### Mann-Whitney U-Test

 P(U = u)  = P(U ≤ u)  = P(U ≥ u)  = $n_1 =$ $n_2 =$ $u=$

This applet computes probabilities for the Mann-Whitney $U$ Test. Let $X_1,\ldots,X_{n_1}$ and $Y_1,\ldots,Y_{n_2}$ be independent random samples. The Mann-Whitney statistic $U$ is the number of pairs $(X_i,Y_j)$ where $X_i > Y_j$.

Note: Although the Wilcoxon Rank Sum test uses a different statistic, it will yield an identical $p-$value as a test based on the Mann-Whitney U-statistic. Both tests are equivalent; the statistics simply differ by a shift in location.

#### Directions

• Enter the size of the first sample in the $n_1$ box.
• Enter the size of the second sample in the $n_2$ box.
• Hitting "Tab" or "Enter" on your keyboard will plot the probability mass function (pmf) of the Mann-Whitney $U$ statistic.

To compute a probability, select $P(U=u)$ from the drop-down box, enter a numeric $u$ value, and press "Enter" on your keyboard. The probability $P(U=u)$ will appear in the pink box. Select $P(U \leq u)$ from the drop-down box for a left-tail probability (this is the cdf).

#### Details

The Mann-Whitney $U$ statistic has:

• Probability mass function (pmf):
$$f(u)=P(U=u)=\frac{n_1!n_2!}{(n_1+n_2)!}c(u|n_1,n_2)$$ where $c(u|n_1,n_2)$ is the total number of sequences where the $X_i$'s exceed the $Y_j$'s exactly $u$ times given $n_1$ and $n_2$.
• Support: $U=0,1,\ldots,n_1 n_2$
• $\mu=E(U)=\frac{n_1 n_2}{2}$
• $\sigma^2=Var(U)= \frac{n_1 n_2(n_1+n_2+1)}{12}$
• $\sigma=SD(U)= \sqrt{\frac{n_1 n_2(n_1+n_2+1)}{12}}$