Question

Let X1, X2,...,Xn represent n random draws from a population with standard deviation σ and variance σ^2 , so that V ar[X1] = V ar[X2] = ... = V ar[Xn] = σ^ 2 . Define the sample average taken from a sample of size n as follows: X¯ n ≡ (X1 + X2 + ... + Xn)/ n .

a) Derive an expression for the standard deviation of X¯ n. [Hint: Your answer should depend only on σ and n]

b) Suppose we need to compute the standard deviation of a sample average using the formula you derived in (a) (which should depend on n and σ if you have answered the question correctly). Unfortunately, while we usually know the sample size n, we usually do not know σ, and are thus unable to use a formula that depends on σ. How might you make an educated guess about σ in order to compute the standard deviation?

Answer 1

We will use the property