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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 "theoretical value". The error (or disturbance) of an observed value is the deviation of the observed value from the (unobservable) ''true'' value of a quantity of interest (for example, a population mean), and the residual of an observed value 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 it leads to the concept of studentized residuals. ==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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Errors and residuals」の詳細全文を読む スポンサード リンク
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