One-Way ANOVA Calculator
Compare three or more group means at once — one-way ANOVA, with the full sum-of-squares table and the effect size.
ANOVA splits the total variation into between-group and within-group parts.
How the one-way anova calculator works
ANOVA splits the total variation into between-group and within-group parts. If the groups really are alike, both estimate the same variance and their ratio F lands near 1; a large F means the group means are further apart than the noise within groups can explain.
A significant F says *some* pair differs, not which. That takes a post-hoc test — Tukey's HSD is the usual choice — because running every pairwise t-test inflates the error rate.
Formula: F = MS_between / MS_within, df = (k − 1, N − k)
Worked examples
| Inputs | F statistic | Note |
|---|---|---|
| Three groups, one clearly higher | 14.965517 | strongly significant |
| Identical groups | 0 | F = 0 |
| Four groups | 49.779762 | the fourth is higher still |
FAQFrequently asked questions
What does ANOVA test?
Whether any of several group means differ. The null hypothesis is that they are all equal.
Why not just run t-tests on every pair?
Because each test carries its own 5% error rate. With four groups that is six tests and about a 26% chance of at least one false positive.
A significant F — now what?
A post-hoc test to find which pairs differ, with a correction for multiple comparisons. Tukey's HSD is the standard.
What is eta squared?
The share of total variation explained by group membership — the effect size. It answers "how much?" where F only answers "is there any?".
What does ANOVA assume?
Independent observations, roughly normal residuals and similar variances. It is fairly robust to the first two with balanced groups; unequal variances with unequal sizes cause real trouble.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.