By Kardi Teknomo, PhD .

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Minkowski Mean

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

Minkowski mean is defined as

Minkowski Mean


When Minkowski Mean , we have Arithmetic mean Minkowski Mean


When Minkowski Mean , we have Harmonic mean

Minkowski Mean

Example :

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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