F Distribution Calculator (p-value & critical F)

Q: Can F be less than 1?

Yes. Values below 1 are common when there is no effect, and they give large p-values.

Q: How is F related to t?

With d₁ = 1, F equals t² with d₂ degrees of freedom, so a two-sided t-test and the F test agree.

Q: Why is only the right tail used?

Group differences can only make the between-group variance larger, so only large F values count as evidence.

Understanding the F distribution calculator — concept and principle

An F distribution calculator turns an F statistic and its two degrees of freedom into a right-tail p-value and gives the critical F value for a significance level α. The F test appears in one-way and factorial ANOVA, in the overall significance test of a regression model, and when comparing two variances.

F compares two variances; 3.2 passes the critical 3.098, so p = 0.0455 is below 0.05.

How it is calculated

p-value (right tail) = 1 − F(x; d₁, d₂).
Critical F is the value c with F(c; d₁, d₂) = 1 − α, found by numerical search.

The CDF is computed with the regularized incomplete beta function: F(x; d₁, d₂) = I_u(d₁/2, d₂/2), where u = d₁x ÷ (d₁x + d₂). d₁ is the numerator df and d₂ the denominator df. In one-way ANOVA with k groups and N observations in total, d₁ = k − 1 and d₂ = N − k. In regression with p predictors and n observations, d₁ = p and d₂ = n − p − 1.

The principle

F is a ratio of two independent chi-square variables, each divided by its degrees of freedom: F = (χ²₁/d₁) ÷ (χ²₂/d₂). In ANOVA the numerator is the variance between group means and the denominator is the variance within groups. If all group means are equal, both estimate the same error variance, so F tends to be near 1. Real differences between groups inflate the numerator, so large F values, in the right tail, are evidence against the null hypothesis. The order of d₁ and d₂ matters: swapping them gives a different distribution.

Worked example

Inputs: F = 3.2, d₁ = 3, d₂ = 20, α = 0.05 (for example, 4 groups and 24 observations).
u = 3 × 3.2 ÷ (3 × 3.2 + 20) = 9.6 ÷ 29.6 ≈ 0.3243.
I_u(1.5, 10) ≈ 0.95453, so the p-value = 1 − 0.95453 ≈ 0.0455.
Solving F(c; 3, 20) = 0.95 gives critical F ≈ 3.0984.

Each point is the calculator re-run with only “F statistic” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

F statistic	p-value (right tail)
0.8	0.508404646
1.6	0.2208598872
2.4	0.0980479459
3.2	0.0454672299
4	0.0220769997
4.8	0.01119752
6.4	0.0032358633

How to read the result

Since 3.2 > 3.098 and p ≈ 0.045 < 0.05, the group means differ significantly at the 5% level. The F test only says that at least one mean differs; post-hoc comparisons such as Tukey's test show which. Critical F values shrink as d₂ grows, because the denominator variance is estimated more precisely.

Common mistakes

Swapping the numerator and denominator degrees of freedom.
Concluding that every group differs after a significant F.
Using the F test for two variances on clearly non-normal data, where it is very sensitive.

Keep in mind: ANOVA assumes independent observations, roughly normal residuals and similar variances across groups. A p-value measures surprise under the null hypothesis, not the size or importance of the effect.

Frequently asked questions

Can F be less than 1?