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288 Part IV: Quality Assurance
If economic factors and time allow, one should take samples or inspection
units large enough so that the lower control limit is positive. The lower control
limit can be positive only if c – > 9. This means that the sample size should be such
that it can catch nine or more nonconformities with high probability. An advantage
of having a positive lower control limit is that it will allow us to see the conditions
under which nonconformities are very low and consequently will give us
the opportunity to perpetuate these conditions on-site and implement them elsewhere
in the industry.
As noted earlier, the size of the inspection unit is usually determined based
on what is convenient. However, to determine the actual inspection unit size,
one should also take into consideration the statistical characteristics of the
process, such as the average run length, the state of the process (that is, whether
the process has deteriorated or improved), and other factors that may require
us to increase or decrease the sample size. Thus, when using control charts for
nonconformities, particularly in phase I, situations may arise where the sample
size may vary. Since in c control charts the sample size is fixed, when the
sample size is variable we use the u control chart instead of the c control chart.
We discuss the u control chart in the following section.
If the samples consist of n inspection units, then the control limits for the c
control chart are given by
UCL = nc +
3 nc
(19.49)
CL = nc (19.50)
LCL = nc 3 nc . (19.51)
Part IV.B.4
The u Chart
The u control chart is essentially the c control chart except that the u control chart
is always based on the number of nonconformities per inspection unit. In other
words, the actual sample size may not be equal to one, or may vary, but the control
limits of the u chart are always determined based on one inspection unit. Thus, if
n is constant then one can use either a c chart or a u chart. For the u chart the center
line is determined by u – = – c /n and the three-sigma control limits are given by
UCL = u+
3 u/ n
(19.52)
LCL =
If the sample size varies, then we define u – as
u
3 u/ n. (19.53)
c + c + ... + c
1 2
u =
n + n + ... + n
1 2
m
m
=
c
n
(19.54)