When should I use Friedman instead of repeated measures ANOVA?

Use the Friedman test when your data are not normally distributed, measured on an ordinal scale (e.g., Likert items), have very small sample sizes where normality cannot be verified, or contain outliers that would distort parametric results. It is specifically designed for repeated measures or matched designs where the same subjects are measured under three or more conditions.

How do I interpret Kendall's W?

Kendall's W (coefficient of concordance) is the effect size measure for the Friedman test, ranging from 0 to 1. W = 0 indicates no agreement among rankings (no difference), while W = 1 indicates perfect agreement (maximum difference). Generally, W 0.5 is a large effect.

How do I report Friedman test results in APA format?

Report the chi-square statistic, degrees of freedom, p-value, and Kendall's W following APA 7th edition. For example: 'A Friedman test indicated a significant difference among conditions, chi-square(2) = 12.40, p = .002, W = .62.' When significant, also report post-hoc pairwise comparison results with Bonferroni correction.

Friedman Test Calculator

Non-parametric alternative to repeated measures ANOVA. Compare ranks across three or more related conditions measured on the same subjects.

What is the Friedman Test?

The Friedman test is a non-parametric statistical test used to detect differences across three or more related groups (repeated measures). It is the non-parametric alternative to repeated measures ANOVA. Developed by Milton Friedman in 1937, the test ranks observations within each subject across conditions and tests whether the mean ranks differ significantly among conditions. It is widely used in medicine, psychology, and education for pre-post-follow-up designs and within- subject experiments.

When to Use the Friedman Test

Use the Friedman test when you have a repeated measures or matched design with three or more conditions and one or more of the following apply: your data are measured on an ordinal scale, the assumption of normality is violated, your sample sizes are small, or your data contain outliers. Common applications include comparing treatment effects over time, evaluating product preferences from the same judges, and analyzing questionnaire responses measured at multiple time points.

Friedman Test vs. Repeated Measures ANOVA

Feature	Friedman Test	RM ANOVA
Type	Non-parametric	Parametric
Data level	Ordinal or continuous	Continuous (interval/ratio)
Normality required	No	Yes (or large n)
Design	Repeated measures / matched	Repeated measures / matched
Effect size	Kendall's W	Partial η²
Post-hoc test	Nemenyi / Bonferroni	Bonferroni pairwise

Worked Example: Friedman Test

A researcher measures pain levels of 10 patients at three time points: before treatment, 1 week after, and 4 weeks after. Since pain ratings are ordinal and the design is repeated measures, a Friedman test is appropriate.

Baseline (n=10)

72, 85, 91, 68, 77, 83, 95, 88, 74, 79

Mdn = 80.5

1 Week (n=10)

78, 89, 95, 73, 82, 87, 98, 92, 79, 83

Mdn = 85.0

4 Weeks (n=10)

82, 93, 99, 78, 86, 91, 102, 96, 84, 88

Mdn = 89.5

Results

χ²(2) = 20.00, p < .001, W = 1.00

There was a significant difference across time points, with a large effect size. Post-hoc comparisons revealed significant improvement from baseline to both follow-up time points.

Assumptions of the Friedman Test

While the Friedman test is less restrictive than repeated measures ANOVA, it still has assumptions:

1. Ordinal or Continuous Data

The dependent variable must be measured on at least an ordinal scale so that values can be meaningfully ranked within each subject.

2. Related Groups (Repeated Measures)

The same subjects must be measured under all conditions. For independent groups, use the Kruskal-Wallis H test instead.

3. Equal Sample Sizes

Each condition must have the same number of observations since each subject provides one observation per condition.

4. Random Sample

Subjects should be randomly selected from the population of interest. Non-random selection may limit the generalizability of results.

Understanding Kendall's W

Kendall's W (coefficient of concordance) is the effect size for the Friedman test. It ranges from 0 to 1, where 0 indicates no agreement in rankings and 1 indicates complete agreement.

W	Interpretation	Practical Meaning
< 0.1	Negligible	Conditions are nearly identical
0.1 - 0.3	Small	Slight consistent difference across conditions
0.3 - 0.5	Medium	Noticeable and consistent pattern
> 0.5	Large	Strong consistent difference across conditions

How to Report Friedman Test Results in APA Format

According to APA 7th edition guidelines, report the chi-square statistic, degrees of freedom, p-value, and Kendall's W:

Example Report

A Friedman test indicated a statistically significant difference in pain levels across the three time points, χ²(2) = 20.00, p < .001, W = 1.00. Post-hoc pairwise comparisons with Bonferroni correction revealed significant improvement from baseline (Mdn = 80.5) to both 1 week (Mdn = 85.0) and 4 weeks (Mdn = 89.5).

Note: Report χ² to two decimal places, degrees of freedom as an integer, and p to three decimal places. Use p < .001 when the value is below .001. Always include Kendall's W as the effect size measure.

Common Mistakes to Avoid

Using repeated measures ANOVA on non-normal data: If your repeated measures data are ordinal or clearly non-normal with small samples, use the Friedman test instead.
Using Kruskal-Wallis for related groups: The Kruskal-Wallis test is for independent groups. For repeated measures or matched designs, always use the Friedman test.
Unequal observations per condition: The Friedman test requires the same number of subjects in each condition. Missing data must be handled before analysis (e.g., listwise deletion or imputation).
Skipping post-hoc tests: A significant Friedman result only tells you that at least one condition differs. Always follow up with pairwise comparisons to identify specific differences.
Ignoring effect size: A significant p-value alone does not indicate practical importance. Always report Kendall's W alongside the test result.

Calculation Accuracy

StatMate's Friedman test calculations have been validated against R (friedman.test function) and SPSS output. The implementation uses chi-square approximation for the p-value and the jstat library for probability distributions. Tied ranks within subjects are handled using the average rank method. All results match R output to at least 4 decimal places.

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