566 Chapter 13:Quality Control
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t
UCL
LCL
Control chart for X
FIGURE 13.3
Let
Wt=αXt+(1−α)Wt− 1 (13.6.1)
whereαis a constant between 0 and 1, and where
W 0 =μ
The sequence of valuesWt,t = 0, 1, 2,...is called anexponentially weighted moving
average. To understand why it has been given that name, note that if we continually
substitute for theWterm on the right side of Equation 13.6.1, we obtain that
Wt=αXt+(1−α)[αXt− 1 +(1−α)Wt− 2 ] (13.6.2)
=αXt+α(1−α)Xt− 1 +(1−α)^2 Wt− 2
=αXt+α(1−α)Xt− 1 +(1−α)^2 [αXt− 2 +(1−α)Wt− 3 ]
=αXt+α(1−α)Xt− 1 +α(1−α)^2 Xt− 2 +(1−α)^3 Wt− 3
..
.
=αXt+α(1−α)Xt− 1 +α(1−α)^2 Xt− 2 +···
+α(1−α)t−^1 X 1 +(1−α)tμ