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Photo: Michael J. Cleary, Ph.D.


The Long and Short of Kurtosis
Simsack strives for Black Belt glory.

Michael J. Cleary, Ph.D.


Story update 9/17/2018: The formula for kurtosis was missing "n" in the denominator. That has been fixed.

Hartford Simsack’s failed attempt to harness the power of p-charts in his quest for the elusive Black Belt hasn’t daunted him. Premature boasting about his accomplishments—and then falling short—has renewed his determination to save Greer Grate & Gate enough money so that he’ll receive the acknowledgment he so richly deserves for his demonstrated statistical prowess. In the meantime, his son, who achieved black belt status at age 11, has given up his martial arts and gone back to watching cartoons and eating jelly doughnuts.

Line five in Simsack’s plant, which assembles wrought-iron fencing for interior use, had 421 defects last month, each of which cost GG&G $150.22. Although Simsack had originally aimed for savings in the millions, his sights have come down slightly, and a savings of $63,242.62 that could be realized from this process piques his interest. Besides, he thinks he has the answer that will, in fact, generate these cost savings.

Simsack takes unprecedented action: He discusses the defect with the line process operator, who has a fairly clear idea about the defect’s source. Simsack sees an immediate opportunity to take credit for fixing something. With his reputation for genius, who would doubt that it was his own idea?

Armed with a histogram showing that, with a Cpk of 1.0, the process is barely capable, he mentions to his boss Rock deBote what a great job he’s done using the power of statistics to single-handedly improve the company. As he spreads out the histogram, deBote notices other statistics relating to the process, including a kurtosis of -0.62. Noticing his boss staring intently at the kurtosis figure, Simsack attempts to pass over the number’s meaning because he has no idea what significance it holds. “Of course the kurtosis is negative,” he says, shaking his head in dismay, “but my plan is to improve that by 100 percent.” Does this make sense in the ongoing improvement of the process?

No, it doesn’t make sense.

As anyone who hasn’t fallen asleep in statistics class knows, kurtosis is a measure of the combined weight of the tails relative to the rest of the distribution. Sometimes it’s referred to as the “fourth movement.” The formula for kurtosis is:

A normal distribution has a kurtosis of zero and is known as “mesokurtic.”

On the other hand, if a distribution is tighter and taller than a normal distribution, the kurtosis would be a positive number. This distribution is referred to as “leptokurtic” because it has a long tail like a kangaroo.

Finally, for a distribution that is flatter than a normal curve, the kurtosis would be negative, or “platykurtic” with a short tail—like a platypus.

If you want to hurl a sophisticated invective at a co-worker, accuse him or her of platykurtic (i.e., non-normal) tendencies.

About the author

Michael J. Cleary, Ph.D., founder and president of PQ Systems Inc., is a noted authority in the field of quality management and a professor emeritus of management science at Wright State University in Dayton, Ohio. A 29-year professorship in management science has enabled Cleary to conduct extensive research and garner valuable experience in expanding quality management methods. He’s published articles on quality management and statistical process control in a variety of academic and professional journals. Note: Kangaroo and platypus drawings are adapted from similar drawings in Donald L. Harnett’s, Statistical Methods, Third Edition (Addison-Wesley Publishing Co., 1982).

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