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If you measure a sample from a wider population, then the average (or mean) of the sample will be an approximation of the population mean. But how accurate is this?
If you measure multiple samples, their means will not all be the same, and will be spread out in a distribution (although not as much as the population). Due to the central limit theorem, the means will be spread in an approximately Normal, bell-shaped distribution.
The standard error, or standard error of the mean, of multiple samples is the standard deviation of the sample means, and thus gives a measure of their spread. Thus 68% of all sample means will be within one standard error of the population mean (and 95% within two standard errors).
What the standard error gives in particular is an indication of the likely accuracy of the sample mean as compared with the population mean. The smaller the standard error, the less the spread and the more likely it is that any sample mean is close to the population mean. A small standard error is thus a Good Thing.
When there are fewer samples, or even one, then the standard error, (typically denoted by SE or SEM) can be estimated as the standard deviation of the sample (a set of measures of x), divided by the square root of the sample size (n):
SE = stdev(xi) / sqrt(n)
This shows four samples of increasing size. Note how the standard error reduces with increasing sample size.
The standard error gives a measure of how well a sample represents the population. When the sample is representative, the standard error will be small.
The division by the square root of the sample size is a reflection of the speed with which an increasing sample size gives an improved representation of the population, as in the example above.
An approximation of confidence intervals can be made using the mean +/- standard errors. Thus, in the above example, in Sample 4 there is a 95% chance that the population mean is within +/- 1.4 (=2*0.70) of the mean (4.78).
Graphs that show sample means may have the standard error highlighted by an 'I' bar (sometimes called an error bar) going up and down from the mean, thus indicating the spread, for example as below:
And the big