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Classical averages studied in ancient Greece
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:
AM
(
x
1
,
…
,
x
n
)
=
1
n
(
x
1
+
⋯
+
x
n
)
GM
(
x
1
,
…
,
x
n
)
=
|
x
1
×
⋯
×
x
n
|
n
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 {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}{\displaystyle {\frac {1}{x_{1}}}+\;\cdots \;+{\frac {1}{x_{n}}}}}\end{aligned}}}
Properties
Each mean,
M
{\textstyle \operatorname {M} }
, has the following properties:
First order homogeneity
M
(
b
x
1
,
…
,
b
x
n
)
=
b
M
(
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,
…
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)
{\displaystyle \operatorname {M} (bx_{1},\,\ldots ,\,bx_{n})=b\operatorname {M} (x_{1},\,\ldots ,\,x_{n})}
Invariance under exchange
M
(
…
,
x
i
,
…
,
x
j
,
…
)
=
M
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…
,
x
j
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x
i
,
…
)
{\displaystyle \operatorname {M} (\ldots ,\,x_{i},\,\ldots ,\,x_{j},\,\ldots )=\operatorname {M} (\ldots ,\,x_{j},\,\ldots ,\,x_{i},\,\ldots )}
for any
i
{\displaystyle i}
and
j
{\displaystyle j}
.
Monotonicity
a
<
b
→
M
(
a
,
x
1
,
x
2
,
…
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)
<
M
(
b
,
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…
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)
{\displaystyle a<b\rightarrow \operatorname {M} (a,x_{1},x_{2},\ldots x_{n})<\operatorname {M} (b,x_{1},x_{2},\ldots x_{n})}
Idempotence
∀
x
,
M
(
x
,
x
,
…
x
)
=
x
{\displaystyle \forall x,\;M(x,x,\ldots x)=x}
Monotonicity and idempotence together imply that a mean of a set always lies between the extremes of the set.
min
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)
≤
M
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)
≤
max
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)
{\displaystyle \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:
HM
(
1
x
1
,
…
,
1
x
n
)
=
1
AM
(
x
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,
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)
{\displaystyle \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:
GM
(
1
x
1
,
…
,
1
x
n
)
=
1
GM
(
x
1
,
…
,
x
n
)
{\displaystyle \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
Geometric proof without words that max (a ,b ) > root mean square (RMS ) or quadratic mean (QM ) > arithmetic mean (AM ) > geometric mean (GM ) > harmonic mean (HM ) > min (a ,b ) of two distinct positive numbers a and b [note 1]
There is an ordering to these means (if all of the
x
i
{\displaystyle x_{i}}
are positive)
min
≤
HM
≤
GM
≤
AM
≤
max
{\displaystyle \min \leq \operatorname {HM} \leq \operatorname {GM} \leq \operatorname {AM} \leq \max }
with equality holding if and only if the
x
i
{\displaystyle 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 ,
AM
≤
max
{\displaystyle \operatorname {AM} \leq \max }
, and reciprocal duality (
min
{\displaystyle \min }
and
max
{\displaystyle \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
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