Friedman Test

The Friedman Test compares three or more related measurements on the same subjects — the nonparametric counterpart of repeated measures ANOVA. Example: rating the same group of patients on three different treatment protocols administered sequentially.

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Step 1 — Provide your repeated measures

The Data Input panel takes one column per measurement. You can:

• Add Group / Remove Group — add as many measurements as your study has
• Upload from Excel — reads your file and fills columns
• Manual entry — paste values directly
• Generate Random Data — fills measurements with random values

Each measurement column must have the same number of values (one per subject).

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Step 2 — Calculate

Click Calculate Friedman Test.

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Step 3 — Read the results

Field	What it shows
Chi-Square (χ²)	Friedman chi-square statistic
Degrees of Freedom	k − 1
P-value (NEJM)	Significance
Kendall's W	Coefficient of concordance with interpretation
Summary Statistics	Median, Q1, Q3, IQR per measurement
Mean Ranks	Mean rank per measurement

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Step 4 — Post-hoc analysis

If the overall test is significant (p < 0.05), pairwise comparisons are shown listing significant differences with Z-statistics and p-values.

A UniversalChatBot is available for discussion.

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Statistical methods used

Test statistic — Friedman χ²

Ranks within each subject (row-wise), then:

χ² = (12 / (n × k × (k + 1))) × Σⱼ Rⱼ² − 3 × n × (k + 1)

p-value: 1 − χ²_CDF(χ², k − 1)

Hypothesis statement

• H₀: All k measurements come from the same distribution
• H₁: At least one measurement differs
• Significance level: α = 0.05

Effect size — Kendall's W

W = χ² / (n × (k − 1)), clamped to [0, 1]

Kendall's W	Label
< 0.10	Very weak agreement
0.10 – 0.30	Weak agreement
0.30 – 0.50	Moderate agreement
0.50 – 0.70	Strong agreement
≥ 0.70	Very strong agreement

Post-hoc — Pairwise mean-rank comparisons

For each pair (i, j):

• SE = √(k(k + 1) / (12n))
• Z = |R̄ᵢ − R̄ⱼ| / SE
• Only pairs with p < 0.05 are listed (unadjusted p-values)