STAT 101
M08 · L02
Module 8 · Lesson 2
Non-Parametric Tests
Every parametric test has a rank-based twin. The study design — paired or independent, two groups or many — picks the test.
01 / 08
STAT 101
M08 · L02
Match design to test
A Test for Every Design
- Two independent → Mann-Whitney U
- Paired → Wilcoxon signed-rank
- 3+ groups → Kruskal-Wallis
- Repeated measures → Friedman
- Whole distributions → Kolmogorov-Smirnov
02 / 08
STAT 101
M08 · L02
Two independent groups
Mann-Whitney U
The U statistic
U = n_1 n_2 + \tfrac{n_1(n_1+1)}{2} - R_1
Pool, rank, sum group-1 ranks. Asks if one distribution is stochastically larger — not if means differ.
03 / 08
STAT 101
M08 · L02
Paired & multi-group
Wilcoxon, KW, Friedman
- Wilcoxon signed-rank — rank paired differences by size, keep signs
- Kruskal-Wallis — rank-based one-way ANOVA (Module 6, Lesson 3)
- Friedman — ranks within each subject; repeated measures
04 / 08
STAT 101
M08 · L02
Compare whole distributions
Kolmogorov-Smirnov
The KS statistic
D = \max_x |F_1(x) - F_2(x)|
The largest gap between two CDFs — catches differences in shape and spread, not just centre.
05 / 08
STAT 101
M08 · L02
The general engine
Permutation Tests
Permutation p-value
p = \dfrac{\#\,\text{as-extreme perms}}{\#\,\text{all perms}}
Shuffle the labels thousands of times; see where the real statistic falls. Assumption-free, works for almost any statistic.
06 / 08
STAT 101
Summary
Recap
What you learned
- Mann-Whitney (2 groups), Wilcoxon signed-rank (paired)
- Kruskal-Wallis (3+ groups), Friedman (repeated measures)
- Kolmogorov-Smirnov compares whole distributions
- Permutation tests build the null by shuffling labels
08 / 08