Memory

There's some road repair in progress on campus, with the result that the bus I normally take is being detoured from its usual stop. I learned this this morning, on my way to work. By the time I decided to go home again, of course, I had forgotten. I went to the usual stop and, to pass the time, began playing one of my memory games with myself. (Yes, I recognize the irony.) After a while, I heard the bells tolling the half-hour, an indicator that the bus would be there within a few minutes. It was at that point that I remembered the detour. (Off to one side was a sign announcing the detour, which somehow managed to be both large and unobtrusive.) There were two other people at the stop, an older black man and a petite Indian woman; I told them what was going on, and we all hustled over to the correct pickup spot.

As it happens, I missed my bus. On the other hand, the other two were spared that fate, and I got a very pretty smile of gratitude from the Indian, so I'm going to count the whole thing on the plus side of the ledger.

What Do You... Mean?

There's an interesting article by Allen Schwenk in this month's College Mathematics Journal, a nice demonstration of some of the subtleties of statistics.

Among the statistics colleges these days are likely to present - to boast of - are figures regarding average class size. Now, you might think that calculating average class size is an easy matter: count how many students are in each class, add them up, and divide by the number of classes. And, indeed, that is the average class size from the point of view of the faculty; it's the average number of students each teacher will have to deal with, per class.

But from the students' point of view, the matter is rather different. For the student, the important question is the average number of students in the given student's classes - and that is a different story. For that figure, you must ask each student to identify how many students were in each of his/her classes; add those up, and divide by the number of students. [Addendum: The divisor should be the number of "student-classes" - analogous to "man-hours"; e.g., if one student took four classes and another three, that's seven student-classes, even if some of the actual classes are the same.]

Take an artificially simple example. Suppose there are exactly two classes, one with fifty students, one with twenty. Then, by the first method, the average class size is (50+20)/2 = 35. But calculating the second way, we have fifty people saying "50" and twenty saying "20"; the average is then (50*50+20*20)/70, or about 41.4 - rather higher than the first!

In fact, unless all classes are exactly the same size, the second method will always give a larger result than the first. Schwenk gives two proofs of this, but the underlying idea is simple: the larger a class is, the more students will give that larger number, and so a large class gets more weight (50 out of 70 in the example) and pulls the average higher. (The first method gives all classes equal weight, regardless of size.)

Moral: if you're applying to be a student at a university, take the "average class size" with a grain of salt; it's probably too low. (If you're applying to teach there, well, salt is still needed, but for different reasons, which you can probably figure out for yourself...)