Significant Figures

Significant figures are always used when a measurement is made and they express the degree of uncertainty in both measurements and calculations. In any measurement, the number of significant figures always includes one more decimal place than that obtained from the smallest calibration on the measuring instrument. All the digits in a measured number are called significant figures with all but the right-most digit known for sure. The right-most digit has been estimated.

When reporting a measurement that includes the estimated last digit, it is appropriate to make note of the uncertainty of the measurement. Generally, a number with N significant digits has an uncertainty of 1 to the Nth digit. For example 28.5 g has an uncertainty of ± .1 g and would be written as 28.5 ± .1 g

The fractional uncertainty that corresponds to the number of significant digits are as follows:

Number of Significant Figures

Fractional uncertainty between

Or roughly

1

10% to 100%

50%

2

1% to 10%

5%

3

.1 % to 1%

.5%

 

Determining the Significant Figures of a Measurement

Several rules can be used to determine the number of significant figures when the measurement contains at least one zero. However, a very simple technique called DOT: RIGHT, NOT: LEFT, is much easier to remember.

Significant Figure Rule DOT: RIGHT, NOT: LEFT

 

Using Significant Figure Rules in Calculations

Addition and Subtraction

When measured quantities are used in addition or subtraction, the uncertainty is determined by the absolute uncertainty in the least precise measurement (not by the number of significant figures).  Simply Put: Round your answer when you run out of precision.

             Example 1
                                                             32.01               m
                                                               5.325             m
                                                 +         12                     m


                                                              49.335             m            = 49 m

Added together, you will get 49.335 m, but the sum should be reported as '49 m' because you have run out of precision after the '2' in the ones place of 12 m.

             Example 2
                                                               22.01               m
                                                                 4.527             m
                                                   +       130                     m


                                                             156.537             m           = 160 m

Added together, you will get 156.537 m, but the sum should be reported as '160 m' because you have run out of precision after the '3' in the tens place of 130 m.  (The zero in 130 m is a place keeper and not significant.)

Multiplication and Division

When experimental quantities are mutiplied or divided, the number of significant figures in the answer is the same as that in the quantity with the smallest number of significant figures.

If, for example, a density calculation is made in which 25.014 grams is divided by 25 mL, the density should be reported with 2 significant figures as this was the least in the problem.

Your raw answer before rounding was 1.00056 g/mL, but you should report your answer as 1.0 g/mL, rounding it to 2 significant figures.

 

 

Loosely adapted from http://chemistry.about.com/od/mathsciencefundamentals/a/sigfigures.htm
http://www.geocities.com/junebug_sophia/sigfigsP.htm
and An introduction to Error Analysis, By John Robert Taylor