Mann Whitney U Test ( Rank Sum Test ) | Non Parametric Test | How To perform the Mann-Whitney U Test

The Mann-Whitney U test, also known as the Rank Sum Test, is a non-parametric test used to determine whether there is a significant difference between the distributions of two independent groups. Unlike parametric tests such as the t-test, the Mann-Whitney U test does not assume a normal distribution of the data, making it a versatile tool for analyzing data that may not meet the stringent assumptions of parametric methods.

When to Use the Mann-Whitney U Test

The Mann-Whitney U test is appropriate in the following scenarios:

1. Non-Normal Data: When the data does not follow a normal distribution.
2. Ordinal Data: When the data is ordinal, or the measurement scale does not allow for the calculation of means and standard deviations.
3. Small Sample Sizes: When dealing with small sample sizes where the central limit theorem does not apply.
4. Independent Samples: When comparing two independent groups.

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Assumptions

1. The observations are independent within and across groups.
2. The dependent variable is measured at least on an ordinal scale.
3. The two groups being compared should be independent of each other.

How to Perform the Mann-Whitney U Test

Step-by-Step Guide

1. Rank the Data: Combine all the data from both groups, then rank the observations from lowest to highest. If there are ties, assign the average rank to the tied values.
2. Sum the Ranks: Calculate the sum of the ranks for each group.
3. Calculate the U Statistic: Use the formula:
4. Determine the Smaller U Value: The Mann-Whitney U statistic is the smaller of the two U values calculated for each group.
5. Compare U to Critical Value: Refer to the Mann-Whitney U distribution table to find the critical value at the desired significance level (e.g., ).
6. Interpret the Results: If the calculated U is less than or equal to the critical value, reject the null hypothesis, indicating a significant difference between the groups.

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Example 1: Comparing Test Scores

Scenario

A researcher wants to determine if there is a difference in test scores between two teaching methods. Method A is used on Group 1 (8 students), and Method B is used on Group 2 (10 students). The test scores are as follows:

Group 1 (Method A): 56, 67, 45, 55, 49, 62, 58, 61

Group 2 (Method B): 68, 72, 51, 69, 73, 64, 53, 66, 70, 75

Steps

1. Rank the Scores:
2. Assign Ranks to Each Group:
3. Calculate U:
4. Find the Smaller U:
5. Determine the Critical Value: For and , at , the critical value from the Mann-Whitney U table is 23.
6. Conclusion: Since is less than 23, reject the null hypothesis. There is a significant difference in test scores between the two teaching methods.

Example 2: Comparing Customer Satisfaction

Scenario

A company wants to compare customer satisfaction scores between two branches. Branch A has 12 customers, and Branch B has 15 customers. The satisfaction scores are as follows:

Branch A: 78, 85, 82, 88, 76, 81, 80, 84, 79, 77, 83, 87

Branch B: 90, 93, 89, 86, 92, 91, 94, 95, 88, 85, 87, 84, 83, 82, 81

Steps

1. Rank the Scores:
2. Assign Ranks to Each Group:
3. Calculate U:
4. Find the Smaller U:
5. Determine the Critical Value: For and , at , the critical value from the Mann-Whitney U table is 41.
6. Conclusion: Since is less than 41, reject the null hypothesis. There is a significant difference in customer satisfaction scores between the two branches.

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Conclusion

The Mann-Whitney U test is a powerful tool for comparing two independent groups when the assumptions of parametric tests are not met. By ranking data and comparing the rank sums, researchers can effectively determine if there is a statistically significant difference between the groups without relying on distributional assumptions.

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