# A measure for kurtosis that takes into account sizes?

I have a dataset of prescribing on drugs by region, across a calendar year. This is a complete population dataset. The raw data looks like this:

region,drug,prescription_count
C123,020201,340
C124,020201,75
C123,020202,980
C124,020202,400
...


I want to find the drugs that are not prescribed consistently across regions - the ultimate goal is to find drugs with regions that are interesting outliers.

In other words, for the full list of drugs (the two anti-depressants 020201 and 020202 in the example above, though in my real data there's about 3000 drugs) I'm interested in knowing whether the prescribing of 020201 is more or less "normal" than 020202.

To complicate matters, the regions have different population sizes, so we can't just look at the raw prescription_count values (and we can't easily normalise for population, because different regional demographics have different general prescribing needs).

To compensate for this, I've calculated the proportion of prescriptions that the drug accounted for in each region, as a proportion of total prescribing in that region on that drug class (which encompasses many drugs of a similar class, e.g. 0202 is the anti-depressant drug class).

So now the data looks like this:

region,drug,drug_class,prescription_count,prescription_count_as_proportion_of_drug_class_in_region
C123,020201,0202,340,0.22
C124,020201,0202,75,0.23
C123,020202,0202,980,0.33
C124,020202,0202,400,0.46
...


If I compare these proportional values for each drug across regions, the distribution is typically normal.

Now I need to find an appropriate measure for "normalness", using the proportional value. I've been using kurtosis, which does produce generally sensible-looking results.

However, the problem with kurtosis on this measure is that it takes no account of overall region size: for me, a large region that is an outlier is more significant than a small region that is an outlier. (Some regions are 10x larger than others.)

Given all the above, is there a way I can compare "normalness" across different drugs? Basically, I want to find a measure like kurtosis, but that applies more weight to regions with an outlier proportion value when they are large regions, not small ones. Then I want to use this measure to compare each drug.