TL;DR
Non-parametric tests replace your data with their ranks, so they don't assume normality. Use them when a parametric test's assumptions clearly fail and a transform doesn't help, especially with skewed data or strong outliers. The common ones mirror the parametric tests: Mann–Whitney U (unpaired), Wilcoxon signed-rank (paired), Kruskal–Wallis (3+ groups). They are not assumption-free, and at very small n they have little power, sometimes none.
What "non-parametric" actually means
Parametric tests (t-test, ANOVA) assume the data follow a particular distribution (roughly normal) and compare means. Non-parametric tests instead rank all the values from smallest to largest and work on those ranks. Because ranks are insensitive to how extreme the largest value is, these tests are robust to skew and outliers, and they don't require normality.
The trade-off: by throwing away the actual values for their ranks, you lose some information, so non-parametric tests are slightly less powerful than the matching parametric test when the parametric assumptions do hold.
"Non-parametric" does not mean "assumption-free." Mann–Whitney and Kruskal–Wallis assume the groups have similarly shaped distributions; only then do they cleanly test a difference in medians. With very different shapes they test the more general "are these drawn from the same distribution?", still useful, but interpret accordingly.
The rank-based tests and their parametric twins
| Design | Parametric test | Non-parametric counterpart |
|---|---|---|
| Two independent groups | Unpaired t-test | Mann–Whitney U (Wilcoxon rank-sum) |
| Two paired measurements | Paired t-test | Wilcoxon signed-rank test |
| 3+ independent groups | One-way ANOVA | Kruskal–Wallis (+ Dunn's post-hoc) |
| 3+ paired/repeated measures | Repeated-measures ANOVA | Friedman test |
| Relationship between two variables | Pearson correlation | Spearman's ρ (rank correlation) |
- Mann–Whitney U asks whether values in one group tend to be larger than in the other.
- Wilcoxon signed-rank works on the within-pair differences (like the paired t-test, but on ranks of the differences).
- Kruskal–Wallis is the rank-based omnibus test for 3+ groups; a significant result tells you some group differs, so follow it with Dunn's test (with a multiplicity correction), the rank-world equivalent of an ANOVA post-hoc. See multiple comparisons.
- Spearman's ρ is just Pearson's correlation computed on ranks, for monotonic, non-linear, or outlier-prone relationships → correlation and regression.
When to reach for them, and when not to
Going non-parametric is the right move when your data are clearly non-normal (heavy skew, hard outliers) and a transform doesn't fix it, or the outcome is genuinely ordinal (a 1–5 score). But it is not the automatic answer to "my data might not be normal."
Before defaulting to a rank test, try a log transform. Right-skewed biological data (concentrations, fold-changes, expression) often become symmetric on the log scale, letting you keep the more powerful and more interpretable t-test/ANOVA, which then tests the ratio of geometric means, often the biologically natural quantity → checking assumptions.
A second, often-missed caveat: non-parametric tests need a minimum n to detect anything at all. A Mann–Whitney with n = 3 per group can return p ≈ 0.10 at best even when the two groups don't overlap, simply because there aren't enough rank orderings. With very small samples you don't escape the problem by switching tests. You lean on the data type, a Q–Q plot, and prior knowledge instead.
Reporting
Report the median and interquartile range (IQR) rather than mean ± SD, give the test statistic (U, W, or H) and the exact p-value, and (because a p-value still doesn't convey magnitude) add an effect size where you can (e.g., rank-biserial correlation, or a Hodges–Lehmann median difference with its CI) → effect size and confidence intervals.
A biology example
You compare a cytokine concentration in treated vs. control wells, n = 7 per group, and the values are strongly right-skewed with one large outlier. A log transform pulls the distribution close to symmetric, so a Welch's t-test on the log values is valid and powerful, usually the better choice. If the skew survived the transform, a Mann–Whitney U would be the fallback, reported as medians with IQR and the U statistic. What you should not do is run a t-test on the raw, skewed values and let the outlier drive the result.
Common mistakes
- Reflexively going non-parametric whenever normality is in doubt, when a simple transform would keep the more powerful parametric test.
- Expecting power at tiny n. With n = 3–4 per group a rank test may be unable to reach significance no matter how clean the separation.
- Reporting mean ± SD alongside a rank test: give median and IQR instead.
- Skipping the post-hoc after Kruskal–Wallis. A significant omnibus result doesn't say which groups differ; use Dunn's test with correction.
- Forgetting the equal-shape assumption and then interpreting a significant Mann–Whitney strictly as "different medians" when the distributions have very different shapes.
- No effect size. A rank p-value alone tells you nothing about magnitude.
Related
t-tests · ANOVA · Checking assumptions · Correlation and regression