Why The Rule for Significant Digits Gives the Correct Answer

Take a look at the calculation of the density of a liquid, and use the example to see why this rule about significant digits exists.  In this example, we're showing the actual uncertainty we can expect.   This uncertainty tells us how large, or how small our actual measured data might be.  Here's the data:

	Measured Result	Could be as high as	Could be as low as
Mass of liquid	68.24 0.01 g	68.25 g	68.23 g
Volume of liquid	90.2 0.2 mL	90.4 mL	90.0 mL

Since density = mass/volume, we can use this data to calculate the biggest (divide the largest possible mass by the smallest possible volume) and smallest (divide the smallest possible mass by the largest possible volume) densities we could have.

We've highlighted in red the digit at which the two answers are different.  Notice that it is at the third significant digit.  If the numbers are different at the third significant digit, then obviously all the other answers to right of them are totally meaningless.  The rule we have works: always round off the answer to the shortest number of significant digits  –  in this case the three digits in the volume.   So you should calculate the density measured here as (68.24 g)/(90.2 mL) =   0.7565410199557 and then report it rounded off to 0.757 g/mL at the third significant digit.