Kruskal-Wallis ANOVA — Previous Year Questions

📊 Topic Overview

Metric	Value
Papers Appeared	9 of 27 (33.3%)
Priority	🟠 HIGH
Section Distribution	Sec A: ~1
Most Repeated	"Discuss Kruskal-Wallis analysis of variance." / "Explain step-by-step procedure for computation of Kruskal-Wallis ANOVA."

🎯 Must-Know Concepts

1. Kruskal-Wallis is the non-parametric alternative to one-way ANOVA
2. Used when comparing three or more independent groups on ordinal data
3. Based on ranks of all scores combined
4. H statistic follows a chi-square distribution
5. Formula and computation procedure

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🔥 Most Repeated Question

Common phrasings:

• "Discuss Kruskal-Wallis analysis of variance."
• "Explain step by step procedure for computation of Kruskal-Wallis ANOVA with an example."
• "When do we use Kruskal-Wallis Analysis of Variance?"
• "Kruskal-Wallis ANOVA test (short note)."

Answer (Section B — ~250 words):

Kruskal-Wallis One-Way Analysis of Variance is a non-parametric statistical test used to determine whether three or more independent groups differ significantly on a variable. It is the non-parametric equivalent of one-way ANOVA.

When to Use:

1. Data is measured on an ordinal scale
2. The assumptions of one-way ANOVA (normality, homogeneity of variance) cannot be met
3. Sample sizes are small
4. Comparing three or more independent groups

Step-by-Step Computation Procedure:

Step 1: Rank all scores from all groups combined, from lowest (rank 1) to highest. Handle ties with average ranks.

Step 2: Sum the ranks for each group: R₁, R₂, R₃, ...

Step 3: Apply the Kruskal-Wallis H formula: $H = \frac{12}{N(N+1)} \sum_{j=1}^{k} \frac{R_j^2}{n_j} - 3(N+1)$

Where:

• N = total number of observations across all groups
• k = number of groups
• nj = number of observations in group j
• Rj = sum of ranks in group j

Step 4: Degrees of freedom: df = k − 1

Step 5: Compare H obtained with χ² critical at df = k − 1 at α = 0.05

Decision: If H obtained ≥ χ² critical → Reject H₀ → At least one group differs significantly

Note on Ties: When there are many tied ranks, a correction factor is applied to H. For small numbers of ties, the correction has minimal effect.

Advantages over one-way ANOVA:

• Does not require normality
• Suitable for ordinal data
• Valid for small samples

Limitations:

• Less powerful than ANOVA when parametric assumptions are met
• Rank-based, so loses some information from actual values
• Only tells us that groups differ — not which specific groups differ (post-hoc tests required)

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🧠 Memory Aid

Kruskal-Wallis vs ANOVA:

• One-way ANOVA = parametric, interval/ratio data
• Kruskal-Wallis = non-parametric, ordinal/ranked data
• Both test whether 3+ groups differ; H statistic ~ χ²

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📚 Related Content

• Kruskal-Wallis ANOVA: Introduction
• Kruskal-Wallis: Step-by-Step Procedure
• ANOVA vs Kruskal-Wallis Comparison

✅ Self-Assessment

1. When would you use Kruskal-Wallis instead of one-way ANOVA?
2. Write the formula for the H statistic.
3. What distribution does the H statistic follow?
4. What are the degrees of freedom for Kruskal-Wallis with 4 groups?
5. What is the first step in computing Kruskal-Wallis?

💡 Exam Tips

• Section C (short note): Non-parametric alternative to ANOVA → Used for 3+ groups, ordinal data → Based on ranks → H ~ χ² → df = k−1. That is sufficient for 3 marks.
• Section B (6 marks): Add the formula, full procedure steps, and comparison with ANOVA.
• The formula must be written out for Section B — it is a key mark-earner.
• Remember: H is compared to χ² critical, not F critical.