Expected variance of subset of population

[ll777]

I want to calculate expected variance of a randomly selected subset of a population.

The particular problem I am trying to solve is as follows. There is a set of values X = {x1, ... , xn}. Let Y be subset of X with n-1 elements. I think that if Y is selected at random (that is, if is produced by randomly removing an element of X), the expected variance of Y is less than the variance of X. Is this right and if so is there a simply proof?

 

[bpet]

A hard way: write sum(Y)=sum(X)-xj etc.

An easy way: The law of total variance.