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# Control Limits That Are Too Wide

This article discusses how the use of sub-groups can lead to exceedingly wide control limits, reducing the effectiveness of control charts, leading to loss of credibility.

The x-bar chart will be used to demonstrate the problem. Applying standard control chart methodology to the data in Table 1 yields a lower and upper control limit of 18.6 and 41.9 respectively.

Table I

Results Avg Range
1 2 3 4 5
39.5 23.5 31.4 1.3 33.3 29.4 20.2
26.8 40.5 19.0 35.4 30.6 30.5 21.5
19.0 26.9 35.3 41.0 24.5 29.3 22.0
32.5 27.7 21.1 37.7 28.8 29.6 16.6
40.1 21.9 30.8 25.2 28.4 29.3 18.2
27.7 25.0 39.5 24.5 34.2 30.2 15.0
40.0 32.1 24.7 20.8 30.0 29.5 19.2
18.1 33.0 38.0 43.4 28.5 32.2 25.3
32.8 20.4 41.6 33.5 29.7 31.6 21.2
42.5 35.0 31.7 19.4 24.6 30.6 23.1
30.3 23.0 22.4 38.4 41.5 31.1 19.1
39.7 26.5 19.3 27.4 34.0 29.4 20.4
34.0 39.7 22.8 22.3 29.4 29.6 17.4
41.4 35.4 18.0 29.5 25.4 29.9 23.4
24.5 22.0 36.2 40.0 31.3 30.8 18.0
41.7 23.5 24.3 33.1 30.9 30.7 18.2
21.1 35.4 32.4 37.7 36.7 30.7 16.6
27.5 23.0 30.1 34.9 40.2 31.1 17.2
45.9 22.1 30.4 35.5 16.4 30.1 29.5
19.7 42.2 37.8 25.4 25.3 30.1 22.5
FIGURE 1: Control limits too wide

Figure 1 shows that these control limits are too wide. In fact problems with the range chart are also experienced, but this is a topic for a future article. There are no out-of-control points and standard procedure was adhered to. What went wrong?

The explanation is simple. The data was collected by taking across the belt samples from an extruder producing five streams of product. Each stream of product had its own "population average" contributing a "fixed" component of variability to the total variation which is not reflected in the variation of the averages over time. This is highlighted by the artificial extreme situation shown in Table II.

Table II

Extruder Die
1 20 20 20 20 20 20 .......
2 25 25 25 25 25 25 .......
3 30 30 30 30 30 30 .......
4 35 35 35 35 35 35 .......
5 40 40 40 40 40 40 .......
Average 30 30 30 30 30 30 .......

The average range is 20 units indicating considerable variability. However the averages of five are a constants 30 indicating that there is no variability in the averages. Control limits should therefore be set at 30±0 and not based on a range of 20 i.e. control limits for these applications must not be based on the fixed component of variability. Variability used to set up control limits should only include the random component.

An inspection of Table III obtained by unscrambling the data in Table I, shows the fixed effect. Unlike table II there is also a random component (variation within columns) which caused the variation in the sub-group averages.

Table III

 1 2 3 4 5 19.3 23.5 31.4 33.2 39.5 19.0 26.8 30.6 35.4 40.5 19.0 24.5 26.9 35.3 41.0 21.1 28.8 27.7 32.5 37.7 21.9 25.2 28.4 30.8 40.1 25.0 24.5 27.7 34.2 39.5 20.8 24.7 30.0 32.1 40.0 18.1 28.5 33.0 38.0 43.4 20.4 29.7 33.5 32.8 41.6 19.4 24.6 31.7 35.0 42.5 23.0 22.4 30.3 38.4 41.5 19.3 26.5 27.4 34.0 39.7 22.3 22.8 29.4 34.0 39.7 18.0 25.4 29.5 35.4 41.4 22.0 24.5 31.3 36.2 40.0 23.5 24.3 30.9 33.1 41.7 21.1 26.7 32.4 35.4 37.7 23.0 27.5 30.1 34.9 40.2 16.4 22.1 30.4 35.5 45.9 19.7 25.3 25.4 37.8 42.2 Average 20.6 25.4 29.9 34.7 40.8

This type of problem may occur in many ways such as through across the belt sampling of extruded products, sampling each head of multi-head machines, collecting same product samples from different machines or batches, taking top to bottom samples from a kiln, etc., etc. The problem may also be more complex, for example "fixed" effects may vary with time.

How to deal with the problem depends on the situation. Options are to control "streams" individually, removing the fixed component of variation, setting control limits on the basis of "residuals" (estimated with an analysis of variance), setting control limits on the basis of simulation or treating the averages of the subgroups as "individuals".

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