Choosing the Quantity to Measure

In this lesson you will: Decide what question a number has to answer before deciding what to measure; Compare groups of different sizes with a rate, not a total; Give every rate a named denominator: per student, per hour, per dollar, per game.

The total is often the wrong quantity

Two schools ran a food drive. Eastside collected 435 cans; Westbrook collected 216. Eastside wins, obviously.

Eastside has 1,450 students and Westbrook has 600. Per student, Eastside collected 435 / 1450=0.3 cans and Westbrook collected 216 / 600=0.36. The smaller school did more.

Neither number is a lie. They answer different questions. “Which drive filled more shelves?” is a question about totals. “Which student body pitched in harder?” is a question about a rate, and answering it with a total is what makes the comparison unfair.

Name the denominator out loud

A rate is not finished until you can say what it is per. “Efficiency” is not a quantity. “Miles per gallon” is.

Changing the denominator can flip a comparison, and that is not a flaw — it is the point. Each denominator answers its own question, so the honest move is to say which one you chose and why.

Descriptive modeling means choosing what to measure

Ohio's N.Q.2 asks you to define appropriate quantities, which is a different skill from computing with them. A question like “which player is more productive?” has no arithmetic in it until somebody decides what productive means.

Points per game is one choice. Points per minute played is another, and it will rank a substitute differently. A quantity is appropriate when it can actually be measured, and when a change in it means the change you care about.

Common mistake — Comparing raw totals across groups of different sizes

A district reports that Central High produced 1,960 hours of community service and Ridgeway produced 1,120. Central is declared the more service-minded school.

Nothing about the arithmetic is wrong, and the bar chart will look convincing. Totals are also the easiest numbers to find, which is part of why they get used.

But Central enrolls 1,400 students and Ridgeway enrolls 700. Per student that is 1.4 hours against 1.6 hours, and the ranking reverses.

A total measures the size of a group at least as much as it measures the behavior of the group.

How to catch it: Before you compare two numbers, ask whether the two groups they came from are the same size. If they are not, divide each total by its group size and compare those. It is one extra division and it is the whole difference between a fair comparison and an unfair one.