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Chapter 19: B. Statistical Process Control 301
where
k =
( USL + LSL/
2 m )
.
USL LSL / 2
( )
(19.70)
Furthermore, it can easily be seen that 0 k 1, so that C pk is always less than or
equal to C p . Also, note that when the process mean m coincides with the midpoint
between the specification limits, then k = 0, and therefore C pk equals C p .
Note that C pk takes care of the centering problem only if the process standard
deviation remains the same. If the process standard deviation changes, then the
value of C pk may not change even when the process mean moves away from
the center. It can easily be seen that this will always be the scenario provided
that the distance of the process mean from the nearest specification limit in terms
of s remains the same. For example, in Table 19.11 we consider four processes with
the same specification limits but with different process means (m) and different
standard deviations (s) such that the value of C pk in each case remains the same.
Clearly, the value of C pk for each of the four processes remains the same even
though the process mean has been moving away from the center, since in each
case the distance between the process mean and the nearest specification limit (in
this example the LSL) is three times the process standard deviation. Thus, we can
say that in some ways C pk is also not an adequate measure of centering. Assuming
the process characteristic is normally distributed, Table 19.12 below gives the parts
per million (ppm) of nonconforming units for different values of C pk .
From Table 19.12, we can see that each of the processes in Table 19.11 will produce
1350 nonconforming ppm. This is only possible if the process standard deviation
is shrinking while the process mean is shifting and the natural tolerance
limits remain within the specification limits. In fact, in Table 19.11, the process
with standard deviation s = 2 would be of Six Sigma quality if the process mean
Table 19.11 Different processes with same value of C pk .
Process LSL USL Center l r C pk
1 12 36 24 24 4 1.00
2 12 36 24 22 3.33 1.00
3 12 36 24 20 2.67 1.00
4 12 36 24 18 2.00 1.00
Part IV.B.5
Table 19.12
Parts per million of nonconforming
units for different values of C pk .
C pk 1.00 1.33 1.67 2.00
PPM 1350 30 1 .001