Shewhart Control Charts for Healthcare with Other than 3-Sigma Limits
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
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
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
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:
we say, "This process is in control,"
the statement really means, "For practical
purposes, it pays to act as if no assignable
causes of variation were present."
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 á-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
á-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
One method for maintaining a satisfactory á-risk
with any number of subgroups is to use "T-sigma"
limits other than 3ó 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
No standard Given
|# of plotted points
|3 - 4
|5 - 9
|10 - 34
|35 - 199
|200 - 1500
Eugene, and Richard Leavenworth. Statistical
Quality Control, 7th ed. New York: McGraw-Hill,
M and Hart R. Statistical Process Control for
Health Care, Pacific Grove, CA: Duxbury, 2002.
Marilyn, and Robert Hart. "X-Bar Control
Limits for an Arbitrary Number of Subgroups."
Proceedings of the Western Decision Sciences
Institute. Maui, Hawaii, pp. 637 – 639,
March 29 – April 2, 1994.
Walter A. Economic Control of Quality of Manufactured
Product. Princeton, NJ: Van Nostrand Reinhold
Co., 1931. (Republished in 1981 by the American
Society for Quality Control, Milwaukee, WI).
For more information, contact Drs. Robert and
Marilyn Hart at email@example.com