Abstract Representation: Your Ancient Heritage

Variance of the Sample Average

In particular, 0 =Var(SN) N s2 + N(n-N) Cov(X1,X2)

Therefore: Cov(X1 X2) = -s2/(N-1).

And hence Var(Sn) = s2 n(1- (n-1)/(N-1)).

Question: What is the SD for sampling without replacement?

Solution: Let Sn = X1 + X2 + … + Xn. Then .

Var(Sn) = åi Var(Xi) + 2 åj<i Cov(Xi Xj)

By symmetry: Cov(Xi,Xj) = Cov(X1,X2), so

Var(Sn) = ns2 + n(n-1) Cov(X1 X2) .