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

AI Thread Summary
The discussion focuses on calculating the uncertainty in measurements when stacking HDPE pellets for gamma attenuation experiments. To determine the uncertainty for n pellets, one approach is to consider the propagation of errors, using the formula that relates the uncertainty in a function to the uncertainty in thickness. An alternative method suggested involves measuring the total volume displacement of the pellets in water to estimate average diameter, simplifying the uncertainty calculation. The conversation emphasizes the importance of defining "uncertainty" and the assumptions made about the distribution of pellet thickness. Overall, combining uncertainties requires careful consideration of the measurement methods and statistical principles.
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TL;DR Summary
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!
 
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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.
 
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.
 
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##.
 
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