Robert F. Hart, Ph.D.
Marilyn K. Hart, Ph.D.
The most elementary approach to making an Xbar
and s chart always uses 3-sigma limits for something
like 25 time-ordered subgroups and requires
that the subgroup sizes be constant. However,
sometimes a control chart is used for what Shewhart
[1931] called "rational" subgroups.
That is, the control chart is made to compare
the average (mean) and the variability (standard
deviation) on subgroups from different applications
to compare their performance. For example, you
may wish to make a control chart to compare
the performance of two different shifts, three
different machines, four different operators,
etc. So you will be making a control chart with
fewer than 25 subgroups and they probably won't
have the same number of observations in each
subgroup. Fortunately, making Xbar and s charts
with an arbitrary number of subgroups and varying
subgroup sizes is straightforward with the computer.
Adjusting for the Number of Subgroups
The risk of finding a point beyond the control
limits with normally distributed data may be
called the false alarm risk. Using 3-sigma limits,
this false alarm risk will be greater with 200
subgroups than for 25 subgroups. Similarly,
the false-alarm risk will be smaller with only
two subgroups. The usual 3-sigma limits work
well with, for example, 25 subgroups, giving
a false-alarm risk of about 0.06 on the Xbar
chart. In order to keep this false-alarm-risk
close to around 0.06, the number of sigma limits
to use needs to be adjusted for the number of
subgroups similar to the procedure of Analysis
of Means developed by Ellis Ott [1975]. Recommended
values of "T" for "T-sigma"
limits for an arbitrary number of subgroups
are shown in Table 1 [Hart and Hart, 2002].
Table 1. T-Sigma Limits
| # of Plotted Points |
T |
| 2 |
1.5 |
| 3 - 4 |
2.0 |
| 5 - 9 |
2.5 |
| 10 - 34 |
3.0 |
| 35 - 199 |
3.5 |
| 200 - 1500 |
4.0 |
Adjusting for the Varying Sample Sizes
When making an 
and s chart, the standard deviation of the population
σ must be estimated. If sampling from
a population having a normal distribution and
all the subgroups have the same size (n), σ
may be estimated by
where 
is
the mean of the standard deviations from all
the subgroups and
[ASTM, 1990, p .91]. Values of c4
may be found in the ASTM book or most SPC texts.
When making an
and s chart with varying subgroup sizes, there
are three accepted ways of estimating σ,
the population standard deviation [SAS, 1995,
pp.1214 - 1215, Hart and Hart, 2002]. Whichever
method is used, the centerline for the s chart
will vary for different subgroup sizes because
using (1) it follows that
where 
is
the c4 value for the ith
subgroup based upon its subgroup size of ni
and 
will
be the centerline value for the ith
subgroup.
When 3-sigma limits are used, the control limits
for the s for the ith subgroup then
become
where
|
B4[ni]
= B4 for ni
B3[ni]
= B3 for ni
|
|
Values for B4[ni] and
B3[ni] may be found in
the ASTM book [1990] or most SPC texts. This
implies that for unequal subgroup sizes, the
centerline and the control limits for the s
chart will step, that is, the centerline and
control limits will differ with the subgroup
size. The stepped centerline will only be visually
noticeable if the subgroup sizes are small.
For the Xbar chart:
That is,
is
the grand average of all the data from all the
subgroups.
The 3-sigma control limits for each subgroup
i become
where
This implies that for unequal subgroup sizes,
the control limits will step (i.e., the control
limits will differ with the subgroup size).
For a very short illustrative example, consider
two subgroups. The first one contains the measurements
2, 5, 6, and 7. The second subgroup contains
8, 4, 4, 5, 6, 15, 24, 8, and 10. The
and s chart with 1.5-sigma limits made by Statit
Express QC is given in Figure 1. Note the step
in the centerline in the s chart and the step
in the control limits for both the Xbar and
the s chart.
Figure 1. Xbar and s Chart with Stepped Centerline
on the s Chart, 1.5-Sigma Limits
(output created with Statit Custom QC)
It will be noted that these authors have found
very few software packages that correctly step
the centerline on the s chart. (In fact, many
SPC software packages do not even handle varying
subgroup sizes.) Consequently, this has become
one of the tests we use to evaluate the correctness
of SPC software.
Summary
When making an Xbar and s chart with varying
subgroup sizes, there are steps in the centerline
on the s chart (which will only be visually
noticeable for small sample sizes ) and the
control limits on both the Xbar and the s chart
will step. When making an Xbar and s chart with
other than 25 subgroups, T-sigma limits are
recommended.
References
ASTM
(American Society for Testing and Materials).
ASTM Manual on Presentation of Data and Control
Chart Analysis, MNL7, 6th ed. Philadelphia:
American Society for Testing and Materials,
1990.
Hart,
Marilyn, and Robert Hart. Statistical Process
Control for Health Care. Pacific Grove,
California: Duxbury, 2002.
Ott,
Ellis. Process Quality Control. New York:
McGraw-Hill, 1975.
SAS
Institute Inc. SAS/QC Software: Usage and
Reference, version 6, ed. 1, vol. 2. Cary,
NC: SAS Institute Inc., 1995.
Shewhart,
Walter A. Economic Control of Quality of
Manufactured Product. Princeton, NJ: Van
Nostrand Reinhold Co., 1931. (Republished in
1981 by the American Society for Quality Control,
Milwaukee, WI).
Analysis
and output created with Statit Express QC, Statit
Software, Inc.
