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In statistics, Bessel's correction, named after Friedrich Bessel, is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This corrects the bias in the estimation of the population variance, and some (but not all) of the bias in the estimation of the population standard deviation, but often increases the mean squared error in these estimations. That is, when estimating the population variance and standard deviation from a sample when the population mean is unknown, the sample variance estimated as the ''mean'' of the squared deviations of sample values from their mean (that is, using a multiplicative factor ) is a biased estimator of the population variance, and for the average sample underestimates it. Multiplying the standard sample variance as computed in that fashion by (equivalently, using instead of in the estimator's formula) corrects for this, and gives an unbiased estimator of the population variance. In some terminology,〔W.J. Reichmann, W.J. (1961) ''Use and abuse of statistics'', Methuen. Reprinted 1964–1970 by Pelican. Appendix 8.〕〔Upton, G.; Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. ISBN 978-0-19-954145-4 (entry for "Variance (data)")〕 the factor is itself called Bessel's correction. One can understand Bessel's correction intuitively as the degrees of freedom in the residuals vector (residuals, not errors, because the population mean is unknown): : where is the sample mean. While there are ''n'' independent samples, there are only ''n'' − 1 independent residuals, as they sum to 0. ==Caveats== Three caveats must be borne in mind regarding Bessel's correction: firstly, it does not yield an unbiased estimator of standard ''deviation;'' secondly, the corrected estimator often has worse (higher) mean squared error (MSE) than the uncorrected estimator, and never has the minimum MSE: a different scale factor can always be chosen to minimize MSE; thirdly it is only necessary when the population mean is unknown (and estimated as the sample mean). Firstly, while the sample variance (using Bessel's correction) is an unbiased estimate of the population variance, its square root, the sample standard deviation, is a ''biased'' estimate of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality. There is no general formula for an unbiased estimator of the population standard deviation, though there are correction factors for particular distributions, such as the normal; see unbiased estimation of standard deviation for details. An approximation for the exact correction factor for the normal distribution is given by using ''n'' − 1.5 in the formula: the bias decays quadratically (rather than linearly, as in the uncorrected form and Bessel's corrected form). Secondly, the unbiased estimator does not minimize MSE compared with biased estimators, and generally has worse MSE than the uncorrected estimator (this varies with excess kurtosis). MSE can be minimized by using a different factor. The optimal value depends on excess kurtosis, as discussed in mean squared error: variance; for the normal distribution this is optimized by dividing by ''n'' + 1 (instead of ''n'' − 1 or ''n''). Thirdly, Bessel's correction is only necessary when the population mean is unknown, and one is estimating ''both'' population mean ''and'' population variance from a given sample set, using the sample mean to estimate the population mean. In that case there are ''n'' degrees of freedom in a sample of ''n'' points, and simultaneous estimation of mean and variance means one degree of freedom goes to the sample mean and the remaining ''n'' − 1 degrees of freedom (the ''residuals'') go to the sample variance. However, if the population mean is known, then the deviations of the samples from the population mean have ''n'' degrees of freedom (because the mean is not being estimated – the deviations are not residuals but ''errors'') and Bessel's correction is not applicable. 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「Bessel's correction」の詳細全文を読む スポンサード リンク
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