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Departments: SPC Guide

Photo: Michael J. Cleary, Ph.D.

  
   

Averting 100-Percent Inspection

Michael J. Cleary, Ph.D.
mcleary@qualitydigest.com

 


Greer Grate & Gate has secured a major National Park Service contract that involves making iron gates to replace deteriorating older models and installing them in all of the national parks in the United States.

Because this is the largest order in the company’s history, its president, Rock deBote, is determined to ensure that the gates will be produced at the highest quality levels. (Rock deBote is Hartford Simsack’s brother-in-law, who hired Simsack as a family favor after he’d been released from the state penitentiary.)

After consulting with Simsack, deBote decides to hire four new quality inspectors, who will be responsible for charting 10 characteristics of the Park Service’s gates. By turning the project over to the quality department, deBote can return to more important matters. He will, for example, be working seriously on his putting, now that he’s found a carpeted conference room and a sign that reads, “Keep Out: Meeting in Progress” to hang on the closed door.

Left to his own devices, Simsack experiences a moment of panic. “Let’s see: 10 characteristics for thousands of gates.” Stumped by the project’s sheer mathematics, he decides to brush up on X-bar and R charts by taking yet another statistics course from his mentor, Dr. Stan Deviation, at the local community college.

Distracted by his preoccupation with the monumental task that lies ahead, Simsack doesn’t pay attention to Deviation’s lecture on the lower control limit for the range section of a control chart. When he hears the question, “Do you all understand why D3 is equal to zero from a sample size of two to six?” Simsack nods along with the rest of the class because he doesn’t want Deviation to know that he hasn’t been paying attention. He believes he’ll never need to know this, anyway.

In fact, the explanation for the fact that D3 is equal to zero (0) for a sample size of two to six is one of the following:

a) A small sample size will lead to higher R values.

b) As all Black Belts understand, the standard error is equal to the standard deviation of the population, divided by the square root of the same size.

c) Unless at least seven people show up for your next dinner party, nothing (0) will be served.

d) A sample size of seven or more will generate a distribution of range values that looks normal.

(Answer to “Averting 100-Percent Inspection” )

The correct answer is d.

Intuitively, this is logical because with a sample size of two, the probability of having a range of zero is high, but with a sample size of three, the probability decreases. As the sample size increases, the probability of a range equal to zero continues to decrease. Thus, with a small sample size, the distribution of range values will be skewed to the right.

As the sample size increases, the chance of getting zero decreases and the probability of getting both very small and very large values in the same sample increases, making the range values larger and more consistent.

With a small sample size (i.e., n less than seven), there will always be an upper control limit for R, but the lower control limit will be zero.

Increasing the sample size will make the distribution of R’s approach a normal distribution.

Thus, the range section will have a lower control limit that’s greater than zero.

About the author

Michael J. Cleary, Ph.D. is a professor emeritus at Wright State University and founder of PQ Systems Inc. He has published articles on quality management and statistical process control in a variety of academic and professional journals. Letters to the editor regarding this column can be e-mailed to letters@qualitydigest.com.