Applied Statistics and Probability for Engineers

16-10 CUMULATIVE SUM CONTROL CHART 633

average of the jth sample. Then if  0 is the target for the process mean, the cumulative sum

control chart is formed by plotting the quantity

(16-29)

against the sample number i. Now, Siis called the cumulative sum up to and including the

ith sample. Because they combine information from severalsamples, cumulative sum charts

are more effective than Shewhart charts for detecting small process shifts. Furthermore,

they are particularly effective with samples of n1. This makes the cumulative sum con-

trol chart a good candidate for use in the chemical and process industries where rational

subgroups are frequently of size 1, as well as in discrete parts manufacturing with automatic

measurement of each part and online control using a microcomputer directly at the work

center.

If the process remains in control at the target value  0 , the cumulative sum defined in

equation 16-29 should fluctuate around zero. However, if the mean shifts upward to some

value  1  0 , say, an upward or positive drift will develop in the cumulative sum Si.

Conversely, if the mean shifts downward to some  1  0 , a downward or negative drift in Si

will develop. Therefore, if a trend develops in the plotted points either upward or downward,

we should consider this as evidence that the process mean has shifted, and a search for the

assignable cause should be performed.

This theory can easily be demonstrated by applying the CUSUM to the chemical process

concentration data in Table 16-3. Since the concentration readings are individual measure-

ments, we would take in computing the CUSUM. Suppose that the target value for the

concentration is  0 99. Then the CUSUM is

Table 16-7 shows the computation of this CUSUM, where the starting value of the

CUSUM, S 0 , is taken to be zero. Figure 16-19 plots the CUSUM from the last column of Table

16-7. Notice that the CUSUM fluctuates around the value of 0.

The graph in Fig. 16-19 is not a control chart because it lacks control limits. There are

two general approaches to devising control limits for CUSUMS. The older of these two

methods is the V-mask procedure. A typical V mask is shown in Fig. 16-20(a). It is a

V-shaped notch in a plane that can be placed at different locations on the CUSUM chart. The

decision procedure consists of placing the V mask on the cumulative sum control chart with

the point Oon the last value of siand the line OPparallel to the horizontal axis. If all the pre-

vious cumulative sums, s 1 , s 2 ,... , si 1 , lie within the two arms of the V mask, the process is

in control. However, if any silies outside the arms of the mask, the process is considered to

be out of control. In actual use, the V mask would be applied to each new point on the

CUSUM chart as soon as it was plotted. In the example shown in Fig. 16-20(b), an upward

shift in the mean is indicated, since at least one of the points that have occurred earlier than

sample 22 now lies below the lower arm of the mask, when the V mask is centered on the

thirtieth observation. If the point lies above the upper arm, a downward shift in the mean is

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