F-Test Calculator

Q: What is the F-test used for?

The F-test compares two sample variances to determine if the populations have the same variability. It is a prerequisite for the two-sample t-test and forms the basis of ANOVA.

Q: Why is F always greater than or equal to 1?

By convention, the larger variance goes in the numerator. Since variance is always positive, dividing a larger number by a smaller one gives at least 1.

Q: What are the assumptions of the F-test?

Both samples must be independent, drawn from normally distributed populations. The F-test is quite sensitive to the normality assumption. Consider Levene's test as a more robust alternative.

Q: How does the F-test relate to ANOVA?

ANOVA uses the F-test to compare variance between groups to variance within groups. If between-group variance is significantly larger (large F), at least one group mean differs.

Q: What if my F-value is close to 1?

An F near 1 means the two sample variances are similar. You would fail to reject the null hypothesis and conclude the variances are not significantly different.

Use this free online F-test calculator to compare two sample variances and test whether they are significantly different. Enter the variances and sample sizes from both groups to get the F-statistic, degrees of freedom, and hypothesis test decision.

Enter Values

Sample 1 Variance (s1 squared)

The variance of the first sample

Sample 1 Size (n1)

Number of observations in the first sample

Sample 2 Variance (s2 squared)

The variance of the second sample

Sample 2 Size (n2)

Number of observations in the second sample

Significance Level (alpha)

The threshold for statistical significance

Result

Enter values above and click Calculate to see your result.

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Formula

F = s1 squared / s2 squared (larger variance in numerator)

Divide the larger sample variance by the smaller sample variance. The F-statistic follows an F-distribution with (n1 - 1) numerator and (n2 - 1) denominator degrees of freedom under the null hypothesis of equal variances.

Worked Example

Variance 1 = 25 (n1 = 20), Variance 2 = 10 (n2 = 18), alpha = 0.05 Step 1: F = 25 / 10 = 2.50 Step 2: Numerator df = 20 - 1 = 19 Step 3: Denominator df = 18 - 1 = 17 Step 4: For F(19, 17), approximate p-value is around 0.04 Step 5: Since p < 0.05, reject H0 Result: F = 2.50, p = 0.04. Evidence that the two population variances are not equal.

What Is an F-Test for Variance Equality?

The F-test for equality of variances determines whether two populations have the same variance. It is a prerequisite for choosing between the pooled t-test (equal variances) and Welch's t-test (unequal variances). The F-test is also the foundation of ANOVA.

The F-statistic is always the ratio of the larger variance to the smaller variance, so F is always at least 1
Under H0, the F-statistic follows an F-distribution with (n1-1, n2-1) degrees of freedom
The F-test is sensitive to departures from normality, more so than the t-test
Levene's test is a more robust alternative when normality is questionable
The F-distribution is used in ANOVA, regression analysis, and comparing nested models

The F-test is commonly used as a preliminary test before running a two-sample t-test. If variances are equal, use the pooled t-test. If unequal, use Welch's t-test.

You can also calculate changes using our T-Test Calculator, Chi-Square Calculator, Degrees of Freedom Calculator or Test Statistic Calculator.

Frequently Asked Questions

What is the F-test used for?

The F-test compares two sample variances to determine if the populations have the same variability. It is a prerequisite for the two-sample t-test and forms the basis of ANOVA.

Why is F always greater than or equal to 1?

By convention, the larger variance goes in the numerator. Since variance is always positive, dividing a larger number by a smaller one gives at least 1.

What are the assumptions of the F-test?

Both samples must be independent, drawn from normally distributed populations. The F-test is quite sensitive to the normality assumption. Consider Levene's test as a more robust alternative.

How does the F-test relate to ANOVA?

ANOVA uses the F-test to compare variance between groups to variance within groups. If between-group variance is significantly larger (large F), at least one group mean differs.

What if my F-value is close to 1?

An F near 1 means the two sample variances are similar. You would fail to reject the null hypothesis and conclude the variances are not significantly different.

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