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I have percentages of people having an event for 21 regions. Since I want to calculate a Coefficient of Variation, I took the arcsine of the proportions to normalize data. The problem is: every region has a different number of people. Thus I wonder whether I should use weights in calculating the CV (so that both the mean and the variance will be more heavily affected by the most populated regions) or forget them (so that each region is equally relevant to calculate mean and variance)? With original data, the actual national average is given by the weighted average of regional proportions, so I see a clear reason why one may want to use weights. With arcsine, however, such equality doesn't hold anymore, so I wonder whether using weights still make sense: is it still ok that more populated regions will have a lower variance just because the mean will be closer to their values, even if the national average cannot be caluclated using such weights anymore?

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