Let's consider the case of a measurement $x$ and its correspondent error $\Delta x$ (which are both always positive).
The relative error is defined as:
$$E = \frac{\Delta x}{x}$$
The relative error has many useful application in error propagation and so on, and it is frequently used to determine the quality of a measurement. However, when the value of the quantity to measure tends to zero, the relative error can become very large, although the measurement itself might be very reliable, because for example the quantity to measure is zero within the instrumentation error.
What would you recommend to use under such situations?
I have seen: How to calculate relative error when the true value is zero? but since I do not have access to a measurement and its true value I am not sure how to apply the discussion there to this case, and I could not find any reference in the literature.
This other question: How to calculate relative error? is somewhat related but I do not think it really applies.
I had considered using this:
$$\tilde{E} = \frac{x \cdot \Delta x}{x^2 + \Delta x^2}$$
which behaves numerically the way I would expect and it is a dimensionless quantity.
Alternatively, I had considered using the Relative Percent Difference (RPD), defined in terms of $d_1(a, b)$:
$$d_1(a, b) = 2 \frac{a - b}{|a| + |b|}$$
(where $a$ is a measurement and $b$ its true value), or, more precisely the absolute Relative Percent Difference (aRPD):
$$|d_1(a, b)| = 2 \frac{|a - b|}{|a| + |b|}$$
Using the following substitutions:
- $|a| = x$,
- $|b| = x + \Delta x$
- $|a - b| = \Delta x$
I would obtain:
$$ \tilde{d_1}(x, \Delta x) = \frac{2 \Delta x}{2 x + \Delta x} = \frac{\Delta x}{x + \frac{1}{2}\Delta x} $$
However, I cannot find any reference actually using any of these or at least discussing these topics for my case. Any answer substantiated with a peer-reviewed paper is greatly appreciated.