Kruskal-Wallis Test

Created:October 31, 2024

Last Updated:September 11, 2025

This calculator helps you compare three or more independent groups using a non-parametric approach. Instead of analyzing raw values like traditional ANOVA, the Kruskal-Wallis test examines the ranks of your data, making it ideal when your data violates normality assumptions or has unequal variances.

What You'll Get:

Complete Test Results: H-statistic, p-values, and effect sizes
Rank Distribution Plot: Visual showing how ranks spread across groups
Group Comparison Table: Sample sizes, medians, and mean ranks
Effect Size Analysis: Eta-squared with practical interpretation
Post-Hoc Guidance: Recommendations for Dunn's test when significant
APA-Ready Report: Publication-quality results you can copy directly

💡 Pro Tip: If your data meets normality and equal variance assumptions, consider ourOne-Way ANOVA Calculatorinstead for greater statistical power.

Ready to analyze your groups? Start with our sample dataset to see how rank-based analysis works, or upload your own data to discover if your groups have significantly different distributions.

Calculator

1. Load Your Data

2. Select Columns & Options

Select group column:

Select value column:

Significance Level:

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Kruskal-Wallis Test

Definition

Kruskal-Wallis Test is a non-parametric method for testing whether samples originate from the same distribution. It's used to determine if there are statistically significant differences between two or more groups of an independent variable on a continuous or ordinal dependent variable. It extends the Mann–Whitney U test, which is used for comparing only two groups.

Test Statistic

H = \frac{12}{N(N+1)}\sum_{i=1}^k \frac{R_i^2}{n_i} - 3(N+1)

Where:

$N$ = total number of observations
$n_i$ = number of observations in group $i$
$R_i$ = sum of ranks for group $i$
$k$ = number of groups

Tie Correction

T = 1 - \frac{\sum_{j}(t_j^3 - t_j)}{N^3 - N}

where $t_j$ is the number of tied observations at rank $j$ .

The corrected test statistic is then calculated as:

H_{\text{corrected}} = \frac{H}{T}

Key Assumptions

Independent Samples: Groups must be independent

Ordinal Data: Variable should be ordinal or continuous

Similar Shape: Distributions should have similar shapes

Practical Example

Step 1: State the Data

Test scores from three groups:

Group 1: $7, 8, 6, 5$
Group 2: $9, 10, 6, 8$
Group 3: $10, 9, 8$

Step 2: State Hypotheses

$H_0$ : The distributions are the same across groups
$H_a$ : At least one group differs in distribution
$\alpha = 0.05$

Step 3: Calculate Ranks

Value	Rank	Rank (Adjusted for ties)	Group
5	1	1	Group 1
6	2	2.5	Group 1
6	3	2.5	Group 2
7	4	4	Group 1
8	5	6	Group 1
8	6	6	Group 2
8	7	6	Group 3
9	8	8.5	Group 2
9	9	8.5	Group 3
10	10	10.5	Group 2
10	11	10.5	Group 3

Step 4: Calculate Rank Sums

Group 1: $R_1 = 13.5\ (n_1 = 4)$
Group 2: $R_2 = 27.5\ (n_2 = 4)$
Group 3: $R_3 = 25\ (n_3 = 3)$
Total number of observations: $N = 11$

Step 5: Calculate H Statistic

H = \frac{12}{11(12)}\left(\frac{13.5^2}{4} + \frac{27.5^2}{4} + \frac{25^2}{3}\right) - 3(12) = 4.269

Here, tied ranks are:

$t_2 = 2$ (the tie occurs at rank 2)
$t_6 = 3$
$t_8 = 2$
$t_{10} = 2$

The tie correction is:

T = 1 - \frac{(2^3 - 2) + (3^3 - 3) + (2^3 - 2) + (2^3 - 2)}{11^3 - 11} = 0.968

The corrected test statistic is:

H_{\text{corrected}} = \frac{4.269}{0.968} = 4.4092

Step 6: Draw Conclusion

Referring to the Chi-square distribution table, the critical value for $\chi^2$ with $df = 2$ degrees of freedom at a significance level of $\alpha = 0.05$ is $5.991$ .

The p-value can be found using the Chi-square Distribution Calculator with $df = 2$ and $\chi^2 = 4.4092$ , which gives $p = 0.1103$ .

Since $H = 4.4092 < 5.991$ (critical value), we fail to reject $H_0$ . There is insufficient evidence to conclude that the distributions differ significantly across groups.

Effect Size

Eta-squared ( $\eta^2$ ) measures the proportion of variability in ranks explained by groups:

\eta^2 = \frac{H - k + 1}{n - k}

Where:

$H$ = Kruskal-Wallis $H$ statistic
$k$ = number of groups
$n$ = total sample size

Interpretation guidelines:

$\text{Small effect: }\eta^2 \approx 0.01 - 0.06$
$\text{Medium effect: }\eta^2 \approx 0.06 - 0.14$
$\text{Large effect: }\eta^2 > 0.14$

For our example:

\eta^2 = \frac{4.4092 - 3 + 1}{11 - 3} = \frac{2.4092}{8} = 0.301

This indicates a large effect size, suggesting substantial practical significance in the differences between groups, even though the result was not statistically significant.

Kruskal-Wallis Test

What You'll Get:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

One-Way ANOVA Calculator

Mann-Whitney U Test Calculator

Friedman Test Calculator

Dunn's Test Calculator

Learn More

Kruskal-Wallis Test

Definition

Test Statistic

Tie Correction

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Calculate Ranks

Step 4: Calculate Rank Sums

Step 5: Calculate H Statistic

Step 6: Draw Conclusion

Effect Size

Verification

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