Error Analysis

Once $n$ repeated readings $a_1,a_2,\ldots,a_n$ of a quantity have been taken, the arithmetic mean $a_m=\dfrac1n\sum_i a_i$ is treated as the best available estimate of the true value, and four related error measures are built from it.

i) Absolute error -- the magnitude of the difference between the mean (taken as the true value) and each individual reading:

ii) Mean absolute error -- the arithmetic mean of all the individual absolute errors:

The true value is then quoted as lying between $a_m+\Delta a_m$ and $a_m-\Delta a_m$, written $a=a_m\pm\Delta a_m$.

iii) Relative (fractional) error -- the ratio of the mean absolute error to the mean value itself:

It expresses how large the absolute error is compared to the size of the quantity being measured -- e.g. a speedometer reading 60 km h$^{-1}$ when the true speed is 62 km h$^{-1}$ has absolute error $2$ km h$^{-1}$ and relative error $2/60=0.033$.

iv) Percentage error -- the relative error expressed as a percentage:

A percentage error close to zero means the measurement is close to the target/true value -- good and acceptable. It is important to always understand why an error occurred: is it from the limits of the equipment, or from a mistake in how the experiment was carried out? …