Further Study

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Analytical Sciences Digital Library

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Limitations to Significant Figures

Although significant figures are important, there are situations where they do not lead to a correct estimate of uncertainty. Consider the following example:

The mass of water in a sample is found by weighing the sample before and after drying, yielding values of 0.4991 g and 0.4174 g. What is the percent water by mass in the sample? The uncertainty in each mass is ±0.0001 g.

Using significant figures to account for uncertainty gives the mass percent of water as

\[\mathrm{100 \times \dfrac{0.4991\: g - 0.4174\: g}{0.4991\: g} = 100 \times\dfrac{0.0817\: g}{0.4991\: g} = 16.3695\%}\nonumber\]

which, allowing for three significant figures, rounds to 16.4%. Using worst case scenarios, the largest and smallest possible results are

\[\mathrm{100 \times \dfrac{0.4992\: g - 0.4173\: g}{0.4990\: g} = 16.4128\%}\nonumber\]

\[\mathrm{100 \times \dfrac{0.4990\: g - 0.4175\: g}{0.4992\: g} = 16.3261\%}\nonumber\]

Comparing the two worst case results and the exact result, we see that rounding to the hundredth's place is the first instance where there is no agreement between the three calculations. The result of the exact calculation, therefore, is rounded to four significant figures, giving 16.40% as the appropriate result of the calculation.

Worst Case Scenario

Four simple rules will help you make use of this approach to estimating uncertainty.

Rule One. If two values are added, the limits for the result are obtained by adding the upper limits and by adding the lower limits. For example, if X is 7.0 ± 0.1 and Y is 5.0 ± 0.1, then the upper limit is

\[\mathrm{X + Y = 7.1 + 5.1 = 12.2}\nonumber\]

and the lower limit is

\[\mathrm{X + Y = 6.9 + 4.9 = 11.8}\nonumber\]

Rule Two. If two values are multiplied, the limits for the result are obtained by multiplying the upper limits and by multiplying the lower limits. For example, if X is 7.0 ± 0.1 and Y is 5.0 ± 0.1, then the upper limit is

\[\mathrm{X \times Y = 7.1 \times 5.1 = 36.21}\nonumber\]

and the lower limit is

\[\mathrm{X \times Y = 6.9 \times 4.9 = 33.81}\nonumber\]

Rule Three. If two values are subtracted, the limits for the result are obtained by subtracting the lower limit of one number from the upper limit of the other number and by subtracting the upper limit of one number by the lower limit of the other number. For example, if X is 7.0 ± 0.1 and Y is 5.0 ± 0.1, then the upper limit is

\[\mathrm{X - Y = 7.1 - 4.9 = 2.2}\nonumber\]

and the lower limit is

\[\mathrm{X - Y = 6.9 - 5.1 = 1.8}\nonumber\]

Rule Four. If two values are divided, the limits for the result are obtained by dividing the upper limit of one number by the lower limit of the other number and by dividing the lower limit of one number by the upper limit of the other number. For example, if X is 7.0 ± 0.1 and Y is 5.0 ± 0.1, then the upper limit is

\[\mathrm{\dfrac{X}{Y} = \dfrac{7.1}{4.9} = 1.44898}\nonumber\]

and the lower limit is

\[\mathrm{\dfrac{X}{Y} = \dfrac{6.9}{5.1} = 1.35294}\nonumber\]

A useful article discussing this approach is Gordon, R.; Pickering, M.; Bisson, D. "Uncertainty Analysis the 'Worst Case' Method" J. Chem. Educ. 1984, 61, 780-781.

Propagation of Uncertainty

A more rigorous approach to determining the uncertainty in a result is called a propagation of uncertainty. You can read more about this approach here. The following applet provides a useful calculator for determining uncertainty (note that you can download this applet to your computer).