Statistics 101 — Standard Error of the Mean

The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the parameter or the statistic is the mean, it is called the standard error of the mean (SEM).

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

Therefore, the relationship between the standard error and the standard deviation is such that, for given sample size, the standard error 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, e.g., confidence intervals).

Population
The standard error of the mean (SEM) can be expressed as:

σ is the standard deviation of the population.
n is the size (number of observations) of the sample.
Estimate
Since the population standard deviation is seldom known, the standard error of the mean is usually estimated as the sample standard deviation divided by the square root of the sample size (assuming statistical independence of the values in the sample).

s is the sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population), and
n is the size (number of observations) of the sample.
Sample
In those contexts where the standard error of the mean is defined not as the standard deviation of the sample mean, but as its estimate, this is the estimate typically given as its value. Thus, it is common to see the standard deviation of the mean alternatively defined as:

The standard deviation of the sample means is equivalent to the standard deviation of the error in the sample mean with respect to the true mean, since the sample mean is an unbiased estimator. Therefore, the standard error of the mean can also be understood as the standard deviation of the error in the sample mean with respect to the true mean (or an estimate of that statistic).

Note: the standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and standard deviation: the standard error of the mean is a biased estimator of the population standard error. With n = 2 the underestimate is about 25%, but for n = 6 the underestimate is only 5%. Gurland and Tripathi (1971) provide a correction and equation for this effect. Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20.

A practical result: Decreasing the uncertainty in a mean value estimate by a factor of two requires acquiring four times as many observations in the sample. Or decreasing the standard error by a factor of ten requires a hundred times as many observations.