What is the Friedman test used for?

The Friedman test is a nonparametric alternative to one-way repeated-measures ANOVA. It compares three or more related conditions or treatments measured on the same subjects or blocks when the assumptions of normality or sphericity are doubtful. It works on within-block ranks rather than raw values.

What are the assumptions of the Friedman test?

The main assumptions are: (1) the dependent variable is at least ordinal; (2) each block or subject is measured under each treatment level; (3) blocks are independent of each other; and (4) the ranking within each block is meaningful. The test does not require normally distributed residuals.

How do I interpret the Friedman test p-value?

The null hypothesis states that all treatments have the same distribution. A small p-value (typically below 0.05) suggests that at least one treatment differs in central tendency compared to the others. The calculator reports the Friedman chi-square statistic, degrees of freedom, and the associated p-value.

What is Kendall’s W in the context of the Friedman test?

Kendall’s W is an effect size measure for the Friedman test. It ranges from 0 (no agreement between rankings) to 1 (perfect agreement). It can be computed from the Friedman statistic as W = Q / (n × (k − 1)), where Q is the Friedman chi-square, n is the number of blocks, and k is the number of treatments.

When should I use Friedman instead of repeated-measures ANOVA?

Use the Friedman test when your outcome is ordinal, you have clear outliers, or the normality/sphericity assumptions for parametric repeated-measures ANOVA are not tenable. If those assumptions are reasonably met and the outcome is continuous, a parametric repeated-measures ANOVA can be more powerful.

Friedman Test Calculator

Run the Friedman test for repeated-measures designs. Enter either raw data (one row per subject/block, one column per treatment) or rank sums, and get the Friedman chi-square statistic, p-value and Kendall’s W effect size.

Suitable for nonparametric analysis of k ≥ 3 related conditions when assumptions for repeated-measures ANOVA are doubtful.

Input mode

Blocks / subjects (n)

Each row corresponds to one subject, block or matched set.

Treatments / conditions (k)

Number of related conditions to compare (k ≥ 3).

Raw repeated-measures data

Enter numeric responses only. Each row = one block/subject; each column = one treatment. Higher values will receive higher ranks within each row.

Significance level (α)

Used for the reject / do not reject decision.

Use dot or comma as decimal separator. Empty cells are treated as missing values and will cause an error.

Friedman test results

Test statistics

Decision

Treatment rank sums and mean ranks

Treatment	Rank sum R_j	Mean rank R_j/n

What is the Friedman test?

The Friedman test is a nonparametric alternative to one-way repeated-measures ANOVA. It tests whether k related treatments (e.g. three or more time points, experimental conditions or methods) have the same distribution when measured on the same blocks or subjects.

Instead of using the raw values directly, the Friedman test:

Ranks observations within each block.
Sums ranks for each treatment \( R_j \).
Compares these rank sums via a chi-square statistic.

Friedman test formula

Let \(n\) be the number of blocks (subjects) and \(k\) the number of treatments. Denote by \(R_j\) the sum of ranks for treatment \(j\) across all blocks.

\( Q = \dfrac{12}{n k (k+1)} \sum_{j=1}^{k} R_j^2 - 3n(k+1) \)

Under the null hypothesis of equal treatment distributions and for reasonably large \(n\), \(Q\) approximately follows a chi-square distribution with \(k - 1\) degrees of freedom.

Kendall’s W effect size

A convenient effect size associated with the Friedman test is Kendall’s W, defined as:

\( W = \dfrac{Q}{n (k - 1)} \)

The coefficient \(W\) ranges from 0 (no agreement / no effect) to 1 (perfect agreement / maximal effect). Very roughly:

\(W \approx 0.1\): small effect
\(W \approx 0.3\): medium effect
\(W \ge 0.5\): large effect

These thresholds are only guidelines; always interpret effect sizes in the context of your field.

When to use the Friedman test

You typically choose the Friedman test when:

You have k ≥ 3 related conditions (e.g. repeated measures or matched sets).
The outcome is ordinal or strongly non-normal with outliers.
Parametric assumptions (normality, sphericity) of repeated-measures ANOVA are violated.

Friedman vs. Kruskal–Wallis vs. repeated-measures ANOVA

Friedman test: nonparametric, for related samples (within-subjects or blocks), works on within-block ranks.
Kruskal–Wallis test: nonparametric, for independent groups, works on ranks across the full sample.
Repeated-measures ANOVA: parametric, assumes normality of residuals and sphericity.