Dunn’s Test Calculator

Dunn's Test Calculator

Group 1 Data

Group 2 Data

Group 3 Data

P-Value Adjustment Method

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).

Non-Parametric Post-Hoc Analysis

What Is Dunn's Test?

Dunn's Test is a non-parametric post-hoc multiple comparison procedure performed following a statistically significant Kruskal-Wallis Test ($H$). It evaluates pairwise differences between group mean ranks while controlling for family-wise error inflation.

By pooling all observations across groups, assigning overall ranks, and calculating standard errors with optional tie corrections, Dunn's test identifies specifically which group pairs differ from one another without assuming normality.

When to Use It

Post-Kruskal-Wallis Follow-Up: Pinpointing exact group differences after finding an omnibus $H$ significance ($p \le 0.05$).
Non-Normal Data & Likert Scales: Comparing ordinal survey responses or skewed continuous distributions across $k \ge 3$ groups.
Equal or Unequal Sample Sizes: Robustly handles unbalanced experimental designs.
Multiple Testing Control: Adjusts $p$-values via Bonferroni or Šidák methods to prevent false positives.

The Mathematical Formulas

$$z_{ij} = \frac{\bar{R}_i - \bar{R}_j}{SE_{ij}}, \quad SE_{ij} = \sqrt{ \left( \frac{N(N+1)}{12} - \frac{\sum (t_m^3 - t_m)}{12(N-1)} \right) \left( \frac{1}{n_i} + \frac{1}{n_j} \right) }$$ $$p_{\text{raw}} = 2 \cdot (1 - \Phi(|z_{ij}|)), \quad p_{\text{adj}} = \min(1.0, \, p_{\text{raw}} \cdot m)$$

Where $\bar{R}_i$ is the mean rank of group $i$, $N$ is total observations, $t_m$ is the count of ties at rank $m$, and $m = \frac{k(k-1)}{2}$ is the total number of pairwise comparisons.

Worked Example

Scenario: 3 groups of $n=4$ ($N=12$) with group values: G1 $[15, 18, 20, 25]$, G2 $[22, 27, 30, 32]$, G3 $[35, 38, 40, 42]$ at $\alpha=0.05$ ($m=3$).

Step 1: Pool & Rank All 12 Observations:
Mean Ranks: $\bar{R}_1 = 2.75, \bar{R}_2 = 6.25, \bar{R}_3 = 10.50$.

Step 2: Pairwise Z-Statistic (Group 1 vs Group 3):
$SE_{13} = \sqrt{\frac{12(13)}{12} \left(\frac{1}{4} + \frac{1}{4}\right)} = \sqrt{13 \times 0.5} = \sqrt{6.5} = 2.5495$
$z_{13} = \frac{2.75 - 10.50}{2.5495} = \frac{-7.75}{2.5495} = -3.0398$.

Step 3: P-Value Adjustment ($m=3$):
$p_{\text{raw}} = 0.0024 \implies p_{\text{adj}} = \min(1.0, 0.0024 \times 3) = 0.0072 \le 0.05$.

Conclusion: Group 1 differs significantly from Group 3 ($p_{\text{adj}} = 0.0072$).

Post-Hoc Pairwise Complete

Dunn's Test Results

Overall Pairwise Findings

Significant Differences Detected

Pairwise rank comparisons adjusted for multiple testing (α = 0.05).

Total Groups (k) 3

Comparisons (m) 3

Total Sample (N) 12

Sig Comparisons 1 of 3

Adjustment Method Bonferroni

Family Alpha (α) 0.05

Pairwise Rank Comparisons Table

Comparison	Mean Rank Diff (R̄ᵢ - R̄ⱼ)	Std Error (SE)	z-Statistic	Unadjusted p	Adjusted p (p_adj)	Significant?

Dunn’s Test Calculator