The standard error of the sample mean

In short

• The standard error of a statistic is the standard deviation of its sampling distribution. For the sample mean: $\sigma_{\bar X} = \sigma/\sqrt{n}$.
• When $\sigma$ is unknown, which is almost always, use $s_{\bar X} = s/\sqrt{n}$, with the sample variance computed using $n - 1$.
• Standard deviation describes how spread out the data are; standard error describes how precise an estimate is. The terms are not interchangeable.
• The standard error falls with $\sqrt{n}$: quadrupling the sample halves it. Required sample size: $n = (\sigma/\text{target SE})^2$.
• If a question gives a variance, take its square root before dividing by $\sqrt{n}$.