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Departments: SPC Guide

Photo: Michael J. Cleary, Ph.D.


Jagged Control Limits
A lesson for Simsack in life's ups and downs

Michael J. Cleary, Ph.D.



Hartford Simsack, longing for the Black Belt status inspired by his son's tae kwon do classes, believes that if he can pull off a project that will save Greer Grate & Gate money--preferably lots of money--his boss, Rock DeBote, will consider advancing him to the position of production manager. Simsack begins by trying to think of big-money projects and settles on investigating defects that are often found in railing connectors. This problem has plagued the company for years, and each defect costs $1.12 to rework. Simsack figures that at the current rate of railing production, the defect costs the company $6.2 million per year.

If he could get the defect rate down by even a half of one percent, he could save the company $1.2 million. "Now that's worth doing," he tells himself.

Things are looking up for Simsack, although the problem of jagged control limits that we considered in last month's column continues to plague him--and worry his boss.

DeBote notes that sample 21 would be out of control if the limits hadn't been adjusted, and sample 16 would be considered in control if Simsack hadn't manipulated the control limits. "You can't just move control limits around like pieces of furniture," he tells Simsack. "You've made them fit the outcome you wanted, rather than letting them be set by the data."

Simsack, of course, is speechless--or almost speechless--after his boss's outburst. "This p-chart is based on the binomial theorem," he stammers, remembering that this explanation had worked before. But this time his boss isn't satisfied. "Control limits don't just go up and down on their own like a yo-yo!" he exclaims. Remembering his wife's expression about life's exigencies, Simsack blurts out, "They do if the average is made of ups and downs."

Could it be that Simsack is right in his assessment?

Alas, Simsack has once again struck out.

He's correct about the binomial theorem as the basis for the p-chart. But why do control limits become wider when a sample size is small and tighter when the sample size is large? I've illustrated this in the classroom by selecting two students, whom we'll call Arvid and Noah. Arvid is invited to select 20 students randomly and ask them whether they believe the economy is improving. Then Noah does the same, this time with 200 students.

If Arvid asks 20 students and Noah asks 200 students, who would be more accurate? Of course, the more accurate would be Noah because he has more data. This would relate to tighter control limits on a control chart and would mean wider control limits for Arvid. When I ask students which of the two would be more accurate, they respond Noah because he has a much larger sample size than Arvid.

To demonstrate this mathematically, consult the formula for control limits:

If n becomes larger, the control limits become tighter. When n becomes smaller, the opposite is true.

In the example, n-bar , the average sample size, is 1,194. Sample 16 is 2,306, which is more than 25 percent of n¯ , and sample 21 is 392, which is less than 25 percent of n-bar . The readjustment in control limits can be seen on the chart above.

A large sample size renders greater confidence in the sample results; therefore, control limits would be tighter. A sample size of less than 25 percent of the average sample size would generate less confidence in the sample results.

About the author

Michael J. Cleary, Ph.D., is a professor emeritus at Wright State University and founder of PQ Systems Inc. Letters to the editor regarding this column can be e-mailed to letters@qualitydigest.com.