How to calculate percentage uncertainty

In science, all measurements have an associated uncertainty. This uncertainty communicates the precision to which the measurement was taken. If a scientific result is the product of a series of calculations of measured values, then the result itself will have an associated uncertainty to show the degree of confidence in the result.

The percent uncertainty of a value is the same as the fractional or relative uncertainty of the value expressed as a percentage. It can be found from the following formula:

$\hspace{3em}$ % uncertainty = $\dfrac{\textrm{absolute uncertainty}}{\textrm{measured value}} \times 100 \%$

For example, if the time for a cart to roll down a ramp is measured to be 3.6 s ± 0.2 s we can see the absolute uncertainty on the measurement is 0.2 s. We want to express this as a percentage of the measured value:

$\hspace{3em}$ % uncertainty = $\dfrac{\textrm{absolute uncertainty}}{\textrm{measured value}} \times 100 \% = \dfrac{0.2}{3.6}=5.6 \%$

The method used to find the percentage uncertainty of a calculated value depends on the mathematical operation being performed.

If the values involved are being added or subtracted, then the absolute uncertainties need to be added to find the absolute uncertainty on the result, and then the percentage uncertainty can be found.

For example, if three length measurements are being added together:

$\hspace{3em}$ (2.0 ± 0.1) cm + (8.2 ± 0.2) cm + (2.5 ± 0.5) cm

The result will be the sum of the lengths and the absolute uncertainty will be the sum of the individual uncertainties:

$\hspace{3em}$ Length = 2.0 + 8.2 + 2.5 = 12.7 cm

$\hspace{3em}$ Absolute uncertainty = 0.1 + 0.2 + 0.5 = 0.8 cm

$\hspace{3em}$ The result is expressed as 12.7 cm ± 0.8 cm.

We can now find the percentage uncertainty on the answer as follows:

$\hspace{3em}$ % uncertainty = $\dfrac{\textrm{absolute uncertainty}}{\textrm{measured value}} \times 100 \% = \dfrac{0.8}{12.7}=6.3 \%$

If two values are being divided or multiplied together, their individual percentage uncertainties must be found. The final result's percentage uncertainty will be the sum of the individual percentage uncertainties.

For example, consider the following calculation of the density of an object. The mass $m$ and volume $V$ of the object are measured to be

$\hspace{3em}$ $m$ = 4.5 g ± 0.2 g

$\hspace{3em}$ $V$ = 2.5 cm$^3$ ± 0.5 cm$^3$

The density can be found from the formula

$\hspace{3em}$ $\rho= \dfrac{m}{V}$

Giving

$\hspace{3em}$ $\rho=\dfrac{4.5}{2.5}=1.8$ g cm$^{-3}$

Now we need to find the percentage uncertainty of the result. First, we find the percentage uncertainties on the mass and the volume:

$\hspace{3em} \dfrac{\Delta m}{m} \times 100 \% = \dfrac{0.2}{4.5} \times 100 \% =4.4 \% \hspace{2em} \hspace{2em}$

and

$\hspace{3em}\dfrac{\Delta V}{V} \times 100 \%= \dfrac{0.5}{2.5} \times 100 \%=20 \%$

Because we are dividing the terms, we add the percentage uncertainties:

$\hspace{3em}$ density % uncertainty = mass % uncertainty + volume % uncertainty = 4.4% +20%

$\hspace{10em}$ = 24%