Non-parametric alternative to Repeated Measures ANOVA for ranked or non-normal data across three or more conditions.
The Friedman Test is the nonparametric alternative to one-way Repeated Measures ANOVA. It tests whether three or more related groups (same participants measured under multiple conditions or at multiple time points) differ significantly in central tendency, without assuming normally distributed data. The test ranks each participant's scores across conditions and compares the mean ranks.
Kendall's W (concordance coefficient) measures the strength of agreement across conditions, ranging from 0 (no agreement) to 1 (perfect agreement). Post-hoc pairwise comparisons use Conover's test with Bonferroni correction. The Friedman test is used in psychology, medicine, and social sciences when repeated measures data are ordinal or non-normal.
The Friedman test ranks each participant's scores across the conditions and asks whether some conditions consistently rank higher. It is the nonparametric counterpart of the one-way repeated measures ANOVA and makes no normality assumption, which suits ordinal ratings and small samples. A significant result says the conditions differ somewhere; pairwise Wilcoxon signed-rank tests with a Bonferroni or Holm correction show where. Kendall's W is the effect size, from 0 (no agreement) to 1.
| Time Point | Median | IQR | Mean Rank |
|---|---|---|---|
| Baseline | 6.0 | 4–7 | 2.74 |
| 1 Week | 4.5 | 3–6 | 2.03 |
| 4 Weeks | 2.5 | 1–4 | 1.23 |
A Friedman test indicated significant differences in pain ratings across three time points (baseline, 1 week, 4 weeks), χ²(2) = 18.43, p < .001, Kendall's W = .31. Conover post-hoc tests revealed significant reductions from baseline to 4 weeks (p < .001, Bonferroni corrected).
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Kendall's W (coefficient of concordance) measures overall consistency across conditions. W = 0 means complete disagreement in rankings; W = 1 means perfect agreement. Values of .1, .3, and .5 are small, medium, and large effects.
Report χ²(df), N, p and Kendall's W, then the post-hoc comparisons: "A Friedman test showed that pain ratings differed across time points, χ²(2, N = 30) = 18.43, p < .001, W = .31." Give medians for each condition.
Pairwise Wilcoxon signed-rank tests with a Bonferroni or Holm correction are the most common choice; Conover and Nemenyi tests are alternatives designed for Friedman data. Report the corrected p-values.
Kendall's W runs from 0 to 1. Cohen's guidelines adapted for W treat about .1 as small, .3 as medium and .5 as large.
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