502 Statistical Methods
The lower control limit is 128.336, the average of the subgroup averages is
134.446, and the upper control limit is 140.556. Note that although most of
the points in the mean chart lie between the control limits, four points (obser-
vations 1, 9, 17, and 20) lie outside the limits. This process is not in control.
Because the process is out of control, you should attempt to identify the
special causes associated with each out-of-control point. Observation 1, for
example, has too much coating. Perhaps the coating mechanism became
stuck for an instant while applying the spray to that item. The other three
observations indicate too little coating on the associated products. In talking
with the operator, you might learn that he had not added coating material to
the sprayer on schedule, so there was insuffi cient material to spray.
It is common practice in SPC to note the special causes either on the front
of the control chart (if there is room) or on the back. This is a convenient
way of keeping records of special causes.
In many instances, proper investigation leads to identifi cation of the spe-
cial causes underlying out-of-control processes. However, there might be
out-of-control points whose special causes cannot be identifi ed.The Range Chart
The x chart provides information about the variation around the average
value for each subgroup. It is also important to know whether the range
of values is stable from group to group. In the coating example, if some ob-
servations exhibit very large ranges and others very small ranges, you might
conclude that the sprayer is not functioning consistently over time. To test
this, you can create a control chart of the average subgroup ranges, called a
range chart. As with the x chart, the width of the control limits depends on
the variability within each subgroup. If s is known, the control limits for
the range chart areLCL 5 D 1 sCenter line 5 d 2 sUCL 5 D 2 s
and if s is not known, the control limits areLCL 5 D 3 RUCL 5 D 4 R
where d 2 , D 1 , D 2 , D 3 , and D 4 are the correction factors from Table 12-3, and R
is the average subgroup range. It’s important to note that the x chart is valid
only when the range is in control. For this reason the range chart is usually
drawn alongside the x chart.