Calculating max percentage error using error propagation?

Question

I was given the actual values of the modulus of elasticity of materials, and I calculated the modulus of elasticity myself using experimental data. I was then asked to "calculate the relative maximum percentage error in determining the modulus of elasticity for each material. This is the error propagation caused due to the precision error in the measurement of $P$, $b$, $t$, $δ$, and $L$."

I usually understand percentage errors, but the error propagation part of it is throwing me off completely. I have so far used partial derivatives, and I've found the values for $\frac{\partial E}{\partial P}\Delta P$, $\frac{\partial E}{\partial b}\Delta b$, and so on. I just don't quite understand what you do with these values and how they correspond to the max percent error.

Answer 1

The usual error propagation formula says that to find the error in a calculation, you have to square all the terms like $\Delta x(\partial E/\partial x),$ sum them, and take the square root. Note that's the absolute error (it has the same units as $E$), not the percent error.

But to be honest, I think the person asking you for a maximum percentage error has probably made a conceptual mistake. The error propagation formula is for one-sigma errors, not maximum errors.

Here's a simple example: say we measure $x_1$ and $x_2$ and calculate $y=x_1+x_2.$ If we had maximum errors $\Delta x_1$ and $\Delta x_2$ on the measurements, then the maximum error in the calculated $y$ would be $\Delta x_1+\Delta x_2.$ But the propagation of errors formula gives you $\sqrt{\Delta x_1^2+\Delta x_2^2},$ which is smaller.