I'm strangely stuck on that one. If I take a set of a finite number of numbers (real numbers, let's be simple), would the average value of rounding/flooring ("take the before-dot part", ie floor(3.9) = 3) each number the same as rounding/flooring the average?
I don't see simple (nor even complex actually) proofs for/against this statement.
Would the same be for more abstract cases, like complex numbers, or infinite sets of elements?
No. The rounding/flooring is always an integer while the average is rational.
$$\left\lfloor\frac{0+1}2\right\rfloor\ne\frac{\lfloor0\rfloor+\lfloor1\rfloor}2.$$
This property only holds for linear transforms.