StatReason

One-Way ANOVA Calculator

Compare the means of three or more independent groups in a single test, rather than running many separate pairwise comparisons.

One group per line, comma-separated values. At least 2 groups required, 3 or more recommended.

What this answers

This test answers "are the means of three or more independent groups all consistent with coming from populations with the same true mean, or does at least one group differ?" It is a single omnibus test, not a collection of pairwise comparisons, which keeps your overall false-positive rate under control in a way that running many separate t-tests would not.

When to use it, and when not to

Use this for three or more independent groups with a roughly continuous outcome. For exactly two groups, use the Independent t-Test directly instead. If your groups have clearly unequal variances, the Welch ANOVA variant is the safer choice; if your data is heavily skewed or ordinal, the Kruskal-Wallis test is a rank-based alternative.

How it is calculated

Total variation in the data is split into variation between group means (SS between) and variation within groups around their own means (SS within). Each is converted to a mean square by dividing by its degrees of freedom, and the F statistic is the ratio of the between-group mean square to the within-group mean square. A large F means the group means differ more than would be expected from within-group noise alone. Eta squared reports the proportion of total variation explained by group membership; omega squared is a less biased version of the same idea, correcting for the number of groups.

Worked example

For three groups with values 1-2-3, 4-5-6, and 7-8-9: the group means are exactly 2, 5, and 8, evenly spaced with identical within-group spread. This produces an F statistic of exactly 27 on 2 and 6 degrees of freedom, a strongly significant result, since the between-group differences here are large relative to the tiny within-group variability.

Assumption audit

Calculated from your data: each group's sample size and mean, the observed within-group variance, and whether any group has too few observations to proceed.
Evidence to review: whether the groups have comparable spread. Very different within-group variances suggest the Welch ANOVA correction is more appropriate than the standard equal-variance F-test used here.
You must verify: that your groups are genuinely independent (not repeated measurements on the same subjects, which would need a repeated-measures design instead) and that group assignment was not influenced by the outcome being measured.

What this result does not mean

A significant omnibus F-test means at least one group's mean differs from the others; it does not identify which specific groups differ, and it does not mean every group differs from every other group. Identifying specific pairwise differences requires a dedicated post-hoc comparison run only after a significant omnibus result, never as a substitute for it.

Limitations

This test assumes independent observations, comparable variances across groups, and residuals that are not wildly skewed. With very unequal group sizes or variances, treat the result cautiously and consider the Welch correction.