Average, Percent,

Average, Percent,

and

Other Misunderstood Math Terms

“Misunderstood? Surely, you jest!” you reply…. But hear me out.. Okay, I agree that almost every seventh grader knows how to compute averages, but the question is, do they understand them… Consider the following:
A school is asked to report the average student/teacher ratio. Now we could do that two ways; the school could count how many students there are, then count how many teachers there are, and in the end, divide the number of students by the number of teachers to get the average… and that is how it is normally done. But you could also have each student count the number of students in each class they go to (assuming one teacher per class) and then average the class sizes reported by all the students. That takes a little more work, but it should give us the average number of students per teacher also… except, in most cases the two numbers will be different.

Let’s illustrate with a simple (no,, I mean REALLY simple) example. Suppose there is a school where there are only two teachers and two classes. The total enrollment is 6 kids in one class and 4 in the other. From the school point of view, there are 10 students and two teachers, so the average is 5 students per teacher. But when we survey the students, the class sizes reported by the ten students is {6,6,6,6,6,6,4,4,4,4} for an average of 5.2 students per teacher. Go ahead, make up your own numbers, but the only way to get it to agree is if EVERY class has the same number of students….and now the big question…. Which is the CORRECT average??? (“pssst… say ‘both’ ”)… But if they are not the same number, then they can’t both be the average of the same thing… Now you start thinking…. Tick… tick… tick… So what is is that each is the average of?

Ok, let’s go to something even easier; fifth grade percentages. Here is the problem:
Three weeks ago 87% of the students were in favor of the new football coach. Then the team lost and now only 67% of the people support the football coach. Ok, So which is correct Newspaper headline:
…..Support for Coach drops 20%
……Support for Coach drops 23%
…….Support for Coach drops 30%

All of them are a valid percent.. but the question with percentages, like the one with averages, and like most of the questions in applied math, resolve down to the object of a preposition…..percent of WHAT?
See, Math really is Hard.