Unexpected Values, part 3

And now, the exciting conclusion!


Dear Dr. Math,
My college administration says that the student-teacher ratio is 5:1. But all my classes have like 30 people in them. What gives? Are they just lying?
Sincerely,
A Student at a Local College

Well, ASAALC, that depends on what you mean by "lying." If you mean, Are they miscalculating their student-faculty ratio?, then the answer's probably "no." These things are public information, and anyone with a calculator can divide the number of students by the number of teachers. (Whether they include non-teaching faculty, like that weird guy who lives in the basement and hasn't taught a class since the 60s, is kind of an ethical gray-area.) However, if the question is, Are they misleading people by reporting a somewhat meaningless statistic?, then "yes." Here's how the magic trick works:

It all comes back to the idea of average, and the different things the word "average" means to people. Many people think of "average" as meaning "typical" or "to be expected," a misperception that isn't helped any by the synonym "expected value." So, if a college reports that it has a student-teacher ratio of 5:1, or equivalently, that it has an average class size of 5, people leap to the conclusion that the typical class they'll encounter at the college will have 5 people in it (and how awesome is that?!). However, that may not be the typical experience, and it may even be impossible. Let's take a look at a simple example:

If I told you that my girlfriend and I were going to flip a coin and decide whether to have a baby based on the result,* the average number of babies we would have, i.e., our expected number of babies, would be 1/2. But of course, we would be surprised, even shocked, if we actually had half a baby. In this case, the "expected" value is anything but. The value 1/2 just represents the plausibility we associate to the chance of having one baby. If we repeated the experiment many times, the ratio of babies to attempts would converge to 1/2.

Similarly, suppose that a tiny school had 10 total students and 2 teachers. So, the student-teacher ratio is 5:1. Now, let's say one of the teachers is that awesome guy who lets you call him by his first name, and the other has really bad dandruff or something, so 9 people register for Class #1 and only 1 person registers for Class #2. The average class size is the average of 9 and 1, which is 5--equal to the ratio of students to teachers, as it should be. However, if I picked a random student and asked him (it's an all boys' school) how many people were in his class, 9 times out of 10 he would say 9, and 1 time out of 10 he would say 1. So the average response I would get would be , for a difference of 3.2! In fact, the only way the two numbers can actually be the same is if all the classes are exactly the same size.

If you think about it, it makes perfect sense that I'd get someone in a larger class more often in a random poll, since the larger classes have more people in them to get polled. The question, then, is which of these numbers actually represents the "typical" student experience. Since the student-teacher ratio is generally smaller, the schools are happy to just report that and hope you don't notice the difference. What they really should be reporting is something more like the distribution of class sizes. As it is now, if you want to know what the classes are like, you've just got to go see for yourself.

Also, what's the deal with the vending machine only having Pepsi?

-DrM

*For the record, that's not how we do it. We roll a 20-sided die.