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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 Geometric mean ： ウィキペディア英語版 In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ''n''th root of the product of n numbers, i.e., for a set of numbers \_^N, the geometric mean is defined as \left(\prod_^N x_i\right)^.For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is \sqrt=4. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is \sqrt()=1/2.A geometric mean is often used when comparing different items – finding a single "figure of merit" for these items – when each item has multiple properties that have different numeric ranges.(【引用サイトリンク】url=http://www.tpc.org/tpcd/faq.asp In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ''n''th root of the product of n numbers, i.e., for a set of numbers $\backslash \_^N$, the geometric mean is defined as $\backslash left(\backslash prod\_^N\; x\_i\backslash right)^$. For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is $\backslash sqrt=4$. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is $\backslash sqrt()=1/2$. A geometric mean is often used when comparing different items – finding a single "figure of merit" for these items – when each item has multiple properties that have different numeric ranges.〔(【引用サイトリンク】url=http://www.tpc.org/tpcd/faq.asp#anchor1140017 )〕 For example, the geometric mean can give a meaningful "average" to compare two companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0 to 100 for their financial viability. If an arithmetic mean were used instead of a geometric mean, the financial viability is given more weight because its numeric range is larger—so a small percentage change in the financial rating (e.g. going from 80 to 90) makes a much larger difference in the arithmetic mean than a large percentage change in environmental sustainability (e.g. going from 2 to 5). The use of a geometric mean "normalizes" the ranges being averaged, so that no range dominates the weighting, and a given percentage change in any of the properties has the same effect on the geometric mean. So, a 20% change in environmental sustainability from 4 to 4.8 has the same effect on the geometric mean as a 20% change in financial viability from 60 to 72. The geometric mean can be understood in terms of geometry. The geometric mean of two numbers, $a$ and $b$, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths $a$ and $b$. Similarly, the geometric mean of three numbers, $a$, $b$, and $c$, is the length of one side of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers. The geometric mean applies only to numbers of the same sign.〔The geometric mean only applies to numbers of the same sign in order to avoid taking the root of a negative product, which would result in imaginary numbers, and also to satisfy certain properties about means, which is explained later in the article. Note that the definition is unambiguous if one allows 0 (which yields a geometric mean of 0), but may be excluded, as one frequently wishes to take the logarithm of geometric means (to convert between multiplication and addition), and one cannot take the logarithm of 0.〕 It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment. The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal 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 (see Inequality of arithmetic and geometric means.) ==Calculation== The geometric mean of a data set $\backslash$ is given by: :$\backslash left(\backslash prod\_^n\; a\_i\; \backslash right)^\; =\; \backslash sqrt().$ The geometric mean of a data set is less than the data set's arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the arithmetic-geometric mean, a mixture of the two which always lies in between. The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences ($a\_n$) and ($h\_n$) are defined: :$a\_\; =\; \backslash frac,\; \backslash quad\; a\_0=x$ and :$h\_\; =\; \backslash frac\; +\; \backslash frac\},\; \backslash quad\; h\_0=y$ where $h\_$ is the harmonic mean of the previous values of the two sequences, then $a\_n$ and $h\_n$ will converge to the geometric mean of $x$ and $y$. This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by Bolzano–Weierstrass theorem) and the fact that geometric mean is preserved: :$\backslash sqrt=\backslash sqrt\}\}=\backslash sqrt+\backslash frac\}\}=\backslash sqrt\}$ Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result. ===Relationship with arithmetic mean of logarithms === You can get the same result every time for the geometric mean using a method that involves logarithms. By using logarithmic identities to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication: :$\backslash left(\backslash prod\_^na\_i\; \backslash right)^\; =\; \backslash exp\backslash left(a\_i\backslash right)$ This is sometimes called the log-average (not to be confused with the logarithmic average). It is simply computing the arithmetic mean of the logarithm-transformed values of $a\_i$ (i.e., the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale, i.e., it is the generalised f-mean with $f(x)\; =\; \backslash log\; x$. For example, the geometric mean of 2 and 8 can be calculated as: :$b^\; =\; 4,$ where $b$ is any base of a logarithm (commonly 2, $e$ or 10). The right-hand side formula above is generally the preferred alternative for implementation in computer languages. This is because calculating the product of many numbers can lead to an arithmetic overflow or arithmetic underflow. This is less likely to occur when you first take the logarithm of each number and sum these. 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「In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ''n''th root of the product of n numbers, i.e., for a set of numbers \_^N, the geometric mean is defined as \left(\prod_^N x_i\right)^.For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is \sqrt=4. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is \sqrt()=1/2.A geometric mean is often used when comparing different items – finding a single "figure of merit" for these items – when each item has multiple properties that have different numeric ranges.(【引用サイトリンク】url=http://www.tpc.org/tpcd/faq.asp」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.