StatsCalculators.com

Friedman Test

Created:October 8, 2024
Last Updated:May 13, 2025

The Friedman Test Calculator helps you perform non-parametric analysis for comparing three or more matched or repeated measurements. It determines whether samples originate from the same distribution by analyzing the ranks of the data rather than the raw values. This makes it particularly useful when your data violates the assumptions of repeated measures ANOVA, such as normality or sphericity. Common applications include comparing multiple treatments across the same subjects, analyzing repeated measurements over time, or evaluating preferences across different conditions. Click here to populate the sample data for a quick example.

Calculator

1. Load Your Data

Note: Column names will be converted to snake_case (e.g., "Product ID" → "product_id") for processing.

2. Select Columns & Options

Related Calculators

One-Way ANOVA Calculator

Repeated Measures ANOVA Calculator

Wilcoxon Signed-Rank Test

Chi-Square Test of Independence Calculator

Learn More

Friedman Test

Definition

Friedman Test is a non-parametric alternative to the one-way repeated measures ANOVA. It tests for differences between three or more matched or paired groups, without requiring normality in the data distribution. The test statistic is based on the ranks of the data values within each block.

Test Statistic

Q=12nk(k+1)∑j=1kRj2−3n(k+1)Q = \frac{12}{nk(k+1)}\sum_{j=1}^k R_j^2 - 3n(k+1)

Where:

  • nn = number of blocks (subjects)
  • kk = number of treatments
  • RjR_j = total rank of treatment jj
  • Q∼χk−12Q \sim \chi^2_{k-1} when n>15n > 15 or k>4k > 4
  • Q∼Fk−1,(k−1)(n−1)Q \sim F_{k-1, (k-1)(n-1)} for better accuracy when n≤15n \leq 15 and k≤4k \leq 4

Key Assumptions

Matched Groups: Data must be paired or matched
Ordinal Data: Measurements must be ordinal or continuous
Independent Blocks: Subjects must be independent

Practical Example

Step 1: State the Data

Scores for three treatments across five subjects:

SubjectTreatment ATreatment BTreatment C
1865
2776
3986
4654
5765
Step 2: State Hypotheses
  • H0H_0: No difference between treatments
  • HaH_a: At least two treatments differ
  • α=0.05\alpha = 0.05
Step 3: Rank Within Blocks
SubjectABC
1321
22.52.51
3321
4321
5321
RjR_j14.510.55
Step 4: Calculate Test Statistic

1. Basic Friedman statistic with b = 5 subjects and k = 3 treatments:

Q=125⋅3(3+1)(14.52+10.52+52)−3⋅5(3+1)=9.1Q = \frac{12}{5 \cdot 3(3+1)}(14.5^2 + 10.5^2 + 5^2) - 3 \cdot 5(3+1) = 9.1

2. Correction for ties:

In our data, we have one tie (7,7) in Subject 2's scores.

C=1−∑Tnk(k2−1)C = 1 - \frac{\sum T}{n k(k^2-1)}

where T=(t3−t)T = (t^3 - t) for each group of tt tied ranks

One pair of ties gives T=(23−2)=6T = (2^3 - 2) = 6

C=1−65⋅3(9−1)=0.95C = 1 - \frac{6}{5 \cdot 3(9-1)} = 0.95

3. Adjusted test statistic:

Qadj=QC=9.10.95=9.58Q_{\text{adj}} = \frac{Q}{C} = \frac{9.1}{0.95} = 9.58
Step 5: Draw Conclusion

Critical value for χ2\chi^2 with df=2df = 2 at α=0.05\alpha = 0.05 is 5.9915.991. Since Q=9.58>5.991Q = 9.58 \gt 5.991, we reject H0H_0.

There is sufficient evidence to conclude that there are significant differences between at least two treatments.

With pp-value = 0.008317<0.050.008317 \lt 0.05, there is strong evidence to conclude that there are significant differences between at least two treatments.

Effect Size

Kendall's W

Kendall's coefficient of concordance (WW) ranges from 00 (no agreement) to 11 (complete agreement):

W=Qn(k−1)W = \frac{Q}{n(k-1)}

For our example:

W=9.585(3−1)=0.958W = \frac{9.58}{5(3-1)} = 0.958

Interpretation guidelines:

  • W<0.3W \lt 0.3: Weak agreement
  • 0.3≤W≤0.50.3 \leq W \leq 0.5: Moderate agreement
  • W>0.5W \gt 0.5: Strong agreement

Verification