By Kardi Teknomo, PhD .

< Previous | Next | Contents >

Generalized Mean

In the previous section, we have discussed two generalized means of Minkowski mean and Lehmer mean. However, the two means is not "general enough" because Minkowski mean does not have a specific p to represent Geometric mean while Lehmer mean does not have a specific p to indicate Quadratic mean. This lead to a question whether we have a general mean to represent all the four basic means (Arithmetic mean, Geometric mean, Harmonic Mean and Quadratic Mean).

Phillips (2000) suggests the following quasi-arithmetic generalized mean which has been available through the work of Andrey Kolmogorov in 1930 . The mean is quite general that it can cover arithmetic mean , quadratic mean , harmonic mean and geometric mean and many others. In fact, Kolmogorov Generalzed mean can also cover Minkowski mean as shown in the example below.

Instead of using a parameter, however, Kolmogorov generalized mean uses function and its inverse function to generate mean. Suppose we have a continuous monotonic function Generalized Mean from positive real numbers to positive real number and the inverse function of Generalized Mean exist, then we can make infinite number of means using

Generalized Mean

Extending the generalized mean to Generalized Mean input numbers is straightforward.

Generalized Mean

I hope you remember the definition of inverse function that if Generalized Mean then Generalized Mean . Simple inverse function can be obtained by exchange Generalized Mean and Generalized Mean in the expression. Then the following are two properties of inverse function:

  1. Generalized Mean
  2. Generalized Mean

Example:

Generalized Mean , then Generalized Mean , then we have Generalized Mean which is an arithmetic mean

Example:

Generalized Mean , then Generalized Mean , then we have Generalized Mean which can be simplified into Geometric mean by Generalized Mean Since Generalized Mean only accept positive values of x, Geometric mean also can accept positive values.

Example:

Generalized Mean , then Generalized Mean , then we get Generalized Mean which is a quadratic mean

Example:

Generalized Mean , then Generalized Mean , then we obtain Generalized Mean which is a harmonic mean

Example:

Generalized Mean , then Generalized Mean , then we get Generalized Mean which is Minkowski mean

Example:

Generalized Mean , then Generalized Mean , then we obtain Generalized Mean which is a Harmonic mean

Table below provides the summary of functions to relate Kolmogorov generalized mean and other means.

Name

function

Inverse function

Mean of (a, b)

Arithmetic Mean

Generalized Mean

Generalized Mean

Generalized Mean

Geometric mean

Generalized Mean

Generalized Mean

Generalized Mean

Harmonic mean

Generalized Mean

Generalized Mean

Generalized Mean

Minkowski mean

Generalized Mean

Generalized Mean

Generalized Mean

Quadratic mean

Generalized Mean

Generalized Mean

Generalized Mean

< Previous | Next | Contents >

See also: Minkowski mean , Lehmer mean , arithmetic mean , harmonic mean , geometric mean , quadratic mean
Rate this tutorial or give your comments about this tutorial

This tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi (2015) Mean and Average. https:\\people.revoledu.com\kardi\tutorial\BasicMath\Average\