Significant figures

The correct use and importance of significant figures

Students may lose marks in their Individual Scientific Investigation (and in their written exams and Extended Essays) through the incorrect use of significant figures. Sometimes it is simply due to carelessness. A classic case is giving an answer from an experiment involving a titration to three significant figures when the burette readings have correctly been given to four significant figures (such as 25.35 cm³) but the concentration of the acid or base used in the burette has only been given as 0.1 mol dm⁻³ rather than 0.100 mol dm⁻³. At other times though it may be because students do not understand how to use significant figures correctly and you will need to spend some time teaching them the theory.

Direct student access

Note that students have direct access to a version of this page written specifically for students (as opposed to teachers) at Significant figures in Complete course for students.

Theory

Essentially whenever a measurement of a physical quantity is taken there will be a built in uncertainty in the reading due to the limitations of the measuring instrument used. The measurement recorded by the student should include the first figure that is uncertain. This should include zero if necessary. Thus a reading of 22.70 ^oC indicates that the temperature was taken with a thermometer that is accurate to ± 0.01 ^oC. The reading should be quoted as 22.7 ^oC if a thermometer accurate to only ± 0.1 ^oC was used.

There are two particular areas that can cause problems:

1. The use of zero

a) When is zero significant?

Zero only becomes significant when it comes after a non-zero digit (1, 2, 3, 4, 5, 6, 7, 8, and 9).

For example:

000147.6 (Zero not a significant figure)

0.0001476 (Zero not a significant figure)

1.0476 (Zero is a significant figure - value quoted to 4 sig. figs)

1.4760 (Zero is a significant figure - value quoted to 5 sig. figs)

b) Use of scientific notation

One or more zeros after a non-zero digit but before a decimal point may or may not be significant depending on how the measurement was made. For example 159 000 might mean exactly one hundred and fifty nine thousand or one hundred and fifty nine thousand to the nearest thousand. The best way to avoid this problem is to use scientific notation.

1.59000 x 10⁶ (value quoted to six significant figures)

1.59 x 10⁶ (value quoted to three significant figures)

2. Calculations

(a). Adding or subtracting

When adding or subtracting it is the number of decimal places that is important.

e.g. 8.20 g (3 sig figs) + 3.60 g (3 sig figs) = 11.80 g (4 sig figs)

This answer can be quoted to four significant figures since the balance used in both cases was accurate to ± .01g.

When subtracting, the number of significant figures may decrease.

e.g. 17.82 g (4 sig figs) – 12.43 g (4 sig figs) = 5.39 g (3 sig figs)

(b). Multiplying or dividing

When multiplying or dividing it is the number of significant figures that is important. The number with the least number of significant figures used in the calculation determines how many significant figures should be used when quoting the answer.

e.g. The heat required to raise the temperature of 0.150 kg of water by 8.4 ^oC

= 0.150 kg x 8.4 ^oC x 4.18 kJ kg^-1 ^oC^-1 = 5.2668 kJ.

Since the temperature is only recorded to two significant figures the answer should be given as 5.3 kJ.

Training your students in the use of significant figures

When you are training your students how to record and process their data during the 'scaffolding' experiments it is good practice to be harsh on them initially so that they learn from their mistakes. When you receive the first draft of their Individual Scientific Investigation you cannot point out specific instances of where they may have misused significant figures but you are allowed to tell them to check that all their values are stated to the correct number of significant figures.

In the Exams the IB does not penalise students if the number of significant figures in an answer differs by one from the correct number (unless the question specifically asks for them) but will penalise if they differ by more than one either way. For example, if the correct answer is 3.58 then 3.6, 3.58 or 3.582 would all be accepted without penalty but 4 or 3.5821 would be penalised.

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