Nonparametric Tests When Assumptions Fail

The one idea: T-tests and ANOVA both lean on an assumption — that the underlying data (or at least the sampling distribution of the mean) is close enough to normal for the math to hold. When that assumption clearly fails — small samples with obvious skew, wild outliers, ranked-but-not-numeric data — rank-based nonparametric tests (Mann-Whitney/Wilcoxon in place of a t-test, Kruskal-Wallis in place of ANOVA) answer the same comparison question without needing the normality assumption at all.

⚠️ The trap: "Our sample is tiny and the data is badly skewed, but the t-test is the standard tool, so we'll run it anyway and trust the p-value." Running a normality-dependent test on data that visibly isn't normal doesn't just make the test "a little less precise" — it can invalidate the p-value's meaning entirely, since the whole calibration of that p-value assumed a bell-shaped sampling distribution that isn't there. The fix isn't to distrust statistics and eyeball it instead — it's to switch to the test built for exactly this situation: convert the data to ranks and let a nonparametric test do the comparison.

The t-tests and z-tests lesson and the ANOVA lesson both compare group means, and both are built on a normality assumption (explicit for small samples; softened but not eliminated for larger ones by the Central Limit Theorem). This lesson covers what to do when that assumption doesn't hold: badly skewed data, small samples with visible outliers, or data that's genuinely ordinal (ranked categories, like satisfaction scores) rather than measured on a scale where means are meaningful.

The core trick: trade the actual values for their ranks

Every nonparametric…