Absolute error and relative error

Besides discount and interest, there is a third use of the percentage that we shall use continually in physics and statistics: the percentage as a measure of variation with respect to a reference. It is the way we judge the quality of a measurement.

Definition — Absolute, relative and percentage error

Given a measured quantity $x_{\text{mis}}$ and a reference value $x_{\text{rif}}$ (exact, expected, true): $\text{absolute error} = |x_{\text{mis}}-x_{\text{rif}}|, \quad \text{relative error} = \frac{|x_{\text{mis}}-x_{\text{rif}}|}{|x_{\text{rif}}|},$ and expressed as a percentage it is the percentage error $= \text{rel. err.}\cdot 100\%$.

The absolute error has the same units as the quantity; the relative error is a pure number, and multiplied by $100$ it becomes the percentage error. It is the latter that tells us whether a measurement is “good”, because it relates the error to the scale of the phenomenon.

Links

Topics: Percentages
Concepts: Absolute error · Percentage error · Relative error
Methods: Relative percentage error
Skills: Using formulae