ANOVA Calculator

Paste your groups to get the ANOVA table, effect size and Tukey post-hoc comparisons.

One-way ANOVA tests whether three or more group means differ. It splits total variation into between-group and within-group sums of squares, and F = MS between / MS within is compared with the F distribution on (k − 1, N − k) degrees of freedom. A significant F says at least one mean differs; Tukey's HSD then shows which pairs differ while holding the family-wise error rate at 5%.

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Formula

F = [Σ nᵢ(x̄ᵢ − x̄)² / (k − 1)] / [ΣΣ (xᵢⱼ − x̄ᵢ)² / (N − k)],   Tukey q = |x̄ᵢ − x̄ⱼ| / √[(MSW / 2)(1/nᵢ + 1/nⱼ)]

MS between measures how far the group means sit from the grand mean. MS within is the pooled variance inside the groups. Their ratio F follows the F distribution when all means are equal. Tukey's q is compared with the studentized range distribution for k groups and N − k degrees of freedom (the Tukey-Kramer form handles unequal group sizes). Eta squared is SS between / SS total. Welch's ANOVA weights each group by nᵢ / sᵢ² and does not assume equal variances.

Questions

How do I do a one-way ANOVA?

Paste one group per line (for example Control: 24.5 23.9 26.1) or paste columns from a spreadsheet with the group names in the first row. The calculator returns the ANOVA table with F and p, eta squared and, if you keep it ticked, the Tukey HSD comparisons.

What does a significant ANOVA p-value mean?

It means at least one group mean differs from the others. It does not say which one. Use the Tukey HSD table to see which pairs differ; its p-values are already adjusted for multiple comparisons.

When should I use Welch's ANOVA?

When group variances or group sizes differ a lot. Welch's F does not assume equal variances and keeps the false-positive rate close to 5% where the classic F test can drift. If the largest SD is more than about twice the smallest, report Welch's result.

How do I interpret eta squared?

Eta squared is the share of total variance explained by group membership. Cohen's rough guide is 0.01 small, 0.06 medium and 0.14 large. Omega squared is a less biased version and is usually a little smaller.

What is the difference between ANOVA and a t-test?

A t-test compares two means. ANOVA compares two or more at once. With two groups, ANOVA's F equals t squared and gives the same p-value as the pooled t-test.

What if my data are not normal?

ANOVA tolerates moderate skew when groups are similar in size and have more than about 15 values each. For small, skewed or ordinal data use the Kruskal-Wallis test instead.

References

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.