ANOVA Calculator
Compare the means of two or more groups with one-way ANOVA.
How one-way ANOVA works
One-way analysis of variance (ANOVA) compares the means of two or more independent groups to test the null hypothesis that they are all equal.
It partitions the total variation into between-group variation (how much the group means differ from the grand mean) and within-group variation (how much observations vary within their groups), and forms their ratio, the F statistic.
A significant result means at least one group mean differs from the others. It does NOT identify which groups differ — that requires post-hoc testing, which this tool does not include in V1.
The formulas
F = MS_between / MS_within · MS_between = SS_between/(k−1) · MS_within = SS_within/(N−k)
SS_between = Σ nᵢ(x̄ᵢ − x̄)² and SS_within = ΣᵢΣⱼ(xᵢⱼ − x̄ᵢ)², where k is the number of groups and N the total observations. The p-value is the upper tail of the F distribution with (k−1, N−k) degrees of freedom.
Worked example
Three groups: 1,2,3 · 4,5,6 · 7,8,9
SS_between = 54 and SS_within = 6, so F = 27 with 2 and 6 degrees of freedom, giving a p-value of about 0.001 — highly significant. At least one group differs, but which ones is not determined by ANOVA alone.
Frequently asked questions
What does a significant ANOVA tell me?
It tells you at least one group mean differs from the others. It does not say which groups differ or by how much — use post-hoc tests or planned contrasts for that, which are beyond this tool's V1 scope.
What are the assumptions?
Independent groups, approximately normally distributed data within groups, and roughly equal variances across groups. With strong violations, consider Welch's ANOVA or the Kruskal-Wallis test.
What are the null and alternative hypotheses?
Null: all group means are equal. Alternative: at least one group mean differs. ANOVA is an overall test — a significant result is only the first step.
Related research methods
- Hypothesis Testing
- Comparison of Means
When to use this tool
Use one-way ANOVA when you compare the means of three or more independent groups on a continuous outcome.
Assumptions
- Independence of observations.
- Approximate normality within groups (or large samples).
- Approximate homogeneity of variance across groups.
Limitations
- ANOVA tells you that at least one group differs; it does not say which pairs — follow significant results with appropriate post-hoc comparisons.
- It is sensitive to outliers and unequal variances.
How to interpret the result
The F statistic compares between-group variability to within-group variability; a significant F suggests the group means are not all equal.
How to report the result
Report F, the degrees of freedom (between, within), p-value, and an effect size — for example, 'F(2, 87) = 5.12, p = 0.008, η² = 0.11'.
Next steps in your analysis
- Effect Size →
- Post-hoc comparisons
- Statistical Power →
Related tools
Related guides
- How to Choose the Right Statistical TestA practical framework for picking the right statistical test: start from your research question, data type and study design, then check the assumptions.
- T-Test vs Z-Test vs ANOVAWhen to use a z-test, a t-test or an ANOVA, the assumptions behind each, and how the tests relate — including why F equals t² for two groups.
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