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In a comment thread relative to another question, Nick Cox says:

Population pyramids are over-rated. Subtle shifts in the ratio females to males, or its reciprocal, are important and common and expected but very hard to read off pyramids. (Bigger shifts are those you can spot easily.)

I then asked him what would be a good alternative to population pyramids (my initial idea was to use overlapping bars with some transparency). He replied:

Plot male and female frequencies on the same axis as dots or lines. Log scale optional, but consistent with the idea that the ratio of either is of central interest.

I learned something thanks to these comments, but now I have additional questions:

  1. What are some other advantages of dot/line plots over population pyramids, besides identifying small shifts more easily?

  2. When using a dot/line plot as described above, are there situations where a log scale would be a bad idea?

  3. What are some other good alternatives to population pyramids, besides dot/line plots?

Thank you.

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I can offer comments on #1 and #2. My concrete comments focus on population pyramids for human (or even other) populations with various age groups and conventionally two genders or sexes(*). The simplest examples compare the numbers of people of different age groups and sexes, but the idea can be extended to say numbers of cases of, or deaths from, different diseases or conditions.

Further, it should be clear, or at least you may want to consider, whenever comments also apply to other set-ups with two sets of quantities that are being compared, say imports and exports, votes for and against, and so forth.

What are some other advantages of dot/line plots over population pyramids, besides identifying small shifts more easily?

To compare frequencies of two sexes, as conventionally distinguished, population pyramids require the reader to look at pairs of bars that are juxtaposed and assess whether they are about the same length. The task of comparison would be much easier if the same information were superimposed. It's then a matter of judgment whether you continue to use bars, or switch to dots (point symbols) or lines.

There is also scope to use (e.g.) logarithmic scale.

The advantages of this proposal go further. Often you can see small frequencies more easily too, usually but not always those for much older people.

Dot and line plots can be extended more easily to comparing e.g. the same country in different years or different countries in the same year. Many comparisons of this kind entail looking at small but nevertheless interesting or important changes. Comparing two or more population pyramids placed side by side is even more difficult than comparing bars on left and right sides of one pyramid. (It's not an accident that we're echoing standard critiques of pie charts, and even of bar charts in general.)

There is an extra dimension. If information is available for other gender categories, the use of paired bars can't be sustained easily or effectively, In practice, in societies I know about, numbers identifying or identified as non-binary, transgender, or other categories are likely to be much smaller, a further argument for logarithmic scales.

When using a dot/line plot as described above, are there situations where a log scale would be a bad idea?

Log scales might be unfamiliar to the intended audience. They might seem awkward if some of the frequencies were very small and variations in those were more evident than variations between the more frequent categories. More generally, some people might think that logarithmic scales distort the data. (I have heard this objection even from experienced scientists who seemingly don't have difficulty in thinking about percent changes in prices or wages.)

Otherwise, it is standard that zeros cannot be shown on logarithmic scales. That constraint can be important for small populations.

Turn and turn about, for any society with some of each gender, but very different numbers, say some prisons, monasteries, convents, scientific laboratories, or parliaments, log scale might make the imbalance easier to see.

Various alternatives include square root or cube root scales, or plotting separately total population and a measure such as (# females MINUS # males) / (# females PLUS # males), which is defined for all situations with two genders.

Brinton (1914) had cogent criticisms of population pyramids in this vein. See also Cliff and Haggett (1988).

Brinton, W.C. 1914. Graphic Methods for Presenting Facts. New York: Engineering Magazine Company.

Cliff, A.D. and Haggett, P. 1988. Atlas of Disease Distributions: Analytic Approaches to Epidemiological Data. Oxford: Blackwell.

See this thread on Statalist for some data and an example graph. It doesn't go as far as logarithmic scale, but play (experiment!) should be trivial in your own favourite software. (If not, you need new favourite software.)

EDIT

I downloaded some data from an official source with populations for the United States. For simplicity I ignored the highest age-group of 85 years up.

Here is one of many possible graphical takes. It's conservative in using bars and a little radical in superimposing them. It's a little adventurous in showing sex imbalance as a linked panel.

An advantage of bars is that they do make explicit the age intervals to which they apply.

enter image description here

(*) I am aware of, and attempt to be sensitive to, views and arguments about the words that are best to use in this context. I am focusing on what to do with what is presented as data, while recognising that that raises a whole bundle of other issues.

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