How Many Clues Do I Need for Limits

In: Business and Management

Submitted By zhwel329
Words 1925
Pages 8
How Many Values Do I Need for Limits?
Donald J. Wheeler
What happens when we use only a few values to create an XmR Chart? There are three aspects to the behavior of the chart that are of interest here. What happens to the number of a false alarms? What happens to the ability of the chart to detect a change? And what happens when the baseline data contain a signal? THE LIKELIHOOD OF A FALSE ALARM Mathematical theory tells us that the risk of a false alarm whenever we place a point on an X-Chart is approximately 0.27%. But what happens to this risk when only k values have been used to compute the limits? Simulation studies provide the answers shown in Table 1 and Figure 1.

Effective Alpha-Level of Chart

1.5%

k even Limits Based on Median Moving Ranges

1.0% 0.5%

k odd

Theoretical 0.27% 0.0% k 5 Limits Based on Average Moving Ranges 10 15 20 25 30 Number of Values Used to Compute Limits 35 40

Figure 1: The Effective Alpha-Levels for the Computed Limits The bottom curve in Figure 1 shows the effective alpha-levels for the limits based on the Average Moving Range. These alpha-levels rapidly approach the theoretical value and then plateau. The two upper curves are for limits based on a Median Moving Range. Both curves show that the effective alpha-level drops down toward the theoretical value as k increases. Moreover, while these limits do have larger effective alpha-levels than limits based on the Average Moving Range (as was expected), these effective alpha-levels do not exceed one percent except when k is 4, 6, 8, or 10. Since an alpha-level of 1.7% is still considered to be a conservative alpha-level, we do not loose the conservative nature of the charts even when the limits are based on few data. Based on Figure 1 it should be clear that we can construct an XmR Chart using as few as four original values without an undue risk of a false alarm.…...

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