The Wilcoxon Signed-Rank Test is designed to compare two related groups or repeated measurements when your data doesn't follow a normal distribution. It tests the whether the differences between pairs of observations are symmetrically distributed around zero. It is a non-parametric alternative to the paired t-test. Common applications include analyzing before-and-after measurements like patient recovery scores, treatment outcomes, or paired observations of any kind. This calculator performs the complete analysis, including hypothesis testing, effect sizes, and descriptive statistics, while generating publication-ready reports. To see how it works, click here to load sample data.
Software Implementation Differences
Different statistical software packages report the Wilcoxon Signed-Rank test statistic in distinct ways:
- R(
wilcox.test): Reports , which is the sum of all positive ranks. - Python(
scipy.stats.wilcoxon): Reports , where:- is the sum of positive ranks
- is the sum of negative ranks
- We provide both R and Python code in the results section for your reference.
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Wilcoxon Signed Rank Test
Definition
Wilcoxon Signed Rank Test is a non-parametric alternative to the paired t-test. It tests the null hypothesis that the differences between pairs of observations come from a distribution with zero median, without requiring normality.
Formula
Test Statistic:
Where:
- = sum of positive ranks
- = sum of negative ranks
For small sample sizes, the test statistic is compared to a critical value from the Wilcoxon Signed Rank table:
- : Reject the null hypothesis
- : Fail to reject the null hypothesis
For larger samples (typically n > 20), a z-approximation is used:
Where n is the number of pairs with non-zero differences.
Key Assumptions
Practical Example
Step 1: State the Data
Weight measurements before and after treatment (kg):
| Subject | Before | After | Difference | Absolute Difference | Rank | Signed Rank |
|---|---|---|---|---|---|---|
| 1 | 70 | 68 | +2 | 2 | 3 | +3 |
| 2 | 80 | 78 | +2 | 2 | 3 | +3 |
| 3 | 90 | 91 | -1 | 1 | 1 | -1 |
| 4 | 60 | 58 | +2 | 2 | 3 | +3 |
| 5 | 85 | 85 | 0 (ignored) | 0 | - | - |
Step 2: State Hypotheses
- : median difference = 0
- : median difference ≠ 0
Step 3: Calculate Test Statistic
- = 3 + 3 + 3 = 9
- = 1
Step 4: Find Critical Value
The critical value for a The critical value for a two-tailed test at with (remove 1 tie in the difference) is 0 by using the Wilcoxon Signed Rank Table.
Step 5: Draw Conclusion
The test statistic is greater than the critical value, , we fail to reject the null hypothesis. There is not enough evidence to suggest that the median difference between the two groups is different from zero.
Alternative Tests
Consider these alternatives:
- Paired t-test: When data is normally distributed
- Sign Test: When only direction of difference matters