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Minkowski Mean

By Kardi Teknomo, PhD.


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Minkowski mean is a generalization of arithmetic, quadratic and harmonic mean.

Minkowski mean is defined as


When , we have Arithmetic mean



When , we have Harmonic mean


When p=2, we have Quadratic mean

Minkowski Mean

However, we do not have a specific p in Minkowski mean to represent Geometric mean because Geometric mean is obtained through the limit of parameter p approaches 0, indicated by the formula below

Minkowski Mean

Use the interactive program below to compute Harmonic mean of a list of numbers separated by comma. You may change with your own input values. Try to change the parameter p to a small number but not zero such as 0.0000000001 and compare the result with Geometric Mean.

Input list of numbers separated by comma, then press the button "Get Minkowski Mean". The program will compute directly when you change the input data or parameter.

Parameter p = or

Table below provides the summary of parameter to relate Minkowski generalized mean and other means.


Parameter p

Arithmetic Mean

p = 1

Geometric mean

p approaches 0 (limit)

Harmonic mean

p = -1

Quadratic mean

p = 2


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See also: Lehmer Mean, Minkowski distance, generalized mean
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This tutorial is copyrighted.

Preferable reference for this tutorial is

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

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