# How to compare percentages for populations of different sizes?

I have several populations (of people, actually) which vary in size (from 5 to 6000). I would like to visualize the ratio of women vs. men in each of them so that they can be compared. I will get, for instance

• case 1: 20% of women, size of the population: 6000
• case 2: 20% of women, size of the population: 5
• case 3: ...
• ...

Both percentages in the first cases are the same but a change of one person in each of the populations obviously changes percentages in a vastly different proportion.

Should I take that into account when presenting the data? I am working on a whole population, not samples, so I would tend to say no. I also have a gut feeling that the differences in the population size should still be accounted in some way.

What I am trying to achieve at the end is the ability to state "all cases are similar" or "case 15 is significantly different" - again with the constraint of wildly varying population sizes.

• You could present the actual population size using an axis label on any simple display (e.g. bar chart) of women/men. A quite different plot would just be #women versus #men; the sex ratios would then be different slopes. Provided all values are positive, logarithmic scale might help. Commented May 25, 2015 at 10:28
• @NickCox: this is a good idea. I wanted to avoid using actual numbers (because of the orders of magnitudes), even with a logarithmic scale (about 93% of the intended audience would not understand it :)). I was more looking for a way to signal this size discrepancy by some "uncertainty bars" around results normalized to 100%. It is just that I do not think it is possible to talk about any kind of uncertainty here, as all the numbers are known (no sampling). I will probably go for the logarythmic version with raw numbers then. if you do not mind could you please turn your comment into an answer?
– WoJ
Commented May 26, 2015 at 11:55
• Sure. In turn, if you would give your data, or a larger fraction of it, I could add authentic graphical examples. Commented May 26, 2015 at 12:17

You could present the actual population size using an axis label on any simple display (e.g. bar chart) of women/men.

A quite different plot would just be #women versus #men; the sex ratios would then be different slopes. Provided all values are positive, logarithmic scale might help. On logarithmic scale, lines with the same ratio #women/#men or equivalently the same fraction of women plot as parallel. An audience naive or nervous about logarithmic scale might be encouraged by seeing raw and log scale side by side.

The problem that you have presented is very valid and is similar to the difference between probabilities and odds ratio in a manner of speaking. The percentage that you have calculated is similar to calculating probabilities (in the sense that it is scale dependent). I would suggest that you calculate the Female to Male ratio (the odds ratio) which is scale independent and will give you an overall picture across varying populations.

• The odds ratio is also sensitive to small changes e.g. a shift from 1 to 2 women out of 5. It's difficult to see that this addresses the question at all. Nothing here on graphics. Commented May 25, 2015 at 10:26
• You can try conducting a two sample t-test between varying percentages i.e. number of women expressed as a percent of total population. Commented May 25, 2015 at 11:00
• What set-up here defines two samples? Commented May 25, 2015 at 11:02
• You are working with different populations, I don't see any other way to compare your results. All the populations (5 - 6000) are coming from a population, you will have to trust your instincts to test if they are dependent or independent. But I would suggest that you treat these as separate samples. Commented May 26, 2015 at 10:22
• I can't follow your comments at all. Note that the question is not mine, but that of @WoJ. Perhaps we're reading the word "populations" differently. For the OP, several populations just define data points with differing numbers of males and females. Regardless of that, I don't see that you have addressed my query about what defines precisely two samples in this set-up. Commented May 26, 2015 at 10:29

Although your figures are for populations, your question suggests you would like to consider them as samples, in which case I think that you would find it helpful to illustrate your results by also calculating 95% confidence intervals and plotting the actual results with the upper and lower confidence levels as a clustered bar chart or perhaps as a bar chart for the actual results and a superimposed pair of line charts for the upper and lower confidence levels.