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If ''t'' is a non-zero Real Number , we can define the generalized mean with exponent ''t'' of the positive real numbers ''a''1,...,''a''''n'' as : The case ''t'' = 1 yields the Arithmetic Mean and the case ''t'' = −1 yields the Harmonic Mean . The case ''t'' = 2 yields the Root Mean Square . As ''t'' approaches 0, the Limit of M(''t'') is the Geometric Mean of the given numbers, and so it makes sense to ''define'' M(0) to be the geometric mean. Furthermore, as ''t'' approaches ∞, M(''t'') approaches the maximum of the given numbers, and as ''t'' approaches −∞, M(''t'') approaches the minimum of the given numbers. In general, if −∞ ≤ ''s'' < ''t'' ≤ ∞, then M(''s'') ≤ M(''t'') and the two means are equal if and only if ''a''1 = ''a''2 = ... = ''a''''n''. Furthermore, if ''b'' is a positive real number, then the generalized mean with exponent ''t'' of the numbers ''ba''1,..., ''ba''''n'' is equal to ''b'' times the generalized mean of the numbers ''a''1,..., ''a''''n''. The power mean could be generalized further to the Generalized F-mean : : and again a suitable choice of an invertible f(''x'') will give the arithmetic mean with f(''x'') = ''x'', the geometric mean with f(''x'') = log(''x''), the harmonic mean with f(''x'') = 1/''x'', and the generalized mean with exponent ''t'' with f(''x'') = ''x''''t''. But other functions could be used, such as f(''x'') = e''x''. SEE ALSO EXTERNAL LINKS |
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