Statistical Glossary
Power Mean:
A power mean of order 
of a set of 
values 
is defined by the following expression:
 |
The family of power mean statistics 
is often called the generalized mean - because, for different values of the parameter 
, it is equivalent to various types of descriptive statistics: 
(i.e 
when 
) coincides with the arithmetic mean ; 
is the harmonic mean ; 
is the root mean square .
The greater the parameter 
, the greater the contribution of the largest values 
towards the values of 
. If we increase 
infinitely, the value of the power mean approaches the maximum value in the sample 
, and the reverse - if we decrease 
infinitely, then the value of the power mean approaches the minimum value in the sample. This is often denoted as
 |
and
 |
Strictly speaking, 
is not the arithmetic mean - because this statistic is defined only for non-negative values 
, while the arithmetic mean is defined for both negative and non-negative values 
).
The power mean 
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 
is scale-invariant .
See also Mean values (comparison) and the online short course Basic Concepts in Probability and Statistics