Control Charts for Non-Normal Distribution

[Don Winton]

I get this a lot, so I thought I would share with the group.

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Control Charts for Non-Normal process.
Message posted by Alvin Ang on March 5, 1999

Hi ppl,
I am in the IC assembly industry. I have encounter problems implementing X-Bar R chart for certain process.

I used normality test and discovered that the process is not normal.

Please advise what control chart should i use and where can i get more information on such control charts.

Thanks in advance and best regards,
Alvin

Re: Control Charts for Non-Normal process.
Message posted by Jack Tomsky on March 5, 1999

In response to Control Charts for Non-Normal process.:

Alvin, if your data is only mildly nonnormal, then if each xbar is based on enough points, the xbars will behave as nearly nomal and your limits will be approximately accurate.

If your data is distinctly nonnormal, then often a simple transformation on each data point, such as the log or square root, will result in transforming the data much closer to normality. You can calculate your limits based on the transformed data. At the end, you will have to transform back into the original units. This will result in assymetrical limits for the xbar chart.

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If your data is distinctly nonnormal…

Even though the distribution in the universe is not normal, the distribution of Xbar values tends to be close to normal. The larger the sample size and the more nearly normal the universe, the closer will the frequency distribution of averages approach the normal curve.

However, even if n is as small as 4 and the universe far from normal, the distribution of the averages of the samples will be very close to normal. Shewhart illustrates this by showing the distributions of averages of 1,000 samples of four from each of two bowls of chips, one containing a rectangular and the other a triangular distribution. Neither of these universes even faintly resemble a normal curve. However, the distribution of the samples drawn from these bowls fairly approximates normal.

The main point of Shewhart’s bowl is this. Even with great departures from normality in the universe, the distribution of Xbar values with n=4 is approximately normal. In sampling from most distributions found in nature and industry, the distribution of Xbar values will be even closer to normal.

However, it is of interest to observe that distributions similar to the rectangular and triangular distributions sometimes found in industry. Although they seldom occur as a result of production alone, they may be found as a result of production followed by 100% inspection. For example, if a production operation gives a distribution on a certain dimension which is roughly normal with a standard deviation of 0.001 and the specified tolerances on the dimension are +/- 0.001 cm, it is obvious that only about 68% of the product will meet the specification. If the production operation accurately centers the dimension at its specified nominal value, about 16% of the product will be rejected by the go gage and another 16% by the no-go gage. The distribution of the accepted product will not be far from rectangular. There will be two distributions something like the triangular, one for the product rejected by the go gage and the other by the no-go gage.

The great practical importance of the normal curve arises even more from its uses in sampling theory than from the fact that some observed distributions are described by it well enough for practical purposes. Of great practical significance is the fact that distributions of averages of samples tend to be approximately normal even though the samples were drawn from non-normal universes.

Grant and Leavenworth, Statistical Quality Control, pp. 60-62.

Regards,
Don

[Don Winton]

Yea, as I stated above, I get this a lot. I think (maybe wrongly) this is most probably due to the relatively narrow scope of most SPC or SQC training(?). To think that a one week, if that, course in control charting techniques explains the concept is a very limited view. Do not get me wrong. Most training programs are very good. There are some that suck, but that is another story for another time. I believe this was why Deming advocated having someone trained in advanced statistics on staff. To work through the muck of the less than admirable training courses and to help expound on those that were adequate.

I believe that to completely understand WHY control charts work you also at least have an understanding of HOW they work.

Regards,
Don

[Kevin Mader]

Don,

Another good post. Got me reaching for the 6th edition copy I have here. Interesting to me to see that n=4 is sensitive enough to depict an accurate presentation of the population, even in a nonnormal distribution. I didn't recall this until I reread this section of the book. This is a good point to keep in mind.

Kevin

[bobdoering]

Originally Posted by Don Winton

However, it is of interest to observe that distributions similar to the rectangular and triangular distributions sometimes found in industry. Although they seldom occur as a result of production alone...

On the contrary, in precision machining, when the process is properly controlled, one will find the uniform distribution rather readily!

Originally Posted by Don Winton

For example, if a production operation gives a distribution on a certain dimension which is roughly normal with a standard deviation of 0.001 and the specified tolerances on the dimension are +/- 0.001 cm, it is obvious that only about 68% of the product will meet the specification. If the production operation accurately centers the dimension at its specified nominal value, about 16% of the product will be rejected by the go gage and another 16% by the no-go gage. The distribution of the accepted product will not be far from rectangular.

If you had a precision machining process that was incorrectly controlled - such as using an X-bar R chart - and you truncated it by sorting, you would have a truncated normal distribution - or very close to a U distribution. But, if you have a normal distribution in precision machining to begin with, then you can rest assured it was out of control, anyway. Properly controlled, you would see the continuous uniform distribution - not the normal distribution.

Originally Posted by Don Winton

The great practical importance of the normal curve arises even more from its uses in sampling theory than from the fact that some observed distributions are described by it well enough for practical purposes.

The normal curve is a terrible estimation of a correctly controlled precision machining process, and it is the result of overcontrol. It leads to bad decisions.

Originally Posted by Don Winton

Of great practical significance is the fact that distributions of averages of samples tend to be approximately normal even though the samples were drawn from non-normal universes.

Only if they were discrete, independent distributions. The correct distribution for precision machining is dependent, and the distributions of averages of samples is - go figure - another truncated uniform distribution.

Unfortunately, you won't find that valuable information in the source cited. But you can in Statistical process control for precision machining - and it is free here! It includes the correct X hi/lo-R charting methodology for precision machining.