# Using Significant Figures to Express Uncertainty in Measurements

*Written by Dale T. McGrosky
revised: April 2018*

When performing measurements and calculations how many digits do we extend to in our final answers? This can be explained by using significant figures to express the uncertainty in measurements. Significant figures is a tool that helps determine the number of digits used in a measurement. When performing measurements and calculations, significant figures express the uncertainty in a measurement and determine how many digits are used in the final answer.

The general rules for determining significant figures are as follows:

Any nonzero digit | 123, 1.23, .123 all have 3 significant figures. |

A zero that is between two nonzero digits | 1023, 1.023, .1023 all have 4 significant figures. |

Zeros that occur AFTER the decimal place are significant | 123.0 has 4 significant figures. 123.00 has 5 significant figures. |

All zeros to the left of the first nonzero digit are NOT significant | .00123 has 3 significant figures. |

Zeros that occur without a decimal are NOT significant | 1230 has 3 significant figures |

**Exact and Inexact Numbers**

Exact number are number that have a definite value. Counted numbers and metric conversions are exact numbers. The number of marbles in a jar, the number of people in a room are exact. Inexact numbers are numbers that do not have a definite value. An example would be a measurement taken with a set of vernier calipers. The vernier calipers dial is graduated at a thousandth of an inch. Lets say you measure the thickness of a washer and the indicator falls between 0.062 and 0.063 inches. We can say for certain that it is 0.062 because the indicator is above the 0.062 inch line, however we can not say for certain that it is 0.063 because the indicator falls below the 0.063 inch line. So we estimate the final digit. Lets say it’s around .0627. “0.062” are exact digits because we know for certain its at least 0.062 inches thick. However, the “7” is an estimated digit. One colleague may say it’s closer to 0.0626 and another 0.0628. The last digit is estimated and therefore the measurement is uncertain. All measurements are uncertain and therefore an inexact number.

Now, let say you measure the thickness of the washer with the same vernier calipers and the indicator falls right on the 0.062 line. If you recorded your results as “0.062” you would not be accurately stating the precision of your caliper. You are showing that you are only certain to 0.06 and the “2” is uncertain. In this case you would include the trailing zero and write your measurement as “0.0620” which shows the “0.062” as your certainty and the “0” as your uncertainty.

**Significant Figures in Multiplication**

When multiplying and dividing numbers, the number of significant figures in your answer will be equal to the number with the least significant figures. Let’s divide 123 by 3.14159 which equals 39.152149071. In this example, 123 has 3 significant figures and 3.14159 has 6. Therefore the answer can only have 3 significant figures.

Note that exact numbers like 10mm in a centimeter have an infinite number of significant figures. We do not write the significant figures down and these numbers are not used in determining the number of significant figures to be used in your answer.

**Significant Figures in Addition and Subtraction**

To determine the number of significant figures in an addition or subtraction problem it is necessary to round the number too the same digit as the number with *least* digits to right of the decimal place. The number of decimal places will determine the number of significant figures to be used in the answer. For example, 29.4165 has 4 significant digits to the right of the decimal place and 234.65 has only 2. Therefore your answer will be rounded off to the 2nd digit after the decimal place.

**Expressing Uncertainty of Measurements**

How precise the final answer will be is determined by the least precise measuring instrument used and how the final uncertain digit is expressed. For example on digital readouts my last digit is always inexact and therefore uncertain. A digital readout with a resolution of 0.0005, 4th digit is either a “0” or “5”, the 4th digit is my uncertainty. A digital scale with a resolution of 0.00001 lbs. My 5th digit is the uncertainty.

If a measurement is taken with a set of dial calipers with a resolution to 0.001 inches, the measurement would be recorded to the ten thousandth of an inch to show your uncertainty at the 4th digit because you can estimate the 4th digit when the indicator falls between two lines. Keep in mind that the final digit of any measurement is uncertain. If your measurement with the same instrument read exactly 2.1 inches and you recorded it as ”2.1 inches”, this show an incorrect uncertainty. “.1” being your final digit is the expressed uncertainty and expression shows it was measured with a device that only measures in 1 inch increments. The correct way to record your measurement with this instrument is 2.1000. This shows it was measured with a device that measures to 0.001 inch. Even though “2.1” and “2.1000” are the same measurement, the expressed uncertainty of measurement is not.

**Significant Figures in Measuring**

A simple density calculation is used to illustrate the use of significant figures.* A*n object weighs 16.5 grams and has a volume of 9.3 milliliters. The formula for calculation is . By inserting the appropriate values we’ll illustrate how this works.

In this example, 16.5g has 3 significant figures and 9.3mL has 2 significant figures. Using the rules of multiplying and dividing with significant figures, the answer would have the same number of significant figures as the *least* number in the calculation. Therefore, the answer should be reported using 2 significant figures, 1.7g/mL.

Let’s try another density calculation. We have a rectangular rubber block and want to calculate the density to determine from what type of elastomer it is made. To calculate the volume of a rectangular rubber block I need to use the formula Length X Width X Height (LxWxH). I will substitute this for volume in the density formula.

A scale will be used to measure the mass and calipers to measure the dimensions of the block and calculate the volume. The scale has a resolution of 0.001 grams and the calipers have a resolution to 0.01mm. The results of the measurements are as follows:

Instrument | Measured Value |
---|---|

Scale with .001g Resolution | 372.561g |

Digital Calipers with .01mm Resolution | Length = 10.00mm Width = 10.00mm Height = 30.00mm |

Note that the length, width and height values are recorded to 2 decimal places and the mass to 3 decimal places as these are the resolutions of instruments used and therefore want to show the uncertainty of measurements in the final answer. For example, if the measurement was recorded as 10mm, which has 2 significant figures, as opposed to 10.00mm, which has 4 significant figures, the measurement would not be expressing the proper uncertainty of measurement.

We want our density value in grams per cubic centimeter because the known density’s of rubber is specified in grams per cubic centimeter not grams per cubic millimeter, it is therefore necessary to convert the length, width and height measurements from millimeters to centimeters. 10.00mm divided by 10 mm/cm = 1.000cm. Because 10mm/cm is an exact number with infinite number of significant figures, it’s not used in determining the number of significant figures in the answer. Therefore, the number of significant figures is determined using measurements, which have 4 significant figures, so the final values can only have 4 significant figures, 1.000cm, 1.000cm and 3.000cm.

Let’s plug in the values into the density formula and calculate the results using significant figures. Using the measured values in cm, the density is calculated:

The answer has 4 significant figures because the least number of significant figures in any number used is 4. Note that the numbers in the formula have the unit of measure which are also calculated out in the final answer. For example, in 1.000cm X 1.000cm X 3.000cm, the 3 “cm” are expressed as “cm^{3}”. In the final value we have “grams” divided by “cm^{3”} expressed as “g/cm^{3”}.

It is good practice to include significant figures in your calculations to show the uncertainty of your measurements so other know the precision at which you were able to measure. It is also good practice to always include the unit of measure in your values. These will be calculated out in your formula and used to determine the proper unit of measure in your final answer.