Definition · Plain-language
ANOVA (analysis of variance)
ANOVA, or analysis of variance, is a statistical test that compares the means of three or more groups by partitioning the total variability into between-group and within-group components.
The step most authors miss
Doing CRediT right? Don’t stop at the statement.
A CRediT statement credits you inside one paper. The recognition CRediT was built for happens when those roles are tied to you, persistently. Sign in with your ORCID — free — and claim your CRediT contributions on casrai.org, the home of the standard. They become a verified, portable part of your identity, not a line that disappears into one PDF.
Free: claim your contributions, then export a journal-ready CRediT statement, schema.org structured data, JATS XML, CSV or BibTeX — and preview your public profile. A membership publishes that profile publicly and verifies the journals you serve.
Why ANOVA instead of multiple t tests
When three or more group means must be compared, running a separate t test for every pair inflates the overall chance of a false positive, because each test carries its own error rate. ANOVA avoids this by testing all the groups in a single analysis with one controlled significance level. It answers a global question — do any of the group means differ? — rather than comparing pairs one at a time. This makes ANOVA the standard approach whenever an independent variable has three or more levels, or when several factors are examined together.
How variance is partitioned into the F-statistic
ANOVA splits the total variability in the outcome into two parts: variance between groups (differences among the group means) and variance within groups (random variation among individuals in the same group). The F-statistic is the ratio of between-group variance to within-group variance. If the groups truly share a common mean, this ratio is near one; the more the group means differ relative to the internal noise, the larger F becomes. The F-statistic is then compared against the F-distribution, using its degrees of freedom, to produce a p-value.
One-way, two-way and post-hoc tests
A one-way ANOVA examines the effect of a single factor with three or more levels. A two-way ANOVA examines two factors at once and can also detect an interaction — whether the effect of one factor depends on the level of the other. Because a significant ANOVA only tells you that some group means differ, not which ones, it is followed by post-hoc comparisons such as the Tukey HSD test, which compare pairs of groups while controlling the family-wise error rate. Effect sizes such as eta-squared describe how much variance the factor explains.
Key facts
At a glance
- Definition: compares the means of three or more groups
- Mechanism: partitions variance into between-group and within-group parts
- Test statistic: the F-statistic (between-group ÷ within-group variance)
- One-way: one factor with three or more levels
- Two-way: two factors plus their interaction
- After a significant F: post-hoc tests (e.g. Tukey HSD) locate differences
Common misconceptions
What people often get wrong
Often heard: A significant ANOVA tells you exactly which groups differ from each other.
Actually: It only shows that at least one group mean differs from the others. To identify which specific pairs differ, you run post-hoc tests such as Tukey HSD, which control the error rate across multiple comparisons.
Often heard: ANOVA analyses variances rather than means.
Actually: Despite the name, ANOVA tests whether group means differ. It does so by analysing variance — comparing between-group with within-group variability — but the conclusion is about the means, not the variances themselves.
Often heard: You should just run lots of t tests instead of a single ANOVA.
Actually: Multiple pairwise t tests inflate the family-wise false-positive rate. ANOVA tests all groups together under one controlled significance level, which is why it is the correct test for three or more groups.
Going deeper
