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I'm currently a high schooler, and I'm writing my first scientific paper. The result is fairly simple, and it is nothing too special, but I see it as a nice way to prepare myself for the academic world.

The paper involves a lot of calculations and I'm actually wondering how I should approach rounding to make it as scientific as possible.

My own guess would be scientific rounding in the intermediate steps, but using the 'correct' value in further calculations. Is this correct?

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    $\begingroup$ It is correct, but you should ask yourself why you want to round on intermediate steps in the first place. If some such result isn't for some reason interesting on its own right, then there's not really much point in presenting its numerical value, it just distracts from the "real" results. And provided you properly laid out the calculation formulas and initial values, then anybody will be able to get those intermediates even if you didn't write them out. $\endgroup$ – leftaroundabout Oct 3 '13 at 21:38
  • $\begingroup$ @leftaroundabout I guess you could say that I'm "dumbing it down" for my teachers to read it fluently. I know this wouldn't be the case for a serious paper in a journal, but yeah.. $\endgroup$ – OmnipresentAbsence Oct 3 '13 at 21:40
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    $\begingroup$ In short: you do not round. You state your answer with the number of significant digits your setup and analysis is capable of producing. $\endgroup$ – Alexander Oct 3 '13 at 22:10
  • $\begingroup$ Fantastic effort - all the best in your publishing. Do you ken a scientist capable of error analysis whom you can trust to take a look over your work? Very few papers are published in one name: I should think many scientists would be happy to look over your work and all they would ask is to be acknowledged in an "acknowledgement" section at the end of your paper (i.e. not as a second author). Before you do that, do you have a blog where you could document an outline of what you are doing and where your paper is heading? This keeps others who might look over your work honest - .... $\endgroup$ – Selene Routley Oct 3 '13 at 23:55
  • $\begingroup$ ... I know that sounds cynical and you hopefully can trust your guts and intuition enough that you can grasp the character of those you approach to help you so the blog thing hopefully never will be needed. But, just to be safe! Academia is pretty stressed in these economic "rationalist" times and in my home land I've seen some pretty shabby behaviour from academics begotten by the poisonous competitive climate that goes with endless cuts and restructures. Mind you, my own culture is not known for its honesty - but talk to someone you trust. $\endgroup$ – Selene Routley Oct 3 '13 at 23:57
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Preferably, you should work symbolically for as long as possible and postpone filling in any specific values until you have a final equation. Of course you don't have to overdo it either, but generally you can assess quite easily when it feels natural to write down a (significant) intermediary result.

Concerning the notation of such a result, error propagation would yield the uncertainty on the obtained value. For intermediary results, we usually round (upward) the error to 1 significant figure and then give the value to the same accuracy, accompanied by $\pm$ the rounded error.

So for example, say you find a value $4.358172\,\text{m/s}$ for some speed you calculated from several measurements. And say the uncertainty you find by application of error propagation is $0.01267\,\text{m/s}$. The rounded error for this intermediary result is $0.02\,\text{m/s}$ (remember, we always round the error upward) and therefore we denote the result as $(4.36\pm0.02)\,\text{m/s}$.

For final results, we round the error (upward) to 2 significant figures, but the rest stays the same. So if our example above were a final result, we would denote it as $(4.358\pm0.013)\,\text{m/s}$.

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  • $\begingroup$ Would you really always round uncertainties up? I can imagine that practice is meant to fight off the tendency for people to overestimate how "good" their precision is, but representing $1\sigma$ errors as $0.02$ rather than $0.012$ can be just as misleading, and in many cases would work in the author's favor ("our results are in statistical agreement with those of our previous paper..."). $\endgroup$ – user10851 Oct 4 '13 at 2:08
  • $\begingroup$ @ChrisWhite True, I gave the sort of standard method in my answer, because the OP is a high schooler. Then again, it is up to the writers of the paper to not present their results in a misleading way and to avoid conclusions that are inconsistent with the actual error. A graphical representation of the results to a higher precision is useful in this respect as well. I'd also say if your conclusion depends on the error rounding, you most likely want and need better measurements anyway. $\endgroup$ – Wouter Oct 4 '13 at 8:02

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