Introduction to Probability and Statistics for Engineers and Scientists

13.6Other Control Charts for Detecting Changes in the Population Mean 563

13.6 OTHER CONTROL CHARTS FOR DETECTING

CHANGES IN THE POPULATION MEAN

The major weakness of theX-control chart presented in Section 13.2 is that it is relatively
insensitive to small changes in the population mean. That is, when such a change occurs,
since each plotted value is based on only a single subgroup and so tends to have a relatively
large variance, it takes, on average, a large number of plotted values to detect the change.
One way to remedy this weakness is to allow each plotted value to depend not only on
the most recent subgroup average but on some of the other subgroup averages as well.
Three approaches for doing this that have been found to be quite effective are based on
(1) moving averages, (2) exponentially weighted moving averages, and (3) cumulative sum
control charts.

13.6.1 Moving-Average Control Charts

The moving-average control chart of span sizekis obtained by continually plotting the
average of thekmost recent subgroups. That is, the moving average at timet, call itMt,
is defined by

Mt=

Xt+Xt− 1 +···+Xt−k+ 1

k

whereXiis the average of the values of subgroupi. The successive computations can be
easily performed by noting that

kMt=Xt+Xt− 1 +···+Xt−k+ 1

and, substitutingt+1 fort,

kMt+ 1 =Xt+ 1 +Xt+···+Xt−k+ 2

Subtraction now yields that

kMt+ 1 −kMt=Xt+ 1 −Xt−k+ 1

or

Mt+ 1 =Mt+

Xt+ 1 −Xt−k+ 1

k

In words, the moving average at timet+1 is equal to the moving average at timet
plus 1/ktimes the difference between the newly added and the deleted value in the