At university, we have been taught to use the following formula to calculate uncertainties, when the uncertainties are independent:

$(\Delta Q)^2 = (\frac{\delta Q}{\delta a})^2(\Delta a)^2+(\frac{\delta Q}{\delta b})^2(\Delta b)^2+...$

where $a, b$ etc are the variables in the formula for $Q$.

However, I am finding an alternative formula given in some physics laboratory manuals. Granted it does not give the error in $Q$ but rather the fractional error. However when I use this formula to calculate the uncertainty in a particular formula, the result I get is different from if I were to use the previous propagation of errors formula. Here is the alternative formula:

If $Q=\frac{a^{\alpha}b^{\beta}...}{p^{x}q^{y}...}$ then

$\frac{\Delta Q}{Q}=\alpha \frac{\Delta a}{a}+\beta \frac{\Delta b}{b}+...+x\frac{\Delta p}{p}+y\frac{\Delta q}{q}+...$.

It is obvious that these formulae don't always give the same uncertainty. Being a beginner in physics and uncertainties, the only explanation I can think of is that the latter formula gives the maximum possible uncertainty, whereas the first one is more refined and closer to the actual uncertainty?

What I find a bit strange though is that even our physics lab manual at university uses the latter formula. However whenever this occurs in the manual, our lab teacher tells us it is wrong and to use the first formula instead to calculate the uncertainty in the result!

Edit: Here is an example of the latter formula being given in a physics laboratory manual (C B Daish and D H Fender, Experimental Physics)

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  • $\begingroup$ "given in some physics laboratory manuals." - do you have any published reference that gives this formula? $\endgroup$
    – ProfRob
    Oct 23, 2022 at 19:07
  • $\begingroup$ Related: Is propagation of uncertainties linear? and links therein. $\endgroup$
    – Qmechanic
    Oct 23, 2022 at 19:12
  • $\begingroup$ @ProfRob Please see the edit. $\endgroup$
    – Andrew Tom
    Oct 23, 2022 at 19:15
  • $\begingroup$ @Qmechanic Did you close the question? Sorry I am not advanced enough to understand the answers given in the related thread. Is there any chance it can be reopened? (I have added pictures of the lab manual I am using to clarify.) Thanks $\endgroup$
    – Andrew Tom
    Oct 23, 2022 at 19:17
  • 2
    $\begingroup$ The answer to the question is included in the text you have included towards the bottom of p.285. one formula is the worst possible error the other is the most likely error. $\endgroup$
    – ProfRob
    Oct 23, 2022 at 19:26


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