Suppose I have three i.i.d. random variables $X_1, X_2, X_3$, and I do "max-min normalization" on them.
$$X_i \mapsto \frac{X_i - \min\limits_i X_i}{\max\limits_i X_i - \min\limits_i X_i} $$
Let $Y$ be the location of the "middle" point after normalization. (It lies in $[0,1]$.) What is the distribution of $Y$?
How can I extend to the case of $n$ i.i.d. variables $X_1, \dotsc, X_n$, asking about the distribution on the $n-2$ middle points?
We could give $X_i$ specific distributions such as $\mathcal{N}(0,1)$ for concreteness.