Given a finite set with real numbers. X = {x1, x2, x3}. There can be a unique order defined for all the subsets using Variance operator.
e.g. X = {1, 2, 4}.
$$ {\displaystyle \operatorname {Var} (X)={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-\mu )^{2},} $$ and $${\displaystyle \mu ={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.}$$ Var({1, 4}) > Var({1, 2, 4}) > Var({2, 4}) > Var({4}) = Var({2}) = Var({1}) = Var({}).
Is there any nontrivial function f(x) such that f(X) = {f(x1), f(x2), f(x3),...} preserve the same order?
f(x) = x and f(x) = -x would work.
The singleton and empty sets are not ordered by the variance.
The variance creates a preorder.
f(x) = ax, a /= 0 and f(x) = a + x preserve the variance preorder.