Standard Error
The standard error of a statistic is the standard deviation of that statistic's sampling distribution: a measure of how much an estimate such as a sample mean, proportion or regression coefficient would be expected to vary from one random sample to the next. For the sample mean it equals the sample standard deviation divided by the square root of the sample size, s/√n, and for a sample proportion it equals √(p × (1 − p)/n). A small standard error means the estimate is precise, in the sense that different samples would give similar answers; a large one means a single sample's estimate could land far from the truth just because of which units happened to be drawn.
The decisive distinction is that standard deviation describes the data while standard error describes an estimate, and they answer different questions. Standard deviation says how far individual observations typically sit from their average and is a stable property of the population: collecting more data makes it better estimated, but not smaller. Standard error says how far the average itself would typically sit from the true mean and depends on both the spread and the sample size, so it keeps shrinking as data accumulate. If adult heights have a standard deviation of about 7 centimeters, that stays about 7 centimeters whether 10 or 10,000 people are measured, while the standard error of the mean height falls from roughly 2.2 centimeters at n = 10 to 0.07 at n = 10,000. The two are linked by the formula SE = standard-deviation / √n, but they should never be used interchangeably.
The square-root law means precision has steep diminishing returns: halving the standard error requires quadrupling the sample. With s = 15 and n = 100 the standard error is 1.5; at n = 400 it is 0.75; at n = 10,000 it is 0.15. Going from 100 to 400 observations costs 300 more units for a gain of 0.75, while going from 400 to 700 costs another 300 for a gain of only about 0.2. This is why sample-size planning for surveys and experiments balances cost against precision, and why a survey of 1,000 people estimating a proportion near 50% has a standard error of √(0.25/1,000) ≈ 0.0158, about 1.6 percentage points, a number that, doubled, gives the familiar 3-point margin-of-error.
Standard errors are the working currency of inference. A 95% confidence-interval is typically the estimate plus or minus about 1.96 standard errors (a t critical value for small samples); a z or t statistic in hypothesis-testing is an estimate divided by its standard error, which is how many standard errors away from a null value it lies; and the standard errors shown beside regression coefficients, A/B test lifts and model metrics determine which differences count as real. When no formula exists, standard errors are obtained by the bootstrap or by sandwich/robust methods, and they are adjusted when observations are clustered or autocorrelated, since the simple s/√n formula assumes independent draws.
Three pitfalls are especially common. The first is the error-bar mix-up: figures often show SD, SE or confidence-interval bars without saying which, and standard-error bars are always narrower, so presenting them where the spread of the data is the point makes variability look smaller than it is. The second is treating a small standard error as proof of accuracy: it measures only random sampling variability, so a biased sample or a mismeasured variable can give a very precise but wrong estimate. The third is applying s/√n to data that are not independent, such as repeated measurements on the same subjects or neighboring points in a time series, where the true standard error is larger than the formula says.
Read the standard error as the standard deviation of an estimate rather than of the data: the typical amount a statistic would wobble across repeated samples, shrinking with the square root of the sample size, and the single number that converts an estimate into a confidence interval or a test statistic.