Chapter 13: Nonparametric Tests

Unlike traditional hypothesis tests that assume normal distributions and equal variances, nonparametric tests make fewer assumptions about the underlying data and are particularly valuable when working with ordinal or ranked data, small samples, or heavily skewed distributions. The chapter introduces the sign test, a straightforward method for testing hypotheses about medians using only the direction of differences rather than their magnitude. The Wilcoxon signed-rank test extends this approach by incorporating the ranks of differences, providing greater statistical power while remaining distribution-free. For comparing two independent samples, the Mann-Whitney U test serves as the nonparametric alternative to the independent samples t-test, ranking all observations and using these ranks to assess whether two groups differ significantly. The chapter also covers the Kruskal-Wallis test for comparing three or more independent groups, functioning as the nonparametric counterpart to one-way analysis of variance. The Spearman rank correlation coefficient is presented as a method for measuring association between two variables when the linear regression assumptions are violated or when data are naturally ranked. Additional topics include the Wilcoxon rank-sum test and various goodness-of-fit tests that evaluate whether observed data follow a specified distribution. Students learn when to apply each test based on research design and data characteristics, how to calculate test statistics using ranks rather than raw values, and how to interpret results in practical contexts. The chapter emphasizes that nonparametric methods sacrifice some power when parametric assumptions are actually met, but provide robust alternatives when standard assumptions fail, making them essential tools for real-world statistical analysis across diverse fields including psychology, medicine, and environmental science.