Class size bias: the Tim Horton’s delusion

If you are an undergraduate university student, it should not take you long to make a list of peeves. Dislike your dorm? Check. Nurse a seething hatred of your roommate? Check. Loathe the food? Check. Even if some of those are not a problem for you, I can guarantee one that is: class size.

In an article on the value and quality of university education in Canada, the Toronto Globe and Mail newspaper stated that first-year students (freshmen to you Americans) reported average class sizes of over 200. The worst offender was McMaster University in Hamilton, Ontario, where first-year students estimated the number of classmates in their lectures to average 392.

The report brought a tart letter to the editor from McMaster’s provost, Ilene Busch-Vishniac, who stated that only 32 per cent of McMaster’s first-year classes exceed 100 students and that 57 per cent have fewer than 60 students. (That leaves 11 per cent of classes with 60 to 100 students.)

Whether that contradicts the students’ estimates is moot. That 32 per cent of classes over 100 could theoretically consist solely of 500-plus-student classes. But that is unlikely. Even if the 32 per cent averaged out to 400 students per class, the overall class size would be much less than the students’ estimate.

So who is right?

I suspect that what is going on here is a phenomenon I call the Tim Horton’s bias. Tim Horton’s is a popular Canadian chain of coffee shops specializing in freshly-baked doughnuts, muffins and strong coffee. I have noticed that at most of my visits to Tim Horton’s the place has been crowded.

“Busy day?” I ask my server.

“Not really.” That’s the usual reply. “We just get these occasional rushes.”

So, whenever I go there, I am part of an occasional rush.

But that is true of most people. Stated simply, most people are there when most people are there. Take a poll of Tim Horton’s customers and you will find that most of them find the shops crowded. The staff, on the other hand, observe long periods of slow business.

That’s likely what’s happening at McMaster University. Large classes contain large numbers of students, so a poll of students will get more responses on average from students in large classes. As a result, student surveys will overestimate class size.

Often in statistics, an absurd example will best improve our grip on a slippery idea. Consider a college in which there are just two classes — Advanced Stuff (A) and Basic Stuff (B). There are 10 students in A and 100 in B. No student takes both classes. Now, what’s the average class size?

Sounds like a no-brainer, no? There are two classes and 110 students. The average is 110/2 = 55 students per class.

But wait. Let’s take the students’ point of view. We ask each student the size of class she or he is in. Ten of them say 10; 100 of them say 100. That’s a total of 10 x 10 + 100 x 100. That’s 100 + 10000 = 10100. There are 110 students, so the average class size reported by students is 10100/110 = 91.8. The unit here is class size per student rather than students per class.

It’s unlikely either side in the McMaster argument was lying. This apparent contradiction, whose roots lie in confusion over units, occurs whenever you get clustering (rushes of crowds for services or students in classes).

It does not occur when the data are more evenly spread. Imagine a university with 1000 classes, each with 20 students. This really is a no-brainer, but let’s do it anyway. From the adminstration point of view, the average class size is 20000/100 = 20 students per class. Now we interview individual students in each class. That’s 20,000 interviews, each one of which reports a class size of 20. The total is 20 x 20000 = 400000. Average class size per student — 400000/20000 = 20. Either way you compute it, the average class size is 20.

The more argumentative of readers might complain that I have not accounted for students taking multiple classes. It’s true that in computing class size per student I have interviewed Mary Chan six times and Simon Smith only once. But that matters not at all. Mary had to have her say in each of her six courses; she gets counted six times, Simon only once.

About aharmlessdrudge

Way back during the late Bronze age -- actually it was the 1950s -- all of us in high school had to take a vocational test to determine our interests and, supposedly, our future careers. I cannot remember the outcome, but I do recall one question that gave me pause. "If you were to win a Nobel prize, would it be in literature or in physics?" I hesitated over the question: although I enjoyed mathematics and science more than English class, I did have a couple of unfinished (and very bad) novels hidden away at home. I cannot remember what I chose back then, but the dilemma followed me to university, where I switched from mathematics to English and -- after a five-year stint in journalism -- back to mathematics. I recently retired as a professor of statistics. Retirement. What a good chance to revive my literary ambitions. I have finished a novel -- more about that in good time -- and a rubble of drafts of articles about mathematics and statistics is taking up space on my hard disk.
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