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I have a powder batch (Starting Batch) with a known particle size distribution (PSD). From this batch, I draw two separate samples to feed into two different methods (Method A and Method B) of a process. I want to compare the output of both methods to see if they produce the powder output PSD result.

However, since the input powder batch A has a particle size distribution (range of particle sizes), each sample I take will only represent a subset of the overall distribution, which might affect the output results of both methods. Given this, how should I account for the differences in particle size between the samples when comparing the outputs of Methods A and B?

Should I adjust the results by considering the overall PSD of the batch (i.e., the mean and standard deviation of the particle size distribution) and somehow factor those into the output PSDs of the two methods?

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    $\begingroup$ When you speak about "the mean and standard deviation of the particle size distribution", this rings some alarm bells (it's not typical to speak about particle distributions in those terms, it is often more complicated with several ways to specify a mean or deviation). What sort of powder, distribution measurements, sampling, and samples are we talking about? $\endgroup$ Commented Jan 9 at 19:35
  • $\begingroup$ "I draw two separate samples to feed into two different methods" what does 'feeding into a method' mean? Is it like you do something to the particles, some treatment or other process, and you want to test whether it influences the psd? $\endgroup$ Commented Jan 9 at 19:41
  • $\begingroup$ "feeding into method" means if I have a sieving process, method A is to have the feeder into the sieve set at speed A and method B is to have it set at speed B and see if the resulting powder PSD output of both the methods are the same. $\endgroup$ Commented Jan 10 at 3:30
  • $\begingroup$ Should the word "same" be added to l. 3 "I want to compare the output of both methods to see if they produce the powder output PSD result"? You are comparing the two methods, right? This is not about comparing a method outcome to the original PSD? Also when talking about "sample", are you talking about a single observation in the statistical sense, or about a number of observations? (This is ambiguous in terminology as "sample" can refer to a single observation or to a number of them.) By the way, add all relevant information to the question, don't just answer in comments. $\endgroup$ Commented Jan 10 at 11:01
  • $\begingroup$ How would you compare the different size distributions? Over the entire range of the distribution, or based on a specific measure that describes the entire distribution? If you do not have replications of the experiments, do you consider some known theoretical distribution of the noise? $\endgroup$ Commented Jan 10 at 12:00

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The problem is not really at the PSD level but at the output of the methods. If I understand correctly, your methods are going to give you random outputs, in part due to the PSD inconsistency between the two batches. You need to address this if you wish to compare them correctly.

The key thing you need to know is the variability of the outputs.

  • If this completely unknown, you would need multiple measurements from both methods in order to compare them in a statistically valid way.

  • If you can somehow know the variability in advance, or deduce it from the PSD, then perhaps a single measure might be enough to compare them.

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  • $\begingroup$ There are no two batches, there is only one batch (batch A with a global PSD) but I am taking two samples from that batch. When I take a sample really I am taking a local subset PSD powder which may not be the same as the global PSD. My question is when I am comparing the outputs, how do I correct for the fact that the input PSD of the two samples won't be the same as global PSD and so can't be assumed to be similar. And thus if the inputs aren't similar it could impact the output PSDs from the two methods, so how can I compensate for the inputs so analysis is only done on impact of the method $\endgroup$ Commented Jan 10 at 3:36
  • $\begingroup$ Unless you have detailed information about how the input PSD is related to the output, you can't. That's why I'm suggesting to instead measure the output variability. Sorry for using the wrong terminology. $\endgroup$ Commented Jan 10 at 9:20
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Edit

Now that I realise particle size distribution refers to a specific concept in this field, and not just to the mean size of particles, this response doesn't really answer the original question, although the points about random sampling, populations, etc., still stand.

It's not totally clear from what you've written, but it sounds like you're describing a classic randomised experimental design, which you can read up on in any introductory statistics textbook:

  • You have a population that you're interested in (all of the powder in Batch A)
  • You're able to randomly draw two samples from this population
  • You apply a different treatment (Method) to each sample and compare the results, and compare the results

Since the samples are drawn at random, they'll always be slightly different, just due to chance. Classical statistical tests like the t-test test whether the difference is big enough that you can reject the hypothesis that the differences are just due to chance.

I'm not sure what you mean by "overall distribution" though. If you're only interested in drawing conclusions about powder from Batch A in general, the design is as I've described. If you're actually interested in generalising to powder more generally, your conclusions might only hold if Batch A doesn't differ from other batches in relevant ways. In this case, you would ideally run an experiment that uses powder from multipe batches.

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  • $\begingroup$ "You apply a different treatment (Method) to each sample and compare the results" This is easier said than done. The question asks for how to perform the comparison and this answer doesn't really give a solution to that. $\endgroup$ Commented Jan 10 at 12:24
  • $\begingroup$ Is it not clear that "classical statistical tests like the t-test" are used to perform the comparison? I can rephrase. Obviously I can't suggest a specific test without knowing what's being measured, but I also wouldn't advise OP to just use, e.g. Chi-squared without reading up on randomised experiments as suggested. $\endgroup$ Commented Jan 10 at 14:08
  • $\begingroup$ The OP is comparing particle size distributions. It is indeed not so clear that performing a t-test is suggested as answer. Or at least, it is not clear how that can be applied to distributions. $\endgroup$ Commented Jan 10 at 14:22
  • $\begingroup$ I see, I didn't realise from the question that "particle size distribution" is a specific, well known concept, I mistook it for an awkward way of referring to the mean particle size. I'll update accordingly. $\endgroup$ Commented Jan 10 at 16:26

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