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.

The formulas presented here assume that the errors are random and not correlated.

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=20.0mm, and the measured mass is m=200.0g.

The propagated uncertainty of the density is:

Finally, the density is:

With respect to the rule of significant figures, the result must be given as

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:

Considering the rule for significant number of figures, the result must be given as