Miscellaneous Types
Other more sophisticated averages are: trimean, trimedian, and normalized mean, with their generalizations.
One can create one's own average metric using the generalized f-mean:
where f is any invertible function. The harmonic mean is an example of this using f(x) = 1/x, and the geometric mean is another, using f(x) = log x.
However, this method for generating means is not general enough to capture all averages. A more general method for defining an average takes any function g(x1, x2, ..., xn) of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value which, when replacing each member of the list, results in the same function value: g(y, y, ..., y) = g(x1, x2, ..., xn). This most general definition still captures the important property of all averages that the average of a list of identical elements is that element itself. The function g(x1, x2, ..., xn) = x1+x2+ ··· + xn provides the arithmetic mean. The function g(x1, x2, ..., xn) = x1x2···xn (where the list elements are positive numbers) provides the geometric mean. The function g(x1, x2, ..., xn) = −(x1−1+x2−1+ ··· + xn−1) (where the list elements are positive numbers) provides the harmonic mean.
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