Non-Parametric Tests: Wilcoxon, Kruskal-Wallis, Friedman and Spearman - When and How
When to switch from a parametric to a non-parametric test, which test replaces which, how to run each in SPSS, what to report (medians, Z, H) and how to compute the effect size.
Part of the series: Statistical Analysis for a Seminar Paper or Thesis: The Complete Guide
Non-parametric tests are the "plan B" for data that doesn't meet the assumptions of t-tests, ANOVA and Pearson. They compare ranks (the order of values) rather than means, so they don't need normality and are less sensitive to outliers.
When to switch
- The outcome is ordinal - a ranking, a single Likert item, a severity grade.
- A small sample (under about 20-30 per group) with a clear departure from normality - see Testing normality.
- Extreme outliers you have no reason to remove.
With a large sample and reasonable distributions, the parametric test is better - it has more power.
What replaces what
| Parametric test | Non-parametric alternative | SPSS (Legacy Dialogs) |
|---|---|---|
| Independent t | Mann-Whitney | 2 Independent Samples |
| Paired t | Wilcoxon | 2 Related Samples |
| One-way ANOVA | Kruskal-Wallis | K Independent Samples |
| Repeated-measures ANOVA | Friedman | K Related Samples |
| Pearson correlation | Spearman | Correlate → Bivariate → Spearman |
All under Analyze → Nonparametric Tests → Legacy Dialogs.
Wilcoxon (before/after)
Compares two measurements of the same people using the ranks of the differences. Output: Z and Asymp. Sig. in Test Statistics, plus the counts of positive and negative ranks (how many improved, how many got worse). Effect size: r = Z / √N (N = number of participants).
Kruskal-Wallis (3+ groups)
The alternative to a one-way ANOVA. Output: H (shown as Kruskal-Wallis H or Chi-Square), df and p. If significant, compare pairs with Mann-Whitney tests and a Bonferroni correction (divide .05 by the number of comparisons), or use the newer Independent Samples dialog, which gives corrected pairwise comparisons automatically.
Friedman (repeated measures, 3+)
The same people in three or more conditions (e.g. three time points). Output: Friedman's χ², df and p; if significant, pairwise Wilcoxon tests with a correction.
Spearman
A rank correlation, for ordinal variables or a monotonic but non-linear relationship. Report rs; as with Pearson, the value itself is the effect size.
What to report
Non-parametric tests are reported with medians (Mdn), not means, sometimes with the interquartile range (IQR):
"Pain scores fell significantly from before treatment (Mdn = 6) to after it (Mdn = 4), Z = −3.41, p < .001, r = .54."
"Satisfaction differed significantly between the three departments, H(2) = 11.86, p = .003. Bonferroni-corrected Mann-Whitney comparisons (α = .017) showed that..."
Common mistakes
- Reporting means alongside a non-parametric test - the reader can't tell what was tested.
- Switching to non-parametric "to be safe" with a large sample and well-behaved data - you lose power.
- Forgetting the multiple-comparison correction after Kruskal-Wallis or Friedman.
- Choosing between parametric and non-parametric by which came out significant.
FAQ
Will my supervisor accept a non-parametric test? Yes, with a reason: "Given the sample size and the departure from normality (Shapiro-Wilk, p = .01), a Wilcoxon test was used."
Where does it fit? Steps 4-5 of 7: Statistical analysis for a seminar paper: the complete guide; choosing the test: How to choose a statistical test. Not sure whether your data is "normal enough"? Tell us.
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