Variance of Sample Mean

Theorem

Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$.

Let:

$\ds \overline X = \frac 1 n \sum_{i \mathop = 1}^n X_i$

Then:

$\var {\overline X} = \dfrac {\sigma^2} n$

Proof

\(\ds \var {\overline X}\)

\(=\)

\(\ds \var {\frac 1 n \sum_{i \mathop = 1}^n X_i}\)

\(\ds \)

\(=\)

\(\ds \frac 1 {n^2} \sum_{i \mathop = 1}^n \var {X_i}\)

repeated application of Variance of Linear Combination of Random Variables: Corollary

\(\ds \)

\(=\)

\(\ds \frac 1 {n^2} \sum_{i \mathop = 1}^n \sigma^2\)

\(\ds \)

\(=\)

\(\ds \frac {\sigma^2 n} {n^2}\)

as $\ds \sum_{i \mathop = 1}^n 1 = n$

\(\ds \)

\(=\)

\(\ds \frac {\sigma^2} n\)

$\blacksquare$