One-Way ANOVA Calculator

ANOVA (Analysis of Variance) tests whether the means of three or more independent groups are significantly different. It partitions the total variability in the data into variation due to group differences and variation within groups.

Despite the name, ANOVA tests differences in means, not variances. It works by comparing how much the group means vary (between-group variance) to how much the data varies within each group (within-group variance).

Running multiple t-tests inflates the Type I error rate (false positive rate). With 3 groups, you would run 3 t-tests. At α = 0.05 each, the overall chance of at least one false positive becomes approximately 1 − (0.95)³ ≈ 14%.

ANOVA controls the overall Type I error at α by testing all group differences simultaneously in a single test.

The F-statistic is the ratio of between-group variance to within-group variance. A large F means the group means vary much more than the data within groups — evidence of a real difference.

The p-value tells you the probability of observing an F this large (or larger) by chance if all group means were equal. If p < 0.05, at least one pair of group means differs significantly.

A significant ANOVA only tells you that at least one group mean differs — not which groups differ. To find out, run a post-hoc test:

• Tukey's HSD: Most commonly used; controls family-wise error rate while maximising power
• Bonferroni correction: Divide α by the number of comparisons — very conservative
• Scheffé's test: Most flexible; suitable for complex contrasts

Post-hoc tests will be added to this tool in a future update.

• Normality: Data within each group should be approximately normally distributed (robust with n > 30 per group)
• Homoscedasticity: The variances of all groups should be approximately equal (check with Levene's test)
• Independence: Observations must be independent — both within and between groups

If normality is seriously violated, use the Kruskal-Wallis test (non-parametric alternative to one-way ANOVA).