f statistic calculator

What is the F statistic?

The F statistic is a ratio used in hypothesis testing. It compares two sources of variability: the variability explained by a model (or differences between groups) and the unexplained variability (random error within groups). If the explained variability is much larger, the F value gets bigger.

You will often see the F statistic in:

• ANOVA (analysis of variance), to compare means across 3+ groups.
• Linear regression, to test whether the model explains a meaningful amount of variance.
• Variance comparison tests, to compare two sample variances.

Core formulas

ANOVA or regression form

F = MS_between / MS_within

• MSbetween: variation due to group differences (or model effects).
• MSwithin: variation due to noise/error.

Two-variance test form

F = s₁² / s₂²

For a standard right-tail interpretation, people typically place the larger variance in the numerator so F ≥ 1. This calculator does that automatically.

How to use this calculator

Method 1: ANOVA / Regression

• Enter MSbetween and MSwithin.
• Enter numerator and denominator degrees of freedom (df1, df2).
• Click Calculate to get F and right-tail p-value.

Method 2: Two-Variance F Test

• Enter both sample variances and sample sizes.
• The tool computes df1 = n_numerator - 1 and df2 = n_denominator - 1.
• You receive F, p-value, and a quick interpretation at α = 0.05.

Worked example

Suppose an ANOVA summary gives MSbetween = 24.8 and MSwithin = 7.2 with df1 = 3 and df2 = 36. Then:

F = 24.8 / 7.2 = 3.444...

A value around 3.44 may indicate significant group differences depending on the exact degrees of freedom. The p-value calculation incorporates df1 and df2, which is why those fields matter.

How to interpret results

• Larger F generally suggests stronger evidence against the null hypothesis.
• p-value is the probability of observing an F this large (or larger) if the null is true.
• If p < 0.05, results are often called statistically significant at the 5% level.

Common mistakes to avoid

• Using standard deviation instead of variance in a two-variance F test.
• Entering incorrect degrees of freedom.
• Interpreting significance as practical importance (they are not the same).
• Ignoring assumptions (independence, normality, equal variances where required).

Final note

This calculator is great for fast checks and study use. For publication-quality analysis, also review assumptions, confidence intervals, effect sizes, and post-hoc tests where appropriate.