# New feature: Wilcoxon-Mann-Whitney and Kruskal-Wallis Tests

We just released an update to our Extreme Optimization Numerical Libraries for .NET that adds some new classes. We’ve added two non-parametric tests: the Mann-Whitney and Kruskal-Wallis tests. These are used to test the hypothesis that two or more samples were drawn from the same distribution. The Mann-Whitney test is used for two samples. The Kruskal-Wallis test is used when there are two or more samples.

For both tests, the test statistic only depends on the ranks of the observations in the combined sample, and no assumption about the distribution of the populations is made. This is the meaning of the term non-parametric in this context.

The Mann-Whitney test, sometimes also called the Wilcoxon-Mann-Whitney test or the Wilcoxon Rank Sum test, is often interpreted to test whether the median of the distributions are the same. Although a difference in median is the dominant differentiator if it is present, other factors such as the shape or the spread of the distributions may also be significant.

For relatively small sample sizes, and if no ties are present, we return an exact result for the Mann-Whitney test. For larger samples or when some observations have the same value, the common normal approximation is used.

The Kruskal-Wallis test is an extension of the Mann-Whitney test to more than two samples. We always use an approximation for the distribution. The most common approximation is through a Chi-square distribution. We chose to go with an approximation in terms of the beta distribution that is generally more reliable, especially for smaller sample sizes. For comparison with other statistical software, the chi-square p-value is also available.

We created some QuickStart samples that illustrate how to use the new functionality:

You can also view the documentation on non-parametric tests, or download the trial version.