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.
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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.
The Friedman test is used when the same subjects are measured under different conditions or treatments, and the data is ordinal or continuous. The test is sensitive to differences in the central tendency of the data, but not to differences in variance.
If the Friedman test indicates significant differences between groups, post-hoc tests can be used to determine which groups differ from each other.
Test Statistic
Where:
- = number of blocks (subjects)
- = number of treatments
- = total rank of treatment
- when or
- for better accuracy when and
Key Assumptions
Practical Example
Step 1: State the Data
Scores for three treatments across five subjects:
| Subject | Treatment A | Treatment B | Treatment C |
|---|---|---|---|
| 1 | 8 | 6 | 5 |
| 2 | 7 | 7 | 6 |
| 3 | 9 | 8 | 6 |
| 4 | 6 | 5 | 4 |
| 5 | 7 | 6 | 5 |
Step 2: State Hypotheses
- : No difference between treatments
- : At least two treatments differ
Step 3: Rank Within Blocks
| Subject | A | B | C |
|---|---|---|---|
| 1 | 3 | 2 | 1 |
| 2 | 2.5 | 2.5 | 1 |
| 3 | 3 | 2 | 1 |
| 4 | 3 | 2 | 1 |
| 5 | 3 | 2 | 1 |
| 14.5 | 10.5 | 5 |
Step 4: Calculate Test Statistic
1. Basic Friedman statistic with b = 5 subjects and k = 3 treatments:
2. Correction for ties:
In our data, we have one tie (7,7) in Subject 2's scores.
where for each group of tied ranks
One pair of ties gives
3. Adjusted test statistic:
Step 5: Draw Conclusion
Critical value for with at is . Since , we reject .
There is sufficient evidence to conclude that there are significant differences between at least two treatments.
With -value = , 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 () ranges from (no agreement) to (complete agreement):
For our example:
Interpretation guidelines:
- : Weak agreement
- : Moderate agreement
- : Strong agreement