# Non-Normal Distribution vs. Gamma Distribution

F

#### fed-up

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

#### Coury Ferguson

##### Moderator here to help
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#### Miner

##### Forum Moderator
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.

Last edited:
• bobdoering
F

#### fed-up

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

#### Attachments

• NonNormal Process Capability Distributions.bmp
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#### Miner

##### Forum Moderator
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.

• Darius and bobdoering
D

#### Darius 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. 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). #### bobdoering

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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). 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). #### Bev D

##### Heretical Statistician
Super Moderator
...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.

• bobdoering

#### bobdoering

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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? #### Bev D

##### Heretical Statistician
Super Moderator
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.

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