I have two constants, $A$ and $B$, with associated uncertainties $\sigma_A$ and $\sigma_B$, from observational errors, for example. I need to perform calculations with these constants, by for example making $f = A/B$. Therefore, I need to propagate the uncertainties from $A$ and $B$ to my function $f$.

Wikipedia provides the following recipe for a similar case, but when $A$ and $B$ are variables, not constants, and the errors are assumed to be their standard deviations.

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How do I compute the standard deviation $\sigma_{AB}$, when $A$ and $B$ are constants? Should I just make $\sigma_A$ times $\sigma_B$? Or moreover, how do I propagate the uncertainties of two constants to their ratio?

  • $\begingroup$ Are Ai and Bi dependent or independent ? $\endgroup$
    – Rodolphe
    Jul 10 '20 at 7:09
  • $\begingroup$ Because if non independent, then the ratio of mean might be different than the mean of ratios. Same reasoning for uncertainty. $\endgroup$
    – Rodolphe
    Jul 10 '20 at 7:17
  • $\begingroup$ They are independent, two measured quantities, not variables. Could be for example mass and speed of something. $\endgroup$
    – ouranos
    Jul 10 '20 at 10:17

Alright, I learned that, when $A$ and $B$ are constants with uncertainties $\sigma_A$ and $\sigma_B$, one just needs to add their relative uncertainties in quadrature, to propagate their uncertainties to a function $f(A, B)$, which may as well be $f(A,B)=\frac{A}{B}$.

$$ \frac{\sigma_f}{f} = \sqrt{\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2} $$


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