F-Distribution

Author

John Robin Inston

Published

August 30, 2026

1 Introduction

The \(F\)-distribution is a continuous probability distribution that arises as the ratio of two independent Chi-Squared Distribution variables, each divided by its degrees of freedom. It is central to the analysis of variance (ANOVA) and to tests comparing two variances.

A random variable \(X\) follows an \(F\)-distribution with \(d_1\) and \(d_2\) degrees of freedom if

\[ X \sim F_{d_1, d_2} \quad \text{or} \quad X \sim F(d_1, d_2). \]

2 Construction from Chi-Squared

If \(U \sim \chi^2_{d_1}\) and \(V \sim \chi^2_{d_2}\) are independent, then

\[ X = \frac{U/d_1}{V/d_2} \sim F_{d_1, d_2}. \]

3 Probability Density Function

The probability density function is

\[ f(x) = \frac{1}{B\!\left(\frac{d_1}{2}, \frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{d_1/2} \frac{x^{d_1/2 - 1}} {\left(1 + \frac{d_1}{d_2} x\right)^{(d_1 + d_2)/2}}, \quad x > 0, \]

where \(B(\cdot, \cdot)\) is the beta function.

Interpretation: Values near \(1\) are typical when the two underlying variances are comparable; large values indicate the numerator variance dominates.

Plotting code 
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.stats import f as fdist

sns.set_style('whitegrid')
sns.set_palette('Set2')

x = np.linspace(0, 5, 400)
fig, ax = plt.subplots(figsize=(7, 4))
for d1, d2 in ((5, 10), (10, 10), (20, 20)):
    ax.plot(x, fdist.pdf(x, d1, d2),
            lw=2, label=rf'$d_1={d1},\ d_2={d2}$')

ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
ax.set_title('$F$ density')
ax.legend()
plt.show()

Density of the \(F\)-distribution for several pairs of degrees of freedom \((d_1, d_2)\).

4 Key Properties

For \(X \sim F_{d_1, d_2}\),

\[ \mathbb{E}[X] = \frac{d_2}{d_2 - 2} \ \ (d_2 > 2), \]

\[ \operatorname{Var}(X) = \frac{2 d_2^2 (d_1 + d_2 - 2)} {d_1 (d_2 - 2)^2 (d_2 - 4)} \ \ (d_2 > 4). \]

Proof. Write \(X = (U/d_1)/(V/d_2)\) with \(U \sim \chi^2_{d_1}\), \(V \sim \chi^2_{d_2}\) independent. Then \(\mathbb{E}[X] = \frac{d_2}{d_1}\mathbb{E}[U]\,\mathbb{E}[V^{-1}]\). Using \(\mathbb{E}[U] = d_1\) and \(\mathbb{E}[V^{-1}] = 1/(d_2 - 2)\) gives \(\mathbb{E}[X] = d_2/(d_2 - 2)\). The variance follows from the second inverse moment of \(V\).

5 Relationship to Other Distributions

  • Chi-Squared Distribution: Defined as a scaled ratio of two independent chi-squared variables.
  • Student's t-Distribution: If \(T \sim t_\nu\) then \(T^2 \sim F_{1, \nu}\).
  • Beta Distribution: If \(X \sim F_{d_1, d_2}\), then \(\frac{d_1 X}{d_1 X + d_2} \sim \text{Beta}(d_1/2, d_2/2)\).
  • Reciprocal: If \(X \sim F_{d_1, d_2}\) then \(1/X \sim F_{d_2, d_1}\).

6 Examples and Applications

In one-way ANOVA the ratio of between-group to within-group mean squares follows an \(F\)-distribution under the null hypothesis of equal group means.

The ratio \(S_1^2 / S_2^2\) of two independent normal sample variances is \(F\)-distributed, giving a test of the hypothesis \(\sigma_1^2 = \sigma_2^2\).

The overall \(F\)-test in linear regression compares a fitted model against the intercept-only model to assess joint significance of the predictors.

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