F-Test

Comparing two variances using the ratio of sample variances, which follows the F-distribution under the null hypothesis of equal population variances.

Chi-square distribution — right-tail p-value

df=4

χ²=6.0

p-value ≈ 0.1992 → fail to reject H₀

Definition

The F-test tests whether two groups have the same variance, or whether a linear regression model has significant explanatory power.

The F-statistic is a ratio of two variance estimates:

$F = \frac{s_1^2}{s_2^2}$

where $s_1^2$ and $s_2^2$ are sample variances from two independent groups. Under $H_0$ (equal variances), $F \sim F_{n_1-1, n_2-1}$ (F-distribution with numerator and denominator degrees of freedom).

The F-distribution is right-skewed and always positive. Large $F$ (or small $F$ ) is evidence against equal variances (two-sided test).

Key properties

The F-statistic is always non-negative — it's a ratio of two non-negative variance estimates
$F_{d_1,d_2}$ and $1/F_{d_1,d_2}$ are related: swapping which variance goes in the numerator inverts the statistic and swaps the degrees of freedom
Assumes both samples come from normal populations — unlike the t-test, the F-test is highly sensitive to this assumption
In regression, the F-test for overall significance and individual t-tests for coefficients are testing related but distinct hypotheses

Common mistakes

Using the classic F-test to check the equal-variance assumption before a t-test: this two-stage approach inflates the overall Type I error rate — Welch's t-test (which doesn't assume equal variances) avoids the issue entirely
Applying the F-test to non-normal data: prefer Levene's test or Brown-Forsythe, which are more robust to violations of normality; Bartlett's test is powerful under normality but even more sensitive to non-normal data

Testing equal variances before a t-test

Two production lines: Line A ( $n=15$ , $s_A^2=8.2$ ), Line B ( $n=12$ , $s_B^2=3.1$ ).

$F = 8.2/3.1 \approx 2.65$ with $(14, 11)$ df. Critical values for two-sided test at $\alpha=0.05$ : 0.35 and 3.09. Since $2.65 < 3.09$ , fail to reject equal variances.

Try it

Why is an F-test for equal variances (Levene's test is often preferred) important before running a pooled two-sample t-test?

Solution

The pooled t-test assumes equal variances — it uses the pooled sample variance $s_p^2$ as a common estimate. If variances actually differ substantially, this pooled estimate is inappropriate, and the t-statistic doesn't follow the t-distribution under $H_0$ . The test becomes unreliable (wrong Type I error rate).

However, many statisticians now recommend using Welch's t-test by default (doesn't require equal variances), avoiding the need to pre-test. Pre-testing for equal variances inflates the overall Type I error rate of the subsequent t-test (two-stage testing).

Related concepts

Statistics· Statistical Tests

ANOVAAnalysis of variance — comparing means across three or more groups by partitioning total variability into between-group and within-group components.

Statistics· Descriptive

VarianceThe average squared distance of each value from the mean — a measure of spread.

Statistics· Inference

Hypothesis TestingA formal framework for deciding whether data provides enough evidence to reject a default assumption.