Pythagorean means

In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. They can now be regarded as special cases of a family of functions appropriately called the generalized means.

Definition The three Pythagorean means are defined by the equations

AM ⁡ ( x 1 , … , x n ) = x 1 + ⋯ + x n n , GM ⁡ ( x 1 , … , x n ) = | x 1 × ⋯ × x n | n , and HM ⁡ ( x 1 , … , x n ) = n 1 x 1 + ⋯ + 1 x n . {\displaystyle {\begin{aligned}\operatorname {AM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\frac {x_{1}+\;\cdots \;+x_{n}}{n}},\\[9pt]\operatorname {GM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\sqrt[{n}]{\left\vert x_{1}\times \,\cdots \,\times x_{n}\right\vert }},{\text{ and}}\\[9pt]\operatorname {HM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\frac {n}{\displaystyle {\frac {1}{x_{1}}}+\;\cdots \;+{\frac {1}{x_{n}}}}}.\end{aligned}}}

Properties Each mean, M {\textstyle \operatorname {M} } , has the following properties for positive real inputs:

First-order homogeneity

M ⁡ ( b x 1 , … , b x n ) = b M ⁡ ( x 1 , … , x n ) {\displaystyle \operatorname {M} (bx_{1},\ldots ,bx_{n})=b\operatorname {M} (x_{1},\ldots ,x_{n})} This ensures the physical value of the mean must be the same, for any choice of a ratio-scale for its units. Invariance under exchange

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