Editing Standard error (section)

=== Derivation ===
The standard error on the mean may be derived from the [[variance]] of a sum of independent random variables,<ref>{{cite book | title=Essentials of Statistical Methods, in 41 pages|last=Hutchinson|first=T. P.| year=1993| publisher=Rumsby| isbn=978-0-646-12621-0|location=Adelaide}}</ref> given the [[Variance#Definition|definition]] of variance and some [[Variance#Properties|properties]] thereof. If <math> x_1, x_2 , \ldots, x_n </math> is a sample of <math>n</math> independent observations from a population with mean <math>x</math> and standard deviation <math>\sigma</math>, then we can define the total <math display="block">T = (x_1 + x_2 + \cdots + x_n)</math>, which due to the [[Bienaymé's_identity|Bienaymé formula]], will have variance

<math display="block">\operatorname{Var}(T) = \operatorname{Var}(x_1) + \operatorname{Var}(x_2) + \cdots + \operatorname{Var}(x_n) = n\sigma^2.</math>

The mean of these measurements <math>\bar{x}</math> (sample mean) is given by <math display="block">\bar{x} = T/n.</math> The variance of the mean is then

<math display="block">\operatorname{Var}(\bar{x}) = \operatorname{Var}\left(\frac{T}{n}\right) = \frac{1}{n^2}\operatorname{Var}(T) = \frac{1}{n^2}n\sigma^2 = \frac{\sigma^2}{n},</math>

where [[Variance#Addition and multiplication by a constant|a propagation in variance]] is used in the 2nd equality. The standard error is, by definition, the standard deviation of <math>\bar{x}</math> which is the square root of the variance:

<math display="block">\sigma_{\bar{x}} = \sqrt{\frac{\sigma^2}{n}} = \frac{\sigma}{\sqrt{n}} .</math>

In other words, if there are a large number of observations per sampling (<math display="inline">n</math> is high compared with the population variance <math display="inline">\sigma</math>), then the calculated mean per sample <math display="inline">\bar{x}</math> is expected to be close to the population mean <math> x </math>.

For correlated random variables, the sample variance needs to be computed according to the [[Markov chain central limit theorem]].