Estimation of error

Suppose a physical quantity is measured repeatedly, giving a set of $n$ readings $a_1, a_2, a_3, \ldots, a_n$. The arithmetic mean of these readings is

and this arithmetic mean is taken as the most probable value of the quantity being measured — it is, in a well-defined statistical sense, our best single estimate given the scatter in the repeated readings.

The magnitude of the difference between the mean value and any one individual reading is called the absolute error in that particular observation. So for the reading $a_1$, the absolute error is $\Delta a_1 = |a_{mean} - a_1|$, and similarly for $a_2$ it is $\Delta a_2 = |a_{mean} - a_2|$, and so on up to $\Delta a_n = |a_{mean} - a_n|$ for the $n$-th reading.

Averaging these individual absolute errors gives the mean absolute error in the measurement:

The measured value of the quantity $a$ can then be reported as $a = a_{mean} \pm \Delta a_{mean}$, which is understood to mean that the true value of $a$ most likely lies somewhere between $a_{mean} - \Delta a_{mean}$ and $a_{mean} + \Delta a_{mean}$.

The ratio of the mean absolute error to the arithmetic mean value is called the relative error:

and when the relative error is expressed as a percentage (i.e. multiplied by 100), it is called the percentage error:

$\text{Percentage error} = \dfrac{\Delta a_{mean}}{a_{mean}} \times 100 \qquad \text{--- (1.6)}$ …