Q: My X-bar control chart is signaling
an out-of-control situation even though all
of the points are very close to the center line.
The rule violation is designated with the letter
"I" and is recorded as "15 points
in a row in Zone C (above and below the center)."
Why is this happening when having all points
close to the center line seems like it would
be desirable?
A: This particular rule violation usually
indicates a problem with the sampling and control
chart procedures, rather than a problem with
the process itself. Typically, an improper sampling
technique called Stratified Sampling is indicated
with this rule violation. It causes the calculated
variability of the data to be inflated, which
causes the control limits to be wider and the
chart to be less sensitive to process changes.
Stratified sampling arises when the measurements
of several processes (machines, workers, tools,
etc.) are combined to form a subgroup for the
control chart. In one example, there are 5 machines
that all produce a particular part. Once an
hour, a single part is taken from each machine
to form a subgroup of size n = 5 to be put on
the control chart. In another example, a painting
machine has 4 heads that simultaneously spray
paint on individual parts. At the end of each
paint cycle, the thickness of the paint coating
is measured on each of the parts and recorded
as a subgroup of size n = 4 for quality control
purposes. This type of sampling is known as
Stratified Sampling, since each subsample includes
an observation from each of the different processes
(strata).
A common problem with stratified sampling situations
is that the measured variance includes the differences
between the strata, in addition to the common
cause process variation. If there is a substantial
difference between the mean value of each machine
or head, this is included in each subgroup range
or standard deviation. This increases the estimate
of the process variation and the calculated
control limits on the X-bar chart become wider.
At the same time, the mean may stay fairly constant
and thus near the center of the control chart.
The following chart shows the control chart
pattern that typically indicates stratified
sampling. This data was simulated with a standard
deviation of 1.0, assuming five components with
means of 1.0, 2.0, 3.0, 4.0, and 5.0, respectively.
The mean of the X-bar chart is around 3.0, as
expected. However, the standard deviation of
the subgroups is 0.927, rather than
as would be expected with a subgroup size of
n = 5. This more than doubling of the subgroup
standard deviation is clearly reflected in the
increased width of the control limits for the
X-bar chart relative to the plotted points.
The control chart signals an out-of-control
situation after there have been 15 points in
a row, all within Zone C (within 1 standard
deviation of the mean). The probability of having
15 consecutive points in a row within 1 standard
deviation of the mean is very, very low for
a process that is in statistical control. Thus,
when this happens, it means that there is something
unusual with either the process or the control
scheme.
Due to the increased width of the control limits,
the X-bar chart created under stratified sampling
will be fairly insensitive to changes in the
overall mean of the samples. It is even less
sensitive to changes in an individual process
mean since typically the overall mean will not
change significantly. Rather than using a stratified
sampling scheme, it is recommended that separate
control charts be made for each process, machine
or head. This will give the most information
about the individual determinants of quality.
