Example 1: The sample mean

The expectation of the sample mean is calculated as follows:

$$E(\overline{X}) = E(\frac{1}{n}\sum_{i=1}^{n} X_i )$$ (5.3)

$$= \frac{1}{n}\sum_{i=1}^{n} E(X_i)$$

$$= \frac{1}{n}\sum_{i=1}^{n} \mu$$

$$= \mu$$

Hence, the bias of the sample mean $E(\overline{X})-\mu$ is zero and the sample mean is an ``unbiased'' estimate of the population mean. As discussed earlier, the variance of the sample mean is given by $\sigma^2/n$ and, therefore, the MSE of the sample mean estimate is simply given by $\sigma^2/n$. As the sample size increases, the MSE tends to zero and the sample mean estimate converges on the true population value. This smooth unbiased convergence is what allows us to use sample means to estimate population means.