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:
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?