Editing Standard error (section)

==Extensions==

=== Finite population correction (FPC) ===

The formula given above for the standard error assumes that the population is infinite. Nonetheless, it is often used for finite populations when people are interested in measuring the process that created the existing finite population (this is called an [[Analytic and enumerative statistical studies|analytic study]]). Though the above formula is not exactly correct when the population is finite, the difference between the finite- and infinite-population versions will be small when [[sampling fraction]] is small (e.g. a small proportion of a finite population is studied). In this case people often do not correct for the finite population, essentially treating it as an "approximately infinite" population.

If one is interested in measuring an existing finite population that will not change over time, then it is necessary to adjust for the population size (called an [[Analytic and enumerative statistical studies|enumerative study]]). When the [[sampling fraction]] (often termed ''f'') is large (approximately at 5% or more) in an [[Analytic and enumerative statistical studies|enumerative study]], the estimate of the standard error must be corrected by multiplying by a <nowiki>''</nowiki>finite population correction<nowiki>''</nowiki> (a.k.a.: '''FPC'''):<ref>{{cite journal
  | first = L.
  | last = Isserlis
  | year = 1918
  | title = On the value of a mean as calculated from a sample
  | journal = [[Journal of the Royal Statistical Society]]
  | volume = 81
  | issue = 1
  | pages = 75–81
  | jstor = 2340569
  | doi = 10.2307/2340569
  | url = https://zenodo.org/record/1449486
 }} (Equation 1)</ref>
<ref>{{ cite journal
| first1 = Warren
| last1 = Bondy
|first2 = William
| last2 = Zlot
| year = 1976
| title = The Standard Error of the Mean and the Difference Between Means for Finite Populations
| journal = [[The American Statistician]]
| volume = 30
| issue = 2
| pages = 96–97
| jstor = 2683803
| doi=10.1080/00031305.1976.10479149
}} (Equation 2)</ref>
<math display="block">
    \operatorname{FPC} = \sqrt{\frac{N-n}{N-1}}
</math>
which, for large ''N'':
<math display="block">
\operatorname{FPC} \approx \sqrt{1-\frac{n}{N}} = \sqrt{1-f}
</math>
to account for the added precision gained by sampling close to a larger percentage of the population. The effect of the FPC is that the error becomes zero when the sample size ''n'' is equal to the population size ''N''.

This happens in [[survey methodology]] when sampling [[Sampling (statistics)#Replacement of selected units|without replacement]]. If sampling with replacement, then FPC does not come into play.

===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.
<!-- uncomment when this is more meaningful
==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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