Q. We are interested in monitoring
production over time and across shifts using
the process capability index Cpk.
How do we determine when a change in the Cpk
is significant?
A. Process capability indices, particularly
Cpk, are increasingly being used
to evaluate the effectiveness of process improvement
techniques. They are used to help reveal possible
quality trends and to compare methods, work
shifts, and even workers. However, Cpk
by itself may not be the best tool to use
for these purposes.
In calculating Cpk, three different
quantities are distilled down into a single
number:
| 1. |
The process mean |
| 2. |
The process standard deviation |
| 3. |
The specification limits |
Just as a control chart needs information
about both the subgroup means and their standard
deviations in order to separate the "common
causes" from the "special causes"
that indicate that something has changed in
the process, Cpk alone will probably
not tell the whole story.
Cpk is easily calculated as:
The resulting value is a point estimate of
the Cpk statistic. It is important
to remember that as with any point estimate,
the calculated value is probably not the same
as the true population value. Sometimes, confidence
intervals are calculated for process capability
indices to capture this uncertainty.
A more direct method of looking at any process
trends or differences is to look at the data
used to calculate the process capability index,
rather than a summary number. This can be
done either graphically or using standard
statistical techniques.
For example, a box plot of the actual data
can be used to evaluate the production over
the past several weeks or to compare the production
over different shifts. The box plot is a simple
graphical representation of the data that
provides an easy visual comparison of the
means as well as the variability. If desired,
the specification lines could be added to
the plot as reference lines.
Analysis of Variance (ANOVA) methods can
be used to determine if any production weeks
or shifts are statistically different from
the others. If only two production periods
are being compared, a two-sample t-test could
be used to look for statistical differences.
The equality of variances can be investigated
using Levene’s test or the F-test in
the ANOVA procedure. To test for trends over
several weeks of production, tools such as
simple linear regression or a nonparametric
runs test can be used.
As an example, consider the following Cpk
data for four weeks of production:
Week 1
1.111
Week 2
1.230
Week 3
1.188
Week 4
0.922
Without more information, Week 4’s Cpk
of 0.922 might be seen as quite troubling.
Week 4’s Cpk value appears
to indicate that an otherwise capable process
with a Cpk greater than 1.0 has
suddenly gone bad. Investigations may be launched
and managers and workers called upon to explain
what went wrong. However, as is recognized
by the principles underlying control charts,
there is always some natural or "common
cause" variation in any process. Individual
points need to be carefully examined to see
whether they, in fact, represent a special
cause of variability that should be investigated.
Instead of just looking at the Cpk
values, it is useful to look at a box plot
(under Graphics -> Distribution Plots ->
Box Plot…) of the underlying data for
the four weeks.
From the plot, it can be seen that the variability
was somewhat larger for week 4, while the
mean was approximately the same as the other
weeks. This increased variability was probably
the cause of the lower Cpk. However,
an ANOVA analysis (under Statistics ->
ANOVA -> Oneway…) reveals that there
is no statistical difference between the means
of the four weeks. In addition, the ANOVA
procedure indicates that the variances of
the weeks can be considered to be equal. Thus,
Week 4 should not be singled out for special
action. If the test had revealed that Week
4 was different because its mean had shifted
and/or its variability had increased significantly,
this would be important diagnostic information
for improving the process. This information
would have been totally hidden when looking
at Cpk values alone!
While Cpk and other process capability
indices are useful one-number summaries of
the data, it is always best to make important
decisions based on as much process data as
possible.
