The standard error of a method of measurement or estimation is the estimated standard deviation of the error in that method. Namely, it is the standard deviation of the difference between the measured or estimated values and the true values. Notice that the true value is usually unknown and the standard error of an estimate is itself an unknown quantity that has to be estimated.
Assumptions and usage
If the data are assumed to be normally distributed, quantiles of the normal distribution and the sample mean and standard error can be used to calculate confidence intervals for the mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where 'x' is equal to the sample mean, 'y' is equal to the standard error of the sample, and 1.96 is the .975 quantile of the normal distribution.
Upper 95% Limit=x+(y*1.96)
Lower 95% Limit=x-(y*1.96). In particular, the standard error of a sample statistic (such as sample mean) is the estimated standard deviation of the error in the process by which it was generated. In other words, it is the standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of <math>SE</math>, <math>SEM</math> (for standard error of measurement or mean), or <math>S_E</math>. Standard errors provide simple measures of uncertainty in a value and are often used because:
- If the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated in many cases;
- Where the probability distribution of the value is known, it can be used to calculate an exact confidence interval; and
- Where the probability distribution is unknown, relationships like Chebyshev’s or the Vysochanskiï-Petunin inequality can be used to calculate a conservative confidence interval
- As the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.
Standard error of the mean
The standard error of the mean (SEM) of a sample from a population is the standard deviation of the sample divided by the square root of the sample size:
- <math>S_E = \frac{\widehat\sigma}{\sqrt{n}}</math>
where
- <math>\widehat\sigma</math> is the standard deviation of the sample, and
- n is the size (number of items) of the sample.
Note: Standard error may also be defined as the standard deviation of the residual error term. (Kenney and Keeping, p. 187; Zwillinger 1995, p. 626)
