What is Normal Theory? - Comes in SPC Topic.

I have seen Central limit Theorem .


Sathish

Without knowing the exact context, I would assume they simply mean theory based on the normal distribution.

Not everything follows a normal distribution, but it is a good approximation to many real-life situations. Furthermore, the mathematics of the normal distribution are well known, so much of statistical analysis is based on the assumption that the data follow the normal distribution.

Normal process: in control with chance variation

In order for a process to be normal, it should be able to be:
1. Set at the mean
2. Will continue to randomly vary about the mean without any operator intervention!

A process in control is in the ideal state 100% conforming and predictable
1. must remain stable over time
2. must operate in a stable and consistent manner
3. must be set at the proper level
4. the natural process spread must not exceed the product’s specified tolerance (capability)

A great example of a normal process is cutting the grass. You set the mower deck to a particular height. As is typical for a quality profession, one would measure each blade of grass after cutting, and you would find most near the mean height (height of the deck setting) – with some a little longer and some a little shorter. Most “natural” variations, such as operations influenced by humans or nature (environmental) that are not unilateral are normally distributed.

I am sure others can give deeper academic explanation, but this is a good overview, I think...:tg:

However, not all processes are normal – and treating them as such generates incorrect decisions. Precision machining is just such an example. It is non-normal, continuous uniform distribution – which means the central limit theory does not apply. For more information on this, see: Statistical process control for precision machining (http://elsmar.com/Forums/blog.php?b=79) :cool:

Also for a variable to be "normal" (Gaussian) ones, there must not be a dependance on time (for example dependance on delays of production line).

Also for a variable to be "normal" (Gaussian) ones, there must not be a dependance on time (for example dependance on delays of production line).

True, good catch! :agree1: By definition it must be random and independent variables!

before we move too far afield can you give us more specifics about your question?

From your post I'm assuming that you are aware that the central limit theorum says that sample averages will tend to approximate a Normal distribution provided that certain conditions inherent in the theorem are met.

Are you asking about specifics of what the Normal Distribution is?

I should point out, since the original question was in the context of SPC, that normality of data is NOT required for use of SPC. Dr. Shewhart established SPC principles based upon the Tchbychev Inequality, and tested the SPC theory for the normal distribution, and for two non-normal (uniform and triangular) distributions.

I should point out, since the original question was in the context of SPC, that normality of data is NOT required for use of SPC. Dr. Shewhart established SPC principles based upon the Tchbychev Inequality, and tested the SPC theory for the normal distribution, and for two non-normal (uniform and triangular) distributions.

Yes, on pages 136 through 137 of "Economic Control of Quality of Manufactured Product", Dr Shewhart analyzed a variety of distributions, and on page 182, he described experiments that described both a triangular and rectangular (uniform) distribution.

So, I agree, you might say that normality is NOT required for use of SPC.

I am kind of sensitive to lumping the "uniform distribution" into that statement. Closer observation of Dr. Shewhart's example of a uniform distribution, it is clear he established a discrete uniform distribution, which is an independent function, and therefore supports Central Limit Theory and Tchbychev Inequality. We must not confuse this with the continuous uniform distribution found in precision machining, which is a dependent function - and not the uniform distribution of Dr. Shewhart's experiment, and not supported by Central Limit Theory and Tchbychev Inequality. But, statistics can be used control the process - they just need to be correctly geared to that distribution (as in NOT Xbar-R). Then it works very nicely.

:topic:
Not to beat a dead horse, but the Central Limit Theorem does indeed apply to both the discreet and continuous uniform distribution.

"the re-averaged sum of a sufficiently large number of identically distributed independent random variables each with finite mean and variance will be approximately normally distributed" is a reasonable statement of the theorem (from Wikipedia).

When the variables are not chosen randomly, then, not surprisingly, the result can and does sometimes fail. In particular, in quality we often sample the output systematically - for example items 1, 11, 21, 31, ... If the process itself is random, then the central limit will still work here. If the process is something like precision machining, the central limit theorem will fail - not because the central limit theorem itself does not apply, but because the sample chosen was not random.

A random sample chosen from a uniform distribution will follow the central limit theorem. It is not the uniform distribution itself that leads to the failure, it is that the uniform distribution was generated in a systematic way AND the sample was chosen in a systematic way.


Tim F

Interesting. Let's assume I am not sampling random. The more sampling that I do, the pool of samples will begin to represent the population. And as the # of samples gets closer to the population, the distribution will more represent a normal distribution, would it not?

I came to know Both Normal Theory and Central Limit theorem are Quite Opposite.

Central Limit Theorem - If the Distribution of measurements is not Normal , the Distribution of Sample Means is NORMAL.

Normal Theory - If the Distribution of measurements is NORMAL , the Distribution of Sample Means is NORMAL.



Sathis

Interesting. Let's assume I am not sampling random. The more sampling that I do, the pool of samples will begin to represent the population. And as the # of samples gets closer to the population, the distribution will more represent a normal distribution, would it not?

Any sample of a continuous uniform distribution taken in order (not random) - such as fifth part - will appear to be uniform (but truncated). It is easy to try. Take the series of numbers from 1 to 30 - in order. Now take every 5th part. Plot the result. It mimics the original distribution (uniform), but to 30 units wide. And as the number of samples gets closer to the population, the distribution will more represent the full uniform distribution.

Not to beat a dead horse...

Might not be a dead enough horse to somebody that has not been following the storyline here for a while. :tg:


When the variables are not chosen randomly....


Which is how one one collect data from an ongoing process - not waiting until you are done - then picking out of the pile. That would be more of the incoming receiving inspection from such a process. Even then, the result of that sampling may lead to incorrect conclusions about the process.


If the process is something like precision machining, the central limit theorem will fail - not because the central limit theorem itself does not apply, but because the sample chosen was not random.


Which is the point of the difference between precision machining and Dr. Shewhart's experiment.

It is not the uniform distribution itself that leads to the failure, it is that the uniform distribution was generated in a systematic way AND the sample was chosen in a systematic way.


Thank you. I hope this further clarifies the issue for the OP. :thanks:

I came to know Both Normal Theory and Central Limit theorem are Quite Opposite.

Central Limit Theorem - If the Distribution of measurements is not Normal , the Distribution of Sample Means is NORMAL.

Normal Theory - If the Distribution of measurements is NORMAL , the Distribution of Sample Means is NORMAL.


Does this clarify your point:

General Classes of Statistics (Oh, I Guess I Do Care)
Ok, so we have these two general categories (i.e., continuous and categorical), what next…? Well this distinction (as fuzzy as it may sound) has very important implications for the type of statistical procedure used and we will be making decisions based on this distinction all through the course. There are two general classes of statistics: those based on binomial theory and those based on normal theory. Chi-square and logistic regression deal with binomial theory or binomial distributions, and t-tests, ANOVA, correlation, and regression deal with normal theory.
(from Types of scales & levels of measurement (http://www.upa.pdx.edu/IOA/newsom/pa551/lecture1.htm))

Interesting. Let's assume I am not sampling random. The more sampling that I do, the pool of samples will begin to represent the population. And as the # of samples gets closer to the population, the distribution will more represent a normal distribution, would it not?

nope. the samples will approximate the population whatever distribution it has.
the central limit theorem states that the sample averages will be approximately Normal, given that the other requirements of the theorem are met.

There are two general classes of statistics: those based on binomial theory and those based on normal theory. Chi-square and logistic regression deal with binomial theory or binomial distributions, and t-tests, ANOVA, correlation, and regression deal with normal theory.


actually if you must limit statistics to only two general classes - which is a GROSS oversimplification that emasculates the science and leads to even more convaluted and uneducated application of bad math - teh two categories are better described as Categorical (counting data) and Continuous (measurement data). Categorical data can also be modeled by the geometric distribution or by the Poisson. Under certain conditions, the Binomial and the Poisson approximate the Normal. "distributions are mathematical models that have utility: they are not truth; they are man made devices. And let us not forget that distributional models are used for 'parametric' testing, distributions are not used for non-parametric testing...

nope. the samples will approximate the population whatever distribution it has.
the central limit theorem states that the sample averages will be approximately Normal, given that the other requirements of the theorem are met.

Yes... but as that sample of data increases, it will begin to approximate normality. The mean of the samples will begin to approximate the mean of the population.

Now... I guess which population we're speaking of can make a big difference. I'm talking the average height of every 9 year old boy in America, not just the population of 9 year old boys in Mrs. Jones's class.

When I see the word random, I think of this definition:

http://cnx.org/content/m13470/latest/

Which is slightly different than suggesting the sampling has to be random.

Yes... but as that sample of data increases, it will begin to approximate normality. The mean of the samples will begin to approximate the mean of the population.

Now... I guess which population we're speaking of can make a big difference. I'm talking the average height of every 9 year old boy in America, not just the population of 9 year old boys in Mrs. Jones's class.

When I see the word random, I think of this definition:

http://cnx.org/content/m13470/latest/

Which is slightly different than suggesting the sampling has to be random.


well that definition is for a random experiment not random sampling - two different things

And I think this may just be a semantics issue.
the theory states: as the number of 'samples' in your subgroups increases, the distribution of subgroup averages will be app. Normal. stated in other words as your sample size increases, teh sample averages will app. a Normal distribution. The individual values within a sample (or subgroup) or even wihtin many samples (or subgroups) will NEVER approximate anything but the actual distribution.

it happens that height has an approx. Normal distribution so the individual values within a sample will aslo be app. Normal.

actually if you must limit statistics to only two general classes - which is a GROSS oversimplification that emasculates the science and leads to even more convoluted and uneducated application of bad math...

We'll have to let Dr. Newsom know that. I was trying to fish for the elusive "Normal Theory" that the OP was citing. In fact, Dr. Shewhart's book has "Normal Theory" cited in the index on page 12, but nothing that I could identify on that page as a theory definition. :confused:

Pretty clear cut, this normal theory...:smokin:

We'll have to let Dr. Newsom know that. I was trying to fish for the elusive "Normal Theory" that the OP was citing. In fact, Dr. Shewhart's book has "Normal Theory" cited in the index on page 12, but nothing that I could identify on that page as a theory definition. :confused:

Pretty clear cut, this normal theory...:smokin:

True...:agree1:

well that definition is for a random experiment not random sampling - two different things

And I think this may just be a semantics issue.
the theory states: as the number of 'samples' in your subgroups increases, the distribution of subgroup averages will be app. Normal. stated in other words as your sample size increases, teh sample averages will app. a Normal distribution. The individual values within a sample (or subgroup) or even wihtin many samples (or subgroups) will NEVER approximate anything but the actual distribution.

it happens that height has an approx. Normal distribution so the individual values within a sample will aslo be app. Normal.

Yeppers. Good writeup, Bev. Things like this would make a good Wiki here.:agree1:

We'll have to let Dr. Newsom know that. I was trying to fish for the elusive "Normal Theory" that the OP was citing. In fact, Dr. Shewhart's book has "Normal Theory" cited in the index on page 12, but nothing that I could identify on that page as a theory definition. :confused:

Pretty clear cut, this normal theory...:smokin:

Well the theory is - the users may not use it correctly. I suspect that the OP is refering to the Normal Distribution model and the theory that it applies to many practical situations. The mathematical model yields occassionally useful relationships. e.g. 95.46% of the values of a data set posessing a Normal distribution will fall within + 2σ of the mean; or more commonly 95% of the values will fall within + 1.96σ of the mean.

As for Dr. Newsom, I'm sure he meant well and I don't have the full context of his discussion, but the mere fact that he has a phd does not confer on him the mantle of absolute truth nor does it exempt him from question, doubt or debate. If that were true then you, Bob, would accept the teachings of Dr. Shewhart and Dr. Deming without question. And we couldn't have that, could we?? :cool:

As for Dr. Newsom, I'm sure he meant well and I don't have the full context of his discussion, but the mere fact that he has a phd does not confer on him the mantle of absolute truth nor does it exempt him from question, doubt or debate. If that were true then you, Bob, would accept the teachings of Dr. Shewhart and Dr. Deming without question. And we couldn't have that, could we?? :cool:

Well, my comment was not meant to infer that because he had a PhD that he was exempt from question, doubt or debate. If anything, it is an open invitation. Just that this must be another poor guy out there trying to make a buck teaching this stuff to his poor minions, and has no idea he got sucked into the maelstrom.:eek: More importantly, it was a reminder that these were someone else's words, not mine. Just trying to root around for the definition of this elusive "Theory".

Me, accept the teachings of Dr. Shewhart and Dr. Deming without question? Yes, that seems unlikely any more, doesn't it? Hey, I used to be like the others. I never really wanted to be different. I used to rubber stamp the concepts, the Xbar-R charts without regard, just like everyone else. It was what we did back in '94. It was how we survived the onset of the rein of the evil empire of QS9000. But, I just couldn't take the pain anymore. No matter how hard I tried to put the rectangular peg into the bell-shaped hole, I couldn't do it. I tried all the tricks to get people to accept my PPAP with bimodal distributions, or Cpk's of 23....but I just couldn't take it any more. As the nights wore on, in the quiet of the empty plant, I scribbled and scratched until the truth finally revealed itself. Dr. Shewhart and Dr. Deming had come so close...but I finally pushed it over the flat edge of SPC. There really was a world beyond! It reminded me of some of the scenes towards the end of the movie "Tommy"!

Can ya dig it?:lol:

You really didn't read all that, did you? :lmao:

Bob,

While I do agree with you that X-bar and R/S charts are not the best solution for a precision machining process, that does not mean that they will not work perfectly well with other processes that do not have an assignable and acceptable cause for a trend, such as tool wear.

Your responses tend to come across as though X-bar and R/S charts are never appropriate. I know this is due to your vigorous defence of your methods, but may be confusing to others that are not in precision machining environments.

Dr. Shewhart and Deming would probably agree with your approach when used in your situation. Shewhart was all about the economic control of quality, and Deming always warned against reading too much into the probabilities around control charts (e.g., 67% inside 1 sigma, etc.).

I don't want to discourage your attempt at educating covers that there is a better way for precision machining, but please don't come across that it is the only way for process control for non-precision machining. There are times when X-bar R/S charts are the best and there are times when IMR charts are best.

The MBB's favorite response is "It depends". The best chart depends on the process, specifically the sources of variation.

While I do agree with you that X-bar and R/S charts are not the best solution for a precision machining process, that does not mean that they will not work perfectly well with other processes that do not have an assignable and acceptable cause for a trend, such as tool wear.


Actually, I agree with that statement 100%. Perhaps my disdain for the opposite notion - that X-bar and R are great for everything, may hint that it totally worthless. On the contrary. For sure, if the process is truly normal, it is nearly perfect. In order for a process to be normal, it should be able to be set at the mean, and will continue to randomly vary about the mean without any operator intervention!

Dr. Shewhart and Deming would probably agree with your approach when used in your situation. Shewhart was all about the economic control of quality, and Deming always warned against reading too much into the probabilities around control charts (e.g., 67% inside 1 sigma, etc.).

My favorite Shewhart quotes used in my presentation:

"Rule No.1
Original data should be presented in a way that will preserve the evidence of the original data for all the predictions assumed to be useful."

"The total information is given by the observed distribution.”

I don't want to discourage your attempt at educating covers that there is a better way for precision machining, but please don't come across that it is the only way for process control for non-precision machining. There are times when X-bar R/S charts are the best and there are times when IMR charts are best.

I am pretty sure by now that you are well aware that the passion of some of us covers can be misinterpreted on occasion. Right? :tg: There are times when X-bar R/S charts are the best and there are times when IMR charts are best. You can't beat it for natural variation like the heights of loaves of bread that come out of the oven at an automated bakery. All about the same height...some a little higher...some a little lower.....

The MBB's favorite response is "It depends". The best chart depends on the process, specifically the sources of variation.

Couldn't have said it better...could I? :tg: