- #1
Niles
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Homework Statement
I have a range of numbers numbers [itex]n_i[/itex], each with a different weight [itex]w_i[/itex] that sum up to 1. To keep things simple, let's take the case where we have three numbers with the following weights:
n_i w_i
------------------------------
100 0.5
30 0.2
20 0.3
Their geometric average is [itex](100^{0.5})*(30^{0.2})*(20^{0.3})=48.4991[/itex]. The arithmetic average of the numbers is [itex]100*0.5 + 30*0.2 + 20*0.3=62[/itex], so it is larger than the geometric average.
How can I find a new set of normalized weights [itex]w_i'[/itex] that sum to 1 that can be used to find the arithmetic average of the numbers such that it is equal to the geometric average? In other words, I would like to find a new set [itex]w_i'[/itex] such that
[itex]100*w_1' + 30*w_2' + 20*w_3' = (100^{0.5})*(30^{0.2})*(20^{0.3})[/itex] given that [itex]w_1'+w_2'+w_3'=1[/itex].
The weights are all nonzero.My best attempt at the moment is
[tex]
\sum_i (\text{GA} \frac{w_i}{n_i}) n_i
[/tex]
where GA is the geometric average. This sum yields GA as expected, but the weights are larger than 1.