Most statistic-s are some kind of integral, no? Or at least they pass through an integrator as well as perhaps other things happening at the same time. So that’s another way of saying what “a statistic” is in fancier terms. I believe the word “operator” holds the same meaning in QM.

To tantalise the computer people you can use the word “reduce” (with perhaps a “map” beforehand). Not so totally meaningless. One takeaway for example is that the mean income of a country, or the mean economic return on a PhD, may not be relevant to you if either [a] you aren’t a member of some important subset (whites, let’s say in the US if we’re discussing US wealth–or numerical methods PhD’s if we’re discussing economic returns on a mathematics degree) or [b] more statistic-s such as skew or variance are high enough to make the first moment less informative as the only number.

PS: I think you could also compute a statistic on a probability distribution. I believe you are thinking of an estimator that gets computed only on measurements of our world.