Applied Statistics and Probability for Engineers

636 CHAPTER 16 STATISTICAL QUALITY CONTROL

EXAMPLE 16-6 A Tabular CUSUM

We will illustrate the tabular CUSUM by applying it to the chemical process concentration data

in Table 16-7. The process target is  0 99, and we will use K1 as the reference value and

H10 as the decision interval. The reasons for these choices will be explained later.

Table 16-8 shows the tabular CUSUM scheme for the chemical process concentration

data. To illustrate the calculations, note that

Therefore, for observation 1 the CUSUMS are

and

as shown in Table 16-8. The quantities nHand nLin Table 16-8 indicate the number of periods that

the CUSUM sH(i) or sL(i) have been nonzero. Notice that the CUSUMS in this example never ex-

ceed the decision interval H10. We would therefore conclude that the process is in control.

When the tabular CUSUM indicates that the process is out of control, we should search

for the assignable cause, take any corrective actions indicated, and restart the CUSUMS at

sL 112 max 3 0, 98x 1 sL 1024 max 3 0, 98102.0 04  0

sH 112 max 3 0, x 1  100 sH 1024 max 3 0, 102.0 100  04 2.0

max 3 0, 98xisL 1 i 124

sL 1 i 2 max 3 0, 1  0 K 2 xisL 1 i 124 max 3 0, 199  12 xisL 1 i 124

max 3 0, xi 100 sH 1 i 124

sH 1 i 2 max 3 0, xi 1  0 K 2 sH 1 i 124 max 3 0, xi 199  12 sH 1 i 124

Table 16-8 The Tabular CUSUM for the Chemical Process Concentration Data

Observation Upper CUSUM Lower CUSUM

ixi xi 100 sH(i) nH 98 xi sL(i) nL

1 102.0 2.0 2.0 1 4.0 0.0 0

2 94.8 5.2 0.0 0 3.2 3.2 1

3 98.3 1.7 0.0 0 0.3 2.9 2

4 98.4 1.6 0.0 0 0.4 2.5 3

5 102.0 2.0 2.0 1 4.0 0.0 0

6 98.5 1.5 0.5 2 0.5 0.0 0

7 99.0 1.0 0.0 0 1.0 0.0 0

8 97.7 2.3 0.0 0 0.3 0.3 1

9 100.0 0.0 0.0 0 2.0 0.0 0

10 98.1 1.9 0.0 0 0.1 0.0 0

11 101.3 1.3 1.3 1 3.3 0.0 0

12 98.7 1.3 0.0 0 0.7 0.0 0

13 101.1 1.1 1.1 1 3.1 0.0 0

14 98.4 1.6 0.0 0 0.4 0.0 0

15 97.0 3.0 0.0 0 1.0 1.0 1

16 96.7 3.3 0.0 0 1.3 2.3 2

17 100.3 0.3 0.3 1 2.3 0.0 0

18 101.4 1.4 1.7 2 3.4 0.0 0

19 97.2 2.8 0.0 0 0.8 0.8 1

20 101.0 1.0 1.0 0 3.0 0.0 0

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