Relative uncertainties

[josephwright]

Split out from #652. Some applications benefit from relative uncertainty, where the first value is factored out to create the form
val \times (1 \pm rel_uncertainty), or with exponentials
val.xxxx \times (1 \pm rel_uncertainty) \times 10^{exponent}.
If uncertainty-mode had another option relative that formatted in this manner, that would be nice!

[Gerald-Meyers]

Perhaps it is worth noting that in literature relative uncertainties are expressed as value \times base^{exponent} \pm x\% where x=uncertainty/value. Having the option for both, including value \times (1+x) may be beneficial, perhaps only the simpler option.

[josephwright]

Can you point to some examples in the published literature? I've never come across this format, so I'd like to see it 'in the wild'.

[Gerald-Meyers]

I'll be honest, outside of a few specific examples that I saw recently, I have only ever seen the standard version. It's fairly niche and not common. I'll look for an example an try to cite it here soon.

[josephwright]

Cool - it's broadly 'policy' to ask for examples as I need to avoid 'well I like it' features

[Gerald-Meyers]

Here is one standard reference: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387884/pdf/cbr_33_2_49.pdf
I had found another a few days ago that I had run across as well. I'll link it when I can get the PDF off of my desktop.

7. Depending on the exact form of the functional relationship y = f (x1, x2, x3, ... , x n), it may be more appropriate to use the relative uncertainties (that is, u(y)/y and Rules for Calculating Uncertainty of Measurement u(xi)/xi) instead of the absolute uncertainty values (u(y) and u(xi)). As a general rule, use relative uncertainties for functions which include terms with multiplication and/or division but not addition or subtraction. Examples of expressions that involve relative uncertainties are given in Table 2. Relative uncertainty is also referred to as coefficient of variation (CV)

I do not claim that there is a standard format for presenting relative uncertainty, so implementation and presentations certainly will vary. I have only come across this in the wild a few times. I mostly have seen relative uncertainty in artificial situations, and a few population statistics. Pick a topic, add the quoted phrase "relative uncertainty" and you can find a few articles that use it in the abstract for different purposes.

[Gerald-Meyers]

Here is the reference that I mentioned: https://www.eurachem.org/index.php/publications/guides/quam
Quantifying Uncertainty in Analytical Measurement, 3rd Edition (2012) from EuraChem.

pg 27:

Rule 2
For models involving only a product or quotient, e.g. y=(p × q × r ×...) or y= p / (q × r ×...), the combined standard uncertainty uc(y) is given by [...latex...] where (u(p)/p) etc. are the uncertainties in the parameters, expressed as relative standard deviations.
NOTE: Subtraction is treated in the same manner as addition, and division in the same way as multiplication.

pg 35 has a table of examples of relative uncertainties

Relative Standard Uncertainty makes a large appearance throughout the document, but the $val \pm rel$% is not present so far as I can tell. I've seen it else where.