I originally created this problem 30 years ago when I taught statistics at Kansas State University. It helps explain the concept of variance.
Question: Random variables x1, x2 …. xn are binary with two outcomes 1 with probability p and 0 with probability 1-p.
What is the Var(nx1)?
What is the Var (x1+x2+…xn)?
What is the meaning of the difference in the variances for the two estimators?
The var(nx1) is n2*var(x1), which is n2*p*(1-p)
The Var(x1+x2+…xn) is var(x1) +var(x2) +… var(xn), which is n*p*(1-p).
The derivation of the variance of the binary variable — Va(xi) =p*(1-p) — can be found at
The point that I was trying to make with this problem is that the variance of the sum of n identical random variables is smaller than the variance of n multiplied by the outcome from one random variable. This is because there is more information in N observations than one observation assuming of course that N is greater than one and all observations are from the same distribution.