Handbook

Uncertainty Propagation []

Sometimes the measurand cannot be measured directly, but rather calculated from other measured quantities or reference values. In these cases, there is an equation that describes how input variables are related to the output variable (i.e., the measurement model). To determine the uncertainty of the output variable, the uncertainties of the input variables need to be propagated correctly.

Rules for propagating uncertainties []

Consider the following example:\ one wants to determine the average speed of a runner. To do so, a certain distance is measured and then the time the runner takes to run this distance is measured. The equation to calculate the speed is the distance divided by the time: \(v=\dfrac{s}{t}\).

The distance and time each have their own uncertainty, which together result in an uncertainty for the speed. This uncertainty has to be calculated by propagating the uncertainty of the distance and time. The rules for propagating uncertainties for some common functions are shown in Tab. 4. In this case, the uncertainty for the speed is given by: \(u_v=|v| \sqrt{\left( \dfrac{u_{s}}{s} \right)^2 + \left( \dfrac{u_{t}}{t} \right)^2}\).

The measurement of the distance is s = (50.0 ± 0.50) m and a time measurement is t = (20.0 ± 1.0) s. This results in a speed of v = (2.50 ± 0.13) m/s.

Table 4: The rules for the propagation of uncertainties for two independent variables x and y. The table shows how to propagate the uncertainties using Gaussian uncertainty propagation and the approximate procedures. There a and b are numbers without uncertainty, σ is the standard deviation and u is the (approximated) uncertainty.

Function Gaussian Propagation Approximation
\(z = ax\) \(\sigma_z = |a|\sigma_x\) \(u_z \approx |a|u_x\)
\(z = ax \pm by\) \(\sigma_z = \sqrt{a^2\sigma_x^2+b^2\sigma_y^2}\) \(u_z \approx au_x + bu_y\)
\(z = xy, z=\dfrac{x}{y}\) \(\sigma_z = |z| \sqrt{ \left( \dfrac{\sigma_x}{\vphantom{y}x} \right)^2+ \left( \dfrac{\sigma_y}{y} \right)^2}\) \(u_z \approx |z| \left( \dfrac{u_x}{|x|} + \dfrac{u_y}{|y|}\right)\)
\(z = ax^b\) \(\sigma_z = \left|z \dfrac{b\sigma_x}{x}\right|\) \(u_z \approx \left|z \dfrac{bu_x}{x}\right|\)
\(z = a \log_n(bx)\) \(\sigma_z = \left|a \dfrac{\sigma_x}{x\ln(n)}\right|\) —
\(z = a^{bx}\) \(\sigma_z = \sigma_z = |z b \ln(a)\sigma_x|\) —
\(z = a \sin(bx)\) \(\sigma_z=|ab\cos(bx)\sigma_x|\) —
\(z = a \cos(bx)\) \(\sigma_z=|ab\sin(bx)\sigma_x|\) —
\(z = a \tan(bx)\) \(\sigma_z = |ab\sec^2(bx)\sigma_x|\) —

Approximations of uncertainty propagation []

The right column of Tab. 4 also shows some approximations for the determination of the uncertainty (approximations taken from [1, p.275–277]).

For the speed measurement from before, the uncertainty can be calculated as follows: $$\displaylines{u_z =& |z| \left( \dfrac{u_x}{|x|} + \dfrac{u_y}{|y|}\right)\\ =& |v| \left(\dfrac{u_s}{|s|}+\dfrac{u_t}{|t|}\right)\\ =& 2.50\text{ m/s}\left(\dfrac{0.50\text{ m}}{50.00\text{ m}}+\dfrac{1.0\text{ s}}{20.0\text{ s}}\right)\\ =& 0.15\text{ m/s}.}$$ Resulting in a measurement result of v = (2.50 ± 0.15) m/s. Which is a slight overestimation of the uncertainty.

For some functions, no standard approximations exist. There is, however, an even easier procedure to approximate the uncertainty propagation that can always be used. In these cases, the uncertainty of the calculated quantity is determined by combining the values of the quantities and their respective uncertainties to calculate the smallest and largest possible outcome, this range is then used as the uncertainty [1, p.277–278].

For the speed measurement, this would be the following procedure. The speed is calculated as: \(v = \dfrac{s}{t} = \dfrac{50.00\text{ m}}{20.0\text{ s}} = 2.50\text{ m/s}\).

The largest possible value would be obtained by combining the measurements and their uncertainties as follows: $$\displaylines{ v =&\dfrac{s+u_s}{t-u_t}\\ =& \dfrac{50.5\text{ m}}{19\text{ s}}\\ =& 2.66\text{ m/s}.}$$ Für den kleinstmöglichen Wert: $$\displaylines{v =& \dfrac{s-u_s}{t+u_t}\\ =& \dfrac{49.5\text{ m}}{21\text{ s}}\\ =& 2.36\text{ m/s}.}$$ The uncertainty is now determined with the procedure of the maximum deviation (see Eq. (9)): the two deviations are: |2.50 m/s – 2.66 m/s| = 0.16 m/s und |2.50 m/s – 2.36 m/s| = 0.14 m/s. The largest of the two deviations is, conservatively, chosen as the uncertainty. The result is now v = (2.50 ± 0.16) m/s, which is slightly larger than the approximation from before.

Online uncertainty calculator []

Although there are indications that calculating the uncertainty by hand is advantageous for students [2]. There are instances where this calculational routine takes away the focus of the data analysis and, hence, the interpretation of the result [3]. Many uncertainty calculators can be found online, one that works quite intuitively can be found here: https://nicoco007.github.io/Propagation-of-Uncertainty-Calculator.

Literature