Question 1

When should I use the Kruskal-Wallis test instead of one-way ANOVA?

Accepted Answer

Use Kruskal-Wallis when: (1) data is clearly non-normal (strongly skewed, heavy tails), (2) groups have unequal or very small sample sizes with non-normal distributions, (3) data is ordinal (e.g., satisfaction ratings), or (4) there are extreme outliers that would distort the F-statistic. If data appears reasonably normal with adequate sample sizes, one-way ANOVA has more statistical power. As a rule of thumb, use ANOVA when n per group ≥ 15 and data looks roughly symmetric.

Question 2

What is the null hypothesis of the Kruskal-Wallis test?

Accepted Answer

H₀: All k populations have the same distribution (equivalent to having the same median for symmetric distributions). H₁: At least one population has a different distribution/location. Rejecting H₀ indicates that at least one group differs, but does not specify which groups. For post-hoc pairwise comparisons after a significant Kruskal-Wallis, use Dunn's test with Bonferroni correction.

Question 3

How is the H-statistic calculated?

Accepted Answer

Step 1: Pool all observations from all groups and rank them from 1 (smallest) to N (largest), averaging tied ranks. Step 2: For each group j, compute the rank sum R_j. Step 3: Compute H = [12/(N(N+1))] × Σ(R_j²/n_j) − 3(N+1). Step 4: Apply a tie correction if needed. Step 5: Compare H to the chi-square distribution with df = k−1 to get the p-value.

Question 4

What does a significant Kruskal-Wallis result tell me?

Accepted Answer

A significant result (p < 0.05) tells you that at least one group distribution differs from the others. It does not tell you which groups differ. To identify specific pairs, run post-hoc pairwise Wilcoxon rank-sum tests between all pairs, applying a Bonferroni correction: use α/k(k−1)/2 as your threshold. For 3 groups, this means using α = 0.05/3 = 0.0167 for each comparison.

Question 5

Does the Kruskal-Wallis test compare medians?

Accepted Answer

Not exactly. The Kruskal-Wallis test is often described as comparing medians, but this is a simplification. Strictly, it tests whether all k populations have the same distribution. It compares medians only when the group distributions have the same shape and spread. If groups have different spreads, H can be significant due to differences in variance, not just location. For median comparisons, report medians (not means) alongside the test.

Question 6

What is the tie correction in the Kruskal-Wallis test?

Accepted Answer

When there are tied values, H is slightly underestimated. The corrected H is: H_c = H / [1 − Σt_i(t_i²−1) / (N³−N)], where t_i is the size of the ith tied group. This correction increases H slightly, making the test more accurate when many ties exist. This calculator applies the tie correction automatically.

Feature	Kruskal-Wallis	One-Way ANOVA
Normality required?	No	Yes (or large n by CLT)
Equal variances?	Not required	Recommended (Levene test)
Tests	Distributions (location)	Means
Works with ordinal?	Yes	No
Statistical power	Slightly lower for normal data	Highest for normal data
Post-hoc test	Dunn's test (Bonferroni)	Tukey's HSD, Bonferroni

Kruskal-Wallis Test Calculator

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Kruskal-Wallis vs One-Way ANOVA

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