One-Way ANOVA Calculator

Compare means across three or more groups. Enter your data, and we calculate the F-statistic, p-value, and pairwise comparisons with Tukey HSD. No signup.

F —
DF (BETWEEN) —
DF (WITHIN) —
P —
η² —
Please enter data for at least two groups (minimum 2 observations per group).

§5Assumptions & when to use this test

Independence

Observations within each group must be independent. Repeated measures or clustered data violate this; use a repeated-measures ANOVA or mixed model instead.

Approximate normality

Each group should be approximately normally distributed. The one-way ANOVA is fairly robust to small departures, especially with larger samples. With severe non-normality, consider the Kruskal–Wallis test (the non-parametric alternative).

Homogeneity of variance

The variance (or standard deviation) should be roughly equal across groups. You can test this with Levene's test (compute the F-statistic for the absolute deviations from each group's median). If variances are very different, consider Welch's ANOVA or a transformation.

Why not just run multiple t-tests?

Each t-test has a false-positive rate (Type I error). If you run three t-tests at α = 0.05, your true family-wise error rate balloons to about 14%, not 5%. ANOVA controls the family-wise error rate and is the correct approach. See the t-test page for pairwise comparisons between just two means.

§6FAQ

What does the p-value mean?

The p-value is the probability of observing a test statistic this large or larger if the null hypothesis (H₀: all group means are equal) were true. A p-value < 0.05 means either the groups truly differ or we have observed an unlikely event. It does not prove that H₀ is false, only that the data provide evidence against it.

What is η² (eta-squared)?

Eta-squared (η²) is the proportion of total variance explained by group membership. η² = SSB / SST. Values near 0.01 are small, 0.06 are medium, and 0.14+ are large (Cohen's conventions). Use the effect-size calculator for interpretation.

When should I use Tukey HSD versus other post-hoc tests?

Tukey HSD is the standard choice when you have many groups and want to compare all pairs while controlling family-wise error. If you have only one or two comparisons of interest, LSD is more powerful. Bonferroni is conservative and works well with few comparisons; Scheffé allows complex contrasts. Most textbooks default to Tukey.

Can I use one-way ANOVA with unequal sample sizes?

Yes. The formulae adjust for unequal ni. The only downside is reduced power (statistical sensitivity) compared to equal-sized groups. Collect equal sample sizes when you can.

What is the critical F-value?

The critical F-value at α = 0.05 is the threshold beyond which we reject H₀. It depends on dfB and dfW. We calculate and mark it on the curve automatically.

§7Sources