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174 P r o c e s s C o n t r o l Q u a n t i f y i n g P r o c e s s Va r i a t i o n 175

of size 3 (leftmost column) would detect a process shift of 1 sigma (from
top row) in 9 subgroups, on average. That is, sometimes it would detect it
more quickly, and sometimes it would take more subgroups; but if you
experienced that condition many times over, the average number of sub-
groups needed to detect the shift is 9. A subgroup of size 5 would detect
the 1 sigma shift in 4 subgroups (on average). A subgroup of size 1 would
need 43 subgroups (on average). Larger subgroups will provide better
sensitivity to smaller shifts, but there is sometimes an unwarranted cost in
obtaining the additional data. The cost implications of failing to detect
that process shift as soon as possible must be weighed against the cost of
the additional data. As a general rule, subgroups of size 3 to 5 are recom-
mended, as they detect reasonable shifts of 1.5 sigma or larger fairly
quickly. When a process has been in control for a period of time, and it is
desirable to detect more subtle shift in the process (e.g., 0.5 sigma shifts),
it is recommended to use EWMA charts, such as described in Keller
(2011b), since larger subgroups are both costly and run the risk of hav-
ing special causes occur in the subgroups collected over a longer period
of time.
As a general rule, it’s best to collect small subgroups more frequently
(than larger subgroups less often). The more frequent subgroups provide
more opportunity to detect process shifts more quickly. This is particu-
larly useful when beginning to analyze a process and there is little infor-
mation concerning the types or frequency of special causes.
It is recommended that a sufficient number of subgroups be collected
to experience the process over a period of time (Keller, 2011b). If the control
chart is limited to only a few days of data, it has hardly experienced the
common cause variation that will predictably occur over longer periods of
time. In some cases, it may be desirable to define the control limits over a
short period of time, such as for a process capability study or prerelease
study for your customer. In those situations, be aware that the control lim-
its may be tighter than what the process will experience over longer peri-
ods, and you may find yourself chasing special causes for several weeks.
An additional consideration is that the constants in the table in
Appendix 1 are really only constants for a “large” number of subgroups.
Although many people quote 25 or 35 subgroups as the minimum num-
ber, this is an appropriate number for a subgroup of size five. Smaller
subgroups require more subgroups before the constants approach con-
stant value down to three decimals or so. For a subgroup of size three,
50 subgroups are recommended (Keller, 2011b). Subgroups of size one
require 150 or more subgroups, which is also recommended so that the dis-
tribution can be verified.
The control limit calculations for the averages and individuals charts
shown above are based on properties of the normal distribution. The use of
three sigma limits provides adequate detection of special causes, without

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