The concept of uncertainty and the reporting of a result

A measurement without uncertainty is meaningless

Measuring always involves estimating a value with a margin of error, known as uncertainty. This uncertainty arises from the instrument (limited accuracy), the operator (reading, reflexes) or the method. Without this margin, it is impossible to know whether two measurements are compatible with one another.

Standardised notation for a result

A measurement result is written in the form:

x = x_measure ± delta x (unit)

where x_measure is the central value (often an average) and delta x is the absolute uncertainty, which is always positive and expressed in the same unit as x_measure.

Example: a length measured with a tape measure (marked in mm) gives L = 15.4 ± 0.1 cm.

Absolute uncertainty and relative uncertainty

Relative uncertainty is calculated as delta x / x_measure, often expressed as a percentage. It allows the quality of two measurements of different quantities to be compared.

Example: for L = 15.4 ± 0.1 cm, the relative uncertainty is 0.1/15.4 = 0.0065, or approximately 0.65 per cent.

Estimating a simple instrumental uncertainty

In the absence of information from the manufacturer, delta x is often taken to be equal to half the instrument’s smallest graduation. For a ruler graduated in mm, delta x = 0.5 mm = 0.05 cm.

A common pitfall

Never state an uncertainty with greater precision than the measurement itself: writing L = 15.437 ± 0.1 cm makes no sense, as the decimal places beyond one-tenth of the central value are unreliable.