# Characterizing the distribution of manufacturing data

I have some data from manufacturing concerning the variability of certain geometric dimensions of a given machine. These are real numbers (diameters, angles, lengths, etc.), so the variables are continuous. I need to perform an uncertainty quantification study of the performance of the machine: I want to propagate the manufacturing process variability across the chain of codes which are used to compute the performance of the machine, in order to estimate the uncertainty in the machine performance. I thus need to perform a Monte Carlo analysis (or Polynomial Chaos, or Gaussian Process - pick your favourite tool). In a way or the other, I need to be able to sample from a distribution. The distribution is not normal, nor it can reasonably be approximated by a normal distribution: see preceding discussion

Anderson-Darling test for normality with estimated parameters

I've been advised to try to characterize the data distribution. How do I do that? Is there a standard procedure, or can you suggest useful references? I'm not an expert (as you probably can tell), so I need something simple. I can code in R, I am familiar with the Monte Carlo standard error and I can run hypothesis tests (for example, I understand when to reject normality based on the result of an Anderson-Darling test). I don't know about more advanced stuff, but I'm a good learner, though :)

PS I imagine that characterizing the data distribution means to fit different distributions to my data and find the best fitting one. However, I've heard that this is a kind of data-dredging (I guess because fitting would favor distributions with more parameters), so I imagine there's more to this problem.

• Can you post (some of) your data, or a plot, such as histogram or dot plot? Aug 7, 2015 at 17:37
• Or, you could just sample from your data ... a kind of bootstrap. Aug 7, 2015 at 17:38
• Hey, thanks for the comment! Unfortunately (well, actually, very fortunately - for me :D I just left for my vacations and I won't access the work pc for some time, so I can't post histograms. BTW, in the end, I'll do something like sampling from my data - I chose to use PCE with orthogonal polynomials numerically generated according to the histograms. And when I'll estimate the pdf I'll actually sample from the histograms and evaluate my response surface, so that's very similar to your suggestion. Hope it works fine! Aug 7, 2015 at 21:24

After failing to fit the distribution that theory suggests the data should follow then you must ask why does this data not fit. For example, the angle on your machine may follow more of an exponential distribution, since they will target a more acute angle and bend until it hits the low end of the tolerance. This is because a plate is much easier to bend than to straighten. If you still cannot find a distribution, then maybe your data is correlated to some other variable outside of your dataset. If this still does not yield an answer then I would let the data determine the distribution.

This is the process I would follow after the above has been exhausted:

• Find the range of the variable
• Divide the range into equal sections
• Calculate the frequency of each section
• Plot the frequencies in a histogram

This should give you a visual representation of the shape of the distribution. Compare this image to several distribution pdfs. Once you have found one that looks like a match then you can test the goodness of fit by using the Chi-squared test. Here is some guidance on the test:

https://onlinecourses.science.psu.edu/stat414/node/259

When checking a distribution, calculate the mean and variance and other parameters of the proposed distribution using your dataset; so you can generate a sample distribution to test. For every parameter you estimate from your data, you must subtract one degree of freedom from your Chi-squated test statistic. Wikipedia is a good source for calculating the parameters of a distribution.

• Concerning the use of a histogram to gauge distributions, you might appreciate Glen_b's analysis at stats.stackexchange.com/questions/51718/…. Note that in your approach, the p-value in the $\chi^2$ GoF test is invalid (due to the initial screening). The p-value is also invalid because the bin cutpoints are determined by the data. Neither of these problems is severe, but one should be aware of them and compensate for them. This is the "data-dredging" referred to by the OP.
– whuber
Aug 7, 2015 at 13:03
• This is a valid concern. I will update my answer. Aug 7, 2015 at 17:25
• @whuber, I read Glen_b's analysis and I was shocked at how much deceptive histograms can be! Ok, got it, don't look at them. But how do I choose the candidate distribution then? In your comment to the question Glen_b answered, you refer to qq plots and GoF tests. I would really appreciate if you could elaborate a bit and write an answer to my question, maybe a sequence of steps that you think I should follow... Aug 7, 2015 at 22:13