What Relative Uncertainty Means and How to Find It

Relative uncertainty or relative error is a measure of the uncertainty of 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:

relative uncertainty = absolute error / known value

Relative uncertainty is often represented using the lowercase Greek letter delta, δ. While absolute error carries the same units as the measurement, relative error has no units or else is expressed as a percent.

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

Three weights are measured at 1.05 g, 1.00 g, and 0.95 g. The absolute error is ± 0.05 g. The relative error is 0.05 g/1.00 g = 0.05 or 5%.

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

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

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 absolute uncertainty in the measurement is:

1.55 hours +/- 14%