# How many significant figures should one give in stating an experimental uncertainty? [duplicate]

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

1. 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].

2. 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].

3. 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].

## marked as duplicate by sammy gerbil, Emilio Pisanty, Jon Custer, Ben Crowell, Kyle KanosSep 22 '17 at 10:03

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)!

• Or, in other words, if you just look at a sheet of log-scale graph paper, the numbers with leading digit 1 take up an inordinate amount of space, a.k.a. Benford's law. – Emilio Pisanty Sep 21 '17 at 17:39
• This correct answer would have been preferable to the incorrect accepted answer to the question that this one duplicates: physics.stackexchange.com/questions/319902/… – Ben Crowell Sep 21 '17 at 22:41

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.

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.

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.

• CODATA is exceptional. It is their business to get a good estimate of the experimental errors. But if you are not in metrology, this would be rare. – Pieter Sep 21 '17 at 20:44
• Why are they exceptional with regard to errors and reporting them? I don't see a difference in determining the error in a vanilla experiment and an experiment determining one of nature's constants. Any experiment should apply the same rigor as is applied there. If you read papers on arXiv.org, they apply the same rigor. They are exceptional in the sense that all of these experiments determine a lot of significant figures and that CODATA determines a consistent set of these constants. – Grimaldi Sep 26 '17 at 15:37
• How large is the error when you measure a length using a meter stick? Maybe 0.3 mm, maybe 0.4 mm. It is not worth the effort to determine this with two significant digits. – Pieter Sep 26 '17 at 16:24
• Yeah, I agree. But very often in an experiment, you calculate errors from error propagation. Why not give 2 digits then? – Grimaldi Sep 28 '17 at 18:35