How many significant figures should one give in stating an experimental uncertainty? How many significant figures should one give in stating a experimental uncertainty? For this purpose I found three rules:


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Experimental uncertainties should be always stated to 1 significant figure.

For example: $3.45 \pm 0.015$ should be $3.45 \pm 0.02$ [doc1].

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The number of significant figures in the experimental uncertainty is limited to one or (if the uncertainty starts with a one, e.g., ± 0.15) to two significant figures.

For example:  $3.45 \pm 0.015$ should be $3.45 \pm 0.015$ [doc2].

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One significant figure should be used to report the uncertainty or occasionally two, especially if the second figure is a five.

For example: $3.45 \pm 0.035$ should be $3.45 \pm 0.035$ [doc3].
 A: The figures reported should reflect your best understanding of the situation without overstating the precision of that understanding. 
Toward that end two figures may make sense some of the time—especially if the leading figure is a small digit, say 1–3. 

Aside: the second rule you found is a special case of the so-called "sliderule convention" in which leading ones simply were not counted as significant in the first place, because a reporting value of of $1 \times 10^\text{whatever}$ would otherwise cover a range from $5 \times 10^{\text{whatever}-1}$ to $1.5 \times 10^\text{whatever}$ which is an astounding factor of three (the situation is less extreme when more value are reported bu the range covered by value with a leading one remains much larger than those associated with larger leading digits)!
A: The sensible rationale for stating uncertainties is given in dmckee's anwer. As for your question (what should you do), the answer is: within those bounds, it's a matter of convention, tradition and choice.
It varies between different fields and subfields and between different laboratories and professors. So if you're writing a future publication, check the journal guidelines; if you are a graduate student, ask your advisor; if you're taking a course, ask your professor; etc.
A: Rounding up to 1 significant figure is fine because it only makes your result more inclusive. Rounding down however is not fine as this will potentially make the truthfulness of the statement false.
There's no real harm in being more accurate than a single digit, but if you are able to then there is probably a way to pull that accuracy into your result before the uncertainty term. So essentially a good rule is to only use 2 significant digits if you believe rounding up would be inappropriate and beyond 2 digits most likely means you can get more accuracy in the base term.
A: I would follow the CODATA TGFC (Task Group Fundamental Constants) practice of giving 2 figures, see the NIST DB of constants (Planck's constant is linked).
In general one should follow the ISO guideline on reporting uncertainty in experiments - JCGM 100:2008, which interestingly enough always uses 2 digits, but seems not to contain the reasoning behind it.
