Random sample approach to rational subgroups (Xbar, R chart)

A fundamental idea in the use of control charts is the collection of sample data according to what Shewhart called the rational subgroup concept.

To illustrate this concept, suppose that we are using an control chart to detect changes in the process mean. Then the rational subgroup concept means that subgroups or samples should be selected so that if assignable causes are present, the chance for differences between subgroups will be maximized, while the chance for differences due to these assignable causes within a subgroup will be minimized.

The rational subgroup concept is very important. The proper selection of samples requires careful consideration of the process, with the objective of obtaining as much useful information as possible from the control chart analysis.

In random sample approach to rational subgroups, random sample (of defined number of units, here 5 units) is collected from all process output produced over the sampling interval.

This method of rational subgrouping is often used when the control chart is employed to make decisions about the acceptance of all units of product that have been produced since the last sample. 

In fact, if the process shifts to an out-of-control state and then back in control again between samples, it is sometimes argued that the snapshot method of rational subgrouping will be ineffective against these types of shifts, and so the random sample method must be used.

Figure 11a, shows process for which the mean experiences series of sustained shifts and the corresponding observations obtained form this process at the points in time along the horizontal axis, a random sample (defined number of units) of all process output over the sampling interval.

Considerable care must be taken in interpreting control charts, when the rational subgroup is a random sample. 

If the process mean drifts between several levels during the interval between samples, this may cause the range of the observations within the sample to be relatively large, resulting in wider limits on the x_bar chart. In fact, one can often make any process appear to be in statistical control just by stretching out the interval between observations in the sample.

It is also possible for shifts in the process average to cause points on a control chart for the range or standard deviation to plot out of control, even though there has been no shift in process variability.

Choice of random sampling is execercised usually when process is expected to be running under control, processes are well understood with all sources of variability known. Finally, one can choose to sample 1 unit for units produced during sampling interval.

Above plot reflect x_bar and R control charts.

Reference:

Chapter 5, of SQC by Montgomery.