Black Belt Bound
Simsack finds himself up against the railings.

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

When he joined Greer Grate & Gate as quality manager, Hartford Simsack told his plant manager and brother-in-law, Rock DeBote, that he'd really prefer the job of production manager. However, because DeBote had given Simsack a job after he was released from prison, he didn't complain too vociferously. Instead, he set about trying to convince his boss of his eminent suitability for the production job.

Simsack's son, who is enrolled in martial arts classes, has quickly moved to Black Belt status, and this gives Simsack an idea. "If I could only be a Black Belt, that would impress DeBote," he thinks. "And it can't be too hard, if an 11-year-old kid can do it."

Simsack selects a project that he believes will save the company almost $1.5 million. He tracks the percentage of defects on extruded wrought-iron railings--the plant's primary output--and, by using a p-chart, anticipates clear analytical support for his assertion about savings.

The railings are produced in lots varying from 50 to 200 capacity. Simsack dismisses the relevance of variable sample sizes for p-charts. "I'll just use the average sample size," he tells himself. But when he looks at the data on his printout, the changing control limits confuse him.

Greer Grate & Gate p-Chart

While Simsack examines the printout, Rock DeBote stops by and finds the charts unclear. "Why do the control limits go up and down like this?" he asks. "Well," Simsack blurts out with mock confidence, "obviously, this can be attributed to application of the binomial theorem." What in the world does the binomial theorem--a term he heard in one of his statistics classes--have to do with the issue of control limits?

The answer is nothing. Simsack is once again bluffing.

However, p-charts are in fact based on the binomial theorem, which has everything to do with the appearance of the charted control limits:

If all samples were the same size--for example 100, the n-bar would equal 100. In Simsack's case, the sizes vary from 50 to 200. The printout shows that the average sample size is 1,194.32. The most common rule is that no adjustment will be made to the control limits if the sample sizes vary by no more than 25 percent. On Simsack's charts, it's clear that sample size varies by more than 25 percent, and an adjustment to the limits must be made.

Mathematically, the adjustment involves substituting the actual n of the sample with n-bar. This is done only when n varies by 25 percent or more from n-bar. Next month's column will offer an intuitive explanation of this methodology.

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. His Web site is www.pqsystems.com. Letters to the editor regarding this column can be e-mailed to letters@qualitydigest.com.