Why is the standard error of the mean also called the standard deviation of the sampling distribution of the mean, when the two values co…

Q: What's the difference between a standard deviation and a standard error?

A: Hard to say what the difference is, but the ratio is the square root of the sample size.

By definition, the standard error of a statistic is its standard deviation (which, again by definition, is the standard deviation of its sampling distribution). The reason for having separate terminology is to reduce the risk of confusion with the population standard deviation. The statement "the latter is calculated by the variance in the sample distribution" is not correct.

Specifically, let $y_1,\dots,y_n$ be independent random variables, with $E(y_j)=\mu, Var(y_j)=\sigma^2.$ Let $\bar{y}$ be their sample mean, which is the natural estimator for $\mu$.

Then the standard deviation of $\bar{y}$ is $\sigma/\sqrt{n}$. This is known as the standard error of the sample mean, to avoid confusion with the population standard deviation $\sigma$.

Why is the standard error of the mean also called the standard deviation of the sampling distribution of the mean, when the two values co…