Standard Error of the Sample Mean

The standard error (SE) of a sample statistic measures how much that statistic would be expected to vary from one random sample to another of the same size — in effect, it is the yardstick that turns sampling error from a vague idea into a precise, computable number. For the sample mean $\bar{x}$, when the population standard deviation $\sigma$ is known (or the sample is large enough that the sample SD $s$ is used as a close approximation), the standard error is:

where $\sigma$ is the population standard deviation and $n$ is the sample size.

Why it matters: a small SE means the sample mean is likely to be close to the true population mean — the estimate is reliable — while a large SE means different samples could give quite different sample means, so the estimate should be trusted less. Because $n$ appears under a square root in the denominator, the standard error falls only as the square root of the sample size — to halve the standard error, the sample size must be quadrupled, not merely doubled. This is the precise, quantitative version of the earlier statement that "sampling error decreases as sample size increases." …