A geometric construction of the quadratic mean and the Pythagorean means (of two numbers

*a* and

*b*). Harmonic mean denoted by

*H*, geometric by

*G*, arithmetic by

*A* and quadratic mean (also known as

root mean square) denoted by

*Q*.

Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are

asymptotes for the harmonic 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^{[1]} because of their importance in geometry and music.

## Definition

They are defined by:

- ${\begin{aligned}\operatorname {AM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\frac {1}{n}}\left(x_{1}+\;\cdots \;+x_{n}\right)\\[9pt]\operatorname {GM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\sqrt[{n}]{\left\vert x_{1}\times \,\cdots \,\times x_{n}\right\vert }}\\[9pt]\operatorname {HM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\frac {n} {\frac {1}{x_{1}}}+\;\cdots \;+{\frac {1}{x_{n}}}}}\end{aligned}}$

## Properties

Each mean, ${\textstyle \operatorname {M} }$, has the following properties:

- Value preservation
- $\operatorname {M} (x,x,\,\ldots ,\,x)=x$
- First order homogeneity
- $\operatorname {M} (bx_{1},\,\ldots ,\,bx_{n})=b\operatorname {M} (x_{1},\,\ldots ,\,x_{n})$
- Invariance under exchange
- $\operatorname {M} (\ldots ,\,x_{i},\,\ldots ,\,x_{j},\,\ldots )=\operatorname {M} (\ldots ,\,x_{j},\,\ldots ,\,x_{i},\,\ldots )$
- for any $i$ and $j$.
- Averaging
- $\min(x_{1},\,\ldots ,\,x_{n})\leq \operatorname {M} (x_{1},\,\ldots ,\,x_{n})\leq \max(x_{1},\,\ldots ,\,x_{n})$

The harmonic and arithmetic means are reciprocal duals of each other for positive arguments:

- $\operatorname {HM} \left({\frac {1}{x_{1}}},\,\ldots ,\,{\frac {1}{x_{n}}}\right)={\frac {1}{\operatorname {AM} \left(x_{1},\,\ldots ,\,x_{n}\right)}}$

while the geometric mean is its own reciprocal dual:

- $\operatorname {GM} \left({\frac {1}{x_{1}}},\,\ldots ,\,{\frac {1}{x_{n}}}\right)={\frac {1}{\operatorname {GM} \left(x_{1},\,\ldots ,\,x_{n}\right)}}$

## Inequalities among means

There is an ordering to these means (if all of the $x_{i}$ are positive)

- $\min \leq \operatorname {HM} \leq \operatorname {GM} \leq \operatorname {AM} \leq \max$

with equality holding if and only if the $x_{i}$ are all equal.

This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The proof follows from the arithmetic-geometric mean inequality, $\operatorname {AM} \leq \max$, and reciprocal duality ($\min$ and $\max$ are also reciprocal dual to each other).

The study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while the arithmetic mean is a linear function of its arguments, so both concave and convex.

## See also

## References

**^** Heath, Thomas. *History of Ancient Greek Mathematics*.

## External links