Let’s calculate the uncertainty propagation for example on the density of a cube.

By definition, the density of the cube is expressed as follow:

with the mass of the cube and its volume, which is calculated from the length of each of its sides .

There are two ways of calculating uncertainty propagation, depending on whether it is a single measurement with known accuracy for each value, or whether it is statistical.

Single measurement

General equation

The general formula for the uncertainty of a function is:

with the variables of the function and the uncertainty of the variable .

Example

Let’s suppose that the accuracy of the scale is Δm=0.2g and the accuracy of the length measurement is Δl=0.1mm. The measured length is l=20mm, and the measured mass is m=200g.

The propagated uncertainty of the density is:

Finally, the density is:

Statistical analysis

General equation

If the accuracy of the measurements is not known but we have multiple measurement datapoints of a similar measurement we can do a statistical analysis of the uncertainty propagation.

As the accuracy of each measurement is not known, we use the corrected sample standard deviation instead (applying Bessel's correction, using N−1 instead of N to yield the unbiased sample variance). The propagated uncertainty is therefore expressed as:

with the corrected sample standard deviation of the variable expressed as:

with the average of all the measurement values for the variable .

Example

In this example, a set of three cubes' density measurements is conducted:

Average of the length:

Standard deviation of the length:

Average of the mass:

Standard deviation of the mass:

Resulting uncertainty of the cube's density:

Finally, the density is: