Undergrad Error Propagation in Measurements

[erobz]

I was imagining trying to construct a rectangle of area $A = xy$

If we give a symmetric error to each dimension $\epsilon_x, \epsilon_y$

$$ A + \Delta A = ( x \pm \epsilon_x )( y \pm \epsilon_y )$$

Expanding the RHS and dividing through by $A$

$$ \frac{\Delta A}{ A} = \pm \frac{\epsilon_x}{x} \pm \frac{\epsilon_y}{y} (\pm)(\pm) \frac{\epsilon_x \epsilon_y}{xy}$$

The first two terms are symmetrical error, but without neglecting the third higher order term should it have a negative bias since $\frac{2}{3}$ of sign ( $\pm$) parings result in a negative third term, and $\frac{1}{3}$ pairings result in a positive third term?

My terminology is probably improper.

 

[erobz]

Never mind! I think I did that wrong... There are only 4 pairings. for some reason I had $C(4,2)$ in my head.

 

[Svein]

The standard term for the error is the relative variation (the square of the standard deviation divided by the measurement). If you have several possible error sources, add the relative variations.

 

[brainpushups]

Three options to consider:
1) Simply evaluate your function using measurements that result in the highest and lowest possible values, in this case calculate area given by the maximum probable measurements and the minimum probable measurements. The difference in these values will be roughly symmetric about the best estimate provided the uncertainties are relatively small. Since the high and low will be roughly symmetric from the best estimate you can get away with just finding either the highest or lowest for

2) What @Svein said. If the relative errors are small you can add them together to find the relative error of the product and then easily find the absolute error. It will match with method 1 when rounded sensibly using standard significant digit 'rules.'

3) Add the relative errors in quadrature (square them, add, then square root). This is likely a more accurate estimate of the uncertainty in the product provided that the uncertainties are not covariant. This method comes from the calculus of probabilities. See Taylor's An Introduction to Error Analysis for an excellent introductory text on this.