Compare the variances of two populations using the F-Test Calculator. Enter each group's sample standard deviation and sample size, choose your significance level (α) and tail type, and get back the F-statistic, p-value, critical F-value, and a clear reject / fail to reject decision on the null hypothesis of equal variances.
F-Statistic
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Degrees of Freedom 1 (df₁)
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Degrees of Freedom 2 (df₂)
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p-Value
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Critical F-Value
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Variance Ratio (S₁²/S₂²)
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Decision
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An F-test for equality of two variances is a statistical hypothesis test that determines whether the variances (spread) of two normally distributed populations are equal. It works by forming an F-statistic — the ratio of the two sample variances — and comparing it to a critical value from the F-distribution. If the ratio is far from 1, there is evidence that the population variances differ.
The F-statistic is calculated as F = S₁² / S₂², where S₁² and S₂² are the sample variances of the two groups. The result follows an F-distribution with degrees of freedom df₁ = n₁ − 1 and df₂ = n₂ − 1. A value close to 1 suggests the variances are similar; values much greater or less than 1 suggest a difference.
If the p-value is less than or equal to your chosen significance level (α), you reject the null hypothesis and conclude that the two population variances are significantly different. If the p-value is greater than α, you fail to reject the null hypothesis, meaning there is insufficient evidence to say the variances differ.
The F-test for equality of variances requires that both populations follow a normal distribution, the two samples are independent of each other, and the data consists of continuous measurements. The test is sensitive to departures from normality, so it's good practice to verify normality before applying it.
A T-test compares the means of one or two populations, while an F-test compares their variances. The F-test is also used in ANOVA (analysis of variance) to simultaneously compare means across multiple groups, and in regression analysis to test whether a model explains a significant portion of variance. The T-test applies to individual coefficients, while the F-test applies to groups of them.
No. Since the F-statistic is a ratio of two variances (squared values), it is always non-negative. Variances are always zero or positive, so their ratio is always ≥ 0. In practice, you will always get a positive F-statistic when at least one sample has some spread.
Use a two-tailed test when you simply want to know if the variances are different (σ₁² ≠ σ₂²) without a prior expectation about direction. Use a right-tailed test when you expect σ₁² > σ₂², and a left-tailed test when you expect σ₁² < σ₂². In most exploratory analyses, the two-tailed version is the standard choice.
A high F-statistic indicates that the variance in one sample is much larger than in the other, suggesting the population variances are not equal. Whether a value counts as 'high' depends on the degrees of freedom and your significance level — that's why comparing the F-statistic to the critical F-value (or looking at the p-value) is essential for drawing a valid conclusion.