F distribution Fisher-Snedecor

F distribution, also called Fisher-Snedecor distribution, is one of fundamental distributions in inferential statistics. Independently introduced by George Snedecor (1934) and Ronald Fisher, named after Fisher who developed analysis of variance (ANOVA). Defined as ratio of two normalized independent chi-squares:

F = (chi^2_{d1} / d1) / (chi^2_{d2} / d2)

where chi^2_{d1} ~ chi^2(d1) and chi^2_{d2} ~ chi^2(d2) independent. Has two degrees of freedom d1 (numerator) and d2 (denominator).

Probability density

F(d1, d2) density:

f(x; d1, d2) = sqrt((d1*x)^d1 * d2^d2 / (d1*x + d2)^(d1+d2)) / (x * B(d1/2, d2/2))

Density positive only for x > 0, non-zero on positive real line. Right-skewed shape.

Moments and properties

• Mean: mu = d2 / (d2 - 2) for d2 > 2.
• Mode: ((d1 - 2) / d1) * (d2 / (d2 + 2)) for d1 > 2.
• Variance: sigma^2 = 2 * d2^2 * (d1 + d2 - 2) / (d1 * (d2 - 2)^2 * (d2 - 4)) for d2 > 4.
• Right skewness.
• Reciprocal: if X ~ F(d1, d2), then 1/X ~ F(d2, d1).
• Limit: F(d1, infinity) -> chi^2(d1) / d1.
• Special: F(1, d2) = (T(d2))^2, Student t squared.

Critical values

• F_0.95(5, 20) = 2.711
• F_0.95(10, 30) = 2.165
• F_0.95(1, 30) = 4.17
• F_0.99(5, 20) = 4.103
• F_0.99(10, 30) = 3.02

F-test for one-way ANOVA

Single-factor analysis of variance: k groups with n_i observations each, test H_0: mu_1 = mu_2 = ... = mu_k. Statistic:

F = MSB / MSW

MSB = mean square between groups = Sigma n_i (x_bar_i - x_bar)^2 / (k-1), MSW = mean square within groups = Sigma_i Sigma_j (x_ij - x_bar_i)^2 / (N-k). Under H_0 F ~ F(k-1, N-k). Reject H_0 if F > F_crit.

Five modes

1. PDF density f(x; d1, d2).
2. CDF P(X <= x; d1, d2).
3. Critical value: F_alpha for one and two-tailed tests.
4. ANOVA F-test: comparing multiple group means.
5. Variance ratio test: comparing sigma_1^2 vs sigma_2^2.

Variance ratio test

For H_0: sigma_1^2 = sigma_2^2 between two normal populations. Two samples with variances s_1^2 and s_2^2:

F = s_max^2 / s_min^2

F ~ F(n_max-1, n_min-1). Reject H_0 if F > F_{alpha/2}(n_max-1, n_min-1) (two-tailed). Often verified before classical Student t-test; if variances very different, use Welch t-test.

Relations with other distributions

• Student t: T(d)^2 ~ F(1, d). t-test is F-test with 1 numerator DOF.
• Chi-squared: F(d1, d2) * d1 -> chi^2(d1) as d2 -> infinity.
• Beta: if X ~ F(d1, d2), then X*d1/(X*d1+d2) ~ Beta(d1/2, d2/2).
• Cauchy: F(1, 1) has heavy tails, undefined moments.
• Non-central: F non-central with parameter lambda for power analysis.

History

Ronald Aylmer Fisher (1890-1962), father of modern statistics, developed analysis of variance in agricultural studies at Rothamsted (1920-1940). His F statistic appears in original 1923 papers on 'Rothamsted Classical Experiments'. George W. Snedecor (1881-1974), American mathematician, popularized F in US in his book 'Statistical Methods' (1934), naming 'F' in honor of Fisher. F distribution is today standard in ANOVA, regression analysis, variance tests.

Applications

• One-way ANOVA: compare means of k groups (most used test in applied statistics).
• Multiple ANOVA (two-way, MANOVA): extensions to multiple factors.
• Linear regression: overall significance test of model (R^2 test).
• Analysis of covariance (ANCOVA): mix of factors and continuous covariates.
• Variance homogeneity test: Bartlett, Levene use related statistics.
• Design of experiments: factorial experimental plan analysis.

The assumptions of the F-test and its robustness

The validity of an F-test rests on three assumptions: observations must be independent, drawn from normally distributed populations and, for ANOVA, have equal variances across groups (homoscedasticity). One-way ANOVA is reasonably robust to moderate violations of normality, especially with large balanced groups, thanks to the central limit theorem. The F-test for comparing two variances, by contrast, is notoriously sensitive to non-normality: even small departures can inflate the false-positive rate.

For this reason, when normality is in doubt, more robust alternatives are preferred: Levene's or Bartlett's test for homogeneity of variances, and Welch's version of ANOVA when variances differ. Checking the assumptions before interpreting the p-value is an essential step: a significant F obtained by violating the assumptions may be an artifact rather than a real effect.

After ANOVA: effect size and post-hoc tests

A significant F value only says that not all group means are equal, but it does not indicate which groups differ nor how large the difference is. To answer the first question you need post-hoc tests, such as Tukey's HSD or the Bonferroni correction, which compare pairs of groups while controlling the type-I error inflation due to multiple comparisons.

To quantify the size of the effect, instead of the p-value alone one reports measures such as eta squared (eta^2), the fraction of total variance explained by the groups, or omega squared (omega^2), a less biased estimate. An effect can be statistically significant yet negligible in practice, especially with large samples: that is why modern statistical good practice recommends always reporting the effect size alongside the p-value and the confidence interval.

Updated July 2026