I am mulling something over for some self directed work. This is not home work that I have been set but I am using it for a dissertation, some post grad work. Happy to cite or be pointed to citations.

We are talking about a Mann Whitney U type situation. Comparing two groups. School running teams, lengths of fish from two ponds, etc.

Or, more mathematically, please imagine two multisets s and t. They are sets which can have more than one instance of each element. They both contain real numbers.

Let’s say there are four in s and four in t. Let’s assume they are two samples from the same big set we can call it B.

Basic question is; are these two contrived orders as likely as each other? Or is one more likely?

Order 1 “four equal pairs”

s = 1,2,3,4

t = 1,2,3,4

s s s s

t t t t

Order 2 “all the same”

s = 1,1,1,1

t = 1,1,1,1

s

s

s

s

t

t

t

t

These have same U statistic, unless I am mistaken.

I can’t put a cig paper between them intuitively either. But it feels like maybe they should be different. I can make arguments either way.

So, if you are interested or have knowledge…Do you think they are as likely as each other under the assumption they are from the same population? Happy for intuitive answers.

I am coding all this in Julia open source soon, for the curious. This is a public facing safe-for-work account.