Determining the Uncertainty in the Results of Arithmetic Calculations

Units And Measurements • Physics

Concept Summary

Every measured value carries an uncertainty, and when measured quantities are combined in a calculation, their uncertainties combine as well. A reading with three significant figures, such as the length 16.2 cm measured with a metre scale, is written as 16.2 ± 0.1 cm, which equals 16.2 cm ± 0.6 %. Similarly a breadth of 10.1 cm is written as 10.1 ± 0.1 cm, or ± 1 %. For a product of two or more experimental values, the combination of errors rule states that the percentage errors add: the area 16.2 × 10.1 = 163.62 cm² has an error of about ± 1.6 %, that is ± 2.6 cm², so the result is quoted as 164 ± 3 cm². Here 3 cm² is the uncertainty in the estimate of area. If data are specified to n significant figures, a result obtained by combining them is also valid to n significant figures; however, when data are subtracted the number of significant figures can reduce, as 12.9 g − 7.06 g can only be given as 5.8 g. The relative error also depends on the size of the number and not only on the number of significant figures: 1.02 g and 9.89 g are both accurate to ± 0.01 g, but their relative errors are ± 1 % and ± 0.1 %. Finally, intermediate results should carry one extra significant figure to avoid accumulated rounding errors.

Key Notes
  • A measurement is quoted as value ± uncertainty, e.g. \(l = 16.2 \pm 0.1\) cm.
  • Relative (percentage) error = (absolute error / measured value) × 100 %.
  • For a product (or quotient), percentage errors add: \(l = 16.2\) cm (± 0.6 %) and \(b = 10.1\) cm (± 1 %) give area ± 1.6 %.
  • Area \(lb = 163.62 \pm 2.6\ \mathrm{cm^2}\), quoted as \(164 \pm 3\ \mathrm{cm^2}\).
  • If the data are specified to n significant figures, the combined result is valid to n significant figures.
  • Subtraction can reduce the number of significant figures: 12.9 g − 7.06 g = 5.8 g.
  • Relative error depends on the number itself: 1.02 g ± 0.01 g is ± 1 %, while 9.89 g ± 0.01 g is ± 0.1 %.
  • Calculate intermediate results to one more significant figure than the least precise measurement.
Important Equations
\[ \text{Relative error} = \frac{\Delta a}{a}, \qquad \text{Percentage error} = \frac{\Delta a}{a}\times100\ \% \] \[ \frac{\Delta l}{l} = \frac{0.1}{16.2}\times100\% \approx 0.6\%, \qquad \frac{\Delta b}{b} = \frac{0.1}{10.1}\times100\% \approx 1\% \] \[ \text{Product } lb:\quad \left(\frac{\Delta (lb)}{lb}\right)\times100\% \approx 0.6\% + 1\% = 1.6\% \] \[ lb = 163.62 \pm 2.6\ \mathrm{cm^2} \;\Rightarrow\; lb = 164 \pm 3\ \mathrm{cm^2} \] \[ \frac{\pm0.01}{1.02}\times100\% = \pm1\%, \qquad \frac{\pm0.01}{9.89}\times100\% = \pm0.1\% \]
Flow Chart
Write each measurement as value ± absolute error
Mind Map
  • Uncertainty in Results
    • Representation
      • Value ± absolute error
      • Value ± percentage error
    • Combining errors
      • Product / quotient: add percentage errors
    • Significant figures
      • Combined result valid to n s.f.
      • Subtraction may reduce s.f.
    • Relative error
      • Depends on n and on the number itself
    • Intermediate steps: one extra digit
Graphic Organiser
QuantityValueAbsolute ErrorRelative / % Error
Length \(l\)16.2 cm± 0.1 cm± 0.6 %
Breadth \(b\)10.1 cm± 0.1 cm± 1 %
Area \(lb\)164 cm2± 3 cm2± 1.6 %
Mass1.02 g± 0.01 g± 1 %
Mass9.89 g± 0.01 g± 0.1 %