What is Normal Theory?

sathis

Involved - Posts
#1
What is Normal Theory? - Comes in SPC Topic.

I have seen Central limit Theorem .


Sathish
 
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Tim Folkerts

Super Moderator
#2
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.
 

bobdoering

Stop X-bar/R Madness!!
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#3
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 :cool:
 

Darius

Quite Involved in Discussions
#4
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).
 

bobdoering

Stop X-bar/R Madness!!
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#5
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!
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#6
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?
 

Steve Prevette

Deming Disciple
Staff member
Super Moderator
#7
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.
 

bobdoering

Stop X-bar/R Madness!!
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#8
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.
 

Tim Folkerts

Super Moderator
#9
: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
 

BradM

Staff member
Admin
#10
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?
 
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