Page 147 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

2:57
August 31, 2006
JWBK119-10
Char Count= 0
132 Process Capability Analysis for Non-Normal Data with MINITAB
PCA is of the utmost importance as it directly affects the management’s decision-
making.
The accuracy of process capability estimation for variable data depends largely
on how well the theoretical distribution fits the actual data. In the past, PCA in the
industrial world was often done by assuming the process to be normally distributed
using the formula

(USL − ¯x) (¯x − LSL)

C pk = min , (10.1)
3σ 3σ
where USL and LSL denote the upper and lower specification limits, respectively
(C pk will be used as the capability measurement in this chapter as it is still the most
widely used index today, even though it is not as good a measure as the z-score or
DPMO). In recent years, however, more and more people have begun to realize that
a significant proportion of processes are not normally distributed, leading to serious
errors in estimates.
When data is skewed and does not fit the normal distribution, the advice is generally
to do a Box--Cox transformation first and use the transformed data to estimate the
process capability. This advice is generally good if the optimum power, λ, for the
transformation is positive. However, if the λ is negative, it can seriously underestimate
the capability of a process. The other approach is to estimate the process capability by
using a statistical distribution that provides a good fit to the data.
In this chapter, a case study data set is used to illustrate in detail the two approaches
mentioned above. This is followed by another case study data set and three sets of
data generated using Monte Carlo simulation with various specification limits. The
following steps were followed throughout all the data sets with the aid of the statistical
software MINITAB 14.

(a) visual assessment of the process capability by plotting the histogram together with
the specification limit;
(b) transformation of the data using the Box--Cox transformation and calculation of
the process capability using the formulas above with the transformed data;
(c) fitting the original data with a best-fit statistical distribution and assessing the
process capability;
(d) comparison of the result from the different approaches.

As most of the calculations and statistical tests are widely available in statistical soft-
ware, this chapter focuses on the discussion of the results obtained from the statistical
software and not on showing the calculations in detail.

10.2 ILLUSTRATION OF THE TWO METHODOLOGIES USING
A CASE STUDY DATA SET

The data set used in this section is taken from a process with mean 12.61 and standard
deviation 1.38. The specification limit of this process is 91. From the distribution of
the data shown in Figure 10.1, we should expect a C pk much higher than 3 as the