What is the variance of the distribution of the average an IID draw of n observations from a population with mean μ and variance σ2.

Question

kirbykirby · Answer

$$\begin{align}Var\left({\overline{X}}\right)&=Var\left( \frac{1}{n}\sum_{i=1}^nX_i\right)\&= \frac{1}{n^2}Var\left( \sum_{i=1}^nX_i\right)\&=\frac{1}{n^2}\sum_{i=1}^nVar(X_i), \text{ by independence}\&= \frac{1}{n^2}\left[Var(X_1)+Var(X_2)+\ldots+Var(X_n) \right]\&=\frac{1}{n^2}\left[\sigma^2+\sigma^2+\ldots+\sigma^2  \right], \text{ since the }X_i \text{ are identically distributed }\&= \frac{1}{n^2}(n\sigma^2)\&=\frac{\sigma^2}{n}\end{align}$$