Is there a term for the average difference between all possible pairs of data points in a set? I guess you could also say "the average expected difference between any pair of data points taken at random from a given set."
Example: Say you have five people, and their heights, in inches, are 62, 66, 66, 72, and 74. The differences in height between all possible pairings of these five people are 4, 4, 10, 12, 0, 6, 8, 6, 8, and 2, so the average height difference between all possible pairings of these people (or the difference you would expect to find on average if you selected two at random) is 6 inches.
I'm wondering what the term is for this kind of average difference. I feel like it has to be some recognized form of deviation or variance, but trying to find the right term has only led me to other terms that don't have quite the meaning I'm looking for.
 A: Let's rewrite what you wrote in a more mathematical way:
Let $X_1, \dotsc, X_n$ be i.i.d. random variables. You are interested in the quantity:
$$
\theta :=\mathbb E\left[ \bigg\vert \frac{X_1 + X_2 - X_3 -X_4}{2} \bigg\vert\right] 
$$
Then the sample estimator (statistic) you propose has the form:
$$
\hat{\theta} = \frac{(n-4)!}{n!}\sum_{\pi \in \Pi(n,4)}\bigg\vert \frac{X_{\pi(1)} + X_{\pi(2)} - X_{\pi(3)} -X_{\pi(4)}}{2} \bigg\vert
$$
Here the set $\Pi(n,4)$ denotes the set of all possible ways to draw $4$ elements (in an ordered way) out of $n$ elements without replacement. (Mathematically: Equivalence classes of permutations in the symmetric group $S_n$ modulo their values evaluated at 1,2,3,4). There are $\frac{n!}{(n-4)!}$ such combinations.
The statistic $\hat{\theta}$ which you proposed is now called the U-Statistic of the parameter $\theta$. Thus, even though I do not know of a specific name for the statistic you are proposing here, if you are just interested in its properties (e.g. asymptotics), then you can as well look up the more general theory of U-Statistics.
