# Confidence Intervals of Weighted Means

Suppose I only have the following summary data:

• "City A" has an adult population of "n1" : a representative sample of 25% of this adult population was asked for their income, and the average income was "x dollars"

• "City B" has an adult population of "n2" : a representative sample of 25% of this adult population was asked for their income, and the average income was "y dollars"

• "City C" has an adult population of "n3" : a representative sample of 25% of this adult population was asked for their income, and the average income was "z dollars"

Based on only this information, I want to estimate the average income of all 3 cities and the standard deviation.

Originally, I had thought that the "Weighted Mean" (https://en.wikipedia.org/wiki/Weighted_arithmetic_mean) was a good approach for this problem. That is, cities with larger populations should have more of an influence on the final estimate, and cities with smaller populations should have less of an influence on the final estimate. I could then calculate the standard deviation as well.

I started looking at references to perform this calculation (e.g. http://seismo.berkeley.edu/~kirchner/Toolkits/Toolkit_12.pdf), and it appears that individual measurements might be required for this calculation. For example, I would need to have the income of every person interviewed in City A, City B and City C. However, I am only provided with the average income from each of these cities - and furthermore, I am not even provided with the standard deviation of these averages.

In such a problem, does it still make sense to calculated the Weighted Mean on essentially three measurements? Or in such cases, is it better to refrain from calculating any statistics, seeing that any estimate generated in such a context is likely to be inherently flawed?

does it still make sense to calculate the Weighted Mean on essentially three measurements?

Yes, I think in this case it would. If all three cities have roughly the same population, your simple mean and weighted mean will be similar; but if one city is 10 times the size of the others, knowing that the sample comes from 25% of each city's population means we know they sampled many more people for the larger city. I would want to weight that city's average income much higher than the other two cities'. Otherwise, you'd be discounting the importance of the individuals in the larger city just because they happened to live somewhere with a lot of other people.

Or in such cases, is it better to refrain from calculating any statistics, seeing that any estimate generated in such a context is likely to be inherently flawed?

I think this really just depends on the context of your problem. Nobody will get hurt if you type in some numbers in a calculator one way vs the other. If policy decisions will be made from this average, then you should disclose up front which way the average was calculated and why you chose to do so that way, and perhaps some followup recommendations for future studies (i.e., data preparers should also include some measure of variance/standard deviation per city).

Finally, this wasn't directly asked in your question, but as far as calculating a weighted standard deviation, I don't think you can do that with the information given. You could calculate one based on the three averages but I doubt it would give you a lot of insight - with just three numbers, it's pretty easy at a glance for someone to understand whether one is much different than the others.

• Welcome to CV. The right weights to use depend not only on sample sizes, but also on the variances within each stratum. (Population doesn't matter, although when sample sizes exceed c. 10% of their populations, a finite-population correction will modify the variance estimates.) A quantitative analysis can be found at stats.stackexchange.com/questions/454120.
– whuber
May 15, 2023 at 18:31