Definition:Pythagorean Mean
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Definition
The Pythagorean means are as follows:
Arithmetic Mean
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers.
The arithmetic mean of $x_1, x_2, \ldots, x_n$ is defined as:
- $\ds A_n := \dfrac 1 n \sum_{k \mathop = 1}^n x_k$
That is, to find out the arithmetic mean of a set of numbers, add them all up and divide by how many there are.
Geometric Mean
Let $x_1, x_2, \ldots, x_n \in \R_{>0}$ be (strictly) positive real numbers.
The geometric mean of $x_1, x_2, \ldots, x_n$ is defined as:
- $\ds G_n := \paren {\prod_{k \mathop = 1}^n x_k}^{1/n}$
Harmonic Mean
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all strictly positive.
The harmonic mean $H_n$ of $x_1, x_2, \ldots, x_n$ is defined as:
- $\ds \dfrac 1 {H_n} := \frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {x_k} }$
That is, to find the harmonic mean of a set of $n$ numbers, take the reciprocal of the arithmetic mean of their reciprocals.
Also see
- Results about the Pythagorean means can be found here.
Source of Name
This entry was named for Pythagoras of Samos.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Pythagorean means