Standard error

The standard error measures how much a sample statistic, most often the sample mean, would vary if the study were repeated many times with new samples drawn from the same population. It is not a measure of spread within one sample, that is the standard deviation, but a measure of how precisely that sample estimates the true population value.

For a sample mean, standard error is calculated as sd / sqrt(n), the sample standard deviation divided by the square root of the sample size. This formula shows why larger samples produce more precise estimates: standard error shrinks as sample size grows, but only at the rate of a square root, so quadrupling the sample size halves the standard error.

Standard error is the building block for a confidence interval and the margin of error reported alongside survey results and experiment estimates; both are typically calculated as the estimate plus or minus some multiple of the standard error. It relies on the central limit theorem to justify treating the sampling distribution as approximately normal. The common pitfall is confusing standard error with standard deviation: a small standard error tells you the mean is precisely estimated, not that individual observations are close together.