Kruskal–Wallis Non Parametric Hypothesis Test

Kruskal-Wallis Test

The Kruskal–Wallis Non Parametric Hypothesis Test (1952) is a nonparametric analog of the one-way analysis of variance. It is generally used when the measurement variable does not meet the normality assumptions of one-way ANOVA. It is also a popular nonparametric test to compare outcomes among three or more independent (unmatched) groups.  

Consider the Mann–Whitney test for just two groups instead of the Kruskal–Wallis test. Like the Mann-Whitney test, this test may also evaluate the differences between the groups by estimating the differences in ranks among the groups.

Generally in the ANOVA test, the assumption is that the dependent variable is drawn from a normally distributed population and also assumes that common variance across groups.  But, in Kruskal-Wallis Test, there is no necessity for these assumptions. Therefore, this test is the best option for both continuous as well as ordinal types of data.

Assumptions of the Kruskal-Wallis Test

• All samples are randomly drawn from their respective population.
• Independence within each sample.
• The measurement scale is at least ordinal.
• Mutual independence among the various samples
• Note: This test evaluates differences in mean ranks and that interpreting these differences as differences in medians requires the assumption of similarly shaped distributions across groups. If you’re not comparing similarly shaped distributions, the test may detect differences due to those inconsistent shapes rather than the differences between the central tendencies.

Uses of Kruskal-Wallis Non Parametric Hypothesis Test Test

The Kruskal-Wallis test can be used for any industry to understand the dependent variable when it has three or more independent groups. For example, this test helps to understand the student’s performance in exams. While the scores are measured on a scale from 0-100, the scores may vary based on the exam anxiety levels (low, medium, high, and severe -in this case, four different groups) of the students.

Procedure to conduct Kruskal-Wallis Test

• First pool all the data across the groups.
• Rank the data from 1 for the smallest value of the dependent variable and the next smallest variable rank 2 and so on… (if any value ties, in that case, it is advised to use mid-point), N being the highest variable.
• Compute the test statistic
• Determine critical value from the Chi-Square distribution table
• Finally, formulate a decision and conclusion

This reliance on ranks can make the Kruskal–Wallis test less powerful than a one-way ANOVA, especially when the assumptions of ANOVA are met.

Calculation of the Kruskal-Wallis Non Parametric Hypothesis Test

The Kruskal–Wallis Non Parametric Hypothesis Test compares differences in mean ranks among k groups (k > 2). Interpreting these differences as differences in medians requires the assumption of similarly shaped distributions across the groups.

The null and alternative hypotheses for the Kruskal-Wallis test are as follows:

• Null Hypothesis H₀: Population medians are equal
• Alternative Hypothesis H₁: Population medians are not all equal

As explained above, the procedure for the Kruskal-Wallis test pools the observations from the k groups into one combined sample, and then ranks from lowest to highest value (1 to N), where N is the total number of values in all the groups.

The test statistic for the Kruskal Wallis test (mostly denoted as H) is defined as follows: 

Where T_i = rank sum for the ith sample i = 1, 2,…,k

The Kruskal-Wallis test is more robust to outliers than parametric tests because it operates on ranked data rather than raw values. As a result, extreme values (high or low) tend to have less impact on the test statistic. However, in smaller datasets, even a few extreme values can still shift the rank distribution enough to influence the results, especially if those outliers affect the relative ordering between groups.

Note on Sample Sizes:

The Kruskal-Wallis test is a non-parametric alternative to one-way ANOVA and relies on ranks rather than raw values. While it doesn’t assume normality, it does require sufficient sample size for reliable results, especially when interpreting the approximate chi-square distribution of the test statistic.

To ensure validity:

1. Minimum per group:
   Aim for at least 5 observations per group, but:
   • Prefer 10+ per group for more stable and reliable p-values.
   • 20+ per group is ideal when possible, especially with tied ranks or unequal variances.
2. Minimum number of groups:
   The test requires at least 3 groups to test for differences in medians.
3. Avoid small group sizes with ties or skewed data:
   The chi-square approximation becomes unreliable with:
   • Heavily tied data
   • Unequal group sizes, especially when some groups are small
4. Small samples (<5 per group):
   • The p-value may be inaccurate under the chi-square approximation.
   • In such cases, consider using exact p-values or Monte Carlo simulations (some statistical packages support this).

Rule of Thumb Summary

Important Note on Interpretation:

While many texts describe the Kruskal-Wallis test as comparing group medians, this is only strictly valid if the distributions across groups are similarly shaped and scaled. In practice, the test detects differences in the distribution of ranks, which can result from differences in location (medians), spread (variability), or shape. Technically, the Kruskal-Wallis test is a test of stochastic dominance, not median equality. That means it’s possible to have equal medians across groups and still reject the null hypothesis if other aspects of the distributions differ. Always inspect distribution shapes before interpreting results solely in terms of medians.

Example of Kruskal-Wallis Non Parametric Hypothesis Test

In a manufacturing unit, four teams of operators were randomly selected and sent to four different facilities for machining techniques training. After the training, the supervisor conducted the exam and recorded the test scores. At 95% confidence level does the scores are same in all four facilities?

• Null Hypothesis H₀: The distribution of operator scores are same
• Alternative Hypothesis H₁: The scores may vary in four facilities

Rank the score in all the facilities

N=16

While for a right-tailed chi-square test with a 95% confidence level, and df =3, the critical χ² value is 7.81

Critical values of Chi-Square Distribution

The calculated χ² value is greater than the critical value of χ²for a 0.05 significance level. χ²_calculated >χ²_critical hence, you reject the null hypotheses

So, there is enough evidence to conclude that difference in test scores exists for four teaching methods at different facilities.

Six Sigma Black Belt Certification Kruskal-Wallis Test Questions:

Question 1: In an organization, management conducted a study comparing Purchase, Marketing, Quality, and Production groups on a measure of leadership skills. Which of the following test would an organization choose?

(A) Mood’Median test
(B) Kruskal-Wallis test
(C) Mann-Whitney U test
(D) Friedman Rank Test

Question 2: Which of the following nonparametric test use the rank sum?

(A) Runs test
(B) Mood’Median test
(C) Sign test
(D) Kruskal-Wallis test

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