How this instrument works
The F-statistic, in its simplest and most common form, is nothing more than one variance divided by another: F = s1² / s2². If the two underlying variances are truly equal, that ratio should land somewhere near 1 — some sampling noise pushes it a bit above or a bit below, but not dramatically. As the ratio drifts further from 1 in either direction, it signals a growing difference in how spread out the two groups actually are.
This ratio is the core test statistic behind the F-test for equality of two variances, and it also underlies analysis of variance (ANOVA), where it compares variance between groups against variance within groups. In every case the logic is the same: F expresses one measure of variability as a multiple of another, and that single number is then compared against a critical value from the F-distribution (which depends on both samples' degrees of freedom) to judge whether the difference is more than chance would explain.
By convention, the larger variance is placed in the numerator, which keeps F at or above 1 and matches the standard one-tailed critical-value tables most textbooks and software use. This instrument computes the ratio exactly as entered — Sample 1 variance divided by Sample 2 variance — so put your larger variance first if you want the conventional F ≥ 1 orientation.
- Enter the first group's variance into Sample 1 variance — for the standard convention, use the larger of your two variances here.
- Enter the second group's variance into Sample 2 variance — it must be greater than zero, since it's the divisor.
- Read F-statistic — the ratio of Sample 1 variance to Sample 2 variance.
- An F near 1 suggests similar variability between the two groups; an F far from 1 (in either direction) suggests one group is meaningfully more variable than the other.
- To judge statistical significance, compare this F-statistic against a critical value from the F-distribution for your two samples' degrees of freedom (each sample size minus 1).
Worked example — variances 225 and 100
Enter 225 into Sample 1 variance and 100 into Sample 2 variance. F-statistic reads 2.25, computed directly as 225 / 100 — no other steps involved.
An F of 2.25 means the first group's variance is 2.25 times larger than the second group's — a meaningfully bigger spread, worth checking against an F-distribution critical value for the two samples' degrees of freedom before concluding the difference is more than sampling noise. If the two variances had instead been equal, F would have come out to exactly 1.0, the boundary case of no difference.
Questions
What does an F-statistic of 1 mean?
It means the two variances entered are exactly equal — the numerator and denominator cancel out to 1. This is the null-hypothesis boundary case for an F-test of equal variances: an F very close to 1 offers no evidence that the two groups differ in variability, while values further from 1 in either direction suggest a real difference in spread.
Why does it matter which variance goes in the numerator?
By convention, placing the larger variance in the numerator keeps F at or above 1, matching the one-tailed critical-value tables most textbooks publish. This instrument simply divides Sample 1 variance by Sample 2 variance as entered — put them in whichever order you like, but if you want the conventional F ≥ 1 form, enter the larger variance as Sample 1.
Can F be less than 1?
Yes, whenever Sample 1 variance is smaller than Sample 2 variance — the ratio simply comes out below 1. That's mathematically valid and sometimes reported directly, but most published F-tables assume F ≥ 1 by convention, so many analysts swap the two variances (putting the larger one first) before looking up a critical value, rather than working with a sub-1 ratio directly.
How is the F-statistic different from Cohen's d?
They measure completely different things. Cohen's d compares two means, standardized by pooled spread, to answer 'how different are the typical values of these two groups?' The F-statistic here compares two variances directly to answer 'how different is the spread (variability) of these two groups?' A pair of groups can have identical means (d = 0) but very different variances (F far from 1), or vice versa.
Do I need equal sample sizes to compute or interpret F?
No — the F-statistic itself is just a ratio of two variance values and doesn't require equal sample sizes to compute. Sample sizes matter for the next step, though: looking up a critical value from the F-distribution requires each sample's degrees of freedom (sample size minus 1), and those can differ between the two groups without any problem.
Why must Sample 2 variance be greater than zero?
Because it's the divisor in F = s1² / s2² — dividing by zero is undefined. A variance of exactly zero would also mean that group's values were all identical, which isn't a meaningful comparison point for a variance ratio in the first place.