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Shifting the Paradigm: Control Charting Issues: Comparing an X-bar and R Control Chart With an Individuals Control Chart

June 27, 2011
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Table 1. Process Time Series Data. Source: Table 12.1 of IEE Volume III

In previous blogs , I described issues with AQL testing and resolution to these problems using a 30,000-foot-level reporting system. In this blog and the next few blogs, I will expand upon the 30,000-foot-level methodology. I will first describe issues with many traditional control charting schemes and present an alternative methodology.

For a given process, do you think everyone would create a similar looking control chart and make a similar statement relative to process control and predictability? What about their statement about its process capability for given specification limits? Not necessarily. Process statements are not only a function of process characteristics and sampling chance differences but can also be dependent upon sampling approach.

This can have dramatic implications:

  • One person could describe a process as out of control, which would lead to activities that immediately address process perturbations as abnormalities, while another person could describe the process as being in control. For this second interpretation, the perturbations are perceived as fluctuations typically expected within the process, where any long-lasting improvement effort involves looking at the whole process.
  • One person could describe a process as not predictable, and another person could describe it as predictable.

    To illustrate how different interpretations can occur, let’s analyze the following process time series data to determine its state of control and predictability and then its capability relative to customer specifications of 95 to 105 (see Table 1).



  • Figure 1: X bar and R control chart, Source: Figure 12.7 of IEE Volume III

    This type of data traditionally leads to an X bar and R control chart, as shown in Figure 1. Whenever a measurement on a control chart is beyond the upper control limit (UCL) or lower control limit (LCL), the process is said to be out of control. Out of control conditions are called special cause conditions, and out of control conditions can trigger a causal problem investigation. Since so many out of control conditions are apparent in Figure 1, many causal investigations could have been initiated. But out of control processes are not predictable, and no process capability statement should be made about how the process is expected to perform in the future relative to its specification limits.

    Figure 2: Individuals control chart of one sample from each subgroup. Source: Figure 12.8 of IEE Volume III

    When creating a sampling plan, we may select only one sample instead of several samples for each subgroup. Let’s say this is what happened and only the first measurement was observed for each of the 10 subgroups. For this situation we would create an individuals control chart like the one shown in Figure 2.

    This control chart is very different from the x̄ and R charts shown in Figure 1. Since the plotted values are within the control limits, we can conclude only common cause variability exists and the process should be considered to be in control or predictable.

    The dramatic difference between the limits of these two control charts is caused by the differing approaches to determining sampling standard deviation, which is a control limit calculation term. To illustrate this, let’s examine how these two control chart limit calculations are made.

    For X bar charts, the UCL and LCL are calculated from the relationships



    where the X double bar is the overall average of the subgroups, A2 is a constant depending upon subgroup size and the R bar symbol is the average range within subgroups.

    For X charts the UCL and LCL are calculated from the relationships



    where the MR bar symbol is the average moving range between subgroups.

    The limits for the X bar chart are derived from within-subgroup variability (the R bar symbol), while sampling standard deviations for XmR charts are calculated from between-subgroup variability ( the MR bar symbol).

    Which is the best approach? In my next blog, I will describe why the individuals control chart is typically a better choice than the X bar and R charting approach.

    Reference: The content of this blog was taken from Chapter 12 of Integrated Enterprise Excellence (IEE Volume III .
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