Choosing the correct subgrouping scheme is
critical to the proper analysis of a Shewhart
chart. An improper subgrouping rationale can
hide process changes or indicate process changes
where in actuality none exist. The wrong subgrouping
scheme can render a chart useless or worse.
Control charts of the type where subgrouping
is used (Xbar, R, S, Median, p) can give erroneous
or misleading results if the method of subgrouping
has not been given a lot of thought or if the
subgrouping scheme is not understood by the
analyst.
For each of these charts, the idea is to subgroup
so that the units measured in each subgroup
are likely to be homogenous and the subgroups
have a higher probability of being unlike. Homogenous
means that the probability is high that the
measurements will be near the same because they
are drawn from the same population.
Sources of Variation
In order to choose the correct subgrouping
scheme we need to understand the sources of
variation. There are several sources of variation
in manufactured product. The first is lot-to-lot
variation. Certainly we would like to minimize
the variation between lots. Lots are usually
manufactured as separate units and as such they
are likely to have some differences in manufacturing.
Minimizing that variation provides for a more
predictable process.
The second is stream-to-stream variation, which
may arise when an inspection is done where several
process streams meet. This variation may also
occur when we are not able to capture the information
identifying which stream a product came from.
For example, the inspected product could be
coming from several machines. If the data contain
no differentiation by machine, stream-to-stream
variation may be incorporated into the control
chart. Other common examples are multiple cavity
molds, different operators, or different inspections.
Third is the time-to-time variation. This is
the primary source of variation that we attempt
to address with control charts. Is our process
changing over time or is it predictably stable?
A fourth source of variation is the piece positional
variation, produced by the choice of location
of the measurement on the part. For example,
diameter of a shaft of the electrical resistivity
of a silicon wafer. The same measurement could
be taken in several different locations on the
part.
The fifth source is the one usually addressed
with Gage R&R studies. Error of measurement
has both instrument and human components that
can be indicated by a Gage R&R study.
One subgrouping scheme may have more than one
source of variation. Understanding of the magnitude
of the variation from different sources helps
to choose subgrouping schemes and to analyze
Shewhart charts.
Questions of a Control Chart
The question that the Shewhart chart asks
is, “Is the pattern of variation among
the subgroups consistent with the averaged pattern
of variation within the subgroup?”.[1]
The question is answered by whether a point
is outside the control limit or not. If there
is a point outside the control limit the question
is answered, “No”, and we conclude
that there is a high probability that an assignable
cause exists in the process.
The control limits of a Shewhart chart are
based on the averaged variation within the subgroups.
Minimizing the variation within the subgroups
keeps the control limits tighter and increases
the sensitivity of the chart to detect process
changes between subgroups.
To properly analyze the chart, we need to understand
what makes up the variation within the subgroup.
If we are grouping by lot, then our variation
is lot-to-lot. However, if the lot is manufactured
by more then one machine, then we have to understand
that machine-to-machine variation is also included
in the subgroup variation.
Example
To illustrate, let’s look at an example.
This example comes from Wheeler[1]. An injection
molding press produces 4 parts with each cycle
of the press via a mold that has 4 cavities.
But there seemed to be a problem with the process.
The QA professional used control charts to analyze
the source of the variation.
Each product sample was collected from 5 consecutive
press cycles and measured. The measurements
were recorded and identified with the hour of
the measurement (1-20), press cycle (A,B,C,D,E)
and mold cavity (I,II,III,IV). The first chart
the analyst produced follows:
(mouse
over the data points for more information)
This chart is asking the questions:
1) Are there hour-to-hour detectable differences?
2) Are there cycle-to-cycle detectable differences?
3) Are the cavity-to-cavity differences consistent?
The answer to all these questions is "No".
The process appears to be in control but the
spread of the range chart indicates that there
could be an issue with subgrouping. We would
expect that the ranges would be spread more
over the 3 sigma range with 2/3 in Zone C, 95%
present within the Zone B boundaries . As it
is, most of the points fall within Zone C. This
chart has changed the color of the "trend
rule" violations, but if you hover over
some of the points on the Range Chart you will
see several violations of the 15 points in a
row in Zone C. The analyst decided to investigate
further thinking perhaps we are asking the wrong
questions.
The analyst then produced this chart. (mouse
over the data points for more information)
As you can see by hovering over a point, each
point is the statistic of a single cavity by
hour and cycle. This chart is asking the questions:
1) are there detectable differences from hour-to-hour,
2) are the cycle-to-cycle differences consistent
3) are there detectable differences from cavity-to-cavity
These two figures show us several things. When
we looked for a signal for changes in the process
on an hour-to-hour basis and a cycle-to-cycle
basis we don't see an alert. But if we look
for a signal with hour-to-hour and cavity-to-cavity
we see several signals. This tells us the we
have some stream-to-stream variation. That is,
each of the cavities (I, II, III, IV) are separate
streams.
We also see that the control limits on the
first Xbar chart are wider then the second Xbar
chart. This illustrates that the cycle-to-cycle
within-group variation is smaller then the cavity-to-cavity
within group variation. A subgrouping scheme
that provides for lower within-group variation
provides for a better chance to detect signals
of a change in process. The within-subgroup
variation provides a limit on the amount of
variation that is needed between subgroups to
signal an alert.
The above chart gives an indication of what
to do next and the analyst put together this
chart: (mouse over the data points
for more information)
This chart definitely indicates the difference
between cavities. But it shows the long-term
hour-to-hour variation as well as the short-term
cavity variation. Because we are getting a number
of out-of-control alerts on each of the cavities,
we see that there is some long-term variation.
The variation is high enough hour-to-hour to
produce a signal. It also signals that one cavity,
I, is consistently higher then the others.
Stream-to-stream variation is not uncommon.
And there are times when it is of little consequence,
but the analyst must be aware of this variation
knowing that it could come into effect. In the
example above, the solution was to clean the
mold more thoroughly and more often. With that
change in procedure, it might not be necessary
to run individual charts on each stream. The
analyst should at least periodically check the
different sources of variation to see if they
have returned to play.
This examples shows the importance of:
Conclusion
The key to choosing a subgrouping scheme is
to understand the sources of variation that exist
in your process. Shewhart charts can help you
to understand the magnitude of the sources of
variation. As we’ve seen, if you have the
data to differentiate other causes of variation,
you can produce Shewhart charts that will give
you insight into the magnitude of the various
sources of variation. With that knowledge you
can choose the subgrouping scheme to use to monitor
the process.
Choose subgroups that minimize variation within
subgroups. If the items in our subgroup are
similar based on time, space or product the
measurements are likely to be similar.
Choose subgroups that maximize the opportunity
for variation between subgroups, so that if there
is a process change we will detect it. The source
of variation indicated between subgroups usually
is time-to-time but can also include one of more
of the other sources as well. That is fine as
long as it is understood by those interpreting
the charts.
Please view other examples of Subgrouping on
our unhosted
Statit e-QC demo. The category Rational
Subgrouping illustrates the concepts. You can
also view
a Statit Webinar on Rational Subgrouping.
References
Florac, W.A., & Carleton, A.D. (1999).
Measuring the software process: Statistical process
control for software process improvement. Boston:
Addison-Wesley. [4]
Grant, E.L., & Leavenworth, R.S. (1996).
Statistical quality control (5th ed.). Boston:
McGraw-Hill. [2]
Shewhart, W.A. (1931). Economic control of quality
of manufactured product. New York: D. Van Nostrand.[3]
Wheeler, D.J. (2004). Advanced topics in statistical
control – The power of Shewhart’s charts
(2nd ed.). Knoxville, TN: SPC Press. [1]
