Introduction to Probability and Statistics for Engineers and Scientists

13.6Other Control Charts for Detecting Changes in the Population Mean 571

13.6.3 Cumulative Sum Control Charts

The major competitor to the moving-average type of control chart for detecting a small-
to moderate-sized change in the mean is the cumulative sum (often reduced to cu-sum)
control chart.
Suppose, as before, thatX 1 ,X 2 ,...represent successive averages of sugroups of sizen
and that when the process is in control these random variables have meanμand standard
deviationσ/

√
n. Initially, suppose that we are only interested in determining when an
increase in the mean value occurs. The (one-sided) cumulative sum control chart for
detecting an increase in the mean operates as follows: Choose positive constantsdandB,
and let

Yj=Xj−μ−dσ/

√

n, j≥ 1

Note that when the process is in control, and soE[Xj]=μ,

E[Yj]=−dσ/

√

n< 0

Now, let

S 0 = 0

Sj+ 1 =max{Sj+Yj+ 1 ,0}, j≥ 0

The cumulative sum control chart having parametersdandBcontinually plotsSj, and
declares that the mean value has increased at the firstjsuch that

Sj>Bσ/

√

n

To understand the rationale behind this control chart, suppose that we had decided
to continually plot the sum of all the random variablesYithat have been observed so far.
That is, suppose we had decided to plot the successive values ofPj, where

Pj=

∑j

i= 1

Yi

which can also be written as

P 0 = 0

Pj+ 1 =Pj+Yj+ 1 , j≥ 0

Now, when the system has always been in control, all of theYihave a negative expected
value, and thus we would expect their sum to be negative. Hence, if the value ofPjever