Ivars Peterson's MathTrek
On the radio program A Prairie Home Companion, host Garrison Keillor starts off each show with the words: "Welcome to Lake Wobegon, where all the women are strong, all the men are good-looking, and all the children are above average."
The tendency to treat all members of a group as above average, especially with respect to numerical values such as test scores, is now often called the Lake Wobegon effect (Word Spy). It can reflect a human propensity to overestimate one's achievements and capabilities in relation to others (Wikipedia).
But there's more to averages than you might think. Although computing averages is straightforward, the results may depend on your point of view, as mathematician Allen Schwenk of Western Michigan University demonstrates in the September College Mathematics Journal.
Suppose that Huxley-Darwin College has precisely 200 students. All 200 are taking the same five courses (English, mathematics, economics, history, and psychology). History and psychology are taught in large lectures of 200, and all the other classes are taught in small sections of 20 students each.
Altogether, there are 32 sections (classes), split among 1,000 enrollees. The average class size is 1000/32 = 31.25.
"This is the figure an administrator would report," Schwenk says. It accurately describes how large, on average, the college's classes are. It also reflects the teaching load experienced by faculty members.
But, this figure doesn't reflect what a student experiences. Each student has five classes, three with 20 students and two with 200. Hence, each one of them computes his or her personal average class size as 460/5 = 92.
So, each student experiences an average class size nearly three times that claimed by the college. Shades of Lake Wobegon! "How can every student have classes that are so much larger than average?" Schwenk asks.
"The answer," he says, "is that when the institution computes average class size it counts each class exactly once, but when students compute their own personal average class size the large lectures get experienced and reported 200 times while each small class gets reported only 20 times."
"Neither the university nor the student is wrong or deceitful, but the averages they report are very different animals," Schwenk says.
Taking care to specify how he computes the average class size experienced by students, Schwenk goes on to prove that the student-experienced average is always greater than the institution average. How much greater depends on the disparity in numbers between the largest and smallest classes.
If the classes are nearly uniform in size, the difference is practically insignificant. When there is a large disparity in size, the overall student experience can be significantly worse than that reflected in figures reported by the college's administrators and admissions officers.
"In all fairness to our students, we must understand this difference," Schwenk concludes, "and we must avoid the temptation to compute the average class size in the customary manner and then shape our opinions and policies on the premise that this average will somehow reflect the typical student experience."
So, parents (and seniors) beware! The average class size reported in a college's brochures may be a far cry from what a freshman would actually experience at that college.
Schwenk, A. 2006. Distortion of average class size: The Lake Wobegon effect. College Mathematics Journal 37(September):293-296.
You can learn more about the Lake Wobegon effect at http://en.wikipedia.org/wiki/Lake_Wobegon_effect and http://www.wordspy.com/words/LakeWobegoneffect.asp.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.