Wilcoxon Signed-Rank Test Calculator
Introduction
The Wilcoxon signed-rank test is a nonparametric statistical method used to compare paired or matched samples. It evaluates whether the median difference between paired observations is zero, providing an alternative to the paired t-test when the assumption of normality is questionable. This test is commonly applied in before-and-after studies, matched case-control trials, and other scenarios where data naturally form pairs.
Calculation Steps and Formula
Given paired observations , compute the differences . Pairs where are excluded from the analysis.
Next, rank the absolute differences from smallest to largest, assigning average ranks in the case of ties. Then, assign the original sign of each difference to its rank, creating signed ranks.
Define as the sum of the positive signed ranks and as the absolute sum of the negative signed ranks. The test statistic is the smaller of these two sums:
For sample sizes greater than approximately 10, can be approximated by a normal distribution with mean and variance given by:
Using this approximation, a z-score and p-value can be calculated to assess statistical significance. For smaller samples, exact critical values or permutation methods are recommended.
Interpreting Results
The null hypothesis states that the median difference between paired observations is zero. A small p-value (commonly below 0.05) suggests evidence against this null hypothesis, indicating a significant median shift.
Because the test uses ranks rather than raw values, it is less sensitive to outliers and non-normal distributions than the paired t-test. This robustness makes it suitable for data with skewness or heavy tails.
Worked Example
Suppose a researcher measures blood pressure for 8 patients before and after administering a medication. The paired observations (before, after) are:
- (120, 115)
- (130, 128)
- (125, 130)
- (140, 135)
- (135, 133)
- (128, 125)
- (132, 130)
- (138, 140)
Calculate the differences : -5, -2, 5, -5, -2, -3, -2, 2.
Exclude zeros (none here). Rank the absolute differences :
- 2 (ranks 1.5, 1.5, 1.5, 1.5 for four occurrences)
- 3 (rank 5)
- 5 (ranks 6.5, 6.5 for two occurrences)
Assign signs to ranks:
- -2 → -1.5
- -2 → -1.5
- -2 → -1.5
- 2 → +1.5
- -3 → -5
- -5 → -6.5
- -5 → -6.5
- 5 → +6.5
Sum positive ranks , sum negative ranks . The test statistic .
Using the normal approximation or exact tables, calculate the p-value to determine significance.
Comparison with Related Tests
| Test | Data Type | Assumptions | Use Case | Sensitivity |
|---|---|---|---|---|
| Wilcoxon Signed-Rank | Paired, continuous or ordinal | Symmetric distribution of differences (less strict) | Paired samples, non-normal differences | Moderate; uses magnitude and sign |
| Paired t-test | Paired, continuous | Normal distribution of differences | Paired samples with normal differences | High; sensitive to outliers |
| Sign Test | Paired, ordinal or continuous | None (only sign of differences) | Paired samples, minimal assumptions | Low; ignores magnitude |
Limitations and Assumptions
- The test assumes that the differences are symmetrically distributed around the median.
- Pairs with zero difference are excluded, which can reduce statistical power if many zeros occur.
- The normal approximation for is less accurate for small sample sizes (typically fewer than 10 pairs); exact methods are preferred in such cases.
- Ties in differences require average ranking, which can affect the distribution of the test statistic.
- Independence of pairs is assumed; correlated pairs violate test assumptions.
Frequently Asked Questions
What is the minimum sample size for the Wilcoxon signed-rank test?
While the test can be applied with small samples, fewer than 10 pairs may require exact critical values rather than normal approximations for accurate inference.
How do I interpret the p-value from this test?
A small p-value indicates evidence against the null hypothesis that the median difference is zero, suggesting a significant effect or change between paired observations.
When should I use the Wilcoxon signed-rank test instead of the paired t-test?
Use the Wilcoxon test when the differences between pairs are not normally distributed or when outliers may distort the paired t-test results.
Can I use this test with ordinal data?
Yes, the Wilcoxon signed-rank test can be applied to ordinal data where the magnitude and direction of differences are meaningful.
Does this calculator handle zero differences?
Pairs with zero difference are excluded from the calculation, as they do not contribute to the ranking process.
Is the normal approximation accurate for all sample sizes?
No, for small samples (less than about 10 pairs), the normal approximation may be inaccurate. In such cases, exact tables or permutation tests are recommended.