### Wilcoxon Signed Rank Test

 P(W = w)  = P(W ≤ w)  = P(W ≥ w)  = $n =$ $w=$

This applet computes probabilities for the Wilcoxon signed rank test. Let $w_1,\ldots,w_n$ be a random sample of size $n$ from a symmetric, continuous distribution centered at the origin. The Wilcoxon signed rank statistic $W$ is the sum of the ranks of the absolute values of the $w$'s for which $w$ is positive. The statistic $W$ has support $0,1,\ldots,\frac{n(n+1)}{2}$.

#### Directions

• Enter the sample size in the $n$ box.
• Hitting "Tab" or "Enter" on your keyboard will plot the probability mass function (pmf) of the Wilcoxon signed rank statistic $W$.

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

#### Details

• Probability mass function (pmf):
$$f(w)=P(W=w)=\frac{c(w|n)}{2^n}$$ where $c(w|n)$ is the number of subsets of $\{1,\ldots,n\}$ that sum to $w$ (using the convention that the empty set sums to 0). See the example below.
• Support: $w=0,1,\ldots,\frac{n(n+1)}{2}$
• $\mu=E(W)=\frac{n(n+1)}{4}$
• $\sigma^2=Var(W)= \frac{n(n+1)(2n+1)}{24}$
• $\sigma=SD(W)= \sqrt{\frac{n(n+1)(2n+1)}{24}}$

#### Example

If n=3, there are two subsets of $\{1,2,3\}$ that sum to 3: $\{3\}$ and $\{1,2\}$. Thus, $$c(w=3|n=3)=2$$ and $$f(3)=P(W=3)=\frac{c(3|3)}{2^3}=\frac{2}{8}=0.25.$$ Also, $c(w|n=3)=1$ for $w=0,1,2,4,5,6$. Thus, $$f(w)=P(W=w)=\frac{1}{8}=0.125$$ for $w=0,1,2,4,5,6$. This distribution is generated and plotted simply by entering $3$ in the $n$ box in the application.