Shewhart Control Charts for Healthcare with Other than 3Sigma 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 timeordered 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 commoncause variation that is common
to the whole process and specialcause variation
(also known as assignablecause 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 specialcause variation and commoncause
variation.
This is most often done with control limits
calculated from the data to estimate the actual
extent of commoncause 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
"3sigma" 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 withinsubgroup 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 commoncause variation
is occurring, and the process is said to be
"in control."
If the control chart shows no evidence of specialcause
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 incontrol data. As Grant and
Leavenworth [1996, p. 46] put it:
When
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 specialcause 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 falsealarm
risk or alpha risk.
Conventional 3sigma control limits are usually
used for data sets with about 20 subgroups.
For such applications the 3sigma limits provide
a good balance between providing sufficient
power for the detection of specialcause variation
and maintaining a satisfactory total árisk
(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, 3sigma limits do not maintain a satisfactory
árisk 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 3sigma control
limits without a special cause would be much
lower than with 20 points. Similarly, the probability
of a false alarm with 3sigma limits would be
much higher for 200 timeordered subgroups than
for 20.
One method for maintaining a satisfactory árisk
with any number of subgroups is to use "Tsigma"
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 TSigma
Limits

No standard Given

Standard Given

# of plotted points 
T

T

1 
NA

2

2 
1.5

2

3  4 
2.0

2

5  9 
2.5

2.5

10  34 
3.0

3.0

35  199 
3.5

3.5

200  1500 
4.0

4.0

References
Grant,
Eugene, and Richard Leavenworth. Statistical
Quality Control, 7th ed. New York: McGrawHill,
1996.
Hart
M and Hart R. Statistical Process Control for
Health Care, Pacific Grove, CA: Duxbury, 2002.
Hart,
Marilyn, and Robert Hart. "XBar 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.
Shewhart,
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 robthart@aol.com
or (541)4120425.
