Editing Standard error (section)

==Assumptions and usage==
{{further|Confidence interval}}

An example of how <math>\operatorname{SE}</math> (Standard Error) is used to make confidence intervals of the unknown population mean is shown. If the [[sampling distribution]] is [[normal distribution|normally distributed]], the sample mean, the standard error, and the [[quantile]]s of the normal distribution can be used to calculate confidence intervals for the true population mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where <math>\bar{x}</math> is for the sample mean, <math>\operatorname{SE}</math> is for the standard error for the sample mean (the standard deviation of sample mean values), and [[1.96]] is the approximate value of the 97.5 [[percentile]] point of the [[normal distribution]]:
{{unbulleted list | style = padding-left: 1.5em
|1 = Upper 95% limit = <math>\bar{x} + (\operatorname{SE}\times 1.96) </math>, and
|2 = Lower 95% limit = <math>\bar{x} - (\operatorname{SE}\times 1.96) </math>.
}}
In particular, the standard error of a [[sample statistic]] (such as [[sample mean]]) is the actual or estimated standard deviation of the sample mean in the process by which it was generated. In other words, it is the actual or estimated standard deviation of the [[sampling distribution]] of the sample statistic. The notation for standard error can be any one of SE, SEM (for standard error of ''measurement'' or ''mean''), or S<sub>E</sub>.

Standard errors provide simple measures of uncertainty in a value and are often used because:
*in many cases, if the standard error of several individual quantities is known then the standard error of some [[function (mathematics)|function]] of the quantities can be easily calculated;
*when the [[probability distribution]] of the value is known, it can be used to calculate an exact [[confidence interval]];
*when the probability distribution is unknown, [[Chebyshev's inequality|Chebyshev]]'s or the [[Vysochanskiï–Petunin inequality|Vysochanskiï–Petunin inequalities]] can be used to calculate a conservative confidence interval; and
* as the [[sample size]] tends to infinity the [[central limit theorem]] guarantees that the sampling distribution of the mean is asymptotically [[normal distribution|normal]].

===Standard error of mean versus standard deviation===
In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are [[descriptive statistics]], whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.<ref>{{cite journal | first = M. | last = Barde | year = 2012 | title = What to use to express the variability of data: Standard deviation or standard error of mean? | journal = [[Perspect. Clin. Res.]] | volume = 3 | issue = 3 | pages = 113–116 | doi = 10.4103/2229-3485.100662 | pmid = 23125963 | pmc = 3487226 | doi-access = free }}</ref>

Put simply, the '''standard error''' of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the '''standard deviation''' of the sample is the degree to which individuals within the sample differ from the sample mean.<ref>{{cite book |first=Sylvia |last=Wassertheil-Smoller |author-link=Sylvia Wassertheil-Smoller |title=Biostatistics and Epidemiology : A Primer for Health Professionals |location=New York |publisher=Springer |edition=Second |year=1995 |isbn=0-387-94388-9 |pages=40–43 |url=https://books.google.com/books?id=-PHiBwAAQBAJ&pg=PA40 }}</ref> If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases.