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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 Standardized moment ： ウィキペディア英語版 Standardized moment In probability theory and statistics, the ''k''th standardized moment of a probability distribution is $\backslash frac\backslash !$ where $\backslash mu\_k$ is the ''k''th moment about the mean and σ is the standard deviation. It is the normalization of the ''k''th moment with respect to standard deviation. The power of ''k'' is because moments scale as $x^k$, meaning that $\backslash mu\_k(\backslash lambda\; X)\; =\; \backslash lambda^k\; \backslash mu\_k(X)$: they are homogeneous polynomials of degree ''k'', thus the standardized moment is scale invariant. This can also be understood as being because moments have dimension; in the above ratio defining standardized moments, the dimensions cancel, so they are dimensionless numbers. * The first standardized moment is zero, because the first moment about the mean is zero * The second standardized moment is one, because the second moment about the mean is equal to the variance (the square of the standard deviation) * The third standardized moment is the skewness * The fourth standardized moment is the historical kurtosis (used in older works) Note that for skewness and kurtosis alternative definitions exist, which are based on the third and fourth cumulant respectively. ==Other normalizations== Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, $\backslash frac$. However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because $\backslash mu$ is the first moment about zero (the mean), not the first moment about the mean (which is zero). See Normalization (statistics) for further normalizing ratios. 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「Standardized moment」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.