If Mann-Whitney U is a median test? Then what the heck is "unpaired pairwise median"?

Header: I am really sorry for the confusing title. I am going to edit it so it fits the responses I get based on Mann Whitney U.

So I had this exact question on stack overflow about is the Mann Whitney U Test a test of medians or means. I really liked Glen_b's response, but it lead to a bit of new confusion about "pairwise" and the meaning of "rank sum". In particular here is what my notes say:

Mann-Whitney Test

Assumptions: Samples are independent from each other

And this nonparametric is for samples where the outcomes are not paired. Before we were assuming that both yi and zi were observed for the same data point. For example let's say we have the same virus [data that we did for the two sample paired Wilcoxian signed rank test]. Now we're just assuming that all the observations are independent of each other. This test is called the Mann-Whitney test. In the Mann-Whitney test we rank all of the yi and zi together. And then, add up the ranks of all the samples that come from the first set. All the ranks of the yi and the ranks of the samples in the second set, all the ranks of the zi. Whichever sum is smaller is compared again against the table that gives the significance of the difference."

What do people mean by "pairwise"? (See the link above for context) How can one have an unpaired sample test that uses "pairwise"?

What the heck is a median difference? Do we mean the median of the rank? Is the median difference a measure of central tendency or measure of location?

• So I thought I should post the original question I had that Glen_b managed to answer. (Part 1 of 4) I was studying some notes I had for my statistics class, and I was confused about the Mann Whitney U Test. I have heard that an "unpaired two sample median test" is a Mann Whitney U Test. However, the lecture notes I watch suggest that a Mann Whitney U Test is also called a Wilcoxian Signed Rank SUM Test. My confusion is that some articles say that it is a median test and some articles say that is not a median test, but a mean test. Jul 30, 2019 at 22:03
• (Part 2 of 4) This is an article that does say its a median test: stats.idre.ucla.edu/other/mult-pkg/faq/general/… ""Consider the following example dataset of 120 observation (60 in each group) that has equal medians and a significant Mann-Whitney-Wilcoxon test."" Jul 30, 2019 at 22:04
• (Part 3 of 4) Another article: sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/… ""A popular test to compare outcomes between two independent groups is the Mann Whitney U test. The Mann Whitney U test, sometimes called the Mann Whitney Wilcoxon Test or the Wilcoxon Rank Sum Test, is used to test whether two samples are likely to derive from the same population (i.e., that the two populations have the same shape). Some investigators interpret this test as comparing the medians between the two populations."" Jul 30, 2019 at 22:10
• (Part 4 of 4) While this says it is not the medians but the means: graphpad.com/guides/prism/7/statistics/… ""The Mann-Whitney test compares the mean ranks -- it does not compare medians and does not compare distributions. More generally, the P value answers this question: What is the chance that a randomly selected value from the population with the larger mean rank is greater than a randomly selected value from the other population?"" So is Mann-Whitney a comparison of means or medians? If its means is it really non-parametric? Jul 30, 2019 at 22:24
• Glen clarified his original post, I think you are sorted now? :) Also check the original publication here it says nothing about means or medians. It concerns medians only if we assume that the alternative hypothesis is restricted to a shift in location. (Which we implicitly do in many cases but that's not what it says on the (test's) tin!) Jul 30, 2019 at 23:03