Confused about proper way to normalize two variables

Question

I have two variables of interest:

• Residential Vacancies (res_vac)
• Commercial Vacancies (com_vac)

I also have two variables with which I might normalize the above:

• Total Residences (res_tot)
• Total Businesses (bus_tot)

Commercial Vacancies

I get the general theory of how to normalize data, but the dichotomous split is giving my brain a headache.

If I wanted to normalize res_vac (say, to compare it against a normalized com_vac variable), for example, which is correct:

• res_vac / res_tot
• res_vac / (res_tot + bus_tot)

I'm leaning towards the first one. A great answer will also help me understand why it's one or the other.

Answer 1

If, as you say in your question, you want to compare res_vac to a normalized com_vac variable, you'll want to use the first approach. Similarly, if you want to see if residential vacancy rates are correlated with commercial vacancy rates, you'd use the same approach--that way your normalized data wouldn't be affected by the fact that cities will differ in the proportion of properties which are residential v. commercial.

However, if for example you wanted to compare cities to see what proportion of ALL vacancies in each city were residential, you could use the second approach (or even a third normalization: res_vac / (res_vac + com_vac).

In short, it's the question that you're trying to answer with the data that guides which normalization approach to use--there's no single "right" answer for all purposes.

Answer 2

Since commercial and residential properties are not substitutes for most purposes, I would favor the first vacancy rate definition. I believe this separation is fairly standard. For example, in the US, the vacancy rate that the Postal Service calculates uses this break-out. Depending on your geography and time, you might also worry about seasonality (vacation homes) and owner vs. renter.