Wilcoxon Signed Rank Test

$n = $

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 $0,1,\ldots,\frac{n(n+1)}{2}$.


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).



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.