# Why is standard error of the mean always calculated from the population variance?

The standard error of the mean is often calculated with the formula $$\sqrt{\sigma^2 / n}$$, which uses population variance (or an estimate of it).

However, standard error of the mean is defined as the standard deviation of the sample means. So why don't we ever directly calculate standard error of the mean as $$\sqrt{E[(X_i - \overline X_n)^2]}$$ where $$X_i$$ is a sample mean and $$\mu$$ is the mean of the sample means? Since the formula for standard deviation is $$\sqrt{E[(X - \mu)^2]}$$.

\begin{align} \text{True standard error} &= \mathbb{S}(\bar{X}_n) = \frac{\sigma}{\sqrt{n}}, \\[12pt] \text{Estimated standard error} &= \hat{\mathbb{S}}(\bar{X}_n) = \frac{s_n}{\sqrt{n}}. \\[6pt] \end{align}