Mann Whitney U Test Calculator

What is the Mann-Whitney U Test?

The Mann-Whitney U test, also known as the Wilcoxon rank-sum test, is a popular non-parametric statistical hypothesis test. It is used to determine whether there is a significant difference between the medians of two independent groups. Unlike the parametric t-test, the Mann-Whitney U test does not assume that the underlying data follows a normal distribution, making it ideal for ordinal data or skewed continuous data.

The null hypothesis ($H_0$) states that the distributions of both groups are equal. The alternative hypothesis ($H_a$) suggests that one population tends to have larger values than the other (one-tailed) or that they differ in some way (two-tailed).

How the Mann-Whitney U Statistic is Calculated

The calculation involves combining the data from both groups, sorting them, and assigning ranks from $1$ to $N$ (where $N = n_1 + n_2$). If there are tied values, they are assigned the average of the ranks they would have otherwise occupied.

Next, the sum of ranks for each group is calculated ($R_1$ for group 1 and $R_2$ for group 2). The U statistics are then calculated as: $$U_1 = n_1 n_2 + \frac{n_1(n_1 + 1)}{2} - R_1$$ $$U_2 = n_1 n_2 + \frac{n_2(n_2 + 1)}{2} - R_2$$ The final Mann-Whitney U statistic is the smaller of these two values: $$U = \min(U_1, U_2)$$

Normal Approximation for Larger Samples

When both sample sizes ($n_1$ and $n_2$) are greater than 8 or 10, the distribution of $U$ approaches a normal distribution. We can calculate the z-score using: $$z = \frac{U - m_U}{\sigma_U}$$ Where the expected mean $m_U$ is: $$m_U = \frac{n_1 n_2}{2}$$ And the standard deviation $\sigma_U$ (adjusted for ties) is: $$\sigma_U = \sqrt{\frac{n_1 n_2}{N(N-1)} \left( \frac{N^3 - N}{12} - \sum \frac{t_i^3 - t_i}{12} \right)}$$ Here, $t_i$ represents the number of tied values in the $i$-th tie group. The calculated z-score is then used to find the corresponding p-value.