What's the meaning of dividing percentages?

Question

I have a question regarding the meaning of dividing percentages. Imagine the following situation:

We have two cities: city A and city B. In city A there are 58,000 residents, of which 29,000 are women. In city B there are 120,000 residents, of which 96,000 are women. So, in city A 50% of the residents are women whereas in city B 80% are women.

If we want to compare the percentages of women in both populations (A vs B) and we do the following:

100 x $\frac{0.8}{0.5}$ = 160%

What does this mean? I guess it is that for every woman in city A there is 1.6 woman in city B, but I don't know if there is another useful interpretation, especially in terms of percentage. I would like to say something like: "there are about 60% more women in city B than in city A", but I'm not sure if that's correct.

Thanks!

Answer 1

If you are just comparing percentages, then you cannot really say anything that compares absolute values. So, statements like "there are this many more women in A compared to B" are right out. This is because, with only percentages, you don't really know the sizes of the two cities. For exmaple, one city could be 1000 times bigger, and have many more women even if only 10% of the population are women.

The figure 160% you calculated simply tells you that the percentage of women in city B is 60% higher than it is in city A. But that is confusing, because when you say "80% is 60% bigger than 50%", it is hard to follow which percentage pertains to what (the first and last % sign speak of the ratio of women, while the second % sign speaks of the ratio of the percentages, so it's "ratio of ratio"). Also, it's even more confusing since you could also say that "80% is 30% bigger than 50%".

So it really pays to not just blindly plug numbers into equations, but think about what the numbers mean. In your case, the best explanation I can think of would be

If the two cities were equally sized, then city B would have 60% more women than city A.

Notice the "if" in the beginning: assuming the two cities are equally sized is needed to avoid comparing absolute values.

Alternatively, we could also say:

A representative sample of city B's population shall have 60% more women than an equally sized representative sample of city A's population.

Answer 2

Percentages of percentages often cause confusion, and it is sensible to avoid them.

• You might say there are $\frac{96000}{29000} \approx 3.31$ times as many women in City B as in City A. Calling this "$231\%$ more women" may lead to misunderstanding.

• You might say the proportion of women in City B is $80\%-50\%=30$ percentage points higher than in City A. Calling this "$30\%$ more women" may lead to misunderstanding.

• To use a technical term, you might say the relative risk of being a woman in City B is $\frac{80\%}{50\%}=1.60$ times that of being a woman in City A. Calling this "$60\%$ more women" may lead to misunderstanding.

There are more alternative descriptions. For example the odds ratio for women between the two cities is $\dfrac{\frac{96000}{120000-96000}}{\frac{29000}{58000-29000}} =4$; this has the nice property that the odds ratio for men is $0.25$, which is its reciprocal, but the disadvantage that few people understand odds ratios.