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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Harmonic mean ： ウィキペディア英語版 Harmonic mean In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average, and in particular one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals. As a simple example, the harmonic mean of 1, 2, and 4 is $\backslash frac+\backslash frac+\backslash frac\}\; =\; \backslash frac(\backslash frac+\backslash frac+\backslash frac)\}\; =\; \backslash frac\backslash ,.$ The harmonic mean ''H'' of the positive real numbers $x\_1,\; x\_2,\; ...,\; x\_n$ is defined to be :$H\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; +\; \backslash frac\}\; =\; \backslash frac\}\; =\; \backslash frac^n\; x\_j\}\}.$ From the third formula in the above equation, it is more apparent that the harmonic mean is related to the arithmetic and geometric means. It is the reciprocal dual of the arithmetic mean for positive inputs: :$1/H(1/x\_1\; \backslash ldots\; 1/x\_n)\; =\; A(x\_1\; \backslash ldots\; x\_n)$ The harmonic mean is a Schur-concave function, and dominated by the minimum of its arguments, in the sense that for any positive set of arguments, $\backslash min(x\_1\; \backslash ldots\; x\_n)\; \backslash leq\; H(x\_1\; \backslash ldots\; x\_n)\; \backslash leq\; n\; \backslash min(x\_1\; \backslash ldots\; x\_n)$. Thus, the harmonic mean can not be made arbitrarily large by changing some values to a bigger one (while having at least one value unchanged). ==Relationship with other means== The harmonic mean is one of the three Pythagorean means. For all ''positive'' data sets ''containing at least one pair of nonequal values'', the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between. (If all values in a nonempty dataset are equal, the three means are always equal to one another; e.g. the harmonic, geometric, and arithmetic means of are all 2.) It is the special case ''M''−1 of the power mean: :$H(x\_1,\; x\_2,\; ...,\; x\_n)\; =\; M\_(x\_1,\; x\_2,\; ...,\; x\_n)\; =\; \backslash frac\; +\; ...\; +\; x\_n^\}$ Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones. The arithmetic mean is often mistakenly used in places calling for the harmonic mean.〔 *''Statistical Analysis'', Ya-lun Chou, Holt International, 1969, ISBN 0030730953〕 In the speed example below for instance, the arithmetic mean of 50 is incorrect, and too big. The harmonic mean is related to the other Pythagorean means, as seen in the third formula in the above equation. This is noticed if we interpret the denominator to be the arithmetic mean of the product of numbers ''n'' times but each time we omit the ''j''th term. That is, for the first term, we multiply all ''n'' numbers except the first; for the second, we multiply all ''n'' numbers except the second; and so on. The numerator, excluding the ''n'', which goes with the arithmetic mean, is the geometric mean to the power ''n''. Thus the ''n''th harmonic mean is related to the ''n''th geometric and arithmetic means. Three positive numbers ''H'', ''G'', and ''A'' are respectively the harmonic, geometric, and arithmetic means of three positive numbers if and only if〔''Inequalities proposed in “Crux Mathematicorum”'', ().〕 :$\backslash frac+\backslash frac+1\; \backslash leq\; \backslash frac\; \backslash left(1+\backslash frac\; \backslash right)^2.$ If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.〔Mitchell, Douglas W., "More on spreads and non-arithmetic means," ''The Mathematical Gazette'' 88, March 2004, 142-144.〕 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「Harmonic mean」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.