


The Use of ThreeSigma for Control Limits



Robert F. Hart, Ph.D.
Marilyn K. Hart, Ph.D.
Threesigma (σ) limits are generally
used for control limits on a Shewhart control
chart. Why? Well, Shewhart [1931] and Deming
[1975] would tell you that they have been shown
to work well in practice, that they minimize
the total cost from both overcorrecting and
undercorrecting. But you have to remember that
control charts traditionally have had 25 subgroups.
The standard American Society for Quality (ASQ)
paper that people used when we were still doing
control charts by hand were set up for 25 subgroups.
For normal data with a known mean and standard
deviation (the case of standard given), the
probability of getting one or more of the 25
points outside the control limits by pure chance
is around 0.065. That’s because the probability
of a given point being inside the 3σ control
limit would be 0.9973. The probability of all
25 points being in the limits is 0.9973^{25}
= 0.9346. Hence, the probability that they are
not all in is 1 – 0.9346 = 0.0654. We call
this probability the falsealarm risk, the αrisk,
or the probability of a type 1 error. This is
the probability that the process really is in
control, i.e. stable (and in our case normally
distributed), and we still get an indication
of lack of control by one or more points outside
the control limits. Note that this is close
to the typical value of α= 0.05 that is
used in hypothesis testing. When the control
limits and the corresponding mean and standard
deviation are estimated from the data (the case
of no standard given), the calculations
become even more complicated.
When control charts are maintained over an
extended period of time, the number of subgroups
grows. With more points, the probability of
points outside the control limits by pure chance
grows, i.e., the α risk grows. This problem
has been acknowledged in the American National
Standards Institute (ANSI) Standard Z1.3 [1958,
1975, p. 18]. It states that
it
is usually not safe to conclude that a state
of control exists unless the plotted points
for at least 25 successive subgroups fall within
the 3sigma control limits. In addition, if
not more than 1 out of 35 successive points,
or not more than 2 out of 100, fall outside
the 3sigma control limits, a state of control
may ordinarily be assumed to exist.
No mathematical basis is given and no suggestions
are given for the evaluation of more than 100
subgroups. Similarly, the number of subgroups
may be smaller than 25. For instance, you may
be comparing the results of three shifts. This
is Shewhart’s concept of rational
subgroups.
One method for keeping a satisfactory arisk
(close to 0.05) is to use control limits other
than 3s limits, which may vary with the number
of subgroups and possibly with the subgroup
size. The objective is to maintain the highest
possible power in detecting specialcause variation
without getting an arisk that is too high.
One approach developed by Ellis Ott [Ott, 1975;
Schilling, 1973] is called Analysis of Means.
It is a more sophisticated method of selecting
the control limits to be used for a given alpha
risk depending on both subgroup size and the
number of subgroups. It requires extensive tables
and is seldom used in practice. Studies done
by Hart and Hart [1994, March 1996, April 1996,
1997] have simplified the methodology by the
use of "Tsigma" limits, which keep
the total arisk to less than 0.09. The results
for no standard given are provided in Table
1. [Hart, 2002]
Table 1. Recommended Values of T for TSigma
Limits, No Standard Given

# of plotted points 
T 
2 
1.5 
3  4 
2.0 
5  9 
2.5 
10  34 
3.0 
35  199 
3.5 
200  1500 
4.3 

Note that 3σ limits are consistent with the
usual use of 25 subgroups. For ongoing process
improvement, 3σ limits are often used to project
into the future for use as standard given to
continuously monitor the process and seek out
specialcause variation. However, with rational
subgroups (which may use very few subgroups)
to compare only a few groups, control limits
tighter than 3σ limits are used. And to simply
judge whether a process is in control where
you have many plotted points, use control limits
that are wider than 3σ limits or you are too
likely to falsely indicate lack of control.
References
ANSI
(American National Standards Institute) Standards
Z1.1, Z1.2, Z1.3 (American Society for Quality
Control Publications B1 and B2). "Guide
for Quality Control; Control Chart Method for
Analyzing Data; Control Chart Method for Controlling
Quality during Production." New York: American
National Standards Institute, 1958, 1975.
Deming,
W. Edwards. “On Some Statistical Aids toward
Economic Production.” Interfaces, vol.
5 (4), pp. 1 – 15, August 1975.
Hart,
Marilyn and Robert Hart. Statistical Process
Control in Health Care. Duxbury, Pacific Grove,
California, 2002.
Hart,
Marilyn. "C Control Chart Limits for a
Given Number of Subgroups." Proceedings
of the Midwest Business Administration Association,
Chicago, pp. 71 – 74, March 13 – 15,
1996.
Hart,
Marilyn. "R and s Control Chart Limits
for a Given Number of Subgroups." Proceedings
of the Western Decision Sciences Institute,
Seattle, WA, pp. 335 – 339, April 2 –
6, 1996.
Hart,
Marilyn. "NP Control Chart Limits for a
Given Number of Subgroups." Proceedings
of the Decision Sciences Institute, San Diego,
vol. 2, pp. 994 – 996, November 22 –
25, 1997.
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.
Ott,
Ellis. Process Quality Control. New York: McGrawHill,
1975.
Schilling,
Edward G. "A Systematic Approach to the
Analysis of Means." Journal of Quality
Technology, pt. 1, vol. 5(3), pp. 93 –
108, July 1973, and pts. 2 and 3, vol. 5(4),
pp. 147 – 159, October 1973.
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.


