Applied Statistics and Probability for Engineers

16-10 CUMULATIVE SUM CONTROL CHART 635

1

3 A

234 ... i

(a)

2 A

A

si

θ

O

L

U

d P

1

• 4

si

51015202530

• 2

0

+2

+4

+6

(b)

Observation, i

K

Figure 16-20 The cumulative sum control chart. (a) The V-mask and scaling. (b) The cumulative

sum control chart in operation.

The tabular procedure is particularly attractive when the CUSUM is implemented on a

computer.

Let SH(i) be an upper one-sided CUSUM for period iand SL(i) be a lower one-sided

CUSUM for period i. These quantities are calculated from

(16-30)

and

(16-31)

where the starting values sH 102 sL 102 0.

sL 1 i 2 max 3 0, 1  0 K 2 xi sL 1 i 124

sH 1 i 2 max 3 0, xi 1  0 K 2 sH 1 i 124

CUSUM

Control Chart

In Equations 16-30 and 16-31 Kis called the reference value,which is usually chosen

about halfway between the target  0 and the value of the mean corresponding to the out-of-

control state,  1  0 . That is, Kis about one-half the magnitude of the shift we are in-

terested in, or

Notice that SH(i) and SL(i) accumulate deviations from the target value that are greater than

K, with both quantities reset to zero upon becoming negative. If either SH(i) or SL(i) exceeds

a constant H, the process is out of control. This constant His usually called the decision

interval.

K



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