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## What is the Wilcoxon Rank-Sum Test?

The Wilcoxon rank-sum test, also known as the Mann-Whitney U test, is a non-parametric statistical test used to assess whether there is a difference between two independent groups. It is often employed when the assumptions of a parametric test, such as the t-test, are not met.

Here’s a basic overview of how the Wilcoxon rank-sum test works:

**Ranking the Data:** Combine the data from both groups into a single dataset. Rank the data from smallest to largest, ignoring the group labels.

**Calculating the Test Statistic (U):** The test statistic U is calculated based on the ranks. It represents the sum of the ranks for one of the groups. The formula for U depends on whether you are interested in determining if one group tends to have higher values than the other or if the groups tend to differ.

**Hypothesis Testing:** The null hypothesis (H0) typically posits that there is no difference between the two groups. The alternative hypothesis (H1) is that there is a difference.

**Decision Rule:** Compare the calculated U value to critical values from statistical tables or use it directly to determine the significance of the results.

**Interpretation:** If the test is statistically significant, you reject the null hypothesis and conclude that there is a significant difference between the two groups.

The Wilcoxon rank-sum test does not assume that the data come from a particular distribution, making it robust in situations where the assumptions of parametric tests may be violated. However, it generally requires larger sample sizes to achieve the same level of power as a parametric test when assumptions are met.

It’s worth noting that there is also a related test called the Wilcoxon signed-rank test, which is used for comparing two related groups or matched pairs. The Wilcoxon rank-sum test, on the other hand, is for independent groups.

## Example of Wilcoxon Rank-Sum

Let’s consider an example scenario where you want to compare the scores of two groups of students (Group A and Group B) on a math test. The scores are not assumed to be normally distributed, so you decide to use the Wilcoxon rank-sum test to assess whether there is a significant difference between the two groups.

Here are the scores for each group:

Group A: 55, 62, 58, 70, 48

Group B: 40, 52, 45, 65, 55

**Combine and Rank the Data:**

Combine the scores from both groups and rank them: Scores: 40, 45, 48, 52, 55, 55, 58, 62, 65, 70

Ranks: 1, 2, 3, 4, 5, 5, 7, 8, 9, 10**Calculate the Test Statistic (U):**

Group A’s U value is the sum of the ranks for Group A (7 + 8 + 9 + 10 = 34). If you’re interested in whether Group A tends to have higher values, you might use the U value for Group B (1 + 2 + 3 + 4 + 5 = 15).**Hypothesis Testing:**

Null Hypothesis (H0): There is no significant difference between the two groups. Alternative Hypothesis (H1): There is a significant difference between the two groups.**Decision Rule:**

Compare the U value to critical values from statistical tables or use it directly to determine statistical significance.**Interpretation:**

If the test is statistically significant, you reject the null hypothesis and conclude that there is a significant difference between the two groups.

Note: The critical values for U depend on the sample sizes, and you would compare your calculated U value to these critical values to make a decision about the null hypothesis.

Keep in mind that this is a simplified example of a Wilcoxon rank-sum test, and in practice, statistical software or statistical tables are often used to obtain the exact p-value associated with the test statistic for a more accurate assessment of statistical significance.