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Departments: SPC Guide

Photo: Michael J. Cleary, Ph.D.


E2 in the ER
Which stat should be used?

Michael J. Cleary, Ph.D.


Dr. Cal S. Terol, emergency room administrator for St. Recover in the Long Run Hospital, loves data. His favorite data displays are those that no one can understand because their confusion gives him an opportunity to demonstrate his own statistical erudition. He loves the language of statistics even more than the clinical jargon that surrounds him in the ER. After all, everyone in the ER understands medical vocabulary, but statistics remain an arcane mystery for which he’s the self-appointed high priest.

Terol frequently points to displays of clinical data in the ER administrative office, mumbling terminology no one comprehends. “Oh, that chart demonstrates kurtosis and skewness in the process,” he declares. Whenever necessary, he obfuscates data’s meaning further by mixing terminology that has little to do with analysis.

So, when he has an opportunity to attend a seminar about control charts and their application in health care, he leaps at the chance. The instructor introduces control limits and teaches participants how to calculate them. Terol is considering ways to confuse his colleagues with this formula when the instructor introduces individual moving-range charts and the role E2 plays in calculating control limits. “Bingo!” he thinks.

Upon his return to the hospital, Terol holds forth to whomever will listen, dropping X-bar and R chart terminology and sprinkling E2 throughout his conversation. He’s caught short, however, when one of the nurses asks him to explain why he’d use E2 for an individual moving range chart rather than A2, which is used for X-bar and R charts in calculating control limits. His resourcefulness provides an immediate response--one that leaves the nurse shaking her head--but Terol himself doesn’t understand the distinction between the two.

Which of the following is a correct statement about E2 and A2 with respect to creating an individual moving range chart?

a) A2 is always the correct factor to use because it comes alphabetically before E2, and one should use the first appropriate factor.

b) E2 is the correct factor because it estimates 3 sigma above and below the mean (X-bar).

c) For individual moving range charts, A2 and E2 will give the same answer.

Answer b is the correct response.

In manufacturing, with its emphasis on mass production, X-bar and R charts are the most common form of control charts. Each sample is at least of the size of two, and five is the most common. In these cases, upper and lower control limits are calculated by these formulas:

A2 is a weighting factor that’s used to estimate upper and lower control limits. They will be three standard errors above the process average and three standard errors below it.

Standard error =

Applications of X-bar and R charts occur in health care, but individual moving range charts are far more common (sample size equals one).

One might track a patient’s blood pressure, or any other critical measure, on a daily basis. In charting this data, a different weighting factor E2 is used.


UCLX = upper control limit for the individuals

LCLX = lower control limit for the individuals

k = number of samples

E2 equals the weighting factor used to estimate three standard deviations above and three below the process average.

Dr. Cal S. Terol might be a great health care administrator, but when it comes to statistics, his number is up.

About the author

Michael J. Cleary, Ph.D., founder and president of PQ Systems Inc., is a professor emeritus of management science at Wright State University in Dayton, Ohio. Letters to the editor regarding this column can be sent to letters@qualitydigest.com.