# Finding Control Limits for a Non-Normal Distribution

A

#### anirban

Hi All,

Broad Problem Statement: I have a set of data (prices of electronics goods at different stores). I want to design UCL and LCL from that data set so that if someone comes with a price, I can estimate whether that price might be correct or not.

Explanation: I have prices (20 data points, say) of a handycam with some specification. The distribution is not normal, which means distribution's skewness and kurtosis should be taken care of while formulating UCL and LCL. With a given confidence interval, how can I derive the formula for UCL and LCL.

Please note, we can not assume normal distribution or t-distribution.

Anirban

#### bobdoering

Trusted Information Resource
Re: Finding Control Limits for non-normal distribution

I would run the Distribution Analyzer to determine which distribution most closely matches the data set, then decide on the control limits based on that distribution's characteristics. One question may be are the data points truly independent, and if so, why would they not be normal? If they are not a natural variation, then you need to understand the underlying contributing factors to the distribution - as in preparing the Total Variation Equation.

I hope you weren't looking for a plug and chug....

#### Steve Prevette

##### Deming Disciple
Super Moderator
Re: Finding Control Limits for non-normal distribution

If you look at Dr. Shewhart's original work on SPC, you will find that he did consider skewness and kurtosis, but established that average and standard deviation were sufficient for SPC to work. He did invoke the Tchybechev Inequality, which holds for ANY distribution that no more than 11% of the data will be outside 3 standard deviations from the average. Standard UCL and LCL's should work sufficiently regardless of the distrubution.

#### bobdoering

Trusted Information Resource
Re: Finding Control Limits for non-normal distribution

He did invoke the Tchebysheff's Inequality, which holds for ANY distribution ...

From RANDOM data. Holds limited to no value for correlated or dependent distributions. All of Dr. Shewharts's work was done with many random cause distributions - as in bark thickness. Real life processes do not always exhibit pure random "chance cause" variation, causing conditions where the simplistic application of standard deviations is inadequate.

It is still critical to understand the underlying causes of the variation to assure yourself that the data related to the process is separated from any variation from measurement, etc.- as they will mask the true process distribution with a tidy, yet inconvenient normal distribution.

Also, as you collect data, you will develop one distribution. However, it is a result of many sub-distributions contributed by the factors in the Total Variance Equation. Shewhart states the more of these sub-components there are, the closer to normal the resulting distribution will be. True...masking the size and effect of the original causes and the appropriate control of each - since you can control one cause - not many - with your adjustment. Understanding the distribution of each of the contributors, as will as making the non-process contributors statistically insignificant, will put you in a better position to develop adequate control.

In fact, in some cases the more the resulting distribution appears normal, the less in control the process is!

Again, I hope you weren't looking for a plug and chug....

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