MACHINE-POWER
# Robustness of Distribution Optimized X-Bar Charts (or How and why a standard X-bar chart can fail with perfectly normal data)

Consider the following distribution of individual data which is perfectly normal and came from a process mean that did not change

One can therefore expect the X-Bar char to be in control. The reality is however far from that as shown in the X-bar chart below

Even the sampling distribution for the averages appears normal, supported by the probability plot.

So how can normally distributed data be so much out of control. The individual data is normal, the sampling distribution is normal. The standard deviation of averages approaches the standard deviation of individuals divided by the square root of the subgroup size as the subgroup size becomes large. In this instance the subgroup size was 25 which can be considered large. One other possibility is that the ‘biasing’ constant d2 for the Range Chart is wrong when subgroup sizes are large. Control limits for a range chart are computed as Centerline +/– 3* Average Range/(d2*sqrt(subgroup size)) where d2 is dependent on the subgroup size. But since this constant has proven this theory must be discarded.

The explanation is very simple. The subgroup data consisted of 5 repeats all of which are the same. For example

Subgroup 1 consisted of 5 repeats as shown below

-1.027453376 | 0.912685542 | 0.135018914 | -0.301030375 | 1.061788024 |

-1.027453376 | 0.912685542 | 0.135018914 | -0.301030375 | 1.061788024 |

-1.027453376 | 0.912685542 | 0.135018914 | -0.301030375 | 1.061788024 |

-1.027453376 | 0.912685542 | 0.135018914 | -0.301030375 | 1.061788024 |

-1.027453376 | 0.912685542 | 0.135018914 | -0.301030375 | 1.061788024 |

The data of the individuals and averages would be normally distributed. But the biasing constant assumes completely random data. Since the data is a repeat of 5 times one set of data the biasing factor applied for subgroup size of 25 is inflated. In this case a biasing factor for a subgroup of 5 would have worked better as the Range of 5 measurements and 25 is the same.

This of course is a cherry picked example of how things can go wrong and one where it is hard to conceive such a problem in practice. However, we have encountered such problems. An example is where operater recorded diameter from 5 locations on each of 5 bolts. The variation with the five locations on the same bolt was small compared to the variation between each bolt. The solution is of course very simple. Take only one reading. Another alternative is to use Distribution Optimized Control charts which make no assumption about normality and do not use formula per se. They work perfectly on not only the x-bar chart but Range chart also

And importantly they are like the standard x-bar chart applied to one set of 5, as is expected. Repeats do not change the sub group averages and ranges. They can’t be exact because they are based on a more robust algorithm.

Distribution Optimized charts should not be used as a substitute for poorly rationalized sampling. Rationalizing sampling must be the first course of action. Similarly Distribution Optimized charts must not be used as a substitute to correcting dirty processes that produce non normal data, not because non normality is inherent to the process but simply because the process is out f control. However, once sampling has bee rationalized and processes in control Distribution Optimized are the safer bet, because no process is perfect and no process is perfectly normal.

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