Robert F. Hart, Ph.D.
Marilyn K. Hart, Ph.D.
The most elementary approach to making an Xbar
and s chart always uses 3sigma limits for something
like 25 timeordered 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 3sigma limits,
this false alarm risk will be greater with 200
subgroups than for 25 subgroups. Similarly,
the falsealarm risk will be smaller with only
two subgroups. The usual 3sigma limits work
well with, for example, 25 subgroups, giving
a falsealarm risk of about 0.06 on the Xbar
chart. In order to keep this falsealarmrisk
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 "Tsigma"
limits for an arbitrary number of subgroups
are shown in Table 1 [Hart and Hart, 2002].
Table 1. TSigma 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 c_{4}
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 c_{4} value for the i^{th}
subgroup based upon its subgroup size of n_{i}
and will
be the centerline value for the i^{th}
subgroup.
When 3sigma limits are used, the control limits
for the s for the i^{th} subgroup then
become
where
B_{4}[n_{i}]
= B_{4} for n_{i}
B_{3}[n_{i}]
= B_{3} for n_{i}


Values for B_{4}[n_{i}] and
B_{3}[n_{i}] 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 3sigma 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 1.5sigma 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, 1.5Sigma 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, Tsigma 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:
McGrawHill, 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.