Range

Ri = max(xi) — min(x{) i = 1, • • • ,N or n

Selection of Control Limits

Three parameters affect the control limit selection:

i. the estimate of average level of the variable, ii. the variable spread expressed as range or standard deviation, and

Hi. a constant based on the probability of Type I error, a.

The "3cr" (a denoting the standard deviation of the variable) control limits are the most popular control limits. The constant 3 yields a Type I error probability of 0.00135 on each side (a = 0.0027). The control limits expressed as a function of population standard deviation cr are:

The x chart considers only the current data value in assessing the status of the process. Run rules have been developed to include historical information such as trends in data. The run rules sensitize the chart, but they also increase the false alarm probability. The warning limits are useful in developing additional rules (run rules) in order to increase the sensitivity of Shewhart charts. The warning limits are established at "2-sigma" level, which corresponds to a/2=0.02275. Hence,

The Mean and Range Charts

Development of the x and R charts starts with the R chart. Since the control limits of the x chart depends on process variability, its limits are not meaningful before R is in-control.

The Range Chart

Range is the difference between the maximum and minimum observer tions in a sample.

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