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I have a set of data, each element has a weight. I need to estimate the mean of this data. I found that there are two ways: A weighted geometric mean and a weighted mean. When should I use each of them and what are the advantages of using each of them?

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All the mentioned mean operations are special cases of the generalized mean operator: f_s = ((x1^s+...+xn^s)/n)^(1/s). For s = -infinit, -1, 0, 1, +infinite, the generalized mean f_s becomes: the minimum, harmonic mean, geometric mean, arithmetic mean and the maximum. Also, f_s depends monotonous on: f_s biases towards larger values for larger s. –  James LI Sep 29 '12 at 16:03
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2 Answers

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The geometric mean and the mean whether weighted or unweighted are different paramaeters of a distribution. So the question of which parameter to estimate depends on which aspect of the distribution you are interested in. For a normal distribution the mean and variance are the natural parameters and it seems that it would make more sense to estimate the population mean.

Now consider a variable Y=exp(X) where X has a normal distribution. Y is siad to have a lognormal distribution. Consider the sample geometric mean for a sample of size n, Y$_1$, Y$_2$,...,Y$_n$.

G$_m$ = Π (Y$_i$)$^1$$^/$$^n$ is the sample geometric mean for the geometric mean parameter of the distribution of Y. ln(G$_m$)=Σln(Y$_i$)/n. Since ln(Y)=X the log of the geometric mean is the sample mean for the corresponding normal random variables X$_i$. So for a log normal distribution the geometric mean may be the natural parameter to esimate since the log of it is the same mean for normal random variables.

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There is a practical difference between the two algorithms. Weighted arithmetic mean allows a lack in one element to be compensated by other elements, but weighted geometric mean better reflects a situation when a shortage in one element limits the result and cannot be compensated by other elements. For example, if you have one element equal to zero or very small, you will get mean equal to zero (or very small value). Weighted geometric mean reflects the "limiting factor" concept in ecology.

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The geometric mean of $0.02$, $2$ and $200$ is $2$, which is rather bigger than $0.02$ –  Henry Sep 17 '12 at 0:05
    
Sure, I mean, it's also smaller than the arithmetic mean 67. Small values just have stronger influence –  nadya Sep 17 '12 at 2:24
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The harmonic mean is even smaller. –  Henry Sep 17 '12 at 6:23
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