# Can RSD be pooled or is it valid to use the uncertainty propagation rule?

This might be a strange case, but I'm sure that somebody in here can help me and that I'm perhaps not the only one who wants to pool RSDs correctly.

Consider this: The mean weight of a powder bed is W mg with a relative standard deviation (RSD) of Wrsd %.

For each weighted powder bed a precise counting of the particles were done and the mean particle mass is calculated to be PM mg/particle and the RSD is found to be PMrsd %.

Can these two RSD (Wrsd and PMrsd) be pooled together and thereby be used to estimate the RSD of the number of particles within a weighted powder bed? If so, what does the equation look like?

I have tried to treat the RSD as relative uncertainties and thereby using the propagation rule,

$e_3 = \sqrt{(e_1)^2 + (e_2)^2 }$

Is this approach violating anything?

## UPDATE

(@jonsca thank you for updating my question so it is more readable)

(@whuber thank you for your interest in my question)

A more detailed description:

A powder die was filled multiple times. Each time it was weighted and the particles counted. Putting it in a table looks like this:

Now this could have been done in a different way:

I could have taken multiple collections of particles and determined the mean particle mass. So ex. take 7 times 100 random particles (no die involved) and determine the mean particle mass from these measurements. I could then just weighted some new powder beds (same type of particles) within the die and divide that with the mean particle mass to estimate the mean number of particles that fits within the die.

I don't just want to know the mean number of particles that fits within the die, but are also interested in the span of this number. Using the latter method I would expect the span to be equal to the RSD of the powder bed mass (Wrsd), which clearly the counting shows are too small.

What I notices was that combining the two RSDs (powder bed and particle mass) gives a quite "good" estimate (7.3 %) of the RSD found by the counting (8.7%).

Does this make sense?

[If you read this far, then thank you for you interest :D]

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Parts of this question are confusing, because initially it sounds like just one "weighted powder bed" is involved, and therefore the number of particles in it is known ("a precise counting of the particles [was] done"). What would be the point of combining the two RSDs then? If multiple powder beds are involved, why not just summarize the counts in each bed? Also, do these data represent a sample of something? What of? –  whuber Aug 10 '12 at 17:52
Please see the updated question. Hope it makes better sense.. –  Norfeldt Aug 11 '12 at 10:41