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What Charts to Show?

Bill Farrell, Ph.D.
Senior Analyst
Sutter Health

Sutter Health is a family of not-for-profit hospitals and physician organizations that share resources and expertise to advance health care quality. Serving more than 100 communities in Northern California, Sutter Health is a regional leader in cardiac care, cancer treatment, orthopedics, obstetrics, and newborn intensive care, and is a pioneer in advanced patient safety technology.

Authors writing about (and people teaching) statistical process control often tend to show SPC charts in pairs: If there’s an Individual (or Item) chart on top, they show a moving range chart on the bottom; if there’s an xBar chart on top, they add an S chart (standard deviation or sigma) on the bottom. Software for SPC abets this tendency: in addition to buttons on the menu bar for I and xBar charts, there will be buttons for I/mR and xBar/S charts; one click and you’ve got your paired charts.

Does the second chart aid in the interpretation of the main chart? The answer is No. Is it necessary for the second chart to be in control before we can accept the main chart? Again, the answer is No. Here is a quote from a respected author on the topic (Donald Wheeler, Advanced Topics in Statistical Process Control, p. 110):

The ultimate argument in favor of the mR chart is not that it improves the ability of the I chart to detect signals, but that it serves as a reminder of the correct way of computing limits for the I chart. When the mR chart is used with the I chart it serves both as a reminder of the correct method, and as a signal to others that the user knows the correct method.

Because there are, in theory, several methods for computing SPC limits (median deviations, root mean square deviations, moving ranges), Wheeler and his academic/industrial colleagues may believe they have to demonstrate that they “know the correct method.” But the correct method is built into modern SPC software, making these secondary charts distracting at best and misleading at worst. To further complicate the issue, rule violations are defined differently for these secondary charts. There is only one way an mR or S chart can be out of control: a single point above 3 sigma. For xBar and I charts, on the other hand, rule violations include two out of three points beyond 2 sigma, six points ascending or descending, etc.

To reiterate: an out-of-control point on an S or mR chart says nothing about how (or whether) we can interpret the main xBar or I chart. The fact that the range went from 5 in Quarter 1 to 10 in Quarter 2 has already been factored into the limits for the I chart. The fact that the standard deviation (actually, the bias-corrected standard error) went from 3.6 in Quarter 1 to 7.5 in Quarter 2 has already been factored into the limits for the xBar chart.

I and xBar charts track the variability of a process. S and mR charts track the variability of the variability of a process. Why would anyone be interested in this? Donald Wheeler might want to show us that he knows what he’s doing and remind us of how SPC limits are calculated, which is fine. Someone on a factory floor might be interested in maintaining a narrow range of output on the fender-stamping machine, which is also fine. In healthcare, though, these secondary charts would seem to raise many more questions than they answer.