Counting Unequal Things

Suppose a school offered 20 classes with 10 students each and one class with 200 students. What would be the average class size? There are a total of 21 classes, and the total class attendance (counting students multiple times if they attend more than one) is 10x20 + 1x200, totaling 400. So the average would be 400/21 or about 19 students per class on the average. Now it could be that there are 200 students, and each attends exactly two classes: one of the size ten classes, and the size 200 class. Now if you asked any of these students what her average class size was, she would say 105, since the average of 10 and 200 is 105. That's very different from the school's average of 19, but the student's view is more likely what we'd care about.

Why did the first average give such a misleading result? Because we were counting things that were unequal. When we count the total number of classes, the huge class only counts as one, as if it were equal to a small class. When we count things, we are expecting to get a number that will tell us some useful quantity. If we count apples, the result is usually useful because it gives an indication of how much food we have, since apples don't usually vary much from the size we're used to.

On the other hand, if we count cakes in a bakery, it makes a big difference if we're counting cupcakes or wedding cakes. It will still make sense to count cupcakes as long as we know that's what they are, but there's not much point in knowing the total number of cakes if we don't know how many are small and how many are large.

This comes up fairly often. A politician might point out that most of the donors to his campaign gave amounts less that $50. That may be true, but if a thousand donors gave twenty dollars each and two donors gave a million each, almost all of his money came from large donors (twenty thousand compared to two million).

When I was young (in the 50's) my father was a Republican. I recall him being pleased after reading that most of the newspaper editors endorsed Republican candidates. It didn't occur to me until a long while later that small town papers tended to support Republicans and large city papers were more likely to support Democrats. Since large city papers have a much larger circulation, it may well have been that many more people actually read newspapers that supported Democrats, so my father's optimism may have been faulty. Again when counting newspapers, counting huge circulation papers as if they were equal to small ones could be very misleading.