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Power Mean

Statistical Glossary

Power Mean:

A power mean of order Math image of a set of Math image values Math image is defined by the following expression:

The family of power mean statistics Math image is often called the generalized mean – because, for different values of the parameter Math image , it is equivalent to various types of descriptive statistics: Math image (i.e Math image when Math image ) coincides with the arithmetic mean ; Math image is the harmonic mean ; Math image is the root mean square .

The greater the parameter Math image , the greater the contribution of the largest values Math image towards the values of Math image . If we increase Math image infinitely, the value of the power mean approaches the maximum value in the sample Math image , and the reverse – if we decrease Math image infinitely, then the value of the power mean approaches the minimum value in the sample. This is often denoted as


Strictly speaking, Math image is not the arithmetic mean – because this statistic is defined only for non-negative values Math image , while the arithmetic mean is defined for both negative and non-negative values Math image ).

The power mean Math image is not a “fair” measure of central location – it does not meet requirements of shift invariance , like other statistics defined only for non-negative values (see explanations of central tendency ). Therefore, it would be more correct to classify the power mean as a measure of “average magnitude” or “effective magnitude”. Any power mean Math image is scale-invariant .

See also Mean values (comparison) and the online short course Basic Concepts in Probability and Statistics

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