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I have to perform outlier detection on population estimates for certain variables at the city level. For example, I might be estimating median income for a city and I want to know if there are any cities where the median income is an outlier with respect to the others.

My problem differs from a traditional outlier detection problem in two ways:

  1. If a city is an outlier, we won't be removing it from our analysis but rather we are just finding outlier cities in order to investigate why they are outliers.
  2. The "records" we are performing outlier detection on are estimates of population totals and not actual records themselves and as such our "records" are just point estimates that have a variance associated with them.

The question is, do we take this variance into account somehow? For instance, if most of our cities had a median income around 50k but one had a median income of 500k we would say the 500k city was an outlier. But what if that 500k city had a variance so large that a 95% CI covered 50k? Is it no longer an outlier? What if all of our cities have giant CIs. Is it possible to determine outliers from the point estimates alone?

One thought I had would be some sort of simulation where we sample estimates from the CIs of each city and perform many outlier tests and then analyse those results.

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Have you considered performing a test for equal means between each city and the general population? This is possible with the data you have and will give you a better idea of how truly extreme the estimated income of a city is, taking into account the uncertainty of the estimate. You can then sort the cities by the magnitude of their corresponding T-statistics or P-values.

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