Today’s post is a short paradox about averages, and how they don’t always match expectations.

Imagine you’re a freshman college student, happily confident of a personalized education—after all, your college boasts an average class size of a mere 10 students per course. But when you arrive, most of your classes have far more students than that. On a hunch, you use the school directory to survey a random sample of your fellow students, and indeed, when they report to you what courses they’re taking and how many students are in each, the average is more like 30 students per class. What gives? How does the college get away with such flagrant false advertising?

The paradoxical answer is that there’s no inconsistency between the college’s claims and the students experiences. Here’s one way the numbers might break down:

Suppose there are 90 classes at this college: 80 of them are small, five-person seminars, but 10 are large lecture courses each containing fifty students. That’s four hundred students in the 80 seminars, and 500 students in the lecture halls, indeed giving an average class size of

$\displaystyle\frac{80 \times 5 + 10 \times 50}{80 + 10} = \frac{400 + 500}{90} = 10$ students per class.

But when all the students get together to compare their experiences, there will be 400 accounts of seminars with 5 classmates and 500 accounts of 50-classmate lectures, giving an “experienced” average of

$\displaystyle\frac{400 \times 5 + 500 \times 50}{400 + 500} = \frac{2000 + 25000}{900} = 30$ classmates per student.

In other words, students tend to experience a greater number of classmates than the true average class size, because a disproportionately large number of those experiences are in the large lectures. Because those classes have more students in them.

If that boggles your brain the way it does mine, consider this thought experiment. Say a college only offers one class: a huge lecture course with 100 students. Now it launches a new class, but just one student switches into it, leaving 99 behind. The “average” class size drops from 100 to 50, but almost everyone’s experiences will have hardly changed, and a survey will still report a perceived average of nearly 100 students per class.

This paradox is related to the phenomenon that your Facebook friends have, on average, more friends than you do, and I think it has some wider philosophical implications as well. But for now, I’ll just leave you with the moral that the average experience isn’t always the experienced average.