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What Are Significant Figures?

Do you know what significant figures are? Or why they matter? And what do significant figures have to do with making measurements and calculating things like speed and miles per gallon? Keep on reading to find out!

By
Jason Marshall, PhD
Episode #169

Did you know that your calculator occasionally lies to you? Okay, perhaps it's a bit melodramatic (in an anthropomorphic kind of way) to say it "lies" to you, but it certainly can mislead you. While I don't want to shatter any relationships with trusty number-crunching companions, it's a good idea to keep this in mind.

What exactly am I talking about? It has to do with something called significant figures—something your calculator has no understanding of. But the good news is that by the end of today you will! And you'll be able to use your new knowledge to make more accurate measurements and calculations.

Ready to see how it all works? Keep on reading to find out!

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How to Estimate Speed

Many moons ago (it was actually many trips around the Sun ago!), we talked about a handy tip that you can use to estimate how fast somebody is running. In particular, we talked about how to quickly estimate how fast an American football player is running.

The important thing to know is that you can calculate the average speed of something or somebody (in units like meters per second, yards per second, or miles per hour) simply by dividing the distance traveled by the time it took to do the traveling.

This is particularly convenient in American football since there are lines painted on the field every 10 yards. All you have to do is measure the time it takes a player to cover the distance between two of those lines, and you can then use the speed equation to find out how fast they're going. As you can check (by doing a little unit conversion), if a player covers 10 yards in about 1 second, they're running about 20 miles per hour.

Exact and Inexact Numbers

What does all of this have to do with today's topic of significant figures? The key to the connection is found in one word from the last paragraph: measure. To understand this, we first need to talk about a funny thing about numbers. Namely that all numbers are not created equal—by which I mean that some numbers are exact and some are not.

Some numbers are exact and some are not.

For example, the number of noses on your face (probably exactly 1), ears on your head (most likely exactly 2), and windows on your house are all exact numbers. The number of people alive on Earth at some instant is also an exact number (although it's a lot harder to know). Why are all of these exact? Because there is no ambiguity about their actual values.

On the other hand, all measurements are inexact—by which I mean there is always some uncertainty about the value you come up with. When you time a football player running 10 yards in order to make your speed calculation, is the time you measure actually the time it took…exactly? And is the player actually covering 10 yards…exactly? No! Both of these measurements are inexact. They may be very good, but they are still inexact.

How to Make Measurements

When making measurements, it's important to know how exact those measurements are. If you're using a stopwatch to measure the time it takes a football player to run 10 yards, how exact do you think your measurement is? The answer depends on things like the quality of the stopwatch, the speed with which your fingers respond to what your eyeballs see, how good you are at using the stopwatch, and so on.

Almost everybody can use a stopwatch to measure something down to the nearest second. Most people can probably measure down to the nearest half second. Perhaps you think you can measure to the nearest tenth of a second. And more sophisticated timing equipment (like they use in the Olympics) can measure much more exactly than that.

Let's say you think you can measure to the nearest few tenths of a second and you measure 1.1 seconds. In that case, what you really know is that the actual time is 1.1±0.1 seconds—in other words, it's somewhere between 1.0 and 1.2 seconds.

What Are Significant Figures?

Here's where the idea of significant figures comes into play. The fact that you can only measure the time it takes a football player to run 10 yards to the nearest few tenths of a seconds puts a limit on the number of digits—aka, "figures"—that actually matter. If your fancy digital stopwatch gives you a time to the nearest hundredth of a second—say 1.13 seconds—only the first two digits on your stopwatch carry any actual meaning. In other words, only the 1.1 part of the 1.13 seconds reading is significant.

All measurements have limited precision and therefore a limited number of significant figures.

After all, you know that you can only measure time down to a few tenths of a second, so the part of the measurement that the stopwatch is giving you to the nearest hundredth of a second are just extra digits that the device—which, I should add, could be a stopwatch, a calculator, a computer, and so on—brings along for the ride because it doesn't know any better. These extra digits are the little "lies" that I mentioned at the outset.

The important thing to know at this point is that all measurements have limited precision and therefore a limited number of significant figures. In the case we talked about today, your measurement of 1.1 seconds had two significant figures.

You might be wondering how to make calculations when combining measurements with different numbers of significant figures. For example, how do you divide a distance measured to the nearest tenth of a yard by a time measured to the nearest few tenths of a second to come up with an estimate of a football player's speed to an appropriate—as determined by significant figures—precision?

That's a great question…that we'll be answering next time. So be sure to check back to see how it works!

Accuracy or Precision?

One final point before wrapping up. A few episodes ago, we talked about the ideas of accuracy and precision. Now that we've learned about the general idea behind significant figures, it's time for a little brain teaser for you to think about: Does the idea of significant figures have more to do with accuracy or precision? (Hint: I might have dropped a hint earlier with some of the words I chose.)

What do you think? We'll talk about the answer next time.

Wrap Up

Okay, that's all the math we have time for today. Be sure to check out my mental math audiobook called The Math Dude’s 5 Tips to Mastering Mental MathAnd for even more math goodness, check out my book The Math Dude’s Quick and Dirty Guide to Algebra.

Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via FacebookTwitter, or email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Stopwatch image from Shutterstock.