What the F-statistic is
The F-statistic is a ratio of two variances. In its most basic form — the F-test for equality of two variances — it is defined as F = s1² / s2², where s1² and s2² are the sample variances of two independent groups. The statistic asks a simple question: is the spread in one group meaningfully larger than the spread in the other, or could the difference be chance? Because a variance can never be negative, F is always ≥ 0, and when the two population variances are truly equal the ratio tends to hover around 1.
This same ratio-of-variances idea underlies ANOVA and regression, where F = (variance explained by the model) / (unexplained variance). This calculator implements the foundational two-sample case: you supply the two sample variances (or standard deviations) and their sample sizes, and it returns F together with the two degrees of freedom that identify the correct F-distribution.
The formula
F = s1² / s2². If you enter standard deviations instead of variances, each value is squared first, since variance is the square of the standard deviation (s² = s × s). The degrees of freedom are df1 = n1 − 1 for the numerator group and df2 = n2 − 1 for the denominator group. The pair (df1, df2) selects the specific F-distribution against which you look up a critical value or p-value; unlike a t- or z-statistic, an F value cannot be interpreted without both degrees of freedom.
Which variance goes on top?
For a two-tailed test of "are these variances equal?", the standard convention is to place the larger sample variance in the numerator so that F ≥ 1, then compare it to the upper critical value at α/2. For a directional (one-sided) hypothesis — for example "is machine A's variability greater than machine B's?" — keep the variances in the order your hypothesis specifies, even if that produces F below 1.
Why it is needed
Comparing means gets most of the attention, but many real questions are about spread. A manufacturer may need two production lines to be equally consistent, not just equal on average. A lab may need to confirm that a new measurement method is no noisier than the old one. Before running a two-sample t-test that assumes equal variances, analysts often run an F-test (or Levene's test) to check that assumption. In every case the F-ratio turns "one group looks more variable" into a number you can test.
Common reference points
- F ≈ 1: the two sample variances are nearly identical — no evidence the population variances differ.
- F = 2.45 with df1 = 15, df2 = 20 (worked example): s1² = 24.5, s2² = 10.0 gives F = 24.5 / 10.0 = 2.45. The upper 5% critical value F(0.05, 15, 20) ≈ 2.20, so this F exceeds it and the variances differ significantly at α = 0.05 (one-tailed).
- Critical values grow as samples shrink: with small samples (few degrees of freedom) you need a much larger F to reach significance. For df1 = df2 = 5 the 5% critical value is about 5.05; for df1 = df2 = 30 it drops to about 1.84.
- Reciprocal symmetry: F(1−α, df1, df2) = 1 / F(α, df2, df1). Swapping the two groups inverts the F-ratio and swaps the degrees of freedom.