# Does the data in a control chart have to be normally distributed?

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#### DJN

This may be a silly question, but here goes. Does the data in a control chart have to be normally distributed for the CP and CPk figures to be valid? The process is stable i.e. in statistical control.

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#### Cristi?nC

Here are my 2 cents:

Control charts and Cp and Cpk figures are closely related, but some differences are important to notice.

First: Data set for your capability analysis must follow a normal distribution. It is an underlying assumption.

Second: Control charts assumes normal distribution when de process is stable. If you are using an Xbar / R chart, with a sample size of 4 or 5 at least and your process is stable, then you can be confident with regard to the normality distribution of your data (There is a statistical theorem behind this). Individual and Moving Range charts are more sensitive to departures from normality, therefore is advisable to check for normality (even if your process is stable)

Practical conclusion: If your chart is an I/MR one, then the normality in the control chart is enough for the Cp and Cpk indexes to be valid, but, if your chart is an Xbar/R one, then you must check for normality of your data before analysing the capability indexes.

Hope this helps.

D

#### DJN

Thanks to both. That has cleared up that point.

David

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#### Bill Ryan - 2007

First of all let me state that I am not that strong in the statistical field.

When I use a Control Chart (X-bar & R) with a "natural" limit of zero (True Position, Flatness, Profile, Concentricity, etc.), what "rule" is being violated? I ask because the "underlying data" is NOT normally distributed (typically skewed right), yet the charts sure seem to work!!!!

Bill

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#### M Greenaway

Bill

Dont see why your examples are necessarily skewed, or naturally limited to zero.

Are you using tolerance based control limits or process based ?

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#### Bill Ryan - 2007

Are you using tolerance based control limits or process based ?

I (we) use process based control limits but of what use is a LCL of less than zero when the dimension cannot "naturally" be less than zero? In other words, I might only have room for -1s between my mean and zero and then have room for up to, say, +5s between my mean and the USL (yet my UCL is still only +3s from the mean).

Bill

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#### Laura M

Bill Ryan said:

When I use a Control Chart (X-bar & R) with a "natural" limit of zero (True Position, Flatness, Profile, Concentricity, etc.), what "rule" is being violated? I ask because the "underlying data" is NOT normally distributed (typically skewed right), yet the charts sure seem to work!!!!

Bill

Atul is right. Control charts work because even tho the individuals are not notmally distributed, the subgroup average tend toward normality. I believe its the Central Limit Theorem - but I'm sure another stat geek will correct me if I'm wrong. Of course depending on the amount of skewness, it won't be completely normal unless subgroup sizes are unusually large.

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#### M Greenaway

I get you Bill.

Did you see my complaints response times charts posted in this topic, it was just as you describe.

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#### Bill Ryan - 2007

Martin, thanks for reminding me of your earlier post and I have just reread the responses. You're right in that it closely follows my question.

Laura - I believe you're correct with the Central Limit Theory stating that subgroups tend to approach the Gaussian distribution.

The issue I've got is time. When we enter the production mode of a product's life, we don't have the time to have enough data to approach anything resembling "normalcy" yet I need to understand my process' output. Typically, I will take my 25 subgroups and plot the X-bar & R chart (I personally don't like I & MR charts because they tend to promote over-adjustment). If all is "well" in the R chart, followed by the X-bar chart, I implement the chart into the inspection process.

I guess I'm saying that the X-bar & R chart is the tool of choice on our production floor and I don't go analyzing all the different charts we have when Cpk indeces are greater than 2 (I have too many that are less than 1!! And yes those get some of the other statistical tools "thrown" at them). Typically, the charts with an index of greater than 2 gets "backed off" as far as frequency of inspection (or completely dropped, if not a safety feature or customer requirement).

Here I go rambling again. Bottom line is that I'm "uncomfortable" with my knowledge of transformation techniques. Yes I can substitute numbers into just about any formula but when I don't grasp the concept, I can't justify "playing games" with data to my management or customers. Maybe it's time I took a statistical refresher course or something.

Thanks for the replies!!!!!!

Bill

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#### Laura M

Oops - I answered why Control Charts work - but you're original question was do Cp and Cpk count if distribution isn't normal.
Each type of distribution (exponential, normal, etc) has its own standard deviation formula. Certainly the percentages of "parts out of spec" do not apply - this is based on a normal distribution. So I would plot the individuals against the spec limits. Is it then possible to estimate how much of a mean shift you would need for the highest point to reach the spec? I would want to make sure the control chart is robust enough to pick up that shift.

You say its in control - so I take it it's not 100% inspected, and you don't have "flyers" outside the spec limit?