Tukey HSD Post Hoc Calculator

Tukey HSD Post-Hoc Test

Group 1 Data

Group 2 Data

Group 3 Data

Family-Wise Significance Level (α) Default threshold is 0.05 (5% family-wise error rate)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Post-Hoc Multiple Comparisons

What Is the Tukey HSD Test?

Tukey's Honestly Significant Difference (HSD) Test is a post-hoc statistical procedure performed after a statistically significant One-Way ANOVA to determine exactly which specific pairs of group means differ significantly from one another.

Unlike running multiple individual independent t-tests, Tukey's HSD strictly controls the family-wise error rate ($\alpha$), preventing the inflation of Type I errors (false positives) across all pairwise comparisons using the Studentized Range distribution ($q$).

When to Use It

Post-ANOVA Follow-Up: Identifying specific group differences after a significant omnibus $F$-test ($p \le 0.05$).
All Pairwise Comparisons: Testing all possible pairs of means ($\frac{k(k-1)}{2}$ comparisons) simultaneously.
Equal or Unequal Sample Sizes: Seamlessly handles unbalanced designs via the Tukey-Kramer method.
Controlling False Positives: Maintaining the total experiment-wise error rate at exactly $\alpha = 0.05$.

The Mathematical Formulas

$$q = \frac{\bar{x}_i - \bar{x}_j}{SE_{ij}}, \quad SE_{ij} = \sqrt{\frac{MS_{\text{within}}}{2}\left(\frac{1}{n_i} + \frac{1}{n_j}\right)}$$ $$HSD_{ij} = q_{\text{crit}} \times SE_{ij}$$ $$\text{Simultaneous } (1-\alpha)\text{ CI} = (\bar{x}_i - \bar{x}_j) \pm HSD_{ij}$$

Where $MS_{\text{within}}$ is the mean square error from the one-way ANOVA, $df_{\text{within}} = N - k$, and $q_{\text{crit}}$ is the upper $\alpha$ critical value from the Studentized Range distribution with parameters $(k, df_{\text{within}})$.

Worked Example

Scenario: 3 groups of $n=4$ patients ($N=12$) with means $\bar{x}_1 = 13.0, \bar{x}_2 = 21.0, \bar{x}_3 = 29.0$, $MS_w = 6.6667$, $df_w = 9$, $\alpha = 0.05$.

Step 1: Compute Standard Error ($SE$):
$SE = \sqrt{\frac{6.6667}{4}} = 1.2910$.

Step 2: Obtain Critical Range Value ($q_{\text{crit}}$):
$q_{\text{crit}}(0.05, k=3, df=9) = 3.9486$.

Step 3: Calculate HSD Threshold:
$HSD = 3.9486 \times 1.2910 = 5.0976$.

Step 4: Pairwise Comparisons:
$|\bar{x}_1 - \bar{x}_2| = 8.0 \ge 5.0976 \implies \text{Significant } (p = 0.0034)$
$|\bar{x}_1 - \bar{x}_3| = 16.0 \ge 5.0976 \implies \text{Significant } (p < 0.0001)$
$|\bar{x}_2 - \bar{x}_3| = 8.0 \ge 5.0976 \implies \text{Significant } (p = 0.0034)$

Post-Hoc Computation Complete

Tukey HSD Results

Tukey Critical Threshold (HSD)

HSD = 5.0976

Significant Pairwise Differences Detected (p ≤ 0.05)

Critical q (q_crit) 3.9486

Mean Sq Error (MS_w) 6.6667

Degrees of Freedom df_w = 9

Groups (k) 3

Total Sample (N) 12

Total Comparisons 3

Step-by-Step Calculation

Pairwise Comparisons & Simultaneous 95% Confidence Intervals

Comparison	Mean Diff (x̄ᵢ - x̄ⱼ)	Std Error (SE)	Studentized q	Tukey HSD	Simultaneous 95% CI	p-value	Significant?

Tukey HSD Post Hoc Calculator

Tukey HSD Post-Hoc Test

Tukey HSD Results

Step-by-Step Calculation

Pairwise Comparisons & Simultaneous 95% Confidence Intervals

Interpretation

Simultaneous Confidence Intervals (Differences vs. Zero Line)

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