Q: How will lack of normality affect
my Cpk statistics for a process? Will the calculated
statistics be higher or lower than they should
be?
A: There is no one answer to this question.
It depends on how different from normal the
distribution is, whether the distribution is
skewed to one side, how close to a specification
the distribution is, and other considerations.
Even moderate departures from normality that
affect the tails of the process distribution
may severely impact the validity of the process
capability calculations.
The construction and interpretation of process
capability statistics are based on the process
being distributed as a normal distribution.
For a normal distribution, approximately 99.73%
of the observations should fall within 3 standard
deviations (s) above the mean and 3 standard
deviations below the mean. The Cp statistic
is defined as:
The Cp statistic is designed to be equal to
1.0 when the process spread (± 3σ) is
the same as the specification width. With a
Cp equal to 1.0 for a normally distributed and
centered process, we would expect about 0.27%
of the output (2700 parts per million) to be
beyond the specification limits.
The Cp statistic assumes that the process is
centered, which may not be true. Therefore the
Cpk statistic is typically reported and is calculated
as:
The Cpk statistic should be equal to 1 when
the ± 3σ process spread coincides with
one or both of the specification limits. With
a Cpk equal to 1.0 for a normally distributed
process, we would expect about 0.27% OR LESS
of the output to be beyond the specification
limits.
When the distribution differs significantly
from a normal distribution, the calculated process
capability indices will probably be incorrect.
There may also be significant discrepancies
between the predicted (theoretical) proportion
and actual proportion of the process output
that are beyond the specification limits. This
is because the theoretical percentages are calculated
based on the tail probabilities for a normal
distribution.
Normal Distribution
First, examine the process capability chart
for a normally distributed process. Note how
the superimposed normal distribution line follows
the histogram of observations fairly well. For
this data and specification limits, the Cp is
approximately 1 and the Cpk is a bit smaller
due to the offset of the process from the center
line. Also note that the values of the "theoretical"
and "actual" proportions beyond the
specifications match fairly well.
Uniform Distribution
Next, examine what happens when the process
distribution is uniformly distributed instead
of being normal. The superimposed normal curve
no longer fits the data in the histogram well,
especially the tails. Even though all of the
process output is within the specification limits,
the calculated Cpk only 0.578. In addition,
the estimated proportion of product beyond the
specifications is over 8%, although the actual
proportion is zero.
The above analysis clearly illustrates a situation
in which the Cpk does not truly capture what
is happening in the process. Although the Cpk
calculated appears "lower" than it
"should be" in the above case, it
is also possible to have scenarios where the
calculated Cpk appears to be significantly better
than the actual situation would warrant. Thus,
it is important to be very careful that the
normality condition is met or nearly met when
doing Cpk calculations.
Statit provides nonparametric process capability
indices by using the QC -> Variables Charts
-> Process Capability Analysis. This analysis
looks at the tail probabilities of the process
itself rather than a theoretical normal distribution.
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ANALYSES
BASED ON EMPIRICAL DISTRIBUTION OF THE
DATA:
If the extrapolations in both plots looks
reasonable, then the estimated fraction
of product outside of specification limits
is
Upper spec limit: 0.013042
Lower spec limit: 0.007186
which equate to a total of 20227 parts
per million out-of-spec.
Empirical versions of the Cpk or Ppk
statistics are then
Upper spec limit: CPU = 0.741656
Lower spec limit: CPL = 0.815947
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