The u chart is the optimum choice for analyzing
nonconformities over an area of opportunity.
We count the number of nonconformities in some
defined region of space, time or product. If
our sample contains varying areas of opportunity,
then we certainly use the u chart.
When we are counting these nonconformities,
we may be counting different type of nonconformities,
some more important or critical than others.
If we track the type of nonconformity and count
for each sample, we could run a u chart on each
type to determine if the more important types
are changing over time.
However, there is another method that is often
used that will help with this analysis: the
D Chart or Demerit Chart.
With this chart, each type of nonconformity
is given a weight, or demerit, with those most
critical getting the highest weights. To illustrate,
let’s consider the battery separator data.
100 separators are pulled from a packed box
and checked for TCE Content, Oil Content, Width,
Length, Web Thickness and Rib Thickness. It
was decided that a high TCE content could contaminate
the battery and that Oil Content would affect
the porosity of the separator, both having adverse
and largely undetectable problems at the customer
site. Since TCE is volatile, this issue is not
as problematic as lower porosity. Oil was given
a weight of 500 and TCE a weight of 100. The
others were weighted from 4 to 50.
From this data, a standard u chart would look
like this:
(mouse over chart to see data
tips)
In this chart, the numerator is the sum of
all the nonconformities found in a sample of
100 separators.
If we applied demerits we might get a chart
such as this:
(mouse over chart to see data
tips)
This chart shows some possible issues. If we
look at the point for Batch 000128B, we see
that there were a number of Oil Nonconformities
leading to an out of control point.
The Demerit Chart helps us to find samples
with large numbers of nonconformities as well
as large numbers of critical nonconformities.
The formulas can be somewhat daunting and can
be found in Ryan pp178-180. In words, we first
calculate the demerits as the weighting factor
times the number of that particular nonconformity
in the sample. Then we calculate the Center
Line as the total demerits divided by the total
number of samples.
Then, for each nonconformity type, we calculate
the u as the total number of that nonconformity
divided by the total number of samples.
The final step is to calculate the sigma for
each subgroup. The steps are:
1) Multiply the square of the weight by the
u for each nonconformity type
2) Add those factors together
3) Divide by n, the sample size of each sample
4) Take the square root of the result
For each subgroup, the Upper Control Limit
is Center Line plus 3 times the sigma for that
subgroup. For the lower control limit, subtract
3 times the sigma.
On our live demo site, http://live.statit.com,
we have set up a demonstration of these charts.
The category is Demerit Charts. The two examples
use basically the same data; the only difference
is that the data is arranged differently. Near
the bottom of each display is a short listing
of a portion of the data as well as the weighting
factors used. The link at the bottom, View Macro
Source, pops up a window of the macro source
used to calculate and display the chart.
For the Simple Demerit, the count of types
of nonconformities are in columns, one for each
type. The rows are individual batches. This
type of arrangement can make the calculations
simpler. However, your data may not be in that
form.
The second example has the data arranged as
one row for each batch and nonconformity type
(Parameter), with six rows per batch. The other
interesting piece of information about these
data is that the No_Tested Column lists 100
for each row. But actually, 100 separators were
pulled for each batch and all 6 tests were run
on each of them. So the source code for this
macro had to make some provisions for that issue.
See View Macro Source.
Since the subgroup size in these data are all
the same size, we get constant control limits.
If checked, the toggle Randomize Denom will
produce varying subgroup sizes.
The Demerit Chart can be a useful tool, particularly
for management reporting. However, it is not
necessarily recommended for finding assignable
causes. The alternative would be to group nonconformity
types in just two or three categories, such
as Critical, Major and Minor and produce u charts
for each category.
We should also mention that there is another
alternative to the D chart, called the Q chart
(Quality Score). In this case the weighting
may be a function of the nonconformity type
and of the extent of the nonconformity. For
example, the weighting for Oil may be also be
dependent on the amount of residual oil in the
separator.
References:
Grant, E.L., & Leavenworth, R.S. (1996).
Statistical quality control (5th ed.). Boston:
McGraw-Hill.
Ryan, T.P. (2000). Statistical Methods for Quality
Improvement (2nd ed.). New York:John Wiley and
Sons
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