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.

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).

Step 2 — Calculate

Click Calculate Friedman Test.

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

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.

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)