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)
