This applet computes probabilities for the Wilcoxon Rank Sum Test.
Let $X_1,\ldots,X_{n_1}$ and $Y_1,\ldots,Y_{n_2}$ be independent random samples.
Suppose the $n_1+n_2$ values are ranked. The Wilcoxon rank sum statistic $W_1$
is the sum of the ranks in the *first* sample.

Note: Although the Mann-Whitney U-Test uses a different statistic, it will yield
an identical $p-$value as a test based on the Wilcoxon Rank Sum 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 Wilcoxon ranked sum distribution $W_1$ corresponding to the first sample.

To compute a probability, select $P(W_1=w_1)$ from the drop-down box,
enter a numeric $w$ value, and press "Enter" on your keyboard. The probability
$P(W_1=w_1)$ will appear in the
pink box.
Select $P(W_1 \leq w_1)$ from the drop-down box for a left-tail probability (this is the cdf).

#### Details

The Wilcoxon rank sum statistic $W_1$ has:

- Probability mass function (pmf):

$$f(w_1)=P(W_1=w_1)=\frac{n_1!n_2!}{(n_1+n_2)!}c(w_1|n_1,n_2)$$
where $c(w_1|n_1,n_2)$ is the number of subsets of size $n_1$ of the set
$\{1,\ldots,n_1+n_2\}$ that sum to $w_1$.
- Support: $w_1=\frac{n_1(n_1+1)}{2},\ldots,\frac{n_1(n_1+1)}{2}+n_1 n_2$
- $\mu=E(W_1)=\frac{n_1(n_1+1)+n_1 n_2}{2}$
- $\sigma^2=Var(W_1)= \frac{n_1 n_2(n_1+n_2+1)}{12}$
- $\sigma=SD(W_1)= \sqrt{\frac{n_1 n_2(n_1+n_2+1)}{12}}$