Tukey HSD Post-Hoc Test Calculator

Tukey's Honestly Significant Difference (HSD) test is a post-hoc statistical procedure used after a one-way ANOVA finds a significant overall effect. It performs all pairwise comparisons between group means while controlling the family-wise error rate, telling you exactly which pairs of groups differ significantly.

ANOVA only tells you that at least one group mean differs — it does not say which groups. Tukey HSD is used after a significant ANOVA result to identify the specific pairs of groups that are significantly different. If ANOVA is not significant, post-hoc testing is generally not warranted.

The HSD critical value is calculated as HSD = q(α, k, N−k) × √(MSE / n), where q is the studentized range statistic, k is the number of groups, N is the total number of observations, MSE is the mean square error from ANOVA, and n is the group sample size. When groups have unequal sizes, the Tukey-Kramer modification is used.

If the absolute mean difference between two groups exceeds the HSD critical value, those two groups are considered significantly different at your chosen α level. This means you can reject the null hypothesis of equal means for that specific pair.

The key assumptions are: (1) independence of observations across groups, (2) approximately normal distribution of residuals within each group, and (3) homogeneity of variances (equal variance across groups, also called homoscedasticity). Tukey HSD is relatively robust to mild violations of normality with larger sample sizes.

The most common choice is α = 0.05, meaning you accept a 5% chance of a Type I error (false positive). For more conservative testing — such as in medical research — α = 0.01 is preferred. A value of α = 0.10 is sometimes used in exploratory research where a higher false-positive rate is acceptable.

Both control the family-wise error rate for multiple comparisons, but Tukey HSD is specifically designed for all pairwise comparisons and is generally more powerful than Bonferroni in that context. Bonferroni is more conservative and can be applied to any set of comparisons, not just pairwise ones.

Tukey HSD can be applied to any number of groups (k ≥ 2), though it is most meaningful with 3 or more groups. With k groups, the number of pairwise comparisons is k(k−1)/2. For example, 4 groups yield 6 pairwise comparisons, and 5 groups yield 10.