Van der Waerden Normal Scores Test
Also known as: normal scores test, Van der Waerden k-sample test, Van der Waerden Testi — Normal Skor
The Van der Waerden test is a nonparametric k-sample hypothesis test that converts observations into normal scores — the quantiles of a standard normal distribution — before comparing groups. Introduced by Bartel Leendert van der Waerden in 1952, it can achieve higher statistical power than the Kruskal-Wallis test when the underlying distributions are symmetric, making it a compelling bridge between rank-based and parametric methods.
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When to use it
Use the Van der Waerden test to compare two or more independent groups on a continuous outcome when the normality assumption required by ANOVA cannot be met. The test performs best when the distributions of all groups are approximately symmetric, even if they are not normal. Key assumptions are: (1) groups are mutually independent; (2) the outcome is measured on at least an ordinal continuous scale; (3) each group's distribution is roughly symmetric in shape. The minimum recommended combined sample size is 15. When distributions are markedly skewed or heavily tied, Kruskal-Wallis is more robust. For two groups only, Mann-Whitney U is the standard nonparametric alternative.
Strengths & limitations
- Achieves higher statistical power than the Kruskal-Wallis test under symmetric distributions by using the information content of normal scores rather than plain ranks.
- Does not require normally distributed data, making it suitable when ANOVA assumptions cannot be verified.
- The normal-score transformation is well-grounded in statistical theory and produces a test statistic with a known chi-squared null distribution.
- Requires approximately symmetric distributions to outperform Kruskal-Wallis; under heavy skewness its power advantage diminishes.
- More computationally involved than Kruskal-Wallis because it requires computing inverse-normal (probit) values for every observation.
- Less widely implemented in standard statistical software than Kruskal-Wallis, making replication and reporting less straightforward.
Frequently asked
How does the Van der Waerden test differ from the Kruskal-Wallis test?
Both are nonparametric k-sample tests for location differences, but they use different transformations of the data. Kruskal-Wallis replaces observations with their ranks; Van der Waerden replaces them with normal scores — the inverse-normal transform of the fractional rank. The normal-score transformation preserves more of the information in the data, so when group distributions are symmetric the Van der Waerden test is more powerful. Under strong skewness, Kruskal-Wallis is generally safer.
Can the test be used for just two groups?
Yes, the test extends to the two-sample case (k = 2). In that setting it is closely related to the Wilcoxon rank-sum / Mann-Whitney U test but uses normal scores instead of plain ranks. For k = 2 the chi-squared statistic with 1 degree of freedom is equivalent to a squared normal-score statistic.
What should I do after obtaining a significant result?
A significant A statistic tells you only that the group distributions are not all identical. You need post-hoc pairwise comparisons — for example, pairwise Van der Waerden tests or Dunn's test with appropriate correction for multiple comparisons (e.g. Bonferroni or Holm) — to determine which specific pairs of groups differ.
Is the chi-squared approximation always valid?
The chi-squared approximation for A works well with moderate to large samples. With a total combined sample size below about 15, or with very unequal group sizes, exact permutation-based critical values are more reliable. Check that no single group is too small relative to the total.
Sources
- van der Waerden, B.L. (1952). Order Tests for the Two-Sample Problem and Their Power. Indagationes Mathematicae, 14, 453–458. link ↗
How to cite this page
ScholarGate. (2026, June 1). Van der Waerden Normal Scores Test. ScholarGate. https://scholargate.app/en/statistics/van-der-waerden-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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- Jonckheere-Terpstra TestStatistics↔ compare
- Kruskal-Wallis testStatistics↔ compare
- Mann-Whitney U testStatistics↔ compare
- One-way ANOVAStatistics↔ compare
- Siegel-Tukey testStatistics↔ compare