Friedman test
Also known as: Friedman two-way analysis of variance by ranks, Friedman rank test, Friedman Testi
The Friedman test is a nonparametric hypothesis test that compares three or more related conditions measured on the same blocks or subjects, serving as the rank-based alternative to repeated-measures ANOVA. It was introduced by Milton Friedman in 1937 and works on ordinal or continuous data without assuming normality.
Key highlights
- Makes no normality assumption, working on ordinal as well as continuous data.
- Handles three or more related conditions in a single test using a within-subject blocking design.
- Robust to outliers because it operates on ranks rather than raw values.
Intuition
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How it works
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When to use it
Use it when the same subjects or blocks are measured under three or more conditions (repeated measures or a blocking design) and the outcome is at least ordinal. It is the natural choice when the normality assumption behind repeated-measures ANOVA fails or sphericity is violated. A reasonable number of blocks is needed: with fewer than about 15 blocks the chi-square approximation becomes unreliable and a permutation test is preferable.
Strengths & limitations
- Makes no normality assumption, working on ordinal as well as continuous data.
- Handles three or more related conditions in a single test using a within-subject blocking design.
- Robust to outliers because it operates on ranks rather than raw values.
- Only tells you that conditions differ overall, not which pairs differ — post-hoc tests are required.
- The chi-square approximation needs an adequate number of blocks; with fewer than about 15 it can mislead.
- Being rank-based, it can be less powerful than repeated-measures ANOVA when the parametric assumptions actually hold.
Common pitfalls
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Applications
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Frequently asked
How is the Friedman test different from the Kruskal-Wallis test?
Both are rank-based and nonparametric, but the Kruskal-Wallis test compares three or more independent groups, while the Friedman test compares three or more related conditions measured on the same blocks or subjects. Choose Friedman whenever your design is repeated-measures or blocked.
What post-hoc test should follow a significant result?
A significant Friedman test tells you only that the conditions are not all equal. Follow it with pairwise comparisons such as the Nemenyi test, applying a Bonferroni or Holm correction to control the family-wise error rate, to find which specific conditions differ.
How do I report effect size?
Kendall's W (the coefficient of concordance) is the usual effect-size measure, ranging from 0 to 1, where higher values indicate stronger agreement in how the conditions are ranked across blocks. Report it alongside the test statistic and p-value.
How many blocks do I need?
The chi-square approximation relies on having enough blocks; with fewer than about 15 it can be inaccurate. In that case use a permutation-based version of the test instead.
Sources
- 1.Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association, 32(200), 675–701.
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ScholarGate. (2026, June 1). Friedman test. ScholarGate. https://scholargate.app/statistics/friedman-test