Portfolio Variance

Quick question: Why is portfolio weightings multiplied twice? What I mean is why are the weightings applied to each individual return then one again to the (Ri-R)^2 formula? I feel as if we are weighting it twice.

Because Var(c*y)=c^2*Var(y) (where c is a constant and y is a random variable). Since weights are constants, the rule above applies. Let me prove the rule: Obviously, E(c*y)=c*E(y). Now Var(c*y)=E[(c*y)^2]-(E(cy)]^2=E(c^2*y^2)-(c*E(y))^2= c^2*(E(y^2)-(E(y))^2)=c^2*Var(y)