Sample Mean is Unbiased Estimator of Population Mean

Theorem

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

Then:

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

is an unbiased estimator of $\mu$.

Proof

If $\bar X$ is an unbiased estimator of $\mu$, then:

$\ds \expect {\bar X} = \mu$

We have:

 $\ds \expect {\bar X}$ $=$ $\ds \expect {\frac 1 n \sum_{i \mathop = 1}^n X_i}$ $\ds$ $=$ $\ds \frac 1 n \sum_{i \mathop = 1}^n \expect {X_i}$ Linearity of Expectation Function $\ds$ $=$ $\ds \frac 1 n \sum_{i \mathop = 1}^n \mu$ as $\expect {X_i} = \mu$ $\ds$ $=$ $\ds \frac n n \mu$ as $\ds \sum_{i \mathop = 1}^n 1 = n$ $\ds$ $=$ $\ds \mu$

So $\bar X$ is an unbiased estimator of $\mu$.

$\blacksquare$