Question

Suppose that we use x to estimate the mean of x, when E[x] = µ, Var[x] = σ 2 . Further suppose that both µ and σ 2 have finite values. As the sample size n gets larger, the variance of x gets closer to _______ ?

Answer 1

Variance of X is given as

Var[X] = /n

As the increases, the term /n gets closer and closer to 0

Suppose = 50 and n = 5, then Var[X] = 50/5 = 10

and if n is increased to 50, then Var[X] = 50/50 = 1

and if n is increased to 1000, then Var[X] = 50/1000 = 0.05

Hence, As the sample size n gets larger, the variance of x gets closer to zero or 0