Standard Error - Standard Error of The Mean

Further information: Variance#Sum of uncorrelated variables (Bienaymé formula)

The standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a 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 (sample standard deviation) divided by the square root of the sample size (assuming statistical independence of the values in the sample):

where

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.

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

where

σ is the standard deviation of the population.

This formula may be derived from what we know about the variance of a sum of independent random variables.

• If are independent observations from a population that has a mean and standard deviation, then the variance of the total is
• The variance of must be
• And the standard deviation of must be .
• Of course, is the sample mean .

Note: 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. 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.

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 standard error by a factor of ten requires a hundred times as many observations.