# Theory and Practice Behind Expecting a Distribution for my Data (specially the normal

U

#### Uchiha

Hello everyone,

Cheers for the good work around here

My question is probably quite basic: how do we proceed - theoretically and by experience/practice - to say that the measurements of such characteristic should follow such distribution?

I am particularly interested to know the generic procedure (the actual analysis idea), and would like to see it applied to the case of the normal distribution...

As a background, our customer gave us a table with a list of different characteristics (basically dimensional and geometrical tolerances) and the corresponding distributions expected. Apart from the question whether the table is correct or not and that we could eventually discuss some specific cases if we have solid understanding of our process, I am wondering on which basis such a table was made?

I know that theoretically the central limit theorem is used to expect the normality of some data, but I don't have enough experience to understand its application on specific cases... Example: why would the data from a dimensional tolerance with two specification limits follow a normal distribution (question from the CLT point of view)? What are the independent random variables we're considering in this case?

Many thanks in advance

D

#### Darius

Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

how do we proceed - theoretically and by experience/practice - to say that the measurements of such characteristic should follow such distribution?
Wellcome to the cove

IMHO, the best is to check by practice, but there are some cases where non-gausian distribution is expected, ie. the case where is a physical limit and if the data (time) depend on delays. The problem with defining it theorically is that depends on whe variation we are allowing to happen in our process. I have seen processes that althought the relationship is well documented, the data doesn't show it because of the tight control done upon it.

T

#### TheGoldenBlazer

Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

The central limit theorum states only that the distribution around the sample mean is normal, not that the data itself is distributed normally. So if you have a set of data lets say:

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4

Although it should be 30+ data points according to the central limit theorum, this is just a quick example. Graphing this data by itself will not result in a normal distribution as can obviously be seen, but if you take the mean of the data, 1.67, and graph the deviaiton from the mean, using standard deviations, the curve will approach normality. (With only 15 points it won't be perfect, but at least you can see what it's getting at)

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#### Steve Prevette

##### Deming Disciple
Staff member
Super Moderator
Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

Given that this has been posting in the "SPC Monitoring . . ." thread, I do want to make my standard statement that SPC does not rely upon Normality for it to function. Non-normal data will work fine with SPC.

There are theorectical reasons to use certain distributions, such as Poisson for counts of objects, Binomail for go-no go testing, Exponential for time between failures, Chi Square for contingency tables, Log Normal for measurements of exposure. Beta-Gamma gets even more interesting in Reliability analyses (and the infamous "bathtub" curve).

Now, whether or not the underlying distribution of the data is important depends upon what you are trying to do with the data. Again, SPC is robust enough to be relatively free of effects of the underlying distribution.

U

#### Uchiha

Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

IMHO, the best is to check by practice, but there are some cases where non-gausian distribution is expected, ie. the case where is a physical limit and if the data (time) depend on delays. The problem with defining it theorically is that depends on whe variation we are allowing to happen in our process. I have seen processes that althought the relationship is well documented, the data doesn't show it because of the tight control done upon it.
I agree with you.

The central limit theorum states only that the distribution around the sample mean is normal, not that the data itself is distributed normally.
I see. I admit that I do not have enough knowledge to discuss this item.

But in order not to go in a much wider scope than the original question, let's get back to the specific case of the normal distribution in the context I described.

Let's suppose you are my customer and you provide, like I said, a table with a lot of information, among which you state that if I, your supplier, have a dimensional tolerance with two specification limits, it should follow a normal distribution. Also let's avoid discussing for now special cases in which this assumption may not be correct and focus only on case(s) where IT IS correct .
Now the question is: I, your supplier, want to know how did you proceed to reach such conclusion... There are definitely theoretical and practical reasons and explanations... These are precisely what I am trying to understand... (and I still think that the CLT is probably involved in the theoretical explanation... Else what would explain such assumption?)

Again, SPC is robust enough to be relatively free of effects of the underlying distribution.
Understood

But allow me to ask the following (although this is out of the scope of the original question): does this mean that when using SPC, knowing, or not, the distribution my data should follow and the distribution it is actually following makes no difference as for the way I deal with my control charts (standard deviation, control limits, capability indexes...).

Excuse my ignorance if I am asking a trivial question...

D

#### Darius

Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

Excuse my ignorance if I am asking a trivial question...
Not trivial at all, Donald Wheeler show in his book (and others I think) "Advanced topics in statistical process control", that as Steve said, SPC works fine for many different types of distributions....., but

Beware of patterns, not all patterns will work, giving false alarms if you configure your system to detect and Donald Wheeler (on the same book) told that he recommended not to use transformations, because it could be more difficult to understand for the most of the users.

The question about capability indexes is another wave, as you may know these indicators use the mean and standard deviation, there are two ways that can address the problem of non-gausian distribution best:
* As I readed on some articles is to transform the data using box-cox (comparing the results using montecarlo simulations with different kinds of distributions).
* Non-parametrical indicators.

And remember that indicators are just that, are oversimplifications that permit us to evaluate our process (an histogram and an SPC chart told me more), and as I readed somewhere "You Can't Manage What You Don't Measure", maybe are not perfect but have their use, so..., do you need to have more precision (than using the standard formulas)?

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

Let's suppose you are my customer and you provide...a table with a lot of information, among which you state that if I...have a dimensional tolerance with two specification limits, it should follow a normal distribution...let's focus only on case(s) where IT IS correct.

Now the question is: I, your supplier, want to know how did you proceed to reach such conclusion... There are definitely theoretical and practical reasons and explanations... These are precisely what I am trying to understand... (and I still think that the CLT is probably involved in the theoretical explanation... Else what would explain such assumption?)
to answer this specific question: You are looking for an explanation that doesn't exist.

The central limit theorem only applies to the distribution of sample averages so if your customer is using this to say that any characteristic with two spec limits must have a Normal distribution, they are mistaken. It is a common misunderstanding...but popularity doesn't make it correct.

Depending on the process - and material - used to generate the characteristic there may be practical scientific reasons to expect a distribution that is somewhat symetrical with high frequency of occurences at the median value and low frequency of occurence at the outer 'tails'. This is commonly referred to as the 'bell shaped curve'.

BUT many processes are simply not homogenous by their very nature and these processes will not result in a Normal or bell shaped curve. The sources of non homogeneity are often not 'thought of' by those with only a theoretical knowledge of the process or limited practical experience with the process. These sources are often 'buried' in the specific details and nuances of the process. The ONLY way to really know is to run the process and take the data.

There is NOTHING inherently wrong with a process that is expected to be bell shaped but that isn't. These processes can still be very stable and capable.

There is no inherent value in a process being Normally distributed...bell shaped processes can be unstable and incapable.

The 'Normal distribution' is not a law of physics. It is a man made model. It can be useful, but it is not some magical, ordained, neccessary thing.

Bottom line: There is no reason to state - either theoretically or practically - that a process with two spec limits must or should be Normally distributed.

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#### bobdoering

##### Stop X-bar/R Madness!!
Trusted Information Resource
Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

Again, SPC is robust enough to be relatively free of effects of the underlying distribution.
It is true, the generic term SPC (statistical process control) will work with any distribution. However, Shewhart charts rely on the process variation to come from independent, random data. All of the examples from Shewhart and Wheeler utilize show a vast number of distributions that work - if those requirements are met. When that is not true, those charts may not be appropriate, but other SPC techniques should be.

#### bobdoering

##### Stop X-bar/R Madness!!
Trusted Information Resource
Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

Recognize this, also: measuring what you can't react to doesn't help control your process. When picking the variable to chart, the use of a CNX analysis is handy. You have much more value in charting something you can adjust for, than something that is "noise" and you have no control over. If that is all you are charting, it will just trigger your sorting mechanism.

#### bobdoering

##### Stop X-bar/R Madness!!
Trusted Information Resource
Re: Theory and Practice Behind Expecting a Distribution for my Data (specially the no

A big 3 auto manufacturer has come out with a chart of "expected distributions" for processes that will not be normal even under controlled conditions. One of the items listed is "Grinding process with auto correction", and indicates the expected distribution is "uniform". Wow.....so close (but much closer than before). Actually, it should say "Grinding process with auto correlation", because auto correction actually generates a normal distribution from overcontrol.) Also, more specifically it is the continuous uniform distribution - quite different from discrete uniform distribution.

I wonder how this was called to their attention.....

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