It's an interesting exercise to prove that the "mean of the means is the mean", by which I mean given a finite list $X=[x_1,\dots,x_n]$ we can define the arithmetic mean $$\mu(X)=\frac{1}{n}\sum_{i=1}^nx_i, \tag{1}$$ and we consider the list of means of non-empty sublists of $X$, writing $S_\mu(X)=[\mu(A)\mid A\subseteq X, A\neq\emptyset]$ (Note that $|S_\mu(X)| =2^n-1$ since both $X$ and $S_\mu(X)$ are allowed to have repeated elements). Then we have $$\mu(X) = \mu(S_\mu(X)). \tag{2}$$ Let's call property (2) sample invariance, as we can think of $S_\mu(X)$ as the list of all possible samples from $X$. Once one has proved (2), it's simple to generalise this statement to show the sample invariance of function means: given an invertible function $f:\mathbb{R}\to\mathbb{R}$ we define the $f$-mean by 'conguation' of $\mu$ (where $f$ is considered to act element-wise on $X$): $$\bar{f}(X) = f^{-1}\left(\frac{1}{n}\sum_{i=1}^nf(x_i)\right) = (f^{-1}\circ\mu\circ f)(X).$$ One can show that $\bar{f}(X) = \bar{f}(S_{\bar{f}}(X))$. For expedient choices of $f$ this shows that the geometric and harmonic means are also sample invariant, along with countless others.
However, there are other functions that we might expect to be sample invariant, perhaps the median or the mode? Does this characterise all sample invariant functions? I suspect not - there are clearly other functions such as $\max$ and $\min$ that are sample invariant but it's not immediately clear that they appear as $f$-means. How would one prove this? Can we classify all sample invariant functions, or perhaps just provide equivalent conditions for sample invariance?
