Page 182 - The Handbook for Quality Management a Complete Guide to Operational Excellence

168 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 169

the subgroup averages, the centerline of the averages chart). This is an
important point: using the grand average would introduce special cause
variation if the process were out of control, thereby underestimating the
process capability, perhaps significantly.
The calculated standard deviation for each subgroup is shown in
Table 9.3.

Control Limit Equations for Average and Sigma Charts Control limits for both
the average and the sigma charts are computed such that it is highly
unlikely that a subgroup average or sigma from a stable process would
fall outside of the limits. All control limits are set at plus and minus
three standard deviations from the centerline of the chart. Thus, the
control limits for subgroup averages are plus and minus three stan-
dard deviations of the mean from the grand average. The control lim-
its for the subgroup sigmas are plus and minus three standard
deviations of sigma from the average sigma.
These control limits are quite robust with respect to non-normality in
the process distribution.
To facilitate calculations, constants are used in the control limit equa-
tions. The table in Appendix 1 provides control chart constants for sub-
groups of 25 or less.

Control Limit Equations for Sigma Charts Based on S-Bar

sum of subgroup sigmas
s =
number of subgroups
L LCL = B s
3
UCL = B s
4
To illustrate the calculations and to compare the range method with the
standard deviation results, the data used in the previous example will be
re-analyzed using the subgroup standard deviation rather than the sub-
group range.
The control limits are calculated from the Table 9.3 data as follows:

.
sum of subgroup sigmas 155 45
s = = = 6 218
.
number of subgroups 25
s
0
LCL = B s = × 6 218 = 0
.
s 3
=
UCL = B s = 2 089 6 218. × . = 12 989
.
s 4
Since it is not possible to have a subgroup sigma less than zero, the LCL is
not shown on the control chart for sigma.

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