The Relative Uncertainty Formula and How to Calculate It

The relative uncertainty or relative error formula is used to calculate the uncertainty of a measurement compared to the size of the measurement. It is calculated as:

• relative uncertainty = absolute error / measured value

If a measurement is taken with respect to a standard or known value, calculate relative uncertainty as follows:

• relative uncertainty = absolute error / known value

Absolute error is the range of measurements in which the true value of a measurement likely lies. While absolute error carries the same units as the measurement, relative error has no units or else is expressed as a percent. Relative uncertainty is often represented using the lowercase Greek letter delta (δ).

The importance of relative uncertainty is that it puts error in measurements into perspective. For example, an error of +/- 0.5 centimeters may be relatively large when measuring the length of your hand, but very small when measuring the size of a room.

Examples of Relative Uncertainty Calculations

Example 1

Three 1.0 gram weights are measured at 1.05 grams, 1.00 grams, and 0.95 grams.

• The absolute error is ± 0.05 grams.
• The relative error (δ) of your measurement is 0.05 g/1.00 g = 0.05, or 5%.

Example 2

A chemist measured the time required for a chemical reaction and found the value to be 155 +/- 0.21 hours. The first step is to find the absolute uncertainty:

• absolute uncertainty = 0.21 hours
• relative uncertainty = Δt / t = 0.21 hours / 1.55 hours = 0.135

Example 3

The value 0.135 has too many significant digits, so it is shortened (rounded) to 0.14, which can be written as 14% (by multiplying the value times 100).

The relative uncertainty (δ) in the measurement for the reaction time is:

• 1.55 hours +/- 14%

Sources

•  Golub, Gene, and Charles F. Van Loan. "Matrix Computations – Third Edition." Baltimore: The Johns Hopkins University Press, 1996.
• Helfrick, Albert D., and William David Cooper. "Modern Electronic Instrumentation and Measurement Techniques." Prentice Hall, 1989.