Discussion on question by Carl: Exactly what does the Wilcoxon signed rank test compare?

One way to think of the hypothesis tested is described by Frank Harrell in a comment here: "As stated earlier it tests precisely whether the probability that a randomly chosen pair of values sums to a positive number is equal to 0.5 or not. "stats.stackexchange.com/questions/569439/how-to-interprete-‌​and-describe-the-res‌​ults-of-a-wilcoxon-s‌​igned-rank-test.

I think Wikipedia explains it pretty well: en.wikipedia.org/wiki/…

@Carl, no problem! I have too much experience with links going dead, so I like to have an informative title. I think there is a fairly broad agreement on that on Cross Validated.

@Carl Sorry I can't do better than linking to standard terminology and reasoning in statistics. I don't know enough about your radioactive decay experiments to comment on that. Regarding your original question, I thought the first comment by SalMangiafico citing Frank Harrell nails it pretty well.

.... where there we're measuring nearness to normality in a Shapiro-Francia / Ryan-Joiner type way. However, the signed rank statistic does approach normality (under H0) very rapidly.