Suppose $X_1,...,X_n$ are data from iid sample. We know that $$V\left(\sum_{i=1}^{n}X_i\right)=nV(X)$$
I'm wondering that if we have a function $h$, can we say that $V\left(\sum_{i=1}^{n}h(X_i)\right)=nV(h(X))$?
My reasoning is that they are because if $X,Y$ independent, then $h(X),h(Y)$ are also independent. So can apply equation $1$ with the mapped variables.