Kruskal Wallis Test Calculator

Kruskal-Wallis Test

Group 1 Data

Group 2 Data

Group 3 Data

Significance Level (α) Default threshold is 0.05 (5%)

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

Non-Parametric Multi-Group Test

What Is the Kruskal-Wallis Test?

The Kruskal-Wallis Test (also referred to as the one-way ANOVA on ranks) is a non-parametric statistical hypothesis test used to determine whether there are statistically significant differences among three or more independent groups.

It is the non-parametric extension of the Mann-Whitney U Test for $k \ge 3$ groups, and serves as the non-parametric alternative to the standard One-Way ANOVA when normality or homogeneity of variance assumptions are violated.

When to Use It

Three or More Groups: Comparing independent samples across 3 or more experimental conditions or demographic cohorts.
Non-Normal Distributions: Evaluating continuous datasets with skewed distributions or heavy outliers.
Ordinal & Survey Data: Comparing Likert-scale responses, customer satisfaction ratings, or performance tiers.
Alternative to One-Way ANOVA: When residuals fail Shapiro-Wilk normality tests or Levene's test for homoscedasticity.

The Mathematical Formulas

$$H = \frac{12}{N(N+1)} \sum_{j=1}^k \frac{R_j^2}{n_j} - 3(N+1)$$ $$C = 1 - \frac{\sum (t_i^3 - t_i)}{N^3 - N}, \quad H_{\text{adj}} = \frac{H}{C}$$ $$df = k - 1, \quad p = 1 - F_{\chi^2}(H_{\text{adj}}; df)$$ $$\eta_H^2 = \frac{H_{\text{adj}} - k + 1}{N - k}, \quad \varepsilon^2 = \frac{H_{\text{adj}}}{N - 1}$$

Where $k$ is the number of groups, $N = \sum n_j$ is the total sample size, $R_j$ is the sum of ranks in group $j$, $t_i$ is the number of tied observations in tie cluster $i$, $C$ is the tie correction factor, and $\eta_H^2$ is the rank eta-squared effect size.

Worked Example

Scenario: 3 training methods are evaluated with scores: Group 1: $[15, 18, 20, 25]$, Group 2: $[22, 27, 30, 32]$, Group 3: $[35, 38, 40, 42]$ at $\alpha = 0.05$.

Step 1: Rank All $N = 12$ Observations:
$15(1), 18(2), 20(3), 22(4), 25(5), 27(6), 30(7), 32(8), 35(9), 38(10), 40(11), 42(12)$.

Step 2: Calculate Rank Sums ($R_j$):
$R_1 = 1+2+3+5 = 11, \quad R_2 = 4+6+7+8 = 25, \quad R_3 = 9+10+11+12 = 42$.

Step 3: Compute $H$-Statistic:
$H = \frac{12}{12(13)}\left[\frac{11^2}{4} + \frac{25^2}{4} + \frac{42^2}{4}\right] - 3(13) = \frac{1}{13}[627.5] - 39 = 9.2692$.

Step 4: Chi-Square P-Value ($df = 3 - 1 = 2$):
$p = 1 - F_{\chi^2}(9.2692; 2) = 0.0097$.

Conclusion: Since $p = 0.0097 \le 0.05$, reject $H_0$. There is a statistically significant difference in median performance between the three training methods.

Computation Complete

Kruskal-Wallis Results

Computed Test Statistic & P-Value

p = 0.0097

Reject Null Hypothesis (Statistically Significant at α = 0.05)

H-Statistic 9.2692

Degrees of Freedom df = 2

Eta-Squared (η²_H) 0.8077

Epsilon-Squared (ε²) 0.8427

Total Sample Size N = 12

Number of Groups k = 3

Kruskal Wallis Test Calculator

Kruskal-Wallis Test

Kruskal-Wallis Results

Step-by-Step Calculation

Interpretation

Chi-Square Distribution & Critical Rejection Region

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