Chapter 3. Cumulative Distribution Functions
The Class Size Paradox
At many American colleges and universities, the student-to-faculty ratio is about 10:1. But students are often surprised to discover that their average class size is bigger than 10. There are two reasons for the discrepancy:
Students typically take 4–5 classes per semester, but professors often teach 1 or 2.
The number of students who enjoy a small class is small, but the number of students in a large class is (ahem!) large.
The first effect is obvious (at least once it is pointed out); the second is more subtle. So let’s look at an example. Suppose that a college offers 65 classes in a given semester, with the following distribution of sizes:
size count 5–9 8 10–14 8 15–19 14 20–24 4 25–29 6 30–34 12 35–39 8 40–44 3 45–49 2
If you ask the Dean for the average class size, he would construct a PMF, compute the mean, and report that the average class size is 24.
But if you survey a group of students, ask them how many students are in their classes, and compute the mean, you would think that the average class size was higher.
Build a PMF of these data and compute the mean as perceived by the Dean. Since the data have been grouped in bins, you can use the mid-point of each bin.
Now find the distribution of class sizes as perceived by students and compute its mean.
Suppose you want to find the distribution of class sizes at a college, but you can’t get reliable data from the Dean. An alternative is to choose a random ...