Yes, that would be a perfect *discrete* uniform distribution, which is random and not dependent

clarity: a deck of cards is discrete uniform distribution - unless you've stacked the deck. The distribution of subgroup averages from a series of random samples - with replacement and shuffling to simulate an infinite distribution of course - will approxiamte teh Normal distribution. The Normal distribution is for continuous data of course, but many discrete or categorical distributions will result in subgroup averages that approximate the Normal distribution. The Binomial and Poisson among them. Unless the defect rate is very small, etc.

Yes, the fact that tool wear is a dependent function of time (tool wear rate), it makes it a *continuous* uniform distribution, which has the same shape, but different statistics than the *discrete* uniform distribution.

So? For both distributions the

**random** sample averages will still approximate a Normal distribution. And the individual values will not. The truly discrete nature of the deck of cards will result in actual truncated tails while a continous distribution from the type of continuous uniform distribution you discuss will have tails - not infinite but they will have tails...

Interesting observation. Shewhart's "bowl of chips" experiment certainly will give a discrete distribution, which will support the central limit theorem. If you look in his book, of all the distributions he evaluated, the one that was

*missing *was the uniform distribution. Oddly, I believe it is also missing from the Pearson distributions. Not sure why.

*Too *simple, perhaps?

Well, Shewart wasn't the only the only one who has done good work on SPC or statistics. Theoretical statisticians tend to stick to theoretical models and applied statisticians tend to work with the dirty real world data.

Also, I'm not sure why you are emphasizing the discrete part. For example purposes of a uniform distribution yielding a usefully approximate Normal distribution of random sample averages I chose a deck of cards. It was the first simple thing that came to mind that anyone could confirm for themselves.

If we were to take the output from a process whose individual values have a continuous uniform distribution and were to randomly sample from them (by either throwing them in a big bucket and stirring them up, or by using a random number generator to select be serial number or even sequence number) the resulting sample averages would also approximate a Normal distribution. My point was that SPC doesn't work that way. We sample from a process stream - not randomly. And many processes don't have a sequential random process output. THIS is what many people who are stuck on the Normal distribution can't grasp. The Central Limit theorem is true, but only for the conditions it clearly states.