One-way ANOVA calculator with Tukey HSD

How to use it

Put each group on its own line, optionally starting with a name and a colon (“control: 4.2 5.1 …”). The tool gives the ANOVA table, effect sizes, Welch’s ANOVA for unequal variances and Tukey’s pairwise comparisons.

Formulas

• SS_between = Σ n_i(x̄_i − x̄)², SS_within = ΣΣ(x − x̄_i)²; F = (SS_between/(k − 1)) / (SS_within/(N − k)); p is the right tail of F(k − 1, N − k).
• η² = SS_between/SS_total; ω² = (SS_between − (k − 1) MS_within) / (SS_total + MS_within).
• Welch’s ANOVA weights each mean by n_i/s_i² (Welch 1951), as R’s oneway.test.
• Tukey–Kramer: difference ± q_{α; k, N−k} √(MS_within/2 × (1/n_i + 1/n_j)); adjusted p from the studentized range distribution (Copenhaver and Holland 1988, as R’s ptukey).

Assumptions and limits

• Independent observations, roughly normal residuals within groups, and similar variances. If the largest SD is more than about twice the smallest, rely on Welch’s ANOVA; for clearly non-normal data use the Kruskal–Wallis test.
• Tukey intervals control the family-wise error over all pairs; they are not needed (or appropriate) for planned single contrasts.
• Studentized range probabilities are accurate to about 1e-7 with 10 or more error degrees of freedom and 1e-5 with 3.

Worked example

R’s PlantGrowth data (dried plant weight, control and two treatments, 10 plants each): F = 4.846 on 2 and 27 df, p = 0.0159, η² = 0.264. Tukey: trt2 − trt1 = 0.865, 95% CI 0.174 to 1.556, adjusted p = 0.0120 — the same as R’s TukeyHSD. The calculator also reproduces the NIST StRD certified ANOVA results for SiRstv and AtmWtAg.

References

• NIST/SEMATECH e-Handbook of Statistical Methods: one-factor ANOVA
• NIST/SEMATECH e-Handbook of Statistical Methods: Tukey’s method
• R documentation: TukeyHSD
• R documentation: oneway.test
• Kramer (1956), Extension of multiple range tests to group means with unequal numbers of replications, Biometrics 12(3)
• Copenhaver and Holland (1988), Computation of the distribution of the maximum studentized range statistic, J. Statist. Comput. Simul. 30
• Welch (1951), On the comparison of several mean values, Biometrika 38
• NIST Statistical Reference Datasets (StRD)