# Weighted geometric mean vs weighted mean

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?

• 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

Now consider a variable Y=exp(X) where X has a normal distribution. Y is said 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 lognormal distribution the geometric mean may be the natural parameter to estimate since the log of it is the same mean for normal random variables.
• 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