option if you have an F statistic, the second if you are looking for a table value.
The F value from your ANOVA or regression output. It cannot be negative.
Between groups in ANOVA: k − 1. In regression, the number of predictors.
Within groups in ANOVA: N − k. In regression, n − k − 1.
Sets the false positive risk. In the social sciences 5% is the common choice.

F (Fisher-Snedecor) Distribution Formulas

Definition:    F = (χ²₁ / d₁) / (χ²₂ / d₂),  χ²₁ and χ²₂ independent
Density:       f(x) = [ (d₁x)^d₁ · d₂^d₂ / (d₁x + d₂)^(d₁+d₂) ]^(1/2) / [ x·B(d₁/2, d₂/2) ]
Cumulative:    F(x) = I₍d₁x/(d₁x+d₂)₎(d₁/2, d₂/2)   (regularized incomplete beta)
p-value:       p = 1 − F(F_obs)      (always the right tail)
Critical:      F₍₁−α; d₁, d₂₎
Reciprocity:   F₍₁−α; d₂, d₁₎ = 1 / F₍α; d₁, d₂₎
Mean:          E(X) = d₂ / (d₂ − 2)          (d₂ > 2)
Variance:      Var(X) = 2·d₂²·(d₁ + d₂ − 2) / [d₁·(d₂ − 2)²·(d₂ − 4)]   (d₂ > 4)
Mode:          [(d₁ − 2)/d₁] · [d₂/(d₂ + 2)]   (d₁ > 2)
ANOVA:         F = MS_between / MS_within ,  d₁ = k − 1,  d₂ = N − k
Link to t:     t²(ν) = F(1, ν)

The mean diverges when d₂ ≤ 2 and the variance when d₂ ≤ 4 — that is not a computational error but a property of the distribution. When H₀ is true the numerator and the denominator estimate the same variance, so the F ratio is expected to cluster around 1; large F values are evidence against H₀.

How to Calculate

  1. Choose the direction of the calculation: "F → p" if you have an F statistic, "α → F" if you are looking for a table value.
  2. Enter the numerator degrees of freedom (df₁): the number of groups − 1 in ANOVA, the number of predictors in regression.
  3. Enter the denominator degrees of freedom (df₂): total observations − number of groups in ANOVA, n − k − 1 in regression.
  4. In "F → p" mode, type the value of the F statistic; F cannot be negative.
  5. Choose the significance level (α); the critical value, the decision, and the table all follow this choice.
  6. Check that the lower critical value comes out the same along both routes: the direct computation and the reciprocal identity.

Worked Examples

ANOVA output: F(2; 27) = 4.35

In an ANOVA with three groups and 27 error degrees of freedom, F(2; 27) = 4.35 gives an upper tail of p = 0.023. The critical value at α = 0.05 is 3.354, so 4.35 > 3.354 and the difference between the groups is statistically significant. The mean of the distribution is 1.08 and its median 0.711.

Upper-tail p-value: p = 0.023 · Decision: Significant · F statistic: 4.3500

Critical F for α = 0.05, df₁ = 3, df₂ = 20

The critical value for an ANOVA with four groups is F(0.95; 3; 20) = 3.098, exactly matching the classic F table. The lower critical value is 0.115471 and is confirmed by the reciprocal identity as well: F(0.95; 20; 3) = 8.6602 and 1 / 8.6602 = 0.115471. The mode is 0.303.

Critical F value: 3.0984 · Rejection rule: F > 3.098 · F statistic: 3.0984

A non-significant result: F(4; 30) = 1.80

In an analysis with five groups, F(4; 30) = 1.80 yields p = 0.155; since the critical value is 2.690, 1.80 < 2.690 and the result is not significant. When the F statistic is close to 1, the between-groups variability is no different from the within-groups variability.

Upper-tail p-value: p = 0.155 · Decision: Not significant · F statistic: 1.8000

Frequently Asked Questions

What is the identity F(1−α; d₂, d₁) = 1 / F(α; d₁, d₂) good for?
To save space, printed F tables list only the upper tail (usually α = 0.05 and 0.01). When a lower-tail critical value is needed — for a two-tailed test of the equality of two variances, for instance — you take the reciprocal of the upper-tail value with the degrees of freedom swapped. The identity follows from the observation that swapping the numerator and the denominator of an F ratio gives 1/F: if X ~ F(d₁, d₂), then 1/X ~ F(d₂, d₁).
Why does the order of the degrees of freedom matter?
F(2; 27) and F(27; 2) are two entirely different distributions; the 95% critical value of the first is 3.354, that of the second 19.459. The numerator degrees of freedom come from the variance estimate in the numerator of the F ratio (between groups / effect in ANOVA), the denominator from the estimate in the denominator (within groups / error). Software output always reports this order as "F(df₁, df₂)".
How is the F distribution computed in Excel, SPSS, and R?
In Excel, `=F.DIST.RT(F, df1, df2)` gives the upper-tail p and `=F.INV.RT(alpha, df1, df2)` the critical value. In R, use `pf(F, df1, df2, lower.tail = FALSE)` and `qf(1 - alpha, df1, df2)`. In Python, `scipy.stats.f.sf(F, df1, df2)` and `scipy.stats.f.ppf(1 - alpha, df1, df2)` produce the same results. The "Sig." column of an SPSS ANOVA table is exactly this upper-tail p-value.
Is there a link between the F and t distributions?
Yes: the identity t²(ν) = F(1, ν) holds. When two independent groups are compared, a t-test and a one-way ANOVA give the same p-value, because the F statistic is the square of the t statistic. For example, t(20) = 2.086 gives F(1; 20) = 4.351, and the two-tailed / upper-tail p-value of both is 0.050.
Why does the mean sometimes come out as divergent?
The mean of the F distribution is finite only when df₂ > 2, and its variance only when df₂ > 4. When the denominator degrees of freedom are small, the right tail is so heavy that the integrals do not converge. This is not a software bug but a mathematical property of the distribution. In practice df₂ is usually in the dozens, so you meet this case only with very small samples.
What are the assumptions of the F test?
The F test in ANOVA requires independent observations, approximately normal values within each group, and homogeneous group variances. Homogeneity of variance is checked with Levene's test; if it is violated, Welch ANOVA is preferred. The classic F test that compares two variances directly is extremely sensitive to violations of normality, which is why Levene's or the Brown-Forsythe test is recommended in its place.