In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean). The residual is the difference between the observed value and the estimated value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances.
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). 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 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 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 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, or more generally studentized residuals.
In univariate distributions
If we assume a normally distributed population with mean μ and standard deviation σ, and choose individuals independently, then we have
and the sample mean
is a random variable distributed such that:
The statistical errors are then
with expected values of zero, whereas the residuals are
The sum of squares of the statistical errors, divided by σ2, has a chi-squared distribution with n degrees of freedom:
However, this quantity is not observable as the population mean is unknown. The sum of squares of the residuals, on the other hand, is observable. The quotient of that sum by σ2 has a chi-squared distribution with only n − 1 degrees of freedom:
This difference between n and n − 1 degrees of freedom results in Bessel's correction for the estimation of sample variance of a population with unknown mean and unknown variance. No correction is necessary if the population mean is known.
Remark
It is remarkable that the sum of squares of the residuals and the sample mean can be shown to be independent of each other, using, e.g. Basu's theorem. That fact, and the normal and chi-squared distributions given above form the basis of calculations involving the t-statistic:
where represents the errors, represents the sample standard deviation for a sample of size n, and unknown σ, and the denominator term accounts for the standard deviation of the errors according to: