Consider a sample of size $n=3$. The sample is assumed to come from a distribution which is symmetrical about a median $M$. The absolute differences of the values of the sample are calculated and ranked (where $1$ is assigned to the smallest absolute difference and $n$ to the highest absolute difference, and for simplicity, it is assumed that no two values can have the same absolute difference and no values have an absolute difference of zero). $W^+$ is the sum of the ranks of values which are greater than $M$, therefore $W^+$ can have any of the $8$ following possible combinations:


$W^+=SUM(1), W^+=SUM(2), W^+=SUM(3)$

$W^+=SUM(1,2), W^+=SUM(1,3), W^+=SUM(2,3)$


My question is "why are each of these 8 combinations equiprobable?"

To give you some context, I'm trying to understand how to calculate p values/critical values of Wilcoxon single sample tests. I've visited the following two web pages and they both assert that the different combinations are all equiprobable (e.g. "All we have to do is recognize that under the null hypothesis, each of the above eight arrangements (columns) is equally likely"), but I can't understand why.

https://onlinecourses.science.psu.edu/stat414/node/319 http://vassarstats.net/textbook/ch12a.html

(For both pages, to jump to the bit I'm talking about, use the "find on page" functionality of your browser to find the 1st occurrence of "equally".)

My textbook makes the same assertion.

As a bonus question, could you please explain the effect (or perhaps lack of effect?) of letting two values have the same absolute difference or have an absolute difference equal to zero.


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