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:
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:
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:
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:
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.