The standard error (SE) of a statistic (usually an estimator of a parameter, like the average or mean) is the standard deviation of its sampling distribution. The standard error is often used in calculations of confidence intervals.

The sampling distribution of a mean is generated by repeated sampling from the same population and recording the sample mean per sample. This forms a distribution of different sample means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).

Suppose a statistically independent sample of ${\displaystyle n}$ observations ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ is taken from a statistical population with a standard deviation of ${\displaystyle \sigma }$ (the standard deviation of the population). The mean value calculated from the sample, ${\displaystyle {\bar {x}}}$, will have an associated standard error on the mean, ${\displaystyle {\sigma }_{\bar {x}}}$, given by:

$${\displaystyle {\sigma }_{\bar {x}}={\frac {\sigma }{\sqrt {n}}}.}$$

Practically this tells us that when trying to estimate the value of a population mean, due to the factor ${\displaystyle 1/{\sqrt {n}}}$, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations.

The standard deviation ${\displaystyle \sigma }$ of the population being sampled is seldom known. Therefore, the standard error of the mean is usually estimated by replacing ${\displaystyle \sigma }$ with the sample standard deviation ${\displaystyle \sigma _{x}}$ instead: $${\displaystyle {\sigma }_{\bar {x}}\ \approx {\frac {\sigma _{x}}{\sqrt {n}}}.}$$