Using Other than Three Sigma Limits on a Control Chart
Authors: Marilyn & Robert Hart
Q: Most control charts use three sigma
limits. When should I use other than three sigma
limits?
A: Three sigma limits are used because
in practice they have been shown to minimize
the cost of the two mistakes: tampering with
the process when it should have been left along
(overcorrecting) and failing to take action
on a process that needs corrective action. The
probability of the first mistake is called the
alpha risk or the false alarm risk. When an
Xbar & R control chart is made with three
sigma limits, it is typically for 25 subgroups
of a small size (usually 3 to 5). In this case,
the alpha risk is around 0.07 for the Xbar chart
alone. So when fewer than 25 or more than 25
subgroups are used, it is desirable to "adjust"
the number of "sigma limits" accordingly
to keep the alpha risk similar to the 0.07 value.
Ellis Ott approached thi problem in his 1975
book "Process Quality Control" using
a method he called "Analysis of Means."
His approach depended both on the subgroup size
and the number of subgroups. A simplified method
that we developed is to use the number of sigma
limits (T) given in Table 1.
Table 1. Process Evaluation for Specialcause
Variation: Recommended Values for T for Tsigma
Limits.*
# of subgroups 
T 
2 
1.5 
34 
2.0 
59 
2.5 
1034 
3.0 
35199 
3.5 
2001500 
4.0 
*The tabular values of T may be used for the
usual case of "no standard given"
with all attribute and variables charts.
It should be noted that an Xbar & s chart
is preferred to an Xbar & R chart since
the s chart works with large subgroups as well
as small subgroups whereas the R chart is only
good for small subgroups. Also:
 At
least 25 subgroups of timeordered data should
be used to predict future performance
 If
fewer than 10 subgroups are used, the subgroup
sizes should be 25 or larger.
