The Wilcoxon Signed Rank Test is a non-parametric statistical test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ. It is a useful tool for hypothesis testing when the data cannot be assumed to be normally distributed.

Wilcoxon Signed Rank Test Formula

The formula for the Wilcoxon Signed Rank Test is based on the differences between paired observations. For each pair of observations, the absolute differences are ranked, and the ranks of positive and negative differences are summed separately.

W = min(W+, W-)

Variables:

  • W+ is the sum of ranks for positive differences
  • W- is the sum of ranks for negative differences

The test statistic W is the smaller of W+ and W-. The p-value is then calculated based on the test statistic and the sample size.

Understanding the Wilcoxon Signed Rank Test

When conducting a Wilcoxon Signed Rank Test, the null hypothesis typically states that the median difference between pairs is zero, meaning there is no difference between the paired observations. The alternative hypothesis can be two-sided, indicating a difference exists, or one-sided, indicating a specific direction of difference.

Steps to Calculate the Wilcoxon Signed Rank Test

To calculate the Wilcoxon Signed Rank Test, follow these steps:

  1. Calculate the differences between each pair of observations.
  2. Rank the absolute values of the differences.
  3. Assign signs to the ranks based on the sign of the differences.
  4. Calculate the sum of the positive ranks (W+) and the sum of the negative ranks (W-).
  5. The test statistic W is the smaller of W+ and W-.
  6. Use the test statistic and the sample size to determine the p-value.
  7. Compare the p-value to the significance level to decide whether to reject the null hypothesis.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Sample 1: 10, 12, 8, 14, 13

Sample 2: 11, 9, 10, 15, 12

Significance Level: 5%

FAQ

1. What is the Wilcoxon Signed Rank Test used for?

The Wilcoxon Signed Rank Test is used to compare two related samples to determine if there is a significant difference between them.

2. How does the Wilcoxon Signed Rank Test differ from the paired t-test?

The Wilcoxon Signed Rank Test is a non-parametric test and does not assume a normal distribution of the data, whereas the paired t-test does.

3. Can the Wilcoxon Signed Rank Test be used for small sample sizes?

Yes, the Wilcoxon Signed Rank Test is particularly useful for small sample sizes and non-normal data distributions.

4. What are the assumptions of the Wilcoxon Signed Rank Test?

The test assumes that the data are paired and that the differences between pairs are symmetrically distributed.

5. How is the p-value interpreted in the Wilcoxon Signed Rank Test?

The p-value indicates the probability of observing the test results under the null hypothesis. A low p-value (typically < 0.05) suggests rejecting the null hypothesis in favor of the alternative.