F-Test Calculator for Equal Variances
Compare two sample variances with an F-test: enter both variances and sample sizes to get the F statistic and its degrees of freedom.
F-Test Calculator for Equal Variances
Calculator
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Formula: F = s₁²/s₂² (larger variance on top)
Worked example — F = 1.5625, df = (29, 24) | Below the 1.94 critical value, so no significant difference
Formula
F = s₁²/s₂² (larger variance on top)
Compares two sample variances to test if they are significantly different.
Worked Examples
Example 1: Comparing two sample variances
Problem:Sample 1: s1² = 25 with n1 = 30. Sample 2: s2² = 16 with n2 = 25.
Solution:Put the larger variance on top so F is always at least 1: F = 25 / 16 = 1.5625. Degrees of freedom: df1 = n1 - 1 = 29 (numerator), df2 = n2 - 1 = 24 (denominator). The upper 5 percent critical value F(29, 24) is about 1.94. Since 1.5625 is below 1.94, the variances are not significantly different at alpha = 0.05.
Result:F = 1.5625, df = (29, 24) | Below the 1.94 critical value, so no significant difference
Example 2: A clearly significant variance difference
Problem:Sample 1: s1² = 90 with n1 = 16. Sample 2: s2² = 18 with n2 = 16.
Solution:F = 90 / 18 = 5.00 with df = (15, 15). The upper 5 percent critical value F(15, 15) is about 2.40 and the 1 percent value is about 3.52. F = 5.00 exceeds both, so the two populations do not have equal variances. A Welch t-test rather than the pooled two-sample t-test would be the safer follow-up.
Result:F = 5.00, df = (15, 15) | Significant at alpha = 0.01, variances are unequal
Frequently Asked Questions
What does the F-test measure?
The F-test compares the ratio of two sample variances to determine whether they come from populations with equal variance. The F-statistic equals the larger variance divided by the smaller variance, so it is always ≥ 1 when arranged this way.
How do you interpret the F-statistic and p-value?
Compare the computed F against a critical value from an F-distribution table using your two degrees of freedom (df₁ = n₁ − 1, df₂ = n₂ − 1). If F exceeds the critical value at your chosen α (commonly 0.05), you reject the null hypothesis of equal variances and conclude the variances are significantly different.
When should I use an F-test vs. Levene's test?
Use the F-test for variance equality when both samples are approximately normally distributed. Use Levene's test when normality cannot be assumed, as Levene's is more robust to departures from normality and less sensitive to outliers that would inflate the F-ratio.
What are the assumptions of the F-test for equality of variances?
The two samples must be independent of each other, each sample must be drawn from a normally distributed population, and the data should be measured on a continuous ratio or interval scale. Violations of normality, especially in small samples, can substantially inflate the Type I error rate.
Why is the larger variance always placed in the numerator?
Putting the bigger sample variance on top forces F to be at least 1, so you only ever need the upper tail of the F distribution and a single critical value. The cost is that the test becomes effectively two-sided even though you read one tail, because you chose the ordering after seeing the data. To keep the true significance level at alpha for a two-sided question about equality of variances, look up the critical value at alpha/2 rather than alpha.
Which sample supplies df1 and which supplies df2?
The numerator degrees of freedom always belongs to the sample whose variance ended up on top, and the denominator degrees of freedom to the other one, each computed as that sample's size minus one. F-Test Calculator for Equal Variances reorders them automatically, so if sample 2 has the larger variance then df1 comes from n2 and df2 from n1. Reading the F table with the two values transposed is one of the most common errors and can flip your conclusion.
How does this F-test differ from the F-test used in ANOVA?
Both compute a ratio of two variance estimates and both are referred to the same F distribution, but the quantities differ. Here the ratio is of two independent sample variances and the question is whether the populations are equally dispersed. In ANOVA the numerator is the between-groups mean square and the denominator the within-groups mean square, so the same machinery is used to ask whether group means differ. Only the ANOVA version is inherently one-tailed.
What does the calculator's note about F above 2.5 actually mean?
It is a rough flag, not a decision. The genuine cut-off depends entirely on your two degrees of freedom and your chosen significance level: with large samples an F barely above 1 can be significant, while with very small samples an F of 3 or 4 may not be. Take the F value and the df pair from the output and read the exact critical value from an F table or software before you report anything.
What is the reciprocal property of the F distribution used for?
The lower and upper tails of the F distribution are mirror images with the degrees of freedom swapped: the lower critical value for df1 and df2 equals one divided by the upper critical value for df2 and df1. Older tables only printed upper-tail values, so this identity was how you obtained a lower bound. It also explains why inverting the variance ratio simply exchanges df1 and df2 and leaves the conclusion unchanged.
Should I run this test before choosing a two-sample t-test?
It was long taught as a preliminary screen to decide between the pooled-variance t-test and the Welch version. The difficulty is that the screening test itself is sensitive to non-normality and has low power in small samples, exactly where the decision matters most, and running it changes the error rate of the t-test that follows. Many statisticians now recommend simply using Welch's t-test by default, since it performs well whether or not the variances are equal.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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