Friedman Test Calculator

Friedman Test

Condition 1 (Time 1 / Treatment A)

Condition 2 (Time 2 / Treatment B)

Condition 3 (Time 3 / Treatment C)

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Non-Parametric Repeated Measures

What Is the Friedman Test?

The Friedman Test is a non-parametric statistical hypothesis test used to detect differences across three or more related, matched, or repeated conditions tested on the same group of subjects.

It serves as the direct non-parametric alternative to the One-Way Repeated Measures ANOVA and an extension of the paired Wilcoxon Signed-Rank Test for $k \ge 3$ conditions, ranking observations within each subject/block.

When to Use It

Repeated Measures with Skewed Data: Comparing subjects measured across 3+ timepoints when normality or sphericity fails.
Ordinal & Likert Scales: Analyzing customer ratings, taste-test rankings, or jury scores across multiple conditions.
Randomized Complete Block Designs (RCBD): Evaluating treatments when response distributions are non-normal.
Alternative to Repeated Measures ANOVA: When residuals violate Gaussian normality or Mauchly's sphericity.

The Mathematical Formulas

$$Q = \frac{12}{N k (k + 1)} \sum_{j=1}^k R_j^2 - 3 N (k + 1)$$ $$C = 1 - \frac{\sum_i \sum_m (t_{i,m}^3 - t_{i,m})}{N k (k^2 - 1)}, \quad Q_{\text{adj}} = \frac{Q}{C}$$ $$W = \frac{Q}{N (k - 1)}, \quad df = k - 1, \quad p = 1 - F_{\chi^2}(Q_{\text{adj}}; df)$$

Where $N$ is the number of subjects (blocks), $k$ is the number of repeated conditions, $R_j$ is the sum of ranks for condition $j$, $C$ is the within-block tie correction factor, and $W$ is Kendall's Coefficient of Concordance effect size.

Worked Example

Scenario: 5 subjects ($N=5$) are tested across 3 conditions ($k=3$) at $\alpha=0.05$: Cond 1: $[4, 3, 6, 2, 5]$, Cond 2: $[7, 5, 8, 4, 7]$, Cond 3: $[8, 9, 7, 6, 9]$.

Step 1: Rank Values Within Each Subject (Row-wise):
Subj 1: $[4, 7, 8] \to [1, 2, 3]$
Subj 2: $[3, 5, 9] \to [1, 2, 3]$
Subj 3: $[6, 8, 7] \to [1, 3, 2]$
Subj 4: $[2, 4, 6] \to [1, 2, 3]$
Subj 5: $[5, 7, 9] \to [1, 2, 3]$

Step 2: Calculate Rank Sums ($R_j$):
$R_1 = 5.0, \quad R_2 = 12.0, \quad R_3 = 13.0$
$\sum R_j^2 = 5^2 + 12^2 + 13^2 = 25 + 144 + 169 = 338.0$.

Step 3: Compute Friedman Statistic ($Q$):
$Q = \frac{12}{5 \times 3 \times 4}(338.0) - 3(5)(4) = \frac{12}{60}(338) - 60 = 67.6 - 60 = 7.6000$.

Step 4: P-Value & Effect Size ($W$):
$p = 1 - F_{\chi^2}(7.6000; df=2) = 0.0224$
$W = \frac{7.6000}{5(2)} = 0.7600 \text{ (Strong Concordance)}$.

Conclusion: Since $p = 0.0224 \le 0.05$, reject $H_0$. Scores differ significantly across the 3 conditions.

Non-Parametric Analysis Complete

Friedman Test Results

Computed Friedman Test Statistic & P-Value

Q = 7.6000, p = 0.0224

Statistically Significant Difference across Repeated Conditions (p ≤ 0.05)

Friedman Stat (Q) 7.6000

Degrees of Freedom df = 2

Kendall's W 0.7600

p-value 0.0224

Subjects / Blocks (N) 5

Conditions (k) 3

Within-Subject Rank Assignments Matrix

Friedman Test Calculator

Friedman Test

Friedman Test Results

Within-Subject Rank Assignments Matrix

Step-by-Step Calculation

Interpretation

Condition Mean Ranks Profile Plot

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