Hello everybody,
I was wondering whether the formula of a geometric mean can be solved to determine the weights. More specifically:
- I have the weighted geometric mean of two values. I know the two values and the geom. mean.
- The sum of the weights is 2, so if w1 = 1, then w2 = 1.
Changing the formula of the weighted arithmetic mean was fairly easy so I am surprised why conventional tools such as the online Wolfram Alpha tool seem to fail, so I presume that I failed somewhere along the way.
I am looking forwards to your thoughts on this!
Lukas
You can easily convert the weighted geometric mean (WGM) formula to a weighted arithmetic mean formula by taking log. If I remember correctly, the WGM formula for two values is
m = (x1^w1 . x2^w2)^(1/2)
where w1+w2 = 2. Taking log on both sides gives
2 log(m) = w1 log(x1) + w2 log(x2)
Then, substituting w2 = 2-w1 gives
w1 = 2 [ log(m) - log(x2) ] / [ log(x1) - log(x2) ]
and
w2 = 2 [ log(x1) - log(m) ] / [ log(x1) - log(x2) ]
*Please double check the computations.
Let G be the geometric mean of two quantities x1 and x2 which have respective weights w1 and w2. And let exp(y) denote the exponential function with base e and ln(y) the natural logarithm function. Then, in the case you've indicated where the weights sum to 2,
G = exp((w1*ln(x1) + w2*ln(x2))/2).
Doing some algebra, you can get to
2*ln(G) = w1*ln(x1) + w2*ln(x2)
so that
2*ln(G) = w1*ln(x1) + (2 - w1)*ln(x2)
as
w1 + w2 = 2, i.e., w2 = 2 - w1.
I am assuming also that w1, w2 > 0.
Then you get
2*ln(G) = w1*ln(x1) + 2*ln(x2) - w1*ln(x2)
= w1*(ln(x1) - ln(x2)) + 2*ln(x2)
so that
2*ln(G) - 2*ln(x2) = w1*(ln(x1) - ln(x2)).
Then divide both sides by
ln(x1) - ln(x2)
and factor out the 2 on the LHS to get
2 * ((ln(G) - ln(x2)) / (ln(x1) - ln(x2))) = w1.
Then w2 is just 2 - w1.
Observe that this result coincides with the result in the other answer, but perhaps adds some additional insight...
How to Find the Weighted Geometric Mean of a Data Set https://www.dummies.com/education/math/business-statistics/how-to-find-the-weighted-geometric-mean-of-a-data-set/
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