How to calculate measurement error by using other quantities?

[Lotto]

TL;DR

When I measure some quantities and then want to calculate an another quantity using these ones (I have determined their measurement error) how to do it in general?

Let's say I have $F=mg \tan \alpha$ and want to calculate $F$. I know $m=(1.0 \pm 0.5)\,\mathrm{kg}$ and $\alpha=(20.5 \pm 0.5)°$. How to calculate $F=( 3.7 \pm ?)\,\mathrm N$? What is the general method of determining a measurement error in these cases?

 

[BvU]

Hi,

With a 50% uncertainty in $m$, all other errors are unimportant.

For other cases, check here

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[Lotto]

Hi,

With a 50% uncertainty in $m$, all other errors are unimportant.

For other cases, check here

$\$

And can I determine the total measurement error as $\sigma_F=\sqrt {\left(\frac Fm {\sigma_m}\right)^2+\left(\frac {F}{\alpha} {\sigma_{\alpha}}\right)^2}$? Or $\sigma_F=\sqrt {\left(\frac Fm {\sigma_m}\right)^2+\left(\frac {F}{\sin \alpha} {\sigma_{\sin \alpha}}\right)^2}$?

 

[BvU]

Neither of the two is correct. What is the formula to use ?
And what is the derivatve of $\tan\alpha$ ?

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[Lotto]

Neither of the two is correct. What is the formula to use ?
And what is the derivatve of $\tan\alpha$ ?

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My fault, I meant $\tan \alpha$ in the second equation. And I think I could do it this way, since we did at school this for instance: $\sigma_{\rho}=\sqrt {\left(\frac {\rho}{m} {\sigma_m}\right)^2+\left(\frac {\rho}{V} {\sigma_V}\right)^2}$. I know that the general formula contains partial derivatives, but we haven't done them, so I try to do it like at school.

 

[BvU]

I know that the general formula contains partial derivatives, but we haven't done them, so I try to do it like at school.

Then it's a bit awkward indeed! Kudos for trying.
What I meant was that the general formula is indeed $$\sigma_f^2 = \left (\partial f\over \partial m\right )^2 \sigma_m^2 + \left (\partial f\over \partial \alpha\right )^2\sigma_\alpha^2$$ and the derivative of $\tan\alpha$ is ${1\over \cos^2\alpha}\$ as in the examples.

with your values $\sigma_m = 0.5$ kg and $\sigma_\alpha = {0.5\over 180}\pi$, so that the term $\left (\partial f\over \partial m\right )^2 \sigma_m^2$ is 350 times bigger than the term $\left (\partial f\over \partial \alpha\right )^2\sigma_\alpha^2$ .

For small $\alpha$ we have $\tan\alpha\approx\alpha$ which yields your first expression in post #3.
(the second expression can be seen to be incorrect: it blows up for $\alpha\downarrow 0$ )

Either way (thorough or approximation) is no better than completely ignoring the 2.5% $\sigma$ in $\alpha$ opposite the 50% $\sigma$ in $m$.

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[Claude Bile]

The recommended procedure is outlined in the BIPM (International Bureau of Weights and Measures) "Guide to Uncertainty and Measurement". In very general terms, one would typically linearise the function about the expectation and compute the uncertainty/variance using the appropriate linear transformation of variable.

In cases where the uncertainty is large relative to the principal value, you will need to use a second order (or higher) expansion and things get a bit more complicated.

Also, you need to be mindful of correlations that might be present in your random variables. The wiki page on "variance" gives a good summary on how to separate correlated variables.