Hypothesis Tests

Kruskal-Wallis Test

Nonparametric alternative to one-way ANOVA for k samples.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Nonparametrics > Kruskal-Wallis

Description

The Kruskal-Wallis test examines whether k ≥ 2 independent samples come from the same distribution. It operates on the ranks of the pooled data and is therefore robust to departures from normality. Under H₀ the test statistic H is approximately χ²-distributed with k−1 degrees of freedom. A tie correction is applied when ties are present. Like ANOVA, Kruskal-Wallis is an omnibus test — it does not identify which group differs.

Formulas

H = \frac{12}{N ( N + 1 )} i = 1 \sum k \frac{R _{i}^{2}}{n _{i}} - 3 (N + 1)

Test statistic from per-group rank sums R_i

C = 1 - \frac{\sum ( t ^{3} - t )}{N ^{3} - N}

Tie correction factor (H is divided by C)

H₀

H_{0} : F_{1} = F_{2} = \dots = F_{k}

Null hypothesis: All distributions are identical

Assumptions

Observations are independent
Continuous (or at least ordinal) data
Distributions have similar shape under H₁ (for location test)

Limitations

χ² approximation inaccurate for very small samples
Omnibus test — does not identify which group differs

References

Kruskal, W. H. & Wallis, W. A. (1952), Use of ranks in one-criterion variance analysis, JASA, 47(260), 583–621
NIST/SEMATECH e-Handbook of Statistical Methods