Editing Standard error (section)

===Correction for correlation in the sample===

[[File:SampleBiasCoefficient.png|thumb|300px|right|Expected error in the mean of ''A'' for a sample of ''n'' data points with sample bias coefficient&nbsp;''ρ''. The unbiased '''standard error''' plots as the ''ρ''&nbsp;=&nbsp;0 diagonal line with log-log slope&nbsp;−{{1/2}}.]]

If values of the measured quantity ''A'' are not statistically independent but have been obtained from known locations in parameter space&nbsp;'''x''', an unbiased estimate of the true standard error of the mean (actually a correction on the standard deviation part) may be obtained by multiplying the calculated standard error of the sample by the factor&nbsp;''f'':
<math display="block">f= \sqrt{\frac{1+\rho}{1-\rho}} ,</math>
where the sample bias coefficient ρ is the widely used [[Prais–Winsten estimation|Prais–Winsten estimate]] of the [[autocorrelation]]-coefficient (a quantity between −1 and +1) for all sample point pairs. This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes. Moreover, this formula works for positive and negative ρ alike.<ref>{{cite journal |first=James R. |last=Bence |year=1995 |title=Analysis of Short Time Series: Correcting for Autocorrelation |journal=[[Ecology (journal)|Ecology]] |volume=76 |issue=2 |pages=628–639 |doi=10.2307/1941218 |jstor=1941218 |bibcode=1995Ecol...76..628B |url=https://zenodo.org/record/1235089 }}</ref> See also [[unbiased estimation of standard deviation]] for more discussion.
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==Standard errors==
===Single sample===
*<math>\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}}</math>
*<math>\sigma_{\widehat p}= \sqrt{\frac{p(1-p)}{n}}</math>
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