Kruskal-Wallis Test Calculator

Calculator

1. Load Your Data

Need to transform your data?

2. Select Columns & Options

Select group column:

Select value column:

Significance Level:

Exclude Outliers

Learn More

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. The parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance(ANOVA).

A significant result indicates that at least one group differs from the others, but it doesn't identify which group is different. Post-hoc tests such as Dunn's test or pairwise Mann-Whitney U test with Bonferroni correction are often used to determine which groups are significantly different.

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	Group
5	1	1
6	2.5	1
6	2.5	2
7	4	1
8	6	1
8	6	2
8	6	3
9	8.5	2
9	8.5	3
10	10.5	2
10	10.5	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

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

\varepsilon^2 = \frac{H}{n^2-1}

Where:

$H$ = Kruskal-Wallis $H$ statistic
$n$ = total sample size

Interpretation guidelines:

$\text{Small effect: }\varepsilon^2 \approx 0.01 - 0.04$
$\text{Medium effect: }\varepsilon^2 \approx 0.04 - 0.16$
$\text{Large effect: }\varepsilon^2 \geq 0.16$

For our example:

\varepsilon^2 = \frac{5.991}{11^2-1} = \frac{5.991}{120} = 0.050

This indicates a medium effect size, suggesting a moderate practical significance in the differences between groups.

Code Examples

1library(tidyverse)
2# Creating the dataset
3data <- tibble(
4  Group = c(rep("Group 1", 4), rep("Group 2", 4), rep("Group 3", 3)),
5  Value = c(7, 8, 6, 5, 9, 10, 6, 8, 10, 9, 8)
6)
7
8# Kruskal-Wallis Test
9kruskal.test(Value ~ Group, data = data)

Python

1import pandas as pd
2from scipy.stats import kruskal
3
4# Creating the dataset
5data = pd.DataFrame({
6    "Group": ["Group 1"] * 4 + ["Group 2"] * 4 + ["Group 3"] * 3,
7    "Value": [7, 8, 6, 5, 9, 10, 6, 8, 10, 9, 8]
8})
9
10# Splitting the data by group
11groups = [group["Value"].values for _, group in data.groupby("Group")]
12
13# Kruskal-Wallis Test
14stat, p_value = kruskal(*groups)
15
16# Displaying the results
17print(f"Kruskal-Wallis H-statistic: {stat}")
18print(f"p-value: {p_value}")

Alternative Tests

Consider these alternatives:

One-way ANOVA: When normality and homogeneity of variance hold
Median Test: Less powerful but more robust to outliers
Jonckheere-Terpstra Test: When groups have natural ordering

Related Calculators

Help us improve

Found an error or have a suggestion? Let us know!