What's the difference between average absolute error and relative error?

Question

I am quite confused by both these terms. I would like to know what's the exact difference between both these terms and which one is more accurate.

Answer 1

The absolute error can be measured using this formula: $$\varepsilon_a=\frac{x_{max}-x_{min}}{2}$$ That is the difference between the highest value and the lowest value that you get after some measurements. The Relative error is: $$\varepsilon_r=\frac{\varepsilon_a}{\bar{x}}$$ where $\bar{x}$ is the average of all your measurements. There is also there is the percent error (relative) that equals to: $$\varepsilon_r\cdot100$$

Answer 2

The absolute error in any one trial (with name or index $k$) is

$$ \varepsilon_a^k = \widetilde x_k - x_k, $$

where $x_k$ is the true value of the quantity under consideration in trial $k$, and $\widetilde x_k$ is the value which is inferred of that quantity in trial $k$, with the techniques and observational data available.

The average of the absolue errors over a set of trials, $k = 1 \, ... \, n$, is accordingly

$$ \varepsilon_a = \frac{1}{n} \, \sum_{k = 1}^n \varepsilon_a^k = \frac{1}{n} \left( \sum_{k = 1}^n \widetilde x_k \right) - \frac{1}{n} \left( \sum_{k = 1}^n x_k \right).$$

If the quantity under consideration happened to have one particular always equal true value $x$ in all trials of this set, i.e. if for all $k$ holds $x_k = x$, then

$$ \varepsilon_a = \frac{1}{n} \left( \sum_{k = 1}^n \widetilde x_k \right) - x.$$

Concerning relative error, there are various definitions to consider.
One, apparently common definition of "relative error" is setting in any one trial

$$ \varepsilon_r^k = \frac{\varepsilon_a^k}{x_k} = \frac{\widetilde x_k - x_k}{x_k}, $$

and correspondingly in a set of trials with equal true value $x$:

$$ \varepsilon_r = \frac{\varepsilon_a}{x} = \frac{1}{n} \left( \sum_{k = 1}^n \frac{\widetilde x_k}{x} \right) - 1.$$

The main drawback of this definition is its apparent failure (divergence) if the true value $x$ over the set of trials under consideration happens to be $0$; or when referring only to one trial, if the true value $x_k$ in that trial happens to be $0$.

Another, arguably more robust definition of relative error is based on noting that the inferred values $\widetilde x_k$ are necessarily elements of an entire range of values which are "technically admissible"; such as the actual practically usable extent on a dial indicator of a measuring instrument, or the actual range of an engineer's scale.

This range, which is due to the technique being used to obtain the inferred values $\widetilde x_k$, should have some non-zero extend (or in other words: it should have more than one element) because otherwise the result of applying that particular technique would be predetermined and known in advance; counter to the meaning of "measuring" and "finding out in dependence of observational data gathered".

Writing this range therefore as $\widetilde x_{max} - \widetilde x_{min}$, the relative error can be defined for any one trial as

$$ \varepsilon_r^k = \frac{\varepsilon_a^k}{\widetilde x_{max} - \widetilde x_{min}} = \frac{\widetilde x_k - x_k}{\widetilde x_{max} - \widetilde x_{min}}, $$

and correspondingly in a set of trials with equal true value $x$:

$$ \varepsilon_r = \frac{\varepsilon_a}{\widetilde x_{max} - \widetilde x_{min}}.$$