Page 228 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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Some Possible Transformations 213
3σ control limits are
√
1 − p 3 1 − p
UCL = + , (14.1)
p p
√
1 − p 3 1 − p
LCL = − . (14.2)
p p
This is because when Z is geometrically distributed we have that
z
F (z; p) = P (Z = z) = 1 − (1 − p) , z = 1, 2,..., (14.3)
and the mean and variance of Z are given by
1 − p 1 − p
E (Z) = , Var [Z] = . (14.4)
p p 2
4
The control limits have been shown by Kaminsky et al. to be better than the tra-
ditional c chart when the underlying distribution is geometric. However, this type
9
of control limit has some serious problems. Xie and Goh noted that for a geometric
distribution the problem with the 3σ LCL is very serious as the limit always takes a
negative value, which is physically meaningless. This can be seen from the fact that
for the LCL in equation (14.2) to be greater than zero it is necessary that

1 − p 1 − p
> 3 (14.5)
p p 2
implying that p < −8, which is impossible as p is always positive. Note that when
a G chart is used to monitor a high-quality process, p is interpreted as the process
fraction nonconforming level.
To solve the problems associated with the 3σ limits of the geometric distribution,
one method is to transform the individual observations so that the transformed data
are more closely modeled by the normal distribution. It is then possible to proceed
with the usual charting and process monitoring using the transformed data.
It should be noted that most authors using geometrically distributed quantities
in process monitoring make use of the probability control limits which are more
appropriate. 1,3 A drawback of this approach is that the control limits will no longer be
symmetrical about the centerline. Furthermore, the advantage of process monitoring
using transformed data is that run rules and other process-monitoring techniques such
as EWMA and CUSUM can be applied directly as most of the procedures assume the
normal distribution.

14.3 SOME POSSIBLE TRANSFORMATIONS

There are a few standard transformations, such as the square root, inverse, arcsine,
inverseparabolicandsquareroottransformations,thathavebeenusedfornon-normal
data. 10−13 For the geometric distribution, Quesenberry’s Q transformation and the
arcsine transformation have been used before. A simple, but accurate, double square
root transformation is proposed in this section following a brief discussion of some
other possibilities.