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Chapter 19: B. Statistical Process Control 269
1. A greater variability or a tendency of perpetuating a greater variability
in the output of the process is occurring. This may be due to new
material of undesirable low quality or a difference between the shifts.
Immediate attention to detect special causes is warranted.
2. The measurement system has changed.
A downward run or a run below the center line in the R chart indicates:
1. A smaller variability or a tendency of perpetuating a smaller variability
in the output of the process is occurring. This is usually a good sign for
the process. A thorough investigation should be made so that similar
conditions are maintained as long as possible. Similar conditions should
be implemented in the process elsewhere.
2. The measurement system has changed.
A run relative to the mean of the X – chart indicates:
1. The process average has changed or is still changing.
2. The measurement system has changed.
Shewhart X – and R Control Chart When Process Mean m and Process
Standard Deviation s Are Known
Step 1. Calculate x – i and R i for the ith sample for i = 1, 2, 3, . . . , m, where
m is the number of samples collected during the study period.
Step 2. Calculate the control limits for the X – control chart:
UCL = m +
LCL =
s
3
n
(19.14)
CL = m (19.15)
m 3 s n . (19.16)
Part IV.B.3
Note: Instead of calculating three-sigma limits, we can also
calculate the probability limits at the desired level of significance
a, simply by replacing 3 with z a/2 in equations (19.14) and (19.16).
Step 3. Calculate the control limits for the R control chart.
Recalling that s = R/d 2 and s R = d 3 s, we have
UCL = d s + 3s
2
= d s + 3d
s
( )
= d + 3d
= D s
2 3
2 3
2
R
s
(19.17)