Hi Guys,

If I have two distributions. These being the Non-Normal and Gamma Distribution. And if both of them have n.s. KS p value. Then why is the gamma distribution chosen over the Non-Normal Distribution.

All I know is that as a rule of thumb that Non-Normal Distribution is chosen as a last resort. But why is that?

Thanks

Does anyone have ideas about this?

The question is unclear. There is no distribution called the "Non-normal distribution". The gamma distribution is one type of many non-normal distributions such as the weibull, exponential, log-normal, uniform, etc. distributions.

In other words, ALL distributions other than the NORMAL distribution is a non-normal distribution.

The question is unclear. There is no distribution called the "Non-normal distribution". The gamma distribution is one type of many non-normal distributions such as the weibull, exponential, log-normal, uniform, etc. distributions.

In other words, ALL distributions other than the NORMAL distribution is a non-normal distribution.

Hi Miner,
Actually I was using the statistica software, and in there it gives you a list of all the distribution that is fit for your data. and one of the option is called Non-Normal (skewness, Kurtosis). So I am not sure what Non-Normal distribution this is refering to. Because what you said makes sense.
So I have attached the photo for the dialogue box.

Thanks

I briefly checked the Statistica website for the "non-normal distribution".

Statistica appears to be using the family of Johnson and Pearson curves to fit a skewed or kurtotic distribution that does not fit the typical distributions and are arbitrarily calling it the "Non-normal distribution". n.s. appears to mean Not Significant, which would mean that it fits both distributions.

I could not find Statistica's rationale for using Non-normal as a last resort. I am a Minitab user, which offers non-normal capability analysis using a wide variety of distributions as well as the Johnson and Box-Cox transformations of the data into Normal distributions for standard capability analysis. I am speculating that Statistica's Non-normal distribution is their equivalent to these transformations.

One reason that I prefer using Minitab's non-normal analysis over the transformations is that you can compare your data directly to the specifications. When you transform data, you also must transform the specification. This makes it difficult to make direct comparisons between them or explain them to non-statistical people.

:applause:Miner

I want also to add about why normal (gaussian) over "non-normal" that

1- Almost all statistics are gaussian based, so as Miner said, transformation is needed unless you use non-parametrical capability indexes.

2- Most of the people think that "non-normal" (non gaussian) is waky behabiur, so something wrong is happening.:bonk:

3- Most of the people think that if you add enought information, your distribution will be "normal" (another lie to the bucket), based on central limit (good tool, wrong application, specially time based measures tend to be non gaussian because is easer to have delays than to do the things faster).:lmao:

3- Most of the people think that if you add enought information, your distribution will be "normal" (another lie to the bucket), based on central limit (good tool, wrong application, specially time based measures tend to be non gaussian because is easer to have delays than to do the things faster).:lmao:

True - and the central limit theorm also does not apply to dependent distributions - such as tool wear. Does not matter how much data you have - unless you have a lot of measurement error, and the more measurements you have the more the error becomes the distribution (because measurement error - not to be confused with gage error - is typically normal). :tg:

...and the central limit theorm also does not apply to dependent distributions - such as tool wear.

Perhaps a minor point of clarification would be helpful here.
The Central Limit theorem applies for "identically distributed, independent random variables". In practice this means that if we sample RANDOMLY from any distribution, the average of the subgroups will be roughly Normal - close enough that the Normal distribution has analytical value.

And the theorem holds very well to emperical studies: randomly deal from a deck of cards - a perfect uniform distribution - and you will get subgroup averages that are Normally distributed.

In practice (for SPC specifically and for some experiments) we don't have a Random OUTPUT. That is a de facto violation of the conditions required for the central limit theorem. And we do SPC sampling by time series sampling as the process produces product and that stream isn't random. (this is what is meant when Bob refers to a dependent process) When the largest component of variation is piece to piece in a process we do have a random output in time series and this situation was very common in Shewhart's day. It's not uncommon today, but it isn't the majority either.

For capability studies we are dealing with individual values not subgroup averages and in this case the central limit theorom doesn't apply either (obviously as it only applies to subgroup averages). Native distributions (of individual values) are rarely Normal.

And the theorem holds very well to empirical studies: randomly deal from a deck of cards - a perfect uniform distribution - and you will get subgroup averages that are Normally distributed.

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

In practice (for SPC specifically and for some experiments) we don't have a Random OUTPUT. That is a de facto violation of the conditions required for the central limit theorem. And we do SPC sampling by time series sampling as the process produces product and that stream isn't random. (this is what is meant when Bob refers to a dependent process)

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.

When the largest component of variation is piece to piece in a process we do have a random output in time series and this situation was very common in Shewhart's day. It's not uncommon today, but it isn't the majority either.

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?:notme:

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?:notme:

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.

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.

I agree. :agree1:


I brought up the issue of discrete not so much for you but others that may be reading this. The issue of a rolling a die and generating a uniform distribution had been brought up before, and I wished to take the opportunity to bring up the difference in the output of that process and a continuous uniform distribution for those that might have missed it.

It does seem like the central limit theorem has been used to "prove" just about everything - maybe even global warming and the big bang. I'm so sure...:cool:

I had more time to search Statistica's website and found the following:

"...the user can choose estimates (e.g., Cpk, Cpl, Cpu based on the percentile method) based on general non-normal distributions (Johnson and Pearson curve fitting by moments), as well as all other common continuous distributions including the Beta, Exponential, Extreme Value (Type I, Gumbel), Gamma, Log-Normal, Rayleigh, and Weibull distributions."

The curve fitted distributions are an approximation of a distribution, so if you have the option of selecting an actual distribution, I would recommend the actual distribution. A process that is inherently non-normal would be more likely to follow a true distribution than an approximated distribution.

thanks for the research on that Miner!


editiorially (:soap:) I would add that too often the individulas who are delegated or assigned the task of performing capability studies are not properly trained in statistical methods (aren't most of us self taught?) to deal with the subtleties beyond the basic theory - or in too many cases the first formula presented in the (AIAG?) manual from the Customer. This is often because their organization doesn't really care, they just need to check the box for the customer. It becomes a rote exercise for both supplier and customer. So frequently the person doesn't have enough data to determine the best fit distribution; tehy only have the requisite minimum number which they may or may not have randomly selected.

<snip>It becomes a rote exercise for both supplier and customer. Neither party having a clue what this is all about, thus not adding a lick of value to "statistical techniques."

Sad...

Stijloor.

I would add that too often the individulas who are delegated or assigned the task of performing capability studies are not properly trained in statistical methods (aren't most of us self taught?) to deal with the subtleties beyond the basic theory - or in too many cases the first formula presented in the (AIAG?) manual from the Customer.

One key missing ingredient to the concept of evaluating process capability is the notion of total variation. Fact is, a process consists of many variations - of which we hope most are statistically insignificant. But, if there are two significant variations, each of a different type of distribution, it makes the idea of finding one third distribution that will describe the behavior of the process a bit suspect. On top of that, if the sample shows a particular fitted distribution, it still may not be the correct one, as another sample later on may have a different relationship between the competing distributions. This can be particularly true with unilateral upper tolerance processes with values close to zero.

So, I agree, most folks just want to make parts, get a PPAP signed and go to the next project. It is so bad, that even when you tell them what the distribution is, they really do not want to believe it. Very cultish. :cool:

Some additional recommendations on the selection of a non-normal distribution:

Several non-normal distribution have a lower boundary of zero. If your process or characteristic has the capability of negative values, these distributions will probably break down over the long term, or as you get closer to zero. Likewise, if your process/characteristic has a lower boundary of zero, you should probably not select a distribution with no lower limit.

This does not mean that you can never use these distributions, but be aware that they will probably break down over the long term.

An example: Flatness has a lower boundary of zero. Negative values are not possible. At large values of flatness, the Normal distribution will typically fit the data. As the process improves and flatness values decrease, the Normal distribution will no longer apply and you will need a non-normal distribution with a lower bound of zero.

Read up on these distributions, and you will find that certain distributions are commonly used for specific types of data. For example, the Gamma distribution is commonly used to model waiting times. The Exponential distribution is used to model product reliability during the useful life (between infant mortality and wear out). If you have two distributions that appear to closely model your characteristic, you should use the distribution that is typically used to model that type of data. The model will probably hold better over time.

The following article (http://www.qualitydigest.com/dec99/html/nonnormal.html) from Quality Digest is a worthwhile read as well as this gallery (http://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm) of distributions from NIST.

Hi All,

I just wanted to thank all you guys. I still am not completely sure about why non-normal is used a last resort. But it certainly did help me a lot in the understanding of it.

So Thank You :)

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 continuous distribution from the type of continuous uniform distribution you discuss will have tails - not infinite but they will have tails...


I found an interesting example of what you are illustrating:

Central Limit Theorem Example: Uniform (http://www.statisticalengineering.com/central_limit_theorem.htm)

What it proves to me is if you attempt to randomly sample out of a uniform distribution population (as in receiving inspection), you will never actually determine the original distribution, you will generate a normal distribution and therefore you will have little understanding of the true variation in the population. In short, CLT is proof you get bad data, in that it has little relationship to the true distribution and decisions that can be made from it. Interesting. :cool:

I found an interesting example of what you are illustrating:

Central Limit Theorem Example: Uniform (http://www.statisticalengineering.com/central_limit_theorem.htm)

What it proves to me is if you attempt to randomly sample out of a uniform distribution population (as in receiving inspection), you will never actually determine the original distribution, you will generate a normal distribution and therefore you will have little understanding of the true variation in the population. In short, CLT is proof you get bad data, in that it has little relationship to the true distribution and decisions that can be made from it. Interesting. :cool:


I don't draw that conclusion at all. The Central Limit theorem tells you about the behavior of sample averages, if you sample randomly from the population or have a 'random' process stream. This doesn't provide bad data; it is real and good data. THis is very helpful for some analyses. The Central Limit Theorem is very useful for hypothesis testing of averages and for Control Charts whan the proper conditions are met. In fact in the example you cite regarding incoming inspection one might find that a (truly) random sample of lots with the sample averages plotted on a control chart would indeed tell you if the vendor process was shifting or drifting with relatively small sample sizes.

Of course if what you need to really understand is the individual values - like for acceptance sampling or capability analyses - or if you don't have a random process stream, you will need to apply some other statistical model.
No approach is useful for everything. If someone takes my screwdriver and uses it as a chisel, that doesn't make the screwdriver a bad tool. Likewise, those who do not really understand the central limit theorem and mis-use a statistical approach based on that understanding are the ones at fault; not the central limit theorem.