Robert F. Hart, Ph. D.
Marilyn K. Hart, Ph.D.

This paper addresses the use of Shewhart control charts for the improvement of healthcare processes using time-ordered continuous data (also known as variables data or measurement data). An example would be to study the variation of a process over time with an Xbar and s chart using 20 monthly subgroups.

The total variation of a process is made up of common-cause variation that is common to the whole process and special-cause variation (also known as assignable-cause variation) that is a signal of an unusual event. The principal use of the control chart is to establish statistically derived "control limits" to discriminate between special-cause variation and common-cause variation.

This is most often done with control limits calculated from the data to estimate the actual extent of common-cause variation in the process. When set in this way, the control limits are said to be calculated with "no standard given," meaning that no outside standard has been used in setting the control limits.

If an outside standard, rather than the data for the chart, is used to set the control limits, these limits are said to be set with "standard given." This standard may be, for example, from earlier process data.

Traditionally, the control limits have been "3-sigma" limits, set at three standard deviations of the variable being plotted. For an Xbar chart, for example, this would be three standard deviation of Xbar (not of X) estimated from the within-subgroup variation (i.e., from sBar) following Shewhart [Shewhart 1931].

When the calculated values for all of the subgroups fall within the limits on the control chart, it is assumed that only the common-cause variation is occurring, and the process is said to be "in control."

If the control chart shows no evidence of special-cause variation, it does not mean that none is present. It may be that not enough data were collected, the data were not collected properly, the data were not subgrouped properly, or chance in sampling yielded only the in-control data. As Grant and Leavenworth [1996, p. 46] put it:

Alternately, if the control chart exhibits evidence of special-cause variation, it does not mean for certain that a special cause of variation exists. Such a false alarm is called a "type I" error. The probability of such a false signal is called a false-alarm risk or alpha risk.

Conventional 3-sigma control limits are usually used for data sets with about 20 subgroups. For such applications the 3-sigma limits provide a good balance between providing sufficient power for the detection of special-cause variation and maintaining a satisfactory total α-risk (generally between 0.05 and 0.09 for an Xbar chart) of getting one or more points outside the control limits by random chance.

However, 3-sigma limits do not maintain a satisfactory α-risk when the number of subgroups differs greatly from 20. If, for example, "year one" is to be compared to "year two" with an Xbar chart, there will be only two subgroups. With only two plotted points, the probability of getting a point outside of 3-sigma control limits without a special cause would be much lower than with 20 points. Similarly, the probability of a false alarm with 3-sigma limits would be much higher for 200 time-ordered subgroups than for 20.

One method for maintaining a satisfactory α-risk with any number of subgroups is to use "T-sigma" limits other than 3σ limits where T increases with the number of subgroups while keeping the false alarm risk to less than 0.09 [Hart and Hart 1994, 2002], as shown in Table 1.

Table 1. Recommended Values of T for T-Sigma Limits

References

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