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 (Two-Sample Variance Test)

Online F-test calculator to compare two variances. Enter sample sizes and sample variances (or standard deviations), choose the tail, and get F statistic, degrees of freedom, and p-value. Includes formulas and FAQ.

Full original guide (expanded)

F-Test Calculator (Two-Sample Variance Test)

Compare two sample variances and test whether the underlying population variances are equal. We automatically put the larger variance on top.

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

—

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.

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

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Variables and units

PMI = private mortgage insurance (monthly) (currency)

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

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Verified by Ugo Candido on 2026-01-19
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Formulas

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Version 0.1.0-draft

Citations

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Changelog

0.1.0-draft — 2026-01-19: Initial draft (review required).