Applied Statistics and Probability for Engineers

16-10 CUMULATIVE SUM CONTROL CHART 637

zero. It may be helpful to have an estimate of the new process mean following the shift. This

can be computed from

(16-32)

It is also useful to present a graphical display of the tabular CUSUMS, which are some-

times called CUSUM status charts. They are constructed by plotting sH(i) and sL(i) versus the

sample number. Figure 16-21 shows the CUSUM status chart for the data in Example 16-6.

Each vertical bar represents the value of sH(i) and sL(i) in period i. With the decision interval

plotted on the chart, the CUSUM status chart resembles a Shewhart control chart. We have

also plotted the sample statistics xifor each period on the CUSUM status chart as the solid

dots. This frequently helps the user of the control chart to visualize the actual process per-

formance that has led to a particular value of the CUSUM.

The tabular CUSUM is designed by choosing values for the reference value Kand the de-

cision interval H. We recommend that these parameters be selected to provide good average

run-length values. There have been many analytical studies of CUSUM ARL performance.

Based on these studies, we may give some general recommendations for selecting Hand K.

Define and , where is the standard deviation of the sample variable used

in forming the CUSUM (if n1, ). Using h4 or h5 and k 1 2 will gener-

ally provide a CUSUM that has good ARL properties against a shift of about (or 1X) in

the process mean. If much larger or smaller shifts are of interest, set k2, where is the size

of the shift in standard deviation units. Some practitioners prefer to use a standardized variable

as the basis of the CUSUM. In that case, Equations 16-30 and 16-31 become

sH 1 i 2 max 3 0, yiKsH 1 i 124 and sL 1 i 2 max 3 0, KyisL 1 i 124

yi 1 xi 02 X

1 X

XX

HhX KkX X

ˆμ

 0 K

sH 1 i 2

nH,^ if^ sH^1 i^2 H

 0 K

sL 1 i 2

nL, if^ sL^1 i^2 H

1

(^946)
2 3 4 5 6 7 8 9 101112131415 1617181920
Sample number
H = 5
(^955)
(^964)
(^973)
98 2
(^991)
100 0
(^1011)
(^1022)
(^1033)
104 4
(^1055)
106 6
sH(i)
sL(i)
x
H = 5
Figure 16-21 The
CUSUM status chart
for Example 15-6.
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