Fisher’s F distribution
The F distribution (Fisher–Snedecor) is the distribution of the ratio of two independent variance estimates, each divided by its degrees of freedom. It has two parameters: the numerator degrees of freedom d₁ and the denominator degrees of freedom d₂. It is always positive and skewed to the right.
Where F appears
- ANOVA: F = mean square between groups ÷ mean square within groups, with d₁ = k − 1 and d₂ = N − k.
- Regression: the overall F-test compares the variance explained by the model with the residual variance, with d₁ = number of predictors and d₂ = n − p − 1.
- Comparing two variances: F = s₁² ÷ s₂², with d₁ = n₁ − 1 and d₂ = n₂ − 1.
What the calculator gives
- the p-value P(F ≥ f), the right tail used in ANOVA and regression;
- the critical F for a significance level, as in a printed F table;
- the cumulative probability P(F ≤ f) and the mean of the distribution, d₂ ÷ (d₂ − 2).
Example
An ANOVA with 4 groups and 28 observations gives F = 3.2 with 3 and 24 degrees of freedom:
- p ≈ 0.041: significant at α = 0.05;
- critical F at α = 0.05: 3.009.
Reading F tables
Printed tables are organized by α, with d₁ across the top and d₂ down the side. For example, at α = 0.05:
| d₁ \ d₂ | 10 | 20 | 30 | 60 |
|---|---|---|---|---|
| 1 | 4.96 | 4.35 | 4.17 | 4.00 |
| 2 | 4.10 | 3.49 | 3.32 | 3.15 |
| 3 | 3.71 | 3.10 | 2.92 | 2.76 |
| 5 | 3.33 | 2.71 | 2.53 | 2.37 |
The calculator gives exact values for any degrees of freedom, including fractional ones.
Relation with other distributions
- With d₁ = 1, F is the square of a t statistic with d₂ degrees of freedom: F(1, d₂) = t².
- As d₂ grows very large, d₁ × F approaches a χ² distribution with d₁ degrees of freedom.
Tips
- In ANOVA and regression, the test is always right-tailed: large F values signal an effect.
- For a two-tailed comparison of two variances, use the F-test calculator, which handles both tails.
- Report F with both degrees of freedom, for example F(3, 24) = 3.20, p = 0.041.