Generalized mean

In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).

Definition

If p is a non-zero real number, and

x

1

,

…

,

x

n

{\displaystyle x_{1},\dots ,x_{n}}

are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is

M

p

(

x

1

,

…

,

x

n

)

=

(

1

n

∑

i

=

1

n

x

i

p

)

1

/

p

.

{\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{p}\right)^{{1}/{p}}.}

(See p-norm). For p = 0 we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):

M

0

(

x

1

,

…

,

x

n

)

=

(

∏

i

=

1

n

x

i

)

1

/

n

.

{\displaystyle M_{0}(x_{1},\dots ,x_{n})=\left(\prod _{i=1}^{n}x_{i}\right)^{1/n}.}

Furthermore, for a sequence of positive weights wi we define the weighted power mean as

M

p

(

x

1

,

…

,

x

n

)

=

(

∑

i

=

1

n

w

i

x

i

p

∑

i

=

1

n

w

i

)

1

/

p

{\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {\sum _{i=1}^{n}w_{i}x_{i}^{p}}{\sum _{i=1}^{n}w_{i}}}\right)^{{1}/{p}}}

and when p = 0, it is equal to the weighted geometric mean:

M

0

(

x

1

,

…

,

x

n

)

=

(

∏

i

=

1

n

x

i

w

i

)

1

/

∑

i

=

1

n

w

i

.

{\displaystyle M_{0}(x_{1},\dots ,x_{n})=\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)^{1/\sum _{i=1}^{n}w_{i}}.}

The unweighted means correspond to setting all wi = 1.

Special cases

For some values of

p

{\displaystyle p}

, the mean

M

p

(

x

1

,

…

,

x

n

)

{\displaystyle M_{p}(x_{1},\dots ,x_{n})}

corresponds to a well known mean.

Properties

Let

x

1

,

…

,

x

n

{\displaystyle x_{1},\dots ,x_{n}}

be a sequence of positive real numbers, then the following properties hold:

min

(

x

1

,

…

,

x

n

)

≤

M

p

(

x

1

,

…

,

x

n

)

≤

max

(

x

1

,

…

,

x

n

)

{\displaystyle \min(x_{1},\dots ,x_{n})\leq M_{p}(x_{1},\dots ,x_{n})\leq \max(x_{1},\dots ,x_{n})}

.

M

p

(

x

1

,

…

,

x

n

)

=

M

p

(

P

(

x

1

,

…

,

x

n

)

)

{\displaystyle M_{p}(x_{1},\dots ,x_{n})=M_{p}(P(x_{1},\dots ,x_{n}))}

, where

P

{\displaystyle P}

is a permutation operator.