Kruskal Wallis Test (original) (raw)

Last Updated : 27 Jul, 2025

The **Kruskal-Wallis test (H test) is a nonparametric statistical test used to compare three or more independent groups to determine if there are statistically significant differences between them. It is an extension of the **Mann-Whitney U test, which is used for comparing two groups.

Unlike the **one-way ANOVA, which assumes normality in data distribution, the Kruskal-Wallis test does not require the data to be normally distributed. Instead, it ranks the data, making it suitable for **ordinal or non-normally distributed continuous data.

Key Characteristics

**Nonparametric: Does not assume a normal distribution, making it useful for skewed or ordinal data.
**Rank-based: Converts data to ranks instead of using raw values.
**Independent groups: Assumes that observations in different groups are independent of each other.

When to Use the Kruskal-Wallis Test

This test is applicable in the following scenarios:

When comparing three or more independent groups.
When the data is ordinal or not normally distributed.
When sample sizes are small or unequal across groups.

The Kruskal-Wallis test is useful when the assumptions for one-way ANOVA are not met. Some common examples include: Medical Research, Social Sciences, Psychology Studies.

Hypotheses in the Kruskal-Wallis Test

**Null Hypothesis (H₀): All groups have the same central tendency and come from the same population.
**Alternative Hypothesis (H₁): At least one group has a different central tendency from the others.

How the Kruskal-Wallis Test Works

Instead of analyzing means, the Kruskal-Wallis test examines differences in the rank sums of the groups. The procedure involves the following steps:

Step 1: Rank the Data

Combine all data from different groups.
Assign ranks to the data, with tied values receiving the average of their ranks.

Step 2: Compute the Rank Sum

Sum up the ranks for each group.

Step 3: Calculate the Test Statistic (H)

The Kruskal-Wallis test statistic H is given by:

H = \left[ \frac{12}{n(n+1)} \sum \frac{R_i^2}{n_i} \right] - 3(n+1)

**Where:

n = Total number of observations.
ni = Number of observations in group iii.
Ri = Sum of ranks for group iii.
k = Number of groups.

Step 4: Determine the Critical Value

Compare H to the Chi-square distribution with k − 1 degrees of freedom.
If H exceeds the critical value, reject the null hypothesis.

Interpretation of Results

If the p-value is **less than the significance level (e.g., 0.05), at least one group differs significantly.
If the p-value is **greater than the significance level, we fail to reject the null hypothesis, meaning there is no strong evidence of differences among the groups..

Example: Evaluating Training Methods

A university wants to evaluate the effectiveness of three different training methods on student performance. A sample of **20 students was divided into three groups based on their training method:

**Video Lectures
**Books and Articles
**Classroom Training

The students' examination scores are as follows:

Video Lecture	Books and Articles	Class Room Training
76	80	70
90	80	85
84	67	52
95	59	93
57	91	86
72	94	79
-	68	80

**Step 1: Identify Variables

**Independent Variable: Training method (3 levels)
**Dependent Variable: Examination scores

**Step 2: State Hypotheses

**H₀: The mean exam scores across all training methods are equal (**μ₁ = μ₂ = μ₃).
**H₁: At least one training method leads to different exam scores.

**Step 3: Rank the Data

The scores from all groups are combined, sorted in ascending order and assigned ranks. If there are ties, the average rank is assigned.

Rank	Score	Training Method
1	52	Classroom Training
2	57	Video Lecture
3	59	Books and Articles
4	67	Books and Articles
5	68	Books and Articles
6	70	Classroom Training
7	72	Video Lecture
8	76	Video Lecture
9	79	Classroom Training
10.5	80	Books and Articles
10.5	80	Classroom Training
10.5	80	Books and Articles
13	84	Video Lecture
14	85	Classroom Training
15	86	Classroom Training
16	90	Video Lecture
17	91	Books and Articles
18	93	Classroom Training
19	94	Books and Articles
20	95	Video Lecture

**Step 4: Calculate Sum of Ranks

Each group’s ranks are summed:

Training Method	Sum of Ranks (Rᵒ)
Video Lectures	66
Books and Articles	70
Classroom Training	74

**Step 5: Calculate H-Statistic

Using the formula:

H = \frac{12}{20(21)} \left[\frac{66^2}{6}+\frac{70^2}{7}+\frac{74^2}{7}\right] - 3(21)

H = 0.0938

**Step 6: Determine Critical Value

**Degrees of freedom (df) = K - 1 = 3 - 1 = 2
**Significance Level (α) = 0.10
**From the chi-square table: X^2_{0.10,2} = 4.605

**Step 7: Compare H with Critical Value

**If H_{calc} < X^2, **fail to reject H₀ (no significant difference).
**If H_{calc} > X^2, **reject H₀ (significant difference exists).

Since **0.0938 < 4.605, we **fail to reject H₀, meaning there is **no significant difference in students' performance across the three training methods.