Non-parametric Tests Cheat Sheet

Key Facts

• Use a non-parametric test when data are ordinal, strongly skewed, have major outliers, or do not reasonably satisfy normality assumptions.
• For a sign test with $n$ nonzero paired differences, the number of positive signs follows $X \sim \operatorname{Binomial}(n, 0.5)$ under the null hypothesis.
• The Wilcoxon signed-rank test ranks the absolute paired differences and uses the sum of signed ranks, often written as $W^{+}$ or $W^{-}$.
• For the Mann-Whitney test, one test statistic is $U_1 = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1$, where $R_1$ is the rank sum for group $1$.
• For the Kruskal-Wallis test, $H = \frac{12}{N(N+1)}\sum \frac{R_i^2}{n_i} - 3(N+1)$, where $R_i$ is the rank sum for group $i$.
• For a chi-square goodness-of-fit or independence test, $\chi^2 = \sum \frac{(O-E)^2}{E}$, where $O$ is observed count and $E$ is expected count.
• For a chi-square test of independence in an $r \times c$ table, the degrees of freedom are $df = (r-1)(c-1)$.
• Spearman rank correlation measures monotonic association and can be computed by $\rho_s = 1 - \frac{6\sum d_i^2}{n(n^2-1)}$ when there are no tied ranks.

Vocabulary

Non-parametric test

A statistical test that does not require a specific population distribution such as a normal distribution.

Rank

A rank is the position of a data value after all values are ordered from smallest to largest or largest to smallest.

Ordinal data

Ordinal data are values with a meaningful order, such as ratings, rankings, or survey levels, but unequal or unknown spacing.

Mann-Whitney U test

The Mann-Whitney $U$ test compares two independent groups by ranking all observations together and comparing rank sums.

Wilcoxon signed-rank test

The Wilcoxon signed-rank test compares paired measurements by ranking the sizes of the nonzero differences and keeping their signs.

Chi-square test

A chi-square test compares observed categorical counts with expected counts using $\chi^2 = \sum \frac{(O-E)^2}{E}$.