When multiple measurements are used to calculate a quantity, each of these measurements may carry its own uncertainty. By being careful and taking into account these uncertainties, scientists and engineers can better assess the reliability and precision of their conclusions. Use this resource to learn how to consider, or propagate, uncertainty.
Propagation of uncertainty involves estimating the uncertainty in the calculated value. The way you propagate uncertainty depends on whether you have added, subtracted, multiplied or divided the measured quantities.
Uncertainty in sums and differences
For addition and subtraction of measurements, the uncertainties are treated the same way.
When measurements are added or subtracted, the absolute uncertainties are added.
Example 1 – propagating uncertainty for sums and differences
Consider the measurements:
\[A=4.0\pm0.2\textrm{ cm}\quad B=2.0\pm0.1\textrm{ cm}\]
Find the sum of \(A\) and \(B\), including the absolute uncertainty.
We simply add the measurements together, and then their uncertainties.
\[\begin{align*} A+B & = (4.0+2.0)\pm(0.2+0.1)\textrm{ cm}\\
& = 6.0\pm0.3\textrm{ cm}
\end{align*}\]
The absolute uncertainty is \(0.3\textrm{ cm}\).
Find \(A-B\), including the absolute uncertainty.
Here, we subtract \(B\) from \(A\), but sum their uncertainties.
\[\begin{align*} A-B & = (4.0-2.0)\pm(0.2+0.1)\textrm{ cm}\\
& = 2.0\pm0.3\textrm{ cm}
\end{align*}\]
The absolute uncertainty is \(0.3\textrm{ cm}\).
Uncertainty in products and quotients
For multiplication and division of measurements, the way we find the uncertainty is different. In fact, we don't look at absolute uncertainty, but instead, percentage uncertainty.
When measurements are multiplied or divided, the percentage uncertainties are added.
Example – propagating uncertainty for products and quotients
Calculate the area of a rectangle and its associated absolute uncertainty from the following data:
\[\textrm{Length}=14.26\pm0.02\textrm{ cm}\quad \textrm{Width} = 5.94\pm0.02\textrm{ cm}\]
The area of the rectangle is calculated by length times width.
\[\begin{align*} A & = =14.26\times5.94\\
& = 84.7\textrm{ cm}^{2}
\end{align*}\]
The percentage uncertainty is the sum of the uncertainty in the length and the uncertainty in the width.
\[\begin{align*} \% \textrm{Uncertainty in length} & = \frac{\textrm{Absolute error in length}}{\textrm{Length}} \times100\\
& = \frac{0.02}{14.26}\\
& = 0.14\%
\end{align*}\] \[\begin{align*} \% \textrm{Uncertainty in width} & = \frac{\textrm{Absolute error in width}}{\textrm{Width}} \times100\\
& = \frac{0.02}{5.94}\\
& = 0.34\%
\end{align*}\] \[\begin{align*} \% \textrm{Uncertainty in area} & = 0.14\% + 0.34\%\\
& = 0.48\%
\end{align*}\]
Putting these together, the area of the rectangle is \(84.7\textrm{ cm}^{2}\pm0.48\%\). But, the question is asking for absolute uncertainty, so we should find this using the percentage.
\[0.48\%\times84.7=0.4\textrm{ cm}^{2}\]
So, the area of the rectangle is \(84.7\pm0.4\textrm{ cm}^{2}\).
Uncertainty in powers
For measurements raised to a power, the percentage uncertainty is added as many times as the power. For example, a value raised to the power of \(3\) will have its percentage uncertainty trebled. This is the same as multiplying the value by itself three times, so the three percentage uncertainty values are added.
When measurements are raised to a power \(n\), the percentage uncertainties are multiplied by \(n\).
Example – propagating uncertainty for powers
For a sphere of radius \(5.0\pm0.1\textrm{ cm}\), calculate its volume and associated uncertainty.
The equation for the volume of a sphere is \(\textrm{Volume} = \dfrac{4}{3}\pi r^{3}\). Let's calculate the volume first.
\[\begin{align*} \textrm{Volume} & = \frac{4}{3}\pi (5.0)^{3}\\
& = 524\textrm{ cm}^{3}
\end{align*}\]
The percentage uncertainty for the radius is:
\[\begin{align*} \% \textrm{Uncertainty in radius} & = \frac{\textrm{Absolute uncertainty}}{\textrm{Radius}}\times 100\\
& = \frac{0.1}{5}\times100\\
& = 2\%
\end{align*}\]
To find the percentage uncertainty for the volume, we multiply by \(n\) which is the power. Here, the power is \(3\), so:
\[\begin{align*} \% \textrm{Uncertainty in volume} & = 3\times \% \textrm{Uncertainty in radius}\\
& = 3\times2\%\\
& = 6\%
\end{align*}\]
The final volume is \(524\textrm{ cm}^{3}\pm 6\%\) or \(524\pm31\textrm{ cm}^{3}\).
Exercise – propagating uncertainty
If \(A=6.0\pm0.1\textrm{ cm}\) and \(B=3.4\pm0.2\textrm{ cm}\), calculate:
\(A+B\)
\(A-B\)
\(A\times B\)
\(A\div B\)
The mass of a square cube was found to be \(60.7\pm0.05\textrm{ g}\). A side length of this cube is \(1.5\pm0.05\textrm{ cm}\). Calculate the density of the material and its associated uncertainty. \(\left(\textrm{Density}=\dfrac{\textrm{Mass}}{\textrm{Volume}}\right)\)
A cylinder has a height of \(10.3\pm0.05\textrm{ cm}\). Its radius is \(4.5\pm0.05\textrm{ cm}\). Calculate the volume of the cylinder and its associated error. \(\left( \textrm{Volume}=\pi r^{2}h \right)\).
