F-Test Calculator
Compare two sample variances and calculate the F-statistic, degrees of freedom, critical F-value, p-value, and hypothesis test decision. Enter your sample data below to perform an F-test instantly.
Enter Sample Data
Use sample variances and sample sizes.
What Is an F-Test?
An F-test is a statistical hypothesis test based on the F-distribution. One common use is comparing the variances of two independent populations to determine whether the difference between their sample variances is statistically significant.
The test calculates a ratio between two sample variances. If the resulting F-statistic falls far enough into the appropriate tail of the F-distribution, the null hypothesis can be rejected at the selected significance level.
What Does This Calculator Test?
This calculator performs a two-sample variance F-test. The default null hypothesis assumes that the two population variances are equal: H₀: σ₁² = σ₂²
You can select a two-tailed test when checking whether the variances are different in either direction, or a one-tailed test when your hypothesis specifically predicts that one variance is larger or smaller than the other.
How the F-Test Calculator Works
Enter Both Samples
Provide the sample variance and sample size for each of the two groups you want to compare.
Choose Your Test
Select the significance level and choose a two-tailed, right-tailed, or left-tailed alternative hypothesis.
Review the Result
The calculator returns the F-statistic, degrees of freedom, critical values, p-value, and statistical decision.
F-Test Formula and Degrees of Freedom
F-Statistic Formula
For two sample variances, the basic F-statistic is: F = s₁² / s₂²
Here, s₁² is the variance of Sample 1 and s₂² is the variance of Sample 2.
Degrees of Freedom
The numerator and denominator degrees of freedom are: df₁ = n₁ − 1 and df₂ = n₂ − 1 .
The two degrees of freedom determine the shape of the F-distribution used to calculate the critical values and p-value.
Two-Tailed vs One-Tailed F-Test
| Test | Alternative Hypothesis | Rejection Region | Typical Use |
|---|---|---|---|
| Two-Tailed | σ₁² ≠ σ₂² | Both tails | Testing whether variances differ |
| Right-Tailed | σ₁² > σ₂² | Upper tail | Testing whether Sample 1 variance is larger |
| Left-Tailed | σ₁² < σ₂² | Lower tail | Testing whether Sample 1 variance is smaller |
How to Interpret Your F-Test Result
The p-value provides a convenient way to interpret the test. Compare the calculated p-value with your chosen significance level, usually written as α.
If the p-value is less than or equal to alpha, the calculator reports that the null hypothesis should be rejected. This means the observed variance difference is statistically significant under the assumptions of the selected test.
If the p-value is greater than alpha, the calculator reports that there is not enough evidence to reject the null hypothesis. This does not prove that the population variances are identical; it only means the sample does not provide sufficient evidence of a difference at that significance level.