Standard error of the mean

The standard error (SE) is the standard deviation of the sampling distribution of a statistic, most commonly of the mean.

The standard error of the mean (SEM) can be seen to depict the relationship between the dispersion of individual observations around the population mean (the standard deviation), and the dispersion of sample means around the population mean (the standard error). Different samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and variance). The relationship with the standard deviation is defined such that, for a given sample size, the standard error equals the standard deviation divided by the square root of the sample size. As the sample size increases, the dispersion of the sample means clusters more closely around the population mean and the standard error decreases.

In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the underlying errors. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate.

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 with respect 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

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

where

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