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164 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 165

random data from the process, or a box of parts shipped from your
supplier, since the time sequence of the data has been lost.

Variable Control Charts
In SPC, the mean, range, and standard deviation are the statistics most often
used for analyzing measurement data. Control charts are used to moni tor
these statistics. An out-of-control point for any of these statistics is an indica-
tion that a special cause of variation is present and that an immediate inves-
tigation should be made to identify the special cause.

Average and Range Control Charts
Average charts (usually called X charts in reference to the symbol used to
designate the averages in Table 9.1) are statistical tools used to evaluate the
central tendency of a process over time. Range charts are statistical tools
used to evaluate the dis persion or spread of a process over time.
Average charts answer the question “Has a special cause of variation
caused the central tendency of this process to change over the time period
observed?” Range charts answer the question “Has a special cause of vari-
ation caused the process distribution to become more or less erratic?”
Average and range charts can be applied to any continuous variable like
weight, size, cycle time, error rate, and so on, subject to the conditions
necessary for rational subgroups.
A predefined subgroup size is defined for the given process. Typical
subgroup sizes are three or five observations in the subgroup.
The average and range are computed for each subgroup separately, then
plotted on the control chart. Each subgroup’s statistics are compared with
the control limit, and patterns of variation between subgroups are analyzed.

Subgroup Equations for Average and Range Charts

sum of subgroup measurements
X =
subgroup size

u
R R = largest in subgroup − smallest in subgroup
Table 9.3 contains 25 subgroups of five observations each. The average
and range for each subgroup are shown in the table.
Control Limit Equations for Average and Range Charts Control limits for both the
average and the range charts are computed such that it is highly
unlikely that a subgroup average or range 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 standard
deviations of the mean from the grand average; the control limits for

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