<P> The standard deviation of the sample mean 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). </P> <P> 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 . See unbiased estimation of standard deviation for further discussion . </P> <P> 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 . </P> <P> In many practical applications, the true value of σ is unknown . As a result, we need to use a distribution that takes into account that spread of possible σ's . When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t - distribution . The standard error is the standard deviation of the Student t - distribution . T - distributions are slightly different from Gaussian, and vary depending on the size of the sample . Small samples are somewhat more likely to underestimate the population standard deviation and have a mean that differs from the true population mean, and the Student t - distribution accounts for the probability of these events with somewhat heavier tails compared to a Gaussian . To estimate the standard error of a student t - distribution it is sufficient to use the sample standard deviation "s" instead of σ, and we could use this value to calculate confidence intervals . </P>

The standard error is smaller than the standard deviation of the population