Can you explain why this formula works: Statistics (Variance)?

If X_1, X_2, ... , X_n are independent random variables with the same variance (I'll denote it v), and Xavg is the average then,





Var(Xavg) = v/n





I don't really understand exactly what that's saying.Can you explain why this formula works: Statistics (Variance)?
When variables are independent the variance of the sum is the sum of their variances. So





Var(X_1 + ...+ X_n) = Var(X_1) + ... + Var(X_n) = n v in this case.





Also Var (k Y ) = k^2 Var(Y). Now put these two facts together with k = 1/n and Xbar = (by def) (X_1 + ... + X_n)/ n to get that





Var(Xbar) = n v / n^2 = v / n.