How Sure Are You?

Last lesson you held a pencil against a ruler, read the mark at its tip, and wrote 9 cm.

Now here's an awkward question. Was it exactly 9 centimetres? Not 9.02. Not 8.97. Exactly nine?

Almost certainly not — and that isn't because you were careless. It's because nobody has ever taken an exact measurement. Not you, not your teacher, not the laboratories that weigh atoms.

That sounds like bad news. It isn't. The moment you stop pretending your numbers are perfect, you can start saying something far more useful: how close to perfect they are.

By the end of this lesson you'll be able to:

• explain why no measurement is ever exact, however careful you are
• take several readings and turn them into one best answer
• say how spread out your readings were, and what that tells you
• tell the difference between a measurement that's precise and one that's accurate

Why the number is never exact

Go back to that pencil. Its tip didn't land neatly on the 9. It landed somewhere between the 9 and the 10, nearer the 9 — so you said 9.

That's the first reason. A ruler can only tell you which two marks the tip lies between. Anything finer than that, you're judging by eye. Move to a ruler marked in millimetres and the gap you have to judge gets smaller, but it never closes. There is always one last digit that came from you rather than from the instrument.

Now try something. Measure that same pencil five times, properly, from the 0 mark each time, and write all five answers down.

They won't all match.

The pencil didn't change length. You did. Your hand shifted a little, your eye sat at a slightly different angle, the pencil rolled a hair to one side. Those small differences aren't something to feel bad about. They're a permanent part of measuring, and every scientist handles them the same way — which is what the rest of this lesson is for.

A measurement isn't a point. It's a narrow range, and an honest answer says so.