Mann Whitney Test

TL;DR

• A non-parametric alternative to the t-test for comparing the medians of two independent samples.
• Uses ranks (the test statistic is the sum of ranks in one sample) rather than raw values.
• Useful when data are not normally distributed or when sample variances are unequal; p-value < 0.05 is commonly used to declare significance.

Definition

The Mann-Whitney test, also known as the Wilcoxon rank-sum test, is a non-parametric statistical test used to compare the medians of two independent samples.

Explanation

The test does not assume a specific data distribution, so it is appropriate when samples do not follow a normal distribution or when the variances of the samples are not equal. To perform the test, values from both samples are ranked together; the test statistic is the sum of the ranks in one sample. Significance is determined by comparing the observed test statistic to its expected value under the null hypothesis, which assumes the two samples come from the same population and therefore have the same median. A p-value is used to evaluate significance, with a p-value less than 0.05 commonly indicating a significant difference.

Examples

Medical treatment comparison

A study is conducted with 100 patients, with 50 assigned to Treatment A and 50 assigned to Treatment B. Researchers collect data on the patients’ symptoms before and after treatment, and use the Mann-Whitney test to compare the medians of the two samples. If the test shows a significant difference between the medians, it can be concluded that there is a significant difference in the effectiveness of the two treatments.

Salary comparison between groups

A company has 100 employees, with 50 in Sales and 50 in Marketing. The company collects the salaries of the employees in each group and uses the Mann-Whitney test to compare the medians of the two samples. If the test shows a significant difference between the medians, it can be concluded that there is a significant difference in the salaries of the two groups.

Use cases

• Comparing medians of two independent groups when data are not normally distributed.
• Comparing medians when the variances of the two samples are not equal.

Notes or pitfalls

• The Mann-Whitney test is non-parametric and does not assume a specific distribution for the data.
• Parametric alternatives, such as the t-test, assume normality and equal variances.
• Interpretation of significance commonly uses a p-value threshold of 0.05.

Related terms

• Wilcoxon rank-sum test
• t-test