Significant Figures and Related Concepts

Some numbers are exact. I have $\bf 1$ daughter and $\bf 2$ ears. There are $\bf 100$ centimeters in a meter (by definition). The number  $\bf\pi$  is the ratio of the circumference to the diameter of any circle. The number $\,\bf\sqrt{2}\,$ is the unique nonnegative number which, when multiplied by itself, gives $\bf 2$.

Most of the numbers used in mathematics are assumed to be exact. For example, when a word problem says that ‘Carol bought one-half pound of chocolate’ you don't usually worry about whether it was really $\,0.49\,$ pounds or $\,0.52\,$ pounds.

In the sciences, however (like chemistry, physics, and biology), numbers are often suspect. Whenever a measurement is made, the number that you get as a result has potential error associated with it. The amount of error depends on things like the quality of the measuring instrument and the skill of the person making the measurement.

The reliability of a measurement has two components:  precision and accuracy.

Precision refers to how closely measurements of the same quantity agree with each other. Accuracy refers to how closely measured values agree with the correct (true) value.

Measurements can be both precise and accurate. This happens when the measurements are close to each other (precise), and also close to the true value (accurate).

Measurements can be neither precise nor accurate. This happens when the measurements are not close to each other (not precise), and also not close to the true value (not accurate).

Measurements can be precise, but not accurate. This happens when the measurements are close to each other (precise), but not close to the true value (not accurate).

Measurements can theoretically be accurate, but not precise. This happens when the measurements are close to the true value (accurate), but not close to each other (not precise). (This is hard to visualize, because if points are close to a true value, then they're usually also close to each other!)

Given a set of measurements, the average value is taken as the best value. The range of a set of measurements is the difference between its greatest and least values. Range is a measure of precision, since it is a measure of how close the individual measurements are to each other.

For example, suppose the following five length measurements are made (all units of feet):

The average value is:

The range is the difference between the greatest value ($\,7.3\,$) and the least value ($\,6.6\,$), so the range is $\,7.3 - 6.6 = 0.7\,.$

Scientists communicate the precision of measurements using a concept called significant figures or significant digits. Roughly, the number of significant figures is the number of digits believed to be correct by the person doing the measuring; usually, there is one estimated digit.

The number of significant figures is determined using the following rules :

• All nonzero digits are significant
  Example: The number $\,529.317\,$ has six significant figures.
• A middle (imbedded) zero is always significant
  Example: The number $\,30,567\,$ has five significant figures.
• A leading zero is never significant
  Example: The number $\,0.0000293\,$ has three significant figures.
• A trailing zero is significant only when a decimal point is specified
  
  Example: The number $\,32{,}000{,}000\,$ (no decimal point) has two significant figures.
  
  Example: The number $\,32{,}000{,}000\bf{.}\,$ (notice the decimal point) has eight significant figures.
  
  Example: The number $\,32{,}000{,}000.00\,$ has ten significant figures.
  
  Example: The number $\,5.2170000\,$ has eight significant figures.

When significant figures are used in calculations, it is important that the result reflects the appropriate skepticism of the component numbers! In general, computations are done as if the numbers are exact, and then the answers are rounded according to the following rules:

Addition and Subtraction

The number of decimal places in the answer is the same as the measured quantity with the fewest number of decimal places.

Example:   Report the sum of these measurements using the correct number of significant figures: $\,0.1\,,$ $\,0.22\,,$ $\,0.333\,$

Solution:   $0.1 + 0.22 + 0.333 = 0.653$   (exact arithmetic);   report as $\,0.7$. The fewest number of decimal places is $\,1\,,$ so round to one decimal place.

Example:   Report the sum of these measurements using the correct number of significant figures: $\,0.22\,,$ $\,0.333\,,$ $\,0.4444\,$

Solution:   $0.22 + 0.333 + 0.4444 = 0.9974$   (exact arithmetic);   report as $\,1.00\,.$ The fewest number of decimal places is $\,2\,,$ so round to two decimal places.

Multiplication and Division

The number of significant figures in the answer is the same as in the measured quantity with the fewest number of significant figures.

Example:   Report the product of these measurements using the correct number of significant figures: $\,0.1\,,$ $\,0.22\,,$ $\,0.333\,$

Solution:   $(0.1)(0.22)(0.333) = 0.007326$   (exact arithmetic);   report as $\,0.007\,.$ The fewest number of significant figures is $\,1\,,$ so report the result with $\,1\,$ significant figure.

Example:   Report the product of these measurements using the correct number of significant figures: $\,7.77\,,$ $\,0.2345\,$

Solution:   $(7.77)(0.2345) = 1.822065$   (exact arithmetic);   report as $\,1.82\,.$ The fewest number of significant figures is $\,3\,,$ so report the result with $\,3\,$ significant figures.

Notice that for addition/subtraction, the number of decimal places is the key concept, whereas for multiplication/division, the number of significant figures is the key concept.

For computations with significant figures, you always round as the last step. If you need some practice with rounding, click here.

Some people adopt the convention that if the number being rounded is exactly halfway between the two rounding candidates, then you round UP. With this convention, $\,2.35\,$ rounds to $\,2.4\,,$ and $\,2.65\,$ rounds to $\,2.7\,$ (when rounding to one decimal place). If this type of situation occurs a lot, then it can give an upward bias to the data, so scientists sometimes adopt a slightly different convention:

NO-BIAS ROUNDING CONVENTION

... for the situation when the number being rounded is exactly halfway between the two rounding candidates:

• If the number before the ‘$\,5\,$’ is ODD, then round UP.
• If the number before the ‘$\,5\,$’ is EVEN, then round DOWN.

Examples

Round each quantity to the tenths place (using the no-bias rounding convention):

$2.35\,$ rounds to $\,2.4$
(Notice that $\,2.35\,$ is exactly halfway between $\,2.3\,$ and $\,2.4\,,$ and the digit ‘$\,3\,$’ is odd.)

$2.65\,$ rounds to $\,2.6$
(Notice that $\,2.65\,$ is exactly halfway between $\,2.6\,$ and $\,2.7\,,$ and the digit ‘$\,6\,$’ is even.)

$2.8501\,$ rounds to $\,2.9\,$
(Notice that $\,2.8501\,$ is closer to $\,2.9\,$ than $\,2.8\,$; the special rule does not apply here.)