# Significant Figures a.k.a. SigFigs

INTRODUCTION   What if you asked your friend how tall she is, and she replied "5 feet 6.48328592 inches". Wouldn't that be odd? Yes, it would be. But why would you think it strange? By giving that detailed answer, your friend (let's call her Peculiarly Precise Priscilla) is implying that her height is known to within one hundred-millionth of an inch. That's very unlikely. The precision of a measurement (that is, how 'good' the measurement is) is reflected in the number of significant figures with which the number is written.

In math class, if you see the number 2.3, that is considered an exact number. In other words, "2.3" is the same as "2.3000000000...", with an infinite number of zeroes. Well, that's not the case in science. First off, the scientific number represents a measurement, so it would have to have some kind of units. For example, "2.3 kilograms", or "2.3 meters", or some other unit appropriate to the quantity being measured (like mass or length). Secondly, there is no such thing as an infinitely precise measurement. "2.3 meters" means that the measurement is any number that rounds to 2.3 meters, say between 2.25 and 2.35 meters.

Activities & Practice

Determining the Number of Significant Figures in a Measurement Given to You.   If you are given a number in a problem, you need to be able to determine how many significant figures it has. Here are the rules for counting the number of significant figures in any given measurement, plus examples.

• Any non-zero digit is significant.
• 5483 (4 significant figures)
• 12.2232 (6 sigfigs)
• Any zeros between other significant figure are considered significant.
• 3.00007 (6 sigfigs)
• 2304.4 (5 sigfigs)
• Leading zeros are not significant.
• 0.00475 (3 sigfigs)
• 0.0000000000002 (1 sigfig)
• Any trailing zeros after the decimal point are significant.
• 45.700 (5 sigfigs)
• 23.4320 (6 sigfigs)

The basic idea is that any zero, whose sole purpose is to push the decimal point to the correct place (a "placeholder"), doesn't count when determining the precision of a measurement. For example, the length measurement "23 meters" could just as well be written as "0.023 kilometers". Changing the unit of measurement doesn't make the measurement any more precise.

1. At a natural history museum, the tour guide proudly told visitors that the dinosaur bones on display were "eighty million and five years old." When asked how the age could be known so precisely, the guide said, "I don't know how they do it, but when I started working here five years ago, they told me that the bones were eighty million years old." [from John McGervey, Probabilities in Everyday Life, 1986, Nelson-Hall, Inc.]

What did the tour guide not understand about precision and significant digits?

Determining the Number of Significant Figures in a Measurement that You Make

Let's say you are measuring the length of a piece of paper. You look very carefully at the ruler and see this:

In general, you are allowed to estimate the last digit of a measurement to one-tenth of the smallest division of whatever measurement device you are using. Do it — look very carefully at the picture above. It looks to me like 26.3 mm. Maybe you think it's 26.4 mm, or 26.2 mm. That's OK! When you are estimating at the limit of precision like this, that last decimal place is usually a little uncertain. But the measurement is clearly NOT 26.5 mm, or 26.1 mm. And our measurement would have 3 significant figures.

Let's look at another example...

Look very carefully. To me, this looks like it's right on the "40", to within a tenth of a millimeter. In other words, we would write this as "40.0 mm", NOT "40 mm". Having that ".0" on the end indicates the precision of this measurement, to within a tenth of a millimeter and three significant figures.

2. Practice finding the number of sigfigs in given numbers using this applet.

12-question quiz

Reload these quizzes to get a new set of numbers. Keep practicing until you can get 100% on both. Good luck!

Rules for Finding the Precision of a Calculation. We usually use measurements to calculate other interesting things. For example, from the measurement of the height of an object, we can calculate how much time it will take to fall to the ground if we drop it. But the precision of our measurements determines the precision of the calculated result. There's an old saying that reflects this idea: Garbage In, Garbage Out. A calculation is only as good as the numbers that go into it.

So here are the rules for determining the number of sigfigs in a number that you calculate:

• Any number calculated from multiplying or dividing measurements, has only as many sigifgs as the least number of sigfigs of any the numbers that went into the calculation.
• 5.4cm x 0.23cm = 1.2cm². (Two sigfigs in the measurements mean the result can only have two sigfigs. Your calculator will say 1.242 is the result, but you have to round that to 1.2.)
• 5.4cm x 0.23485929395cm = 1.2cm² (increasing the precision of one measurement doesn't increase the precision of the result, because the first number still only has two sigfigs.)
• Any number calculated from adding or subtracting measurements has only as many decimal places as the least precise of the numbers going into the calculation.
• 5.4cm + 0.23cm = 5.6cm
• 5.4cm + 0.27cm = 5.7cm
• In scientific formulae, there are often numbers that do not represent measurements, but are actually pure numbers. Pure numbers are infinitely precise, so don't affect or limit the precision of the result.

For example, the 1/2 in

is not a measurement that is only approximately one-half. It is exactly 0.500000... with infinite precision. Therefore, it does not limit the precision of the calculated result.

Do the calculations requested below, using these given measurements.

A= 5.40m
B= 0.031123m
C= 0.7m

3. Calculate A×B
4. Calculate A/B
5. Calculate A+B
6. Calculate A-B
7. Calculate A×B×C
8. Calculate A+B+C

9. A chain is only as strong as its weakest link. How is this old proverb relevant to the precision of calculated numbers?

Tricky Cases.   How many sigfigs does the measurement 1200 kg have? There's no easy answer to that, because there's no decimal point to give us any clues. Maybe the person who made that measurement was only precise to within 100 kg, or maybe 10 kg, or within 1 kg. In other words, when written in scientific notation, could we write this measurement as

1.2×103 kg or 1.20×103 kg or 1.200×103 kg

Remember, trailing zeros after the decimal point are considered significant, so these three scientific notation numbers indicate differing levels of precision. Without the decimal point, as in 1200 kg, we can't easily tell what precision is intended. In this case, one solution is to write the measurement in scientific notation. Another solution to this problem is to underline the significant figures. Using this notation, the measurements written above in scientific notation could also be written like this:

1200 kg or 1200 kg or 1200 kg

When do I round? Sometimes you will use measurements to calculate something, and then use that number to calculate something else. When you do a whole series of calculations in order to reach a final answer, carry a few extra sigfigs through the calculations. Then, when you get your final answer, round it to the proper precision. In other words, round once — only the final result.

Precision versus Accuracy.

In everyday language, we tend to think of the words precise and accurate as synonyms, but in fact there is a subtle but important difference between their meanings. Precision describes how closely measurements, taken by different people independently, would agree with each other. For the above example (with the ruler), the precision would be about 0.1 mm, because any measurements by competent observers would agree within that amount.

Accuracy, on the other hand, describes how closely the measurements (or the average of the measurements) agree with The Truth — that is, the physical reality that we are measuring. Again, consider our measure-the-paper example from above. What if we made a mistake, and had the left edge of the paper at the left edge of the ruler, instead of at the zero mark? That would throw off the entire measurement by several millimeters, even though the precision could still be 0.1 mm. That would be an example of a systematic error, a mistake you make that consistently throws your measurements too high, or too low.

Let's consider an analogy: throwing darts. Precision describes how closely the thrown darts all cluster together; accuracy describe how well the darts are centered on the bullseye, which represents The Truth.

10. In a documentary called Sniper: Inside the Crosshairs on the History Channel, I noticed an odd statement. The narrator was talking about the M14 rifle, which was used by U.S. forces during the Vietnam War. The narrator said the M14 was effective against targets up to a range of "1094 yards". That seemed suspiciously precise to me: four significant figures! What do you think happened that led the writer to put that number in the script?

11. In a newspaper article about wildfires in Russia, the writer states "Visibility in some parts of Moscow dropped to below about 164 feet..." (McClatchy-Tribune News Service, 8 August 2010). The author got that information from Russia's Interfax news agency, but quoted it with a high level of precision (three significant figures). What do you think Interfax actually reported?

12. Take this online significant figures exam.

13. I recently (7/2013) noticed something a little odd on this box of stuffing I found in my kitchen pantry. What is odd about it?