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Rational Subgroups

Most control charts, including the X-bar and Individual-X charts, rely upon Rational Subgroups to estimate the short term variation in the process. This short-term variation is then used to predict the longer-term variation defined by the control limits. But what is a Rational Subgroup?

A Rational Subgroup is simply "a sample in which all of the items are produced under conditions in which only random effects are responsible for the observed variation." [Nelson, Lloyd S. "Control Charts: Rational Subgroups and Effective Applications," Journal of Quality Technology. Vol. 20, No. 1, January 1988). As such, it has the following properties.

1. The observations comprising the subgroup are independent. Two observations are independent if neither observation influences, or results from, the other. When observations are dependent on one another, we say the process has Autocorrelation, or Serial Correlation. (These terms mean the same thing). Many processes are subject to Autocorrelation. Examples include:
   • Chemical Processes: When dealing with liquids, particularly in large baths, samples taken close together in time are influenced by one another. The liquid retains the effect of the first observation, such as temperature, which carries over into subsequent temperature observations for a period of time. Subgroups formed over a small time frame from these types of processes are sometimes called homogenous subgroups, since the observations within the subgroups are often nearly identical (except for the effect of measurement variation).
   • Service Processes: Consider the wait time at a bank. The wait time of any person in the line is influenced by the wait time of the person in front of him/her.
   • Discrete part manufacturing: Although this is the "classic" case of independent subgroups, when feedback control is used to change a process based upon past observations, the observations become inherently dependent.

When observations within a subgroup are autocorrelated, the within subgroup variation is often quite small, and not a reflection of the between subgroup process variation. The small within subgroup variation forces the control limits to be too narrow, resulting in frequent out of control conditions. This leads to Tampering .

See Autocorrelation Charts for a discussion of techniques useful for dealing with processes of this sort.  See also Struggling with Independence for further discussion of these types of processes.

2. The observations within a subgroup are from a single, stable process. If subgroups contain the elements of multiple process streams, or if other special causes occur frequently within subgroups, then the within subgroup variation will be large relative to the variation between subgroup averages. This large within subgroup variation forces the control limits to be too far apart, resulting in a lack of sensitivity to process shifts. Run Test 7 (15 successive points within one sigma of center line) is helpful in detecting this condition.
3. The subgroups are formed from observations taken in a time-ordered sequence. In other words, subgroups cannot be randomly formed from a set of data (or a box of parts); instead, the data comprising a subgroup must be a "snapshot" of the process over a small window of time, and the order of the subgroups would show how those snapshots vary in time (like a "movie"). The size of the "small window of time" is determined on an individual process basis to minimize the chance of a special cause occurring in the subgroup.