What this calculator does
The F-test compares two variances by dividing one by the other, always putting the larger on top so the statistic is at least 1. Standard deviations of 12 and 7 from samples of 15 and 20 give F = 2.939.
Note that the statistic works on variances, not standard deviations, so the ratio is squared relative to what you might expect. Standard deviations of 20 and 5 are four times apart, but the F-statistic is 16. That squaring makes the test look dramatic quickly, which is one reason it should be read against the proper critical value rather than by eye.
The formula
Each standard deviation is squared to give a variance, and the larger is divided by the smaller. Degrees of freedom are one less than each sample size, in the order matching numerator and denominator. The test is notoriously sensitive to departures from normality, much more so than tests about means, which limits its practical usefulness.
| Term | Meaning |
|---|---|
| F-statistic | The ratio of the two variances, with the larger on top. |
| Numerator df | One less than the size of the sample providing the larger variance. |
| Homogeneity of variance | The assumption of equal spread, which this tests and which the pooled t-test requires. |
| Levene test | A more robust alternative that does not assume normality. |
The inputs explained
| Field | What to enter |
|---|---|
| Sample 1 standard deviation | Sample 1 standard deviation. |
| Sample 1 size | Sample 1 size. |
| Sample 2 standard deviation | Sample 2 standard deviation. |
| Sample 2 size | Sample 2 size. |
When to use it
Checking a t-test assumption
The pooled two-sample t-test assumes equal variances, and this is the classical way to check.
Comparing process consistency
Two machines or suppliers may produce the same average while differing markedly in variability, which often matters more.
Comparing measurement precision
Whether one instrument or method is more repeatable than another.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the ratio grow?
Increasing the first standard deviation against a fixed second.
| Sample 1 standard deviation | F-statistic | Degrees of freedom (numerator, denominator) | Larger variance used as numerator |
|---|---|---|---|
| s₁ = 7 | 1.000 | 14, 19 | Sample 1 |
| s₁ = 10 | 2.041 | 14, 19 | Sample 1 |
| s₁ = 12 | 2.939 | 14, 19 | Sample 1 |
| s₁ = 20 | 8.163 | 14, 19 | Sample 1 |
Questions
Why is the larger variance always on top?
It makes the statistic at least 1 and lets a single upper-tail critical value be used, which is how F-tables are constructed. Putting the smaller on top would give values below 1 requiring a lower-tail lookup, which tables do not usually provide.
Is the F-test reliable?
It is unusually sensitive to non-normality, far more so than t-tests are. Heavy tails can produce a significant result when the variances are actually equal. Levene test or the Brown-Forsythe variant are more robust and generally preferred in practice.
Should I run this before a t-test?
It is the traditional approach but is now widely discouraged, because running a preliminary test to choose the main test distorts the overall error rate. The common modern recommendation is to use Welch t-test by default and skip the variance test entirely.
What does an F of 1 mean?
That the two sample variances are identical. Under the null hypothesis of equal population variances, F should be near 1, with sampling variation pushing it above or below. How far it can stray before being surprising depends on both degrees of freedom.
For comparing several group means, see the one-way ANOVA calculator. For combining two variances, see the pooled standard deviation calculator.