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I have two statistics from a range of districts obtained by sampling from the total district population e.g. number of doctors per district and the gender of these doctors.

Now I want to see whether there is a correlation between the number of doctors there is in a district and their gender (to know if e.g. there are fewer doctors in a district they are more likely to be male).

Could I just find the Pearson's correlation coefficient? And would the p-value (significance) make any sense? Because the statistics that I have are from a sampled so they have a confidence interval as well?

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I would like to add a comment but can't (no enough Reputation), so I'll just make a short answer that you can delete afterwards.

From experience as a biologist (I'm not a statistician - beware): if data as a percentage, you should transform your data. The problem with percentage is that variance (of a group) increases as mean (of the group) increases. For data $Gender$ the fix is $Arcsin(SquareRoot(Gender))$.

As for the use of Pearson, I'm not sure. $Number of Doctors$ should follow a gaussian distribution. $Gender$, of course, will be problematic. Perhaps switch to Spearman's correlation test, that goes on the ranks rather than the values?

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As doctors are a rare population, you must have a pretty large sample, so the sampling error is unlikely to cause much of a problem, and computing Pearson's correlation and its p-value is going to get you basically the right answer. The expected effect of sampling error in the variables is that the correlation you compute will be an under-estimate though, but presumably you are only interested in testing if there is an effect rather than being super-precise about its magnitude. ´The only hitch I can think of is going to be if you find a result that is borderline not significant.

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