Undergrad Uncertainty in measurements of the thickness of a pile of pellets

[nuclearsneke]

TL;DR

TL;DNR: how to measure an uncertainty for the thickness of a pile of pellets?

Howdie!

We have been playing around with melting and molding HDPE pellets recently. After that, we measured their diameter and thickiness 5 times each to get an uncertainty. In our experiments we put one pellet between gamma-source and detector and measure its attenuation. After that we place the next pellet on top of the previous and carry out the same measurements. And so on.

My question is:
If each pellet had thickness d with uncertainty delta_d, then how would I calculate the uncertainty for n pellets?

Would it be a square root of the sum of uncertainty squares or sth trickier? Thank you in advance!

 

[PeroK]

That depends how you define "uncertainty " and what you assume about the distribution of thickness for each pellet. In theory, you probably need to combine normal distributions.

 

[anorlunda]

How about measuring the total volume, by putting them under water and measuring the displacement. Then divide by the number of pellets N and assume spherical. Then you'll get an estimate for the average diameter with only one uncertainty to deal with instead of N independent uncertainties.

 

[Dale]

If each pellet has a thickness $d$ and if you have some function $f(d)$, then the uncertainty in $f$ is related to the uncertainty in $d$ by the propagation of errors formula: $$\sigma_f^2 = \left( \frac{\partial f}{\partial d}\right)^2 \sigma_d^2$$

So you just write down $f(d)$, take the partial derivative, square that, and multiply by the variance in $d$ and that gives you the variance in $f$.