Editing Errors and residuals (section)

==Introduction==
Suppose there is a series of observations from a [[univariate distribution]] and we want to estimate the [[mean]] of that distribution (the so-called [[location model (statistics)|location model]]). In this case, the errors are the deviations of the observations from the population mean, while the residuals are the deviations of the observations from the sample mean.

A '''statistical error''' (or '''disturbance''') is the amount by which an observation differs from its [[expected value]], the latter being based on the whole [[statistical population|population]] from which the statistical unit was chosen randomly. For example, if the mean height in a population of 21-year-old men is 1.75 meters, and one randomly chosen man is 1.80 meters tall, then the "error" is 0.05 meters; if the randomly chosen man is 1.70 meters tall, then the "error" is −0.05 meters. The expected value, being the [[arithmetic mean|mean]] of the entire population, is typically unobservable, and hence the statistical error cannot be observed either.

A '''residual''' (or fitting deviation), on the other hand, is an observable ''estimate'' of the unobservable statistical error. Consider the previous example with men's heights and suppose we have a random sample of ''n'' people. The ''[[sample mean]]'' could serve as a good estimator of the ''population'' mean. Then we have:
* The difference between the height of each man in the sample and the unobservable ''population'' mean is a ''statistical error'', whereas
* The difference between the height of each man in the sample and the observable ''sample'' mean is a ''residual''.

Note that, because of the definition of the sample mean, the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily ''not [[statistical independence|independent]]''. The statistical errors, on the other hand, are independent, and their sum within the random sample is [[almost surely]] not zero.

One can standardize statistical errors (especially of a [[normal distribution]]) in a [[z-score]] (or "standard score"), and standardize residuals in a [[t-statistic|''t''-statistic]], or more generally [[studentized residuals]].