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Can someone explain to me when do I use this formula

...and when do I use this one?

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  • $\begingroup$ Could you provide some context? Where have you seen these equations? $\endgroup$
    – ChrisM
    Mar 26, 2015 at 12:17
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/170886/2451 , physics.stackexchange.com/q/59628/2451 and links therein. $\endgroup$
    – Qmechanic
    Mar 26, 2015 at 12:19
  • $\begingroup$ that is exactly my problem: there is no major context. The lecture notes my professor has uploaded just throws these formulas in but doesn't explain it $\endgroup$
    – Christian
    Mar 26, 2015 at 12:21
  • $\begingroup$ The only time I've seen the second equation (and I have seen it) is in teacher's lecture notes, when the teacher doesn't want to address the issue of error propagation, and/or doesn't think the students know how to use a calculator. See @pyramids answer. $\endgroup$
    – garyp
    Mar 26, 2015 at 13:17

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You use the first formula you gave when you have (entirely) uncorrelated errors where the standard variances (the squares of the standard deviances) add. Gaussian distributions of errors are usually assumed.

You might use the second formula if your errors are perfectly correlated, but even then only as a worst-case measure (if you know the correlation, you in some cases a minus sign would be technically appropriate when one error acts to partially cancels the other). In practice, it really only makes sense if you use it as an upper bound estimate where you want to cover for the possibility that you have the worst possible correlation between errors. Note that this will mess up your statistics: If your worst-case error estimate does not overestimate your errors, you either have the worst case or you have made an error!

Using such worst-case estimate may be a good idea if you cannot justify that errors are not correlated. But it is important to realize that now you are making very different statistical statements to usual error estimation. The proper way forward, if you want actual error estimates, would be to determine the correlation, which leads you to error ellipses.

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  • $\begingroup$ Note that in certain engineering contexts, the upper bound is regularly used in determining the tolerances that should be demanded. Not exactly a physics answer, but one of some interest to a large subset of people who study physics. $\endgroup$ Mar 26, 2015 at 19:17
  • $\begingroup$ Then that is one case where one should use the second formula! Thank you! $\endgroup$
    – user73762
    Mar 26, 2015 at 22:22

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