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I would like to know if it is possible to calculate the arithmetic mean of a series of ratios:

$$A=\frac{1}{n}\sum_{i=1}^{n} \frac{x_i}{y_i}$$

in a compositional fashion. What I mean by that is that I would like to calculate some function $f(x_i)$ that does not depend on $y_i$ and some function $g(y_i)$ that does not depend on $x_i$ separately, and then obtain the arithmetic mean by operating on the results of those functions.

I know that can easily be done with the geometric mean because the geometric mean of ratios is the ratio of the geometric means of the numerator and denominator (which can be proven by applying the quotient rule of logarithms), so I can calculate the geometric mean of $x_i$ and $y_i$ and just divide one by the other.

Is there a similar property for the arithmetic mean?

EDIT: consider the values positive and non-zero.

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2 Answers

No. Consider the case when one of the $y_i = 0$ and all other values are positive.

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The reason it works with geometric means is that they are multiplicative at heart. For arithmetic means, imagine $\frac{0}{1}$ and $\frac{1}{1}$ compared with $\frac{0}{1000}$ and $\frac{1}{1}$ compared with $\frac{0}{1000}$ and $\frac{1000}{1000}$. Here you are mixing addition and multiplication.

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