Subgroup Sizes for Attribute Charts

Robert F. Hart, Ph.D.
Marilyn K. Hart, Ph.D.

The question always arises as to what subgroup size is necessary for attribute charts so that the results of the chart can be believed. Unfortunately the answer is not a simple one. Take the p chart. The p chart is based primarily on the binomial distribution. The standard deviation for the p chart is calculated by

.

However, we assume the normal distribution approximation to the binomial when we use symmetric limits, typically 3-sigma limits. The normal approximation to the binomial is typically considered valid if np = 4 and n(1-p) = 4, although some authors suggest the limit of 5 rather than 4.

In accordance with ASTM [1990], the following guidelines for minimum subgroup size are recommended so that distribution will not be too skewed. Find pBar. If pBar is less than or equal to 0.5, use the following. [Hart and Hart, 2002, p. 190]

1.	n should always be at least 1/pBar. If not, ignore that subgroup or combine data in such a way as to get large enough subgroups.
2.	A point beyond the control limits is correctly identified as “out of control” if n is at least 4/pBar. Otherwise, ignore that subgroup or combine subgroups to get the minimum subgroup size needed.

In the case that pBar is greater than 0.5, replace pBar with (1 - pBar) in the computation of the minimum subgroup sizes.

The advanced reader can use the binomial probability distribution or the adjustment suggested by the ASTM [1990, p. 58]; however, this goes beyond the scope of this paper.

The following example, taken from Hart and Hart [2002] illustrates the concept of combining subgroups when needed. Figure 1 is the p chart on fraction rejected by month. Note that there is a point outside the control limits for month 23. However, note the small subgroup size of n = 8. This point may be outside of the control limits either because there really is a special cause of variation that month or because the distribution is so skewed. Using the above guidelines, we accept that point as being out of control if n > 4/pBar. Since pBar = 0.17 in this chart, we require that n > (4/0.17) = 23.5. Since the subgroup size of 8 is less than 23.5, the subgroup size is indeed too small to determine whether or not that point is out of control. We also required that all subgroups have a size of at least 1/pBar = 1/0.17 = 5.9. This condition is also not met. We will then subgroup the data into six-month periods. This chart (Figure 2) requires 2-sigma limits since there are only four subgroups [Hart and Hart, 2002; Statit Tips, December 2000].

Figure 1. p Chart on Fraction Rejected by Month

Figure 2. P Chart on Fraction Rejected by Six-Month Periods

Note that the pBar is still 0.17, so the minimum requirements stated earlier still apply. But now these minimum requirements in subgroup sizes are met, so we accept the results of the chart

For the np chart, the requirement would be (when ≤ 0.5):
≥ 1 minimum for all subgroups and
≥ 4 minimum to believe any points beyond the control limit.
(When > 0.5, replace with (1 - ).

The c chart and the u chart use the Poisson distribution, which can be highly skewed. So similar restrictions are placed on their subgroup sizes.

For the c chart, the requirement would be:
≥ 1 minimum for all subgroups and
≥ 05;05; 4 minimum to believe any points beyond the control limits

For the u chart, the requirement would be:
n ≥ 1/ minimum for all subgroups and
n ≥ 4/ minimum to believe any points above the upper control limit

Again, if these subgroup requirements are not met, try combining subgroups until the subgroups become large enough.

References:

For more information, contact Drs. Robert and Marilyn Hart at robthart@aol.com.

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