Editing Average (section)

==Miscellaneous types==
Other more sophisticated averages are: [[trimean]], [[trimedian]], and [[normalized mean]], with their generalizations.<ref>{{cite journal |last1=Merigo |first1=Jose M. |last2=Cananovas |first2=Montserrat |title=The Generalized Hybrid Averaging Operator and its Application in Decision Making |date=2009 |journal=Journal of Quantitative Methods for Economics and Business Administration |volume=9 |pages=69–84 |issn=1886-516X }}</ref>

One can create one's own average metric using the [[generalized f-mean|generalized ''f''-mean]]:

: <math>y = f^{-1}\left(\frac{1}{n}\left[f(x_1) + f(x_2) + \cdots + f(x_n)\right]\right)</math>

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&nbsp;''x''.

However, this method for generating means is not general enough to capture all averages.  A more general method<ref name="Bibby" />{{fv|date=May 2022}} for defining an average takes any function ''g''(''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>) of a list of arguments that is [[Continuous function|continuous]], [[Monotonicity|strictly increasing]] in each argument, and symmetric (invariant under [[permutation]] of the arguments). The average ''y'' is then the value that, when replacing each member of the list, results in the same function value: {{nowrap|1=''g''(''y'', ''y'', ..., ''y'') =}} {{nowrap|''g''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}}.  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 {{nowrap|1=''g''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) =}} {{nowrap|''x''<sub>1</sub>+''x''<sub>2</sub>+ ··· + ''x''<sub>''n''</sub>}} provides the arithmetic mean. The function {{nowrap|1 = ''g''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) =}} {{nowrap|''x''<sub>1</sub>''x''<sub>2</sub>···''x''<sub>''n''</sub>}} (where the list elements are positive numbers) provides the geometric mean. The function {{nowrap|1 = ''g''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) =}} {{nowrap|(''x''<sub>1</sub><sup>−1</sup>+''x''<sub>2</sub><sup>−1</sup>+ ··· + ''x''<sub>''n''</sub><sup>−1</sup>)<sup>−1</sup>)}} (where the list elements are positive numbers) provides the harmonic mean.<ref name=Bibby>{{cite journal | last1 = Bibby | first1 = John |date= 1974 | title = Axiomatisations of the average and a further generalisation of monotonic sequences | journal = [[Glasgow Mathematical Journal]] | volume = 15 | pages = 63–65 | doi=10.1017/s0017089500002135| doi-access = free }}</ref>

===Average percentage return and CAGR===
{{Main|Compound annual growth rate}}

A type of average used in finance is the average percentage return. It is an example of a geometric mean. When the returns are annual, it is called the Compound Annual Growth Rate (CAGR). For example, if we are considering a period of two years, and the investment return in the first year is −10% and the return in the second year is +60%, then the average percentage return or CAGR, ''R'', can be obtained by solving the equation: {{nowrap|1= (1 − 10%) × (1 + 60%) = (1 − 0.1) × (1 + 0.6) = (1 + ''R'') × (1 + ''R'')}}. The value  of ''R'' that makes this equation true is 0.2, or 20%. This means that the total return over the 2-year period is the same as if there had been 20% growth each year. The order of the years makes no difference – the average percentage returns of +60% and −10% is the same result as that for −10% and +60%.

This method can be generalized to examples in which the periods are not equal. For example, consider a period of a half of a year for which the return is −23% and a period of two and a half years for which the return is +13%. The average percentage return for the combined period is the single year return, ''R'', that is the solution of the following equation: {{nowrap|1= (1 − 0.23)<sup>0.5</sup> × (1 + 0.13)<sup>2.5</sup> = (1 + ''R'')<sup>0.5+2.5</sup>}}, giving an average return ''R'' of 0.0600 or 6.00%.