Math & Statistical
Tukey HSD & Tukey-Kramer Post-Hoc ANOVA Pairwise Test Formula
Calculate Tukey Honestly Significant Difference HSD = q_crit * SQRT(MS_W / n) for balanced groups and the Tukey-Kramer Studentized Range statistic q = ABS(M1 - M2) / SQRT((MS_W / 2) * (1/n1 + 1/n2)) for unequal sample sizes.
Pinpoint WHICH treatment pairs differ after a significant One-Way ANOVA!
A significant One-Way ANOVA F-test tells you at least one treatment mean differs from the others—but running separate t-tests on every pair inflates family-wise Type I error! John Tukey's Honestly Significant Difference (HSD) controls family-wise alpha across all k * (k - 1) / 2 pairwise comparisons using the Studentized Range critical value q_crit(alpha, k, df_W)!
For balanced groups of equal size n, any mean gap exceeding HSD = q_crit * SQRT(MS_W / n) (where MS_W is ANOVA Within-Group Mean Square Error) is statistically significant! When group sample sizes n_1 and n_2 are unequal, Clyde Kramer's generalization—the Tukey-Kramer Test—computes Studentized Range test statistic q_stat = ABS(Mean_1 - Mean_2) / SQRT((MS_W / 2) * ((1 / n_1) + (1 / n_2))) and declares significance whenever q_stat > q_crit!
| Pairwise Comparison (MS_W = 18.0, q_crit = 3.53 at alpha = 0.05) | Sample Sizes (n1, n2) | Tukey-Kramer q_stat | Decision vs q_crit = 3.53 |
|---|---|---|---|
| Balanced A (84.2) vs B (76.5) | n1 = 10, n2 = 10 | 7.7 / 1.3416 = 5.739 (HSD = 4.736) | Significant (5.739 > 3.53) |
| Unbalanced A (84.2) vs C (76.5) | n1 = 10, n2 = 6 | 7.7 / 1.5492 = 4.970 | Significant (4.970 > 3.53) |
| Small Gap D (79.0) vs B (76.5) | n1 = 10, n2 = 10 | 2.5 / 1.3416 = 1.863 | Not Significant (<= 3.53) |
Tukey-Kramer Studentized Range Statistic q_stat (Equal or Unequal n)
Evaluates q_stat = ABS(B2 - B3) / SQRT((B6 / 2) * ((1 / B4) + (1 / B5))), reducing identically to balanced Tukey when n1 = n2.
=IF(OR(B4 < 2, B5 < 2, B6 <= 0), "Invalid Sample Size or MS_Within", ROUND(ABS(B2 - B3) / SQRT((B6 / 2) * ((1 / B4) + (1 / B5))), 3))Balanced Tukey Minimum Significant Difference HSD & Pairwise Verdict
Compares Tukey-Kramer q_stat against Studentized Range critical value q_crit in B7.
=IF((ABS(B2 - B3) / SQRT((B6 / 2) * ((1 / B4) + (1 / B5)))) > B7, "Significant Difference (Reject H0)", "Not Significant")Try the example
Use the original values or change them to test your understanding.