Friedman Test (non-parametric RM)

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Independently verified. Every statistic this test reports has been re-derived against an independent reference — never the library the pipeline itself calls — the rendered output was read back in a browser, and the result is locked with a committed regression suite.

Run this test straight away on a free built-in teaching dataset — no data of your own needed — or bring your own. Either opens the guided workspace: variable setup, assumption diagnostics, results with effect sizes and confidence intervals, figures, and APA-ready reporting.

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Non-parametric repeated-measures comparison of a DV across 3+ within-subjects conditions.

Ranks within each subject and compares sums of ranks across conditions via a χ²(k − 1) statistic. The within-subjects analogue of Kruskal-Wallis. Used when RM-ANOVA assumptions (normality, sphericity) are not tenable, the DV is ordinal, or per-subject n is too small to trust the parametric F. Reports χ², df = k−1, p, and per-condition median + IQR. Pair with pairwise Wilcoxon signed-rank with Bonferroni correction for post-hoc.

Worked example

Do raters score three product designs differently?

20 raters each evaluated all three designs (within-subjects, ordinal ratings); a Friedman test compares the ranked scores.

Result

Ratings differed across designs, χ²(2) = 22.9, p < .001, Kendall's W = .57, with Design B ranked highest.

How you'd report it (APA)

A Friedman test showed the three designs were rated significantly differently, χ²(2) = 22.9, p < .001, Kendall's W = .57.

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

  • Repeated measures on an ordinal or non-normal DV
    Each patient rates treatment satisfaction on a 5-point Likert scale at 4 visits; analysed via Friedman rather than RM-ANOVA.
  • Randomised block design (subjects as blocks)
    12 wine tasters each rate 5 different wines; Friedman tests whether the median ratings differ across the 5 wines.

When NOT to — use instead

Hypotheses

H₀: the population distributions of the DV are identical across the k within-subjects levels (equivalently, the median rank is equal across conditions).
Hₐ: at least one population distribution differs across conditions — typically interpreted as at least one median differs from the others.

Parameter tested: set of population distribution functions {F₁, …, Fₖ} across related conditions

Assumptions (and what to do if they fail)

Sufficient n_subjects for the chi-square approximation (n ≥ 10 recommended for k = 3; larger for more levels)medium

Check: n_subjects ≥ 10 for k = 3; n_subjects ≥ 15 for k = 4; larger for more levels.

If violated: With small n_subjects, the χ² approximation to the Friedman statistic can be inaccurate.

Ties are not overwhelming (tie correction can attenuate the statistic)low

Check: Inspect the per-condition boxplot; severe discretisation → many ties.

If violated: Many ties reduce the test's discriminating power.

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