Assume we have taken a sample from some population and have calculated its arithmetic mean `x̄`

and its standard deviation `s`

. Now we want to estimate the mean of the population and its standard deviation on the basis of these values of the sample. The mean of the population is `μ`

and will not be discussed here.

If we were to take more random samples from the same population that we took the present sample from, their means will show some distribution which, in turn, will show a standard deviation around a mean. The **standard error** of a mean `x̄`

of a distribution of size `n`

is the expected value of the standard deviation `s`

of means of several samples of equal size. This is estimated from a single sample by the following formula:

e | = | s |

√n |

Take once more the sample that consists of `{ 2.0, 4.0, 4.0, 5.0 }`

, with `n=4, x̄=3.75, s=1.2583`

. The standard error of this `x̄`

is calculated as follows:

`e = 1.2583 / √4 = 1.2583 / 2 = 0.62915.`

Thus, on the basis of the properties of our sample, we may expect the means of other samples to show a standard deviation of `0.63`

. This is, so to speak, an estimate of the representativity of our sample. The smaller the standard error, the more safely other samples may be expected to show similar properties as ours.