What is the standard way of reporting/interpreting measurement uncertainty in scientific publications?

Question

As an undergraduate student I was often told in lab classes to report every single measured or calculated quantity with its associated uncertainty in lab classes, in the form $(10.1 \pm 0.1 )~\mathrm{g}$ for instance, where 0.1 is the standard deviation or estimated uncertainty. I'm starting to find, however, that this practice is seldom followed in scientific publications.

When reporting uncertainty in a scientific publication, is it good practice to report every measurement in the form $(300.1 \pm 0.1 )~\mathrm{K}$ ? I sometimes see this done in the body of scientific papers, but usually never in tables/lists of data. This worries me because, in my understanding, all measurements have an uncertainty associated with them, so I find these data hard to interpret unambiguously.

Is it usually assumed that numbers written without an explicit uncertainty are accurate to the last expressed digit? (i.e. that $10.5~\mathrm{g}$ implies $(10.5 \pm 0.1)~\mathrm{g}$ ?) sometimes this seems to be the case, but doesn't this negate the difference between $ \pm 0.1$ and $\pm 0.5$ for instance? Is this difference usually considered unimportant?

As a side question: Regardless of the usual practice in academic writing, do you consider that demanding every single number to be reported with an associated uncertainty is a good idea if teaching a lab class?

Answer 1

In my experience, this varies greatly with your field of study.

In organic chemistry, I have yet to experience proper uncertainty reporting in journal articles or lab reports with the notable exception of physical organic chemistry. Once in my undergraduate years, I noted while writing a report that $130~\mathrm{mmol}$ where actually $1.3 \times 10^2~\mathrm{mmol}$ and wrote it that way. Okay, I should have written $0.13~\mathrm{mol}$ in a more correct notation to signify the significant digits. But the way the TA marked the report clearly told me ‘I don’t give a damn about correct error margins or significant digits, just God damn write $130~\mathrm{mmol}$!’ Lesson learnt from that anecdote: synthetic organic chemistry does not care about minor details such as significant figures or uncertainty.

The closer you get to physical chemistry, the more your measurements actually represent data values to be cited in a table and thus the more exact the values determined become, the more people care about reporting error margins and uncertainty. And funnily enough, in biology this is also done much more regularly. I suspect that there are different underlying reasons: One side (physical chemistry/physics) does not want the negative outcome of ‘you pretended your results are more accurate than they are!’, while the other side, biology, knows that it is intrinsically dealing with very error-prone systems that require large repetition counts which can then be analysed statistically.

Although synthetic organic chemistry practically also deals with rather error prone systems, either the idea is that there are too many uncontrollable variables or the idea is $n=1$ (always!); in any case, error bars or uncertainty are deemed superfluous.