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286 Part IV: Quality Assurance
ˆ c + c + c + ... + c
1 2 3
l = c =
m
m
.
(19.45)
Then the three-sigma control limits for the c control chart are defined as follows:
UCL = c +
3 c
(19.46)
CL = c (19.47)
LCL = c 3 c
(19.48)
Note that for small values of c – ( 5) the Poisson distribution is asymmetric; the
value of a type I error (a) above the UCL and below the LCL is usually not
the same. Thus, for small values of c – it may be more prudent to use probability
control limits rather than the three-sigma control limits. The probability control
limits can be found by using Poisson distribution tables.
To illustrate the construction of a c control chart using three-sigma control
limits we consider the data in Table 19.7 of Example 19.9.
Part IV.B.4
EXAMPLE 19.9
A paper mill has detected that almost 90 percent of rejected paper rolls are due to nonconformities
of two types, holes and wrinkles in the paper. The Six Sigma Green Belt
team in the mill decided to set up control charts to reduce the number of or eliminate
these nonconformities. To set up control charts, the team decided to collect some data
by taking random samples of five rolls each day for 30 days and counting the number of
nonconformities (holes and wrinkles) in each sample. The data are shown in Table 19.7.
Set up a c control chart using these data.
Solution:
Using the data in Table 19.7, the estimate of the population parameter is given by
30
c i
ˆ i=
1
222
l = c = = = 74 . .
30 30
Therefore, using equations (19.46) through (19.48), the three-sigma control limits of the
phase I c control chart are given by
UCL = 74 . + 3 74 . = 1556 .
CL = 74 .
LCL = 74 . 3 74 . = – 076 . = 0.
Note that if LCL turns out to be negative, as in this example, then we set LCL at zero
since the number of nonconformities can not be negative. The c control chart for the
data in Table 19.7 is shown in Figure 19.13.
Continued