# ARL and Devor Rules

The following control chart was plotted for a process that has been in control for sometime, and the mean and standard deviation are 100.08 and 1.01. The subgroup size is n=3.

(Please let me know how to insert a chart)

Looking at this chart we see one clear point on the xbar chart that has exceeded the UCL at subgroup 23. However we also know that the mean changed by 0.5 at subgroup 7.

What is “actual” run length (i.e. the number of runs after a change before a UCL alarm is seen)?

What is the theoretical value for ARL in this scenario (NB: note that the mean shift = 0.5 sigma)?

Why are they so different?

a) The run length before a violation is a random number, and the ARL is simply the average of that number.

b) The process also had an increase in the underlying standard deviation.

c) By using n=3 we narrowed the control limits by sqrt(3) and so any earlier alarm was expected.

2) Which of the “Devor Rules” on Slides 33 and 34, (repeated below) would have given the earliest indication of out of control behavior and when would that indication occur?

1. Outside ±3σ (1 at 23)

2. 2 of 3 >±2σ

3. 4 of 5 >± 1σ

4. All points inside ±1σ

5. Runs of 8 or more above or below the centerline

6. 6 or more points in consistent direction

7. Bi-Modal Data

8. 8 successive points outside ±1s

Rule Violation (or none)

At which subgroup can we confirm the violation? (or none).