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| {{merge from|Deviation (statistics)|date=February 2013}}
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| In [[statistics]] and [[mathematical optimization|optimization]], '''statistical errors''' and '''residuals''' are two closely related and easily confused measures of the [[deviation (statistics)|deviation]] of an observed value of an element of a [[Sample (statistics)|statistical sample]] from its "theoretical value". The '''error''' (or '''disturbance''') of an observed value is the deviation of the observed value from the (unobservable) ''true'' function value, while the '''residual''' of an observed value is the difference between the observed value and the ''estimated'' function value.
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| The distinction is most important in [[regression analysis]], where it leads to the concept of [[studentized residual]]s.
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| ==Introduction==
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| 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.
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| 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.
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| A '''residual''' (or fitting error), 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:
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| * The difference between the height of each man in the sample and the unobservable ''population'' mean is a ''statistical error'', whereas
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| * The difference between the height of each man in the sample and the observable ''sample'' mean is a ''residual''.
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| Note that 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.
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| 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]].
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| ===Example===
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| If we assume a [[normal distribution|normally distributed]] population with mean μ and [[standard deviation]] σ, and choose individuals independently, then we have | |
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| :<math>X_1, \dots, X_n\sim N(\mu,\sigma^2)\,</math> | |
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| and the [[arithmetic mean|sample mean]]
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| :<math>\overline{X}={X_1 + \cdots + X_n \over n}</math>
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| is a random variable distributed thus:
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| :<math>\overline{X}\sim N(\mu, \sigma^2/n).</math>
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| The ''statistical errors'' are then
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| :<math>\varepsilon_i=X_i-\mu,\,</math>
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| whereas the ''residuals'' are
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| :<math>\widehat{\varepsilon}_i=X_i-\overline{X}.</math>
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| (As is often done, the "[[Caret|hat]]" over the letter ε indicates an observable ''estimate'' of an unobservable quantity called ε.)
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| The sum of squares of the '''statistical errors''', divided by σ<sup>2</sup>, has a [[chi-squared distribution]] with ''n'' [[Degrees of freedom (statistics)|degrees of freedom]]:
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| :<math>\sum_{i=1}^n \left(X_i-\mu\right)^2/\sigma^2\sim\chi^2_n.</math>
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| This quantity, however, is not observable. The sum of squares of the '''residuals''', on the other hand, is observable. The quotient of that sum by σ<sup>2</sup> has a chi-squared distribution with only ''n'' − 1 degrees of freedom:
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| :<math>\sum_{i=1}^n \left(\,X_i-\overline{X}\,\right)^2/\sigma^2\sim\chi^2_{n-1}.</math>
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| It is remarkable that the [[Squared deviations|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 quotient
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| :<math>{\overline{X}_n - \mu \over S_n/\sqrt{n}}.</math>
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| The probability distributions of the numerator and the denominator separately depend on the value of the unobservable population standard deviation ''σ'', but ''σ'' appears in both the numerator and the denominator and cancels. That is fortunate because it means that even though we do not know ''σ'', we know the probability distribution of this quotient: it has a [[Student's t-distribution]] with ''n'' − 1 degrees of freedom. We can therefore use this quotient to find a [[confidence interval]] for ''μ''.
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| ==Regressions==
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| In [[regression analysis]], the distinction between ''errors'' and ''residuals'' is subtle and important, and leads to the concept of [[studentized residual]]s. Given an unobservable function that relates the independent variable to the dependent variable – say, a line – the deviations of the dependent variable observations from this function are the unobservable errors. If one runs a regression on some data, then the deviations of the dependent variable observations from the ''fitted'' function are the residuals.
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| However, a terminological difference arises in the expression [[mean squared error]] (MSE). The mean squared error of a regression is a number computed from the sum of squares of the computed ''residuals'', and not of the unobservable ''errors''. If that sum of squares is divided by ''n'', the number of observations, the result is the mean of the squared residuals. Since this is a [[bias (statistics)|biased]] estimate of the variance of the unobserved errors, the bias is removed by multiplying the mean of the squared residuals by ''n'' / ''df'' where ''df'' is the number of [[degrees of freedom (statistics)|degrees of freedom]] (''n'' minus the number of parameters being estimated). This latter formula serves as an unbiased estimate of the variance of the unobserved errors, and is called the mean squared error.<ref>{{cite book |last1=Steel |first1=Robert G. D. |last2=Torrie |first2=James H. |title=Principles and Procedures of Statistics, with Special Reference to Biological Sciences |year=1960 |publisher=McGraw-Hill |page=288}}</ref>
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| However, because of the behavior of the process of regression, the ''distributions'' of residuals at different data points (of the input variable) may vary ''even if'' the errors themselves are identically distributed. Concretely, in a [[linear regression]] where the errors are identically distributed, the variability of residuals of inputs in the middle of the domain will be ''higher'' than the variability of residuals at the ends of the domain: linear regressions fit endpoints better than the middle.
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| This is also reflected in the [[Influence function (statistics)|influence functions]] of various data points on the [[regression coefficient]]s: endpoints have more influence.
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| Thus to compare residuals at different inputs, one needs to adjust the residuals by the expected variability of ''residuals,'' which is called [[studentizing]]. This is particularly important in the case of detecting [[outliers]]: a large residual may be expected in the middle of the domain, but considered an outlier at the end of the domain.
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| ===Stochastic error===
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| The stochastic error in a measurement is the error that is random from one measurement to the next. Stochastic errors tend to be [[gaussian]] (normal), in their distribution. That's because the stochastic error is most often the sum of many random errors, and when many random errors are added together, the distribution of their sum looks gaussian, as shown by the [[Central Limit Theorem]]. | |
| A stochastic error is added to a regression equation to introduce all the variation in Y that cannot be explained by the included Xs. It is, in effect, a symbol of our inability to model all the movements of the dependent variable.
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| ==Other uses of the word "error" in statistics==
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| The use of the term "error" as discussed in the sections above is in the sense of a deviation of a value from a hypothetical unobserved value. At least two other uses also occur in statistics, both referring to observable prediction errors:
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| [[Mean square error]] or '''mean squared error''' (abbreviated MSE) and [[Root mean square deviation | root mean square error]] (RMSE) refer to the amount by which the values predicted by an estimator differ from the quantities being estimated (typically outside the sample from which the model was estimated).
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| '''Sum of squared errors''', typically abbreviated SSE or SS<sub>e</sub>, refers to the [[residual sum of squares]] (the sum of squared residuals) of a regression; this is the sum of the squares of the deviations of the actual values from the predicted values, within the sample used for estimation. Likewise, the '''sum of absolute errors''' (SAE) refers to the sum of the absolute values of the residuals, which is minimized in the [[least absolute deviations]] approach to regression.
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| ==See also==
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| {{Portal|Statistics}}
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| {{Colbegin}}
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| * [[Absolute deviation]]
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| * [[Consensus forecasts]]
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| * [[Deviation (statistics)]]
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| * [[Error detection and correction]]
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| * [[Explained sum of squares]]
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| * [[Innovation (signal processing)]]
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| * [[Innovations vector]]
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| * [[Lack-of-fit sum of squares]]
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| * [[Margin of error]]
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| * [[Mean absolute error]]
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| * [[Propagation of error]]
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| * [[Regression dilution]]
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| * [[Root mean square deviation]]
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| * [[Sampling error]]
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| * [[Studentized residual]]
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| * [[Type I and type II errors]]
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| {{Colend}}
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| ==References==
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| {{reflist}}
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| *{{cite book |last1=Cook |first1=R. Dennis |last2=Weisberg |first2=Sanford |title=Residuals and Influence in Regression. |year=1982 |publisher=[[Chapman and Hall]] |location=New York |isbn=041224280X |url=http://www.stat.umn.edu/rir/ |edition=Repr. |accessdate=23 February 2013}}
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| *{{cite book |last=Weisberg |first=Sanford |title=Applied Linear Regression |year=1985 |publisher=Wiley |location=New York |isbn=9780471879572 |url=http://books.google.com/books?id=yRrvAAAAMAAJ&dq=editions:UMM1U2yvYVUC |edition=2nd |accessdate=23 February 2013}}
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| *{{springer|title=Errors, theory of|id=p/e036240}}
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| {{least squares and regression analysis|state=expanded}}
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| {{Statistics|correlation|state=collapsed}}
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| {{DEFAULTSORT:Errors And Residuals In Statistics}}
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| [[Category:Statistical deviation and dispersion]]
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| [[Category:Regression analysis]]
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| [[Category:Statistical theory]]
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| [[Category:Error]]
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| [[Category:Measurement]]
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| [[Category:Statistical terminology]]
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| [[fr:Résidu (statistiques)]]
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| [[id:Galat]]
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| [[it:Errore statistico]]
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| [[pt:Teoria dos erros]]
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| [[zh:误差]]
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