# Significant Figures

The precision of any measurement (length, volume, or mass) is limited by the device used in the measurement.  Precision is given by the number of digits in the reported measurement.  These digits are called significant figures.

The following examples and discussion illustrate measurements with different devices.  The first step to making a measure is to determine the scale of the device.  How many marks are on the scale and what dimension do they represent.

In the first example of length, two different rulers are shown.  The first is marked in 1 cm increments.  The second is marked in 1 mm units.  A mm is a smaller length than a cm so the 1 mm ruler would give a more precise measurement.  Hence the measurement with the mm ruler would have a greater number of significant figures.

LENGTH

Measuring the length with cm Ruler 1.

With this ruler, one knows the length of the paper is more than 10 cm, but definitely less than 11 cm.  An estimate is that it is 0.7 or 0.8 beyond 10 cm.  So one would report the length of the paper to be 10.7 cm knowing the .7 is an estimate (it could be 0.6 or 0.8).  This measurement is reported with 3 significant figures and the length would be 10.7 cm.

Measuring the length with mm Ruler 2.

With this ruler, one knows the length of the paper is more than 10 cm, but definitely less than 11 cm.  One also knows is it greater than 10.6 cm but less than 10.7 cm. An estimate is that it is 0.5 cm beyond 10.6 cm.  So one would report the length of the paper to be 10.65 knowing the 0.05 is an estimate (it could be 0.04 or 0.06).  This measurement is reported with 4 significant figures and the length would be 10.65 cm

Significant figures reflect the precision of a reported measurement.  When any measurement is obtained, there is an uncertainty associated with it.  The fewer the number of significant figures, the more uncertainity (less precision) there is in a reported measurement.  The greater the number of significant figures, the less uncertainity (more precision) there is in a reported measurement.

Regarding the two measurements discussed above, the measurement with Ruler 1 (fewer markings) is less precise (3 significant figures).  The measurement with Ruler 2 (more markings) is more precise (4 significant figures).

VOLUME

Another example would be measuring volume of a liquid in a lab.

Liquids are commonly measured using a graduated cylinder.  A graduated cylinder is a cylinder with markings, usually in milliliters (mL).  As with the ruler in the preceding discussion, different graduated cylinders have different markings.   The student needs to know how the graduated cylinder is marked.

In this example, three graduated cylinders, 10 ml, 25 ml and 50 ml, will be used.

Another factor to consider is the meniscus which occurs due to the interaction between the liquid and the glass of the graduated cylinder.  In the examples shown, the meniscus is concave and the bottom of the meniscus is the volume of the liquid.

This graduated cylinder has white lines going completely around the cylinder to indicate the mL quantities.  The shorter lines indicate tenths of a mL.

With this graduated cylinder, the bottom of the meniscus is more than 8 mL but less than 9 mL.  One also knows it is greater than 8.6 mL but less than 8.7 mL.  An estimate is that it is 0.8 mL beyond 8.6 mL.  So one would report the volume of the liquid to be 8.68 mL knowing the 0.08 is an estimate (it could be 0.07 or 0.09).  This measurement is reported with 3 significant figures and the volume would be 8.68 mL.

This graduated cylinder has white lines going completely around the cylinder to indicate odd number of mL quantities.  There is a line half way around the cylinder to indicate the even number of mL quantities.  There are 5 shorter lines between each whole number.  Each of these shorter lines indicates 0.2 mL.

With this graduated cylinder, the volume of the bottom of the meniscus is more than 10 mL but less than 11 mL.  One also knows it is greater than 10.4 mL but less than 10.6 mL.  An estimate is that it is 0.5 mL beyond 10.4 mL.  So one would report the volume of the liquid to be 10.5 knowing the 0.5 is an estimate.  This measurement is reported with 3 significant figures and the volume would be 10.5 mL

This graduated cylinder has white lines going completely around the cylinder to indicate 10 mL quantities.  The shorter lines indicate 1 mL increments.

With this graduated cylinder, the volume of the bottom of the meniscus is more than 22 mL but less than 23 mL.  An estimate is that it is 0.7 mL beyond 22 mL.  So one would report the volume of the liquid to be 22.7 mL knowing the 0.7 is an estimate (it could be 0.6 or 0.8).  This measurement is reported with 3 significant figures and the volume would be 22.7 mL

MASS
The mass of an object is measured using a balance.  Most balances today are digital balances and very easy to use.  One places the object to be weighted on the pan of the balance and the mass value will appear in the digital read out.

An aluminum cylinder is weighted on three balances as shown below.

Balance 1

This balance indicates the aluminum cylinder has a mass of 34.70 g

Balance 2

This balance indicates the aluminum cylinder has a mass of 34.699g

Balance 3

This balance indicates the aluminum cylinder has a mass of 34.7191 g

In addition to easy use and readout digital balances have another advantage.  They report the measurement to the correct number of significant figures.  Balance 1 has four significant figures and provides a mass value to the nearest + or – 0.01 g,   Balance 2 has five significant figures and provides a mass value to the nearest + or – 0.001 g.  Balance 3 has six significant figures and provides a mass value to the nearest + or – 0.0001 g.  Recalling the discussion above, balance 3 provides the most precision of the mass of the aluminum cylinder.

When making any measurement, the first thing a student needs to determine is the markings (scale) of the device used to make the measurement.  This determines the level of precision and the number of significant figures to be recorded.

There are rules for identifying significant figures and for mathematical operations involving significant figures.  The links below are good resources to check out.  If you want more resources, Google significant figures.

Wikipedia

Significant Figures