The Sample Mean

Suppose we have $n$ observations $X_1, X_2, \ldots, X_n$ drawn independently from the same distribution. We call such observations independent and identically distributed, or i.i.d. for short.

The sample mean is the average of these observations:

$$\bar{X} = \dfrac{1}{n}\sum\limits_{i=1}^n X_i = \dfrac{X_1 + X_2 + \cdots + X_n}{n}.$$

If each $X_i$ has expected value $E[X_i] = \mu,$ then by linearity of expectation,

$$E[\bar{X}] = E\!\left[\dfrac{1}{n}\sum\limits_{i=1}^n X_i\right] = \dfrac{1}{n}\sum\limits_{i=1}^n E[X_i] = \dfrac{1}{n}\cdot n\mu = \mu.$$

The expected value of the sample mean equals the population mean. We say that $\bar{X}$ is an unbiased estimator of $\mu.$

For instance, if $X_1,\ldots,X_8$ are i.i.d. with $E[X_i]=5,$ then $E[\bar{X}]=5.$