Math Lesson of the Day: Quick Averages

When I was a kid, my teachers spent the first two or three weeks of every school year complaining that I didn’t “show my work” when I was doing math. The problem, most of the time, was that I didn’t have any work to show. I looked at the problem and knew the answer.

The rest of the time, the problem was that I did “work,” but it was either the work of ordinary reason, or the work of a quick way to solve the problem that would take longer to explain than the teacher wanted to spend on it, even if I could make the teacher understand what I meant. So they gave up, bless their souls, and let me be myself.

A poor math teacher will instruct the students in one way to solve a problem, often a way that involves a correct but cumbersome series of steps, will not explain why it works, and will let it go at that. Those students who see beyond the series of steps will learn, and those who don’t will forget the steps as soon as the exam is over. That’s the case with averages, my topic today.

People with a sense of number may smile, but it is really true that most of their fellows know only one way to come up with an average – the most cumbersome. Suppose you had to average the following numbers:

347

362

334

380

355

348

359

312

344

366

Most people would add up all the numbers, making a mistake along the way, then divide by the number of items, which in this case is ten. That is slow. It takes a pen and paper, or a calculator, the latter prone to the hard-to-notice error of punching in a wrong digit.

It’s quite unnecessary. Most of the time, when we have to average numbers, it’s because we’re dealing with something for which an average would be significant. In other words, we’re dealing with something that shows a certain regularity. The numbers above, for example, might be the enrollment of students at a private school from year to year. What’s the average?

The first thing to notice is what I call “static” – the stuff that clogs up the mental works. Here, it’s that we always have a class of more than 300. In fact, to average the numbers above is to average how much more than 300 they are. We could, if we like, just eliminate the first digit and average the remaining ones, then adding that average to 300 afterwards.

But once we see that, we notice a second thing. We notice the probable approximate answer. Here, it’s 350. We choose 350 because some of the items are greater and some are smaller, and because 350 is a convenient number to work with. Now we ask, “How much greater or less than 350 is the average of the numbers above?”

That’s the key, there. We scan the numbers, always with 350 in mind. We keep track of how many to the good or the bad we are, from number to number, going down the list. What’s more than 350 is to the good, what’s less than 350 is to the bad. So then, mentally, we tally the numbers so:

347 3 bad

362 9 good

334 7 bad

380 23 good

355 28 good

348 26 good

359 35 good

312 3 bad

344 9 bad

366 7 good

The average is more than 350. It is 7/10 more than 350: 350.7.

Or suppose you have a student with the following grades:

83

79

91

88

93

90

82

93

Choose 90 as your target, for ease. Run through the numbers thus, going down:

83 7 bad

79 18 bad

91 17 bad

88 19 bad

93 16 bad

90 no change

82 24 bad

93 21 bad

There were 8 grades. He’s lower than 90, by 21/8, or 2 and 5/8. His average is 87 and 3/8, or 87.38.

Or bowling scores:

159

132

201

178

220

156

162

Choose 170 for the target. Run through the numbers thus, going down:

159 11 bad

132 49 bad

201 18 bad

178 10 bad

220 40 good

156 26 good

162 18 good

There were 7 games. He’s higher than 170, by 18/7 pins, or 2 and 4/7. His average is 172.57.

Often it will take you only a moment to average a list of numbers, if they are close to one another:

55

56

53

54

52

58

59

60

52

54

Those are temperatures over a ten day period. Use 55 for the target. A very quick scan will show that the average temperature was 55.3.