What is an F-test for two variances?

It is a hypothesis test to check whether two normal populations have the same variance. The test statistic is the ratio of the sample variances and it follows an F distribution under the null hypothesis.

How is the F statistic computed?

F = s1^2 / s2^2, where s1^2 is the larger sample variance and s2^2 is the smaller. Degrees of freedom are n1−1 and n2−1 for the two samples.

What does the p-value mean in the F-test?

The p-value is the probability of observing a ratio of variances as extreme as the one you got if the two population variances were actually equal. A small p-value (often < 0.05) suggests the variances differ.

F-Test Calculator (Compare Two Variances)

1. Sample data

You can enter either the sample variance (s²) or the sample standard deviation (s). If you fill both for a sample, we will use the variance.

Sample 1

n₁ (sample size) s₁² (variance) s₁ (std dev)

Sample 2

n₂ (sample size) s₂² (variance) s₂ (std dev)

2. Test setup

Significance (α)

Tail

Two-tailed is the default for testing equality of variances.

3. Results

F statistic

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df₁

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df₂

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p-value

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How the F-test works

The classical F-test for two variances tests:

H₀: σ₁² = σ₂²
H₁: σ₁² ≠ σ₂² (two-tailed) or σ₁² > σ₂² / σ₁² < σ₂² (one-tailed)

The test statistic is the ratio of sample variances:

F = s₁² / s₂²

To make the test stable, we put the larger sample variance in the numerator so F ≥ 1. The sampling distribution of F under H₀ is the F distribution with df₁ = n₁ − 1 and df₂ = n₂ − 1.

Interpreting the p-value

If the p-value is small (e.g. < 0.05), we reject H₀ and conclude the variances are significantly different. If the p-value is large, the data do not provide evidence that the variances differ.

Note: The F-test assumes both samples come from normal populations. If this is not the case, consider alternative robust tests.