Significant Figures

## Truth in Imperfect Measurement

Only those who have the patience to do simple things perfectly will acquire the skill to do difficult things easily. -- Johann von Schiller

Being honest and telling the truth while avoiding hype is critical for the information age.  But this is nothing new for scientists.  Isaac Newton, Robert Boyle, and Antoine Lavoisier in their quest to abandon alchemy and superstition established a custom of precisely communicating clear information.

The avoidance of cute introductory jokes and antidotes often makes science see dry and impersonal.  But scientists were centuries ahead of most other professions in recognizing the needs of information exchange.

It was no accident that scientists were among the first to adopt computers, that a scientist at CERN (European Nuclear Research) invented WWW, or that the first U.S. Internet hub was at the Stanford Linear Accelerator Center.  The skill of communicating measurements honestly is at the core of effective information exchange.

### Uncertainty in Measurement

Measurements are not as perfect as other aspects of mathematics.  Each of our senses has limited acuity for detecting slight differences. And each measuring tool also has a tolerance for possible errors. Significant Figures are ONE way to tell the truth about the precision of a measurement.

We should distinguish "accurate" from "precise." Precision and accuracy are terms used to describe the quality of a measurement. Some view these terms as synonyms, but in fact they are different.

Precision indicates the degree of reproducibility of a measurement. It depends on how well you make a measurement.

Accuracy describes how close a measured value is to the true value. It depends upon the quality and calibration of the measuring device.

Example

Imagine working in a carnival, guessing people's mass. You have people sit on your lap.  Lets say the same person comes back 4 different times and each time you guess a different mass. If the person's true mass is 60 kg ( about 132 lbs here on earth) and your guesses are:
• 56 kg
• 65 kg
• 70 kg
• 51 kg
Average = 60.5 kg
These represent good accuracy but poor precision.
• 68 kg
• 69 kg
• 67 kg
• 68 kg
Average = 68.0 kg
They would have represented poor accuracy but good precision.
Without God's perspective, it is often difficult to determine accuracy. Scientists rely on repeated measurements using different tools and different methods to determine accuracy.

### Significant Figures

Why worry about significant figures? Since the quality and calibration of our equipment and the care of our measurement can influence the accuracy and precision of our measurement, we need a way to describe what confidence should be placed in our reported measurement. There are several ways to do this: significant figures is the easiest.

Two Examples to Ponder

When you add a quart of oil to your car, do you need to measure out 1.000 quarts in volumetric flask, or is it sufficient to just add about 1 quart?

If promoter sells tickets to a concert, do they need to know if there are 19,852 seats or is it good enough to know that there are 20,000 give or take 5000?

The simplest way to describe the uncertainty of a measurement is to combine information about the quality of the measurement with the number of the measurement. We let the number of digits reported for a measurement tells us about the certainty of a measurement. The rule is that we know all of the digits with certainty up to the last digit, which is estimated.

Example

Lets say you wanted to measure your weight using a bathroom scale with a digital readout. The readout might say you weigh 164 lbs. Providing the scale is well calibrated the measurement tells you that your exact weight is between 163.5 and 164.5 pounds. It would be OK to say you weigh about 160 lbs but it would be misleading to claim that you weigh 163.892 lbs. It is acceptable to discard information if it won't be needed, but it's a lie to claim more information than is known.
Rules for significant figures follow. They are an important part of experimental science, but they don't make for a very exciting lecture. They allow you to properly interpret measured results, and in certain cases they can save you a lot of time.
You will be expected to use the correct number of significant figures in lab reports as well as on quizzes and exams.

Such skill is merely useful in science but will be invaluable for the measurements you rely on during the rest of your life!

### Several Points of Clarification

Significant figures are the easiest way to deal with uncertainty in measurements, but it is not a perfect system. The Mesopotamians invented zero over 2000 years ago. We use it to distinguish when we know there is none. But we also use the zero as a place holder (zero wasn't invented 2 years ago). Zeros used only as place holders are not significant figures while their other use is. Several rules to work around this ambiguity will follow.

Some definitions involve exact numbers. (100 cm = 1 m, 12 = 1 dozen)

Numbers can be exact by count. (5 playing on a basketball team)

Exact numbers have an infinite number of significant figures, so when they are used in a calculation they do not change the number of significant figures.

## Significant Figures Rules

### Quick Reference Section

This section presents the basic rules for significant figures. Read later sections to gain a complete understanding of what these rules really mean!

How does one tell how many significant digits there are in a given number?

• The left most digit (←) which is not a zero is the most significant digit.
• If the number does not have a decimal point, the right most digit (→) which is not a zero is the least significant digit.
• If the number does have a decimal point, the right most significant digit (→) is the least significant digit, even if it's a zero.
Every digit between the least and most significant digits should be counted as a significant digit.
For example,
according to these rules, all of these numbers have three significant digits:
123
123,000
12.3
1.23 x 106
1.00
0.000123
How many significant figures should one retain in the final answer to a problem?
For results obtained using addition or subtraction, the number of places after the decimal point in the result should be less than or equal to the number of decimal places in every term.

For results obtained using multiplication or division, the number of significant figures in the result should be equal to the number of significant digits in the least precise number (the number with the fewest significant digits given).

It is sometimes good practice to give one more significant figure than is required by these rules; this helps prevent rounding errors if the number is used in later calculations. To indicate its lesser significance, this extra digit is usually shown in a smaller font.

## Details and Examples

Introduction

Significant figures are a shorthand way to express how certain one is about one's data and calculations coming from that data. While significant figures are by no means as precise as detailed calculations of the uncertainty of a value, they are a very useful way to estimate uncertainty quickly.

Uncertainty and its Meaning

Any value that is the result of a scientific measurement has some uncertainty. The most precise way to state the uncertainty of a measurement is to write it as a number, plus or minus the expected error in that number. Scientists often use the standard deviation for their expected error. After a measurement has been repeated many times, the standard deviation is determined by averaging the absolute amounts by which each measurement differs from the mean.

Fine Print

For example, if you measured a wire's length 30 times, and got an average length of 28.3 cm, with an average error in that length of about 0.2 cm, you would write the length as (28.3 ± 0.2) cm if you wanted to be precise about your measurement results. This means that the majority of your measurements fall between 28.1 and 28.5 cm. (To be more specific, it means that 68% of the measurements fall between those two values.)

If, for example, we then wished to find the volume of this wire, and we had a measurement of 2.31 ± 0.07 mm for the wire's diameter, we would have to put these numbers into the formula for the volume of a cylinder. This would require doing separate calculations for the largest and smallest errors in addition to the calculation for the average values. To be precise in our treatment of uncertainty, we need some complicated mathematics to see how the errors on the individual quantities translate through the formula to become errors on the volume.

But for most measurements you will ever use, a simpler system of dealing with uncertainty will be adequate. This system is the system of significant figures.

Significant Figures: The Simplest Method for Expressing Uncertainties

If we just want an approximate idea of the extent to which a value is certain, and we either don't want to learn the mathematical techniques, or don't want to spend the time to apply them, we can keep track of the amount of certainty of a piece of data simply by paying attention to the number of digits we use to express it. That is to say, where we choose to round off our number tells where we think uncertainty creeps in. For example, if we have a length of 12.37 ± 0.10 cm, we just call the length 12.4 centimeters, to three significant figures. When we express a number with three significant figures, what we are saying is that the first two digits are essentially exactly correct, and the last one is uncertain by a small amount (generally it is only uncertain by about ± 1). In the example above, we rounded our answer to 12.4 cm because our answer is uncertain to ± 0.1 cm, viz., our answer is uncertain in the last digit by about 1.

How Many Digits to Use?

The question of the greatest practical importance is how many digits to include in your final answer. This is important because, as explained above, the number of digits you include in your answer shows the reader the precision of the data leading to the answer, and the accuracy of the answer. It might be useful to read this section again after reading through the following sections which explain how to determine the number of significant digits.

When adding and subtracting numbers, the rules of significant figures require that the number of places after the decimal point in the answer is less than or equal to the number of decimal places in every term in the sum. (Treat subtraction as adding the same number with a negative sign in front of it.) If some of the numbers have no digits after the decimal point, use the same basic rule, but don't record any digits to the right of the last digit in the least significant number. Clarify these rules, are

some examples:

2355.2342    15600.00    15600        13.7     137000
+ 23.24      + 172.49    + 172.49    + 1.3     + 1330
2378.47      15772.49    15800        15.0     138000
Note it is not unusual for a sum to have more significant figures than the measurements added. This is why finding an average gives greater information than a single measurement.

Also note that a difference often has fewer significant figures. This apparent shortcoming is sometimes used by scientists in reverse!  One of the most sensitive tests is to measure a null difference to verify that the much larger opposing forces are equal to an accuracy not directly measurable. Inverse squared force laws have been verified to a large number of significant figures this way.

Multiplication and Division

When multiplying and dividing numbers, the number of significant digits you use is simply the same number of significant figures as is the number with the fewest significant figures.

Some examples:

13.1    13.10     13.100       1500    15310       1.00
x 2.25   x 2.25   x 2.2500    x 2.315    x 2.3    x 10.04
29.5     29.5     29.475       3400    35000       10.0
Someday calculators may be able to do significant figures; but in the meantime the operators need that wisdom.

Why do Multiplication and Addition Have Different Rules?

When you add two numbers, you add their uncertainties, more or less. If one of the numbers is smaller than the uncertainty of the other, it doesn't make much of a difference to the value (and hence, uncertainty) of the final result. Thus it is the location of the digits, not the amount of digits that is important.

When you multiply two numbers, you more or less multiply the uncertainties. Thus it is the percentage by which you are uncertain that is important -- the uncertainty in the number divided by the number itself. This is given roughly by the number of digits, regardless of their placement in terms of powers of ten. Hence the number of digits is what is important.

Which Digits are "Significant?"

In order to figure out how many significant figures to put into your final answer you must figure out how many significant figures are in each of the numbers you are working with. The rules are best explained separately for fundamental constants, physical constants, numbers not ending in 0 and numbers ending in 0.

Fundamental Constants
Fundamental constants are numbers without units of any kind that come strictly from mathematics; they are not "measured" like most quantities in science. Some examples of these are regular integers such as 2, 10, 14, or 27; fractions such as the 4/3 in the formula:
V = (4/3)πR3,
for the volume of a sphere; and constants such as and e, where e = 2.71828... is the base of the natural logarithms.

When you have one of these numbers, you should never let it determine how many significant figures you have. If the number is an integer or a rational fraction, just assume it has more significant figures that the least accurate of the measurements. When you have an irrational number like π, look up as many digits as you need so it has more digits than required by the least accurate measurement.

(Calculators are not programmed to do significant figures because they have no way to recognize which numbers are constants.)

Physical Constants
When you are dealing with a physical constant such as Planck's constant, the speed of light, or the charge of an electron, you should remember that these numbers are found by experiment and do not have any purely mathematical definition. So there may be some occasions where you have a piece of data that is more certain than the best value of your physical constant. In these cases, it is acceptable to let your physical constant define your uncertainty, and hence your number of significant figures.  To avoid this problem it has been a top priority of some scientists to measure these physical constants very accurately.
Numbers Without Zeroes at the End
Numbers without zeroes at the end are the simplest case. When a given piece of data ends in a digit other than zero, all the digits in that measurement are significant digits.
Numbers with Zeroes at the End
Numbers ending in zero are more complicated because zero has two different meanings. In these cases, you must determine whether a zeroes is a significant digit representing the quantity of "none" or just a place holders.Zeroes right (→) of a decimal point are always significant digits because without them the actual value of the number is no different, so we assume they are placed there to show additional certainty in the value of the number. (Zeroes between other significant digits are also always significant.) Any other zero between the last non-zero digit and the decimal point is not a significant digit.

Examples:

• 130 has two significant digits.
• 130.0 has four significant digits.
• 1.000000 has seven significant digits.
• 100000 has one significant digit.
1.30 x 102 is the way you write 130 if you want to make it clear that there are 3, not 2 significant digits in the number.
Perhaps someday there will be universal adoption of another symbol, perhaps ∅, to represent "none" and end the confusion. For example, 13∅0 would have 3 significant digits and be a briefer way to write the identical 1.30 x 103 = 1300 ± 10.

Exception to the Rules?

For those doing calculations on a calculator (or computer), it is wise to carry through all the digits in the calculator until the very end of the problem, and then truncate your final answer to the correct number of significant digits. In other words, during some intermediate step of the calculation, don't attempt to eliminate the insignificant digits in your calculator or write down an intermediate answer with fewer digits and then re-enter that new number into the calculator. Following the rules above, you might be tempted to round off midway through a problem, but doing so could introduce a small (but sometimes non-negligible) amount of "truncation error."  This is especially prone to happen if you have a complicated calculation involving many steps where truncation errors could accumulate.  In addition, it is possible to make an error re-entering a number.  These types of error are totally avoidable (as opposed to measurement errors, which we are stuck with) and therefore it is best to keep all intermediate digits until the final answer.

As a corollary of the above statement, it is sometimes desirable to add one more significant digit in your answer than the rules presented above would otherwise require. This extra digit should be indicated by a smaller font. One example where this is desirable is if your result is to be plugged into a future calculation where "truncation errors" might occur.

One final exception:  It is acceptable to round off to fewer significant digits when the full information is not needed or helpful. For example one might say their birthday is 6 months away even though it is possible to calculate more precisely.

But generally, stick to the rules above whenever precision and accuracy matters.

Acknowledgement goes to Patrick Woodward and Robert Zinn who contribute insight about these basic mathematical skills on their web sites;  but the content above remains the responsibility of Dave Trapp who can be contacted via the link below.

created 31 July 2000
minor revisions 22 February 2013
by D Trapp