Standard Error

Square root of sample variance, also call sample standard deviation.

(The following is from wikipedia)

## Standard error of the mean

The '''standard error of the mean''' (SEM) is the standard deviation of the [[sample (statistics)|sample]] mean estimate of a [[statistical population|population]] mean. (It can also be viewed as the standard deviation of the error in the sample mean relative to the true mean, since the sample mean is an unbiased estimator.) SEM is usually estimated by the sample estimate of the population [[standard deviation]] ([[standard deviation#Estimating population standard deviation from sample standard deviation|sample standard deviation]]) divided by the square root of the sample size (assuming statistical independence of the values in the sample):

(1)
\begin{align} SE_\bar{x}\ = \frac{s}{\sqrt{n}} \end{align}

where

:''s'' is the [[standard deviation#Estimating population standard deviation from sample standard deviation|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.

This estimate may be compared with the formula for the true standard deviation of the mean:

(2)
\begin{align} SD_\bar{x}\ = \frac{\sigma}{\sqrt{n}} \end{align}

where

:''σ'' is the [[standard deviation]] of the population.

''Note 1:'' Standard error may also be defined as the standard deviation of the [[Errors and residuals in statistics|residual error term]]. (Kenney and Keeping, p. 187; Zwillinger 1995, p. 626)

''Note 2:'' Both the standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and deviations: 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 (see Sokal and Rohlf, Biometry, 2nd ed., 1981, p 53 for an equation of the correction factor for small samples of n < 20). See [[unbiased estimation of standard deviation]] for further discussion.

''A practical result:'' Decreasing the uncertainty in your mean value estimate by a factor of two requires that you acquire four times as many observations in your sample. Worse, decreasing standard error by a factor of ten requires a hundred times as many observations.

page revision: 8, last edited: 04 Sep 2009 01:12