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 The Xbar and s Chart with an Arbitrary Number of Subgroups and Varying Subgroup Sizes

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

 (1)

where is the mean of the standard deviations from all the subgroups and

 (2)

[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.

For more information, contact Drs. Robert and Marilyn Hart at robthart@aol.com or (541)412-0425.