Friedman Test#

The Friedman test is the nonparametric alternative to repeated measures analysis of variance (Repeated Measures ANOVA). It tests whether the distributions of three or more related samples differ significantly.

When to Use#

Use the Friedman test when you want to:

Compare three or more related measurements (e.g., multiple time points)
The dependent variable is at least ordinally scaled
The assumptions of Repeated Measures ANOVA (normality, sphericity) are not met
The sample is small

Assumptions#

Repeated measures (related samples)
At least ordinal scale of measurement for the dependent variable
Random sampling
The blocks (subjects) are independent of one another

Formula#

Within each block (subject), the $k$ measurements are ranked. The test statistic $\chi^2_F$ is calculated as:

\chi^2_F = \frac{12}{nk(k+1)} \sum_{j=1}^{k} R_j^2 - 3n(k+1)

where $n$ is the number of blocks (subjects), $k$ is the number of conditions, and $R_j$ is the rank sum of condition $j$ .

Example#

Practical Example: Wine Tasting

A sommelier has 12 participants rate three different wines on a scale from 1–10:

Wine A: Red wine from France
Wine B: Red wine from Italy
Wine C: Red wine from Spain

Each person rates all three wines (repeated measures). Since the rating scale is ordinal and the normality assumption is questionable, the Friedman test is used. If the result is significant, post-hoc tests follow (e.g., Wilcoxon tests with Bonferroni correction).

Effect Size#

Kendall's coefficient of concordance $W$ as effect size:

W = \frac{\chi^2_F}{n(k-1)}

Effect Size	Kendall's W
Small	0.1
Medium	0.3
Large	0.5