Editing Standard error (section)

{{short description|Statistical property}}
{{for|the computer programming concept|standard error stream}}
[[File:standard deviation diagram.svg|325px|thumb|For a value that is sampled with an unbiased [[normal distribution|normally distributed]] error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above and below the actual value.]]

The '''standard error''' ('''SE''')<ref name=":0">{{Cite journal|last1=Altman|first1=Douglas G|last2=Bland|first2=J Martin| date=2005-10-15|title=Standard deviations and standard errors|journal=BMJ: British Medical Journal| volume=331| issue=7521| pages=903|doi=10.1136/bmj.331.7521.903|issn=0959-8138|pmc=1255808|pmid=16223828}}</ref> of a [[statistic]] (usually an [[estimator]] of a [[Statistical parameter|parameter]], like the average or mean) is the [[standard deviation]] of its [[sampling distribution]]<ref>{{cite book |last=Everitt |first=B. S.  |year=2003 |title=The Cambridge Dictionary of Statistics |publisher=Cambridge University Press |isbn=978-0-521-81099-9 }}</ref> or an estimate of that standard deviation. In other words, it is the standard deviation of statistic values (each value is per sample that is a set of observations made per sampling on the same population). If the statistic is the sample mean, it is called the '''standard error of the mean''' ('''SEM''').<ref name=":0" /> The standard error is a key ingredient in producing [[Confidence interval|confidence intervals]].<ref>{{Cite journal |last=Wooldridge |first=Jeffrey M. |date=2023 |title=What is a standard error? (And how should we compute it?) |url=https://www.sciencedirect.com/science/article/pii/S0304407623002336 |journal=Journal of Econometrics |volume=237 |issue=2, Part A |doi=10.1016/j.jeconom.2023.105517 |issn=0304-4076}}</ref>

The [[sampling distribution]] of a mean is generated by repeated sampling from the same population and recording the sample mean per sample. This forms a distribution of different means, and this distribution has its own [[mean]] and [[variance]]. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the [[square root]] of the sample size.<ref name=":0" /> In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

In [[regression analysis]], the term "standard error" refers either to the square root of the [[reduced chi-squared statistic]] or the standard error for a particular regression coefficient (as used in, say, [[confidence interval]]s).