Monday, November 30, 2009

Why Are My Control Limits So Narrow?

Statistical Process Control (SPC) charts are widely used in engineering applications to help us determine if our processes are predictable (in control). Below are Xbar and Range charts showing 25 subgroup averages and ranges for 5 Tensile Strength values (ksi) taken from each of 25 heats of steel. The Range chart tells us if our within subgroup variation is consistent from subgroup-to-subgroup and the Xbar chart tells us if our subgroup averages are similar. The Xbar chart has 19 out of 25 points outside of the limits. This process looks totally out-of-control, or does it?

Data taken from Wheeler and Chambers (1992), Understanding Statistical Process Control, 2nd edition, table 9.5, page 222.

The limits for Xbar are calculated using the within subgroup ranges, Rbar/d2. In other words, the within subgroup variation, which is a local measure of variation, is used as a yardstick to determine if the subgroup averages are predictable. In the context of our data, the within subgroup variation represents the variation among 5 samples of steel within one heat (batch) of the steel and the between subgroup variation represents the heat-to-heat variation. While the details are limited, we can imagine that every time we have to heat a batch of steel, we may be changing raw material lots, tweaking the oven conditions, or running them on a different shift, which can lead to more than one basic source of variation in the process.

Having multiple sources of variation is quite common for processes which are batch driven and the batch-to-batch variation is often the larger source of variation. For the Tensile Strength data, the heat-to-heat variation accounts for 89% of the total variation in the data. When we form rational subgroups based upon a batch, the control limits for the Xbar chart will only reflect the within batch variation and may result in control limits which are unusually tight and many points will be outside of the control limits.

In order to make the Xbar chart more useful for this type of data we need to adjust the control limits to incorporate the batch-to-batch variation. While there are several ways to appropriately adjust the limits on the Xbar chart, the easiest way is to treat the subgroup averages as individual measurements and use an Individuals and Moving Range chart to calculate the control limits.

The plot below shows the Tensile Strength data for the 25 heats of steel and was created using a JMP script for a 3-Way control chart. The first chart is the Xbar chart with the adjusted limits using the moving ranges for the subgroup averages and the chart below it is the moving range chart for the subgroup averages. The third chart (not shown here) is the Range chart already presented earlier. Note, the limits on the Range chart do not require any adjustments. Now what do we conclude about the predictability of this process?

Indeed, the picture now looks quite different. No points are outside of the limits and there are no violations in runs rule. The Range chart shows 3 points above the upper control limit suggesting that these three heats of steel had higher within subgroup variation. As Wheeler and Chambers point out, "this approach should not be used indiscriminately, and should only be used when the physical situation warrants its use".

1 comment:

  1. My experience has been that rolling the measures is a very powerful
    approach to continuously changing systems. However, one must be carful not
    to 'over-role' as this will result in overfitting and a very low power.

    If False alarms are an issue with the system you are monitoring then you
    might find the papers in useful. They
    deal with situations where more than one attribute or more than one SPC are
    used simultaneously.

    With Regard,
    Eli Kling