Introduction to Probability and Statistics for Engineers and Scientists

564 Chapter 13:Quality Control

moving average. For values oftless thank,Mtis defined as the average of the firstt
subgroups. That is,

Mt=

X 1 +···+Xt

t

ift<k

Suppose now that when the process is in control the successive values come from a
normal population with meanμand varianceσ^2. Therefore, ifnis the subgroup size, it
follows thatXiis normal with meanμand varianceσ^2 /n. From this we see that the average
ofmof theXiwill be normal with meanμand variance given by Var(Xi)/m=σ^2 /nm
and, therefore, when the process is in control

E[Mt]=μ

Var(Mt)=

{

σ^2 /nt ift<k

σ^2 /nk otherwise

Because a normal random variable is almost always within 3 standard deviations of its
mean, we have the following upper and lower control limits forMt:

UCL=

{

μ+ 3 σ/

√

nt ift<k

μ+ 3 σ/

√

nk otherwise

LCL=

{

μ− 3 σ/

√

nt ift<k

μ− 3 σ/

√

nk otherwise

In other words, aside from the firstk−1 moving averages, the process will be declared
out of control whenever a moving average differs fromμby more than 3σ/

√

nk.

EXAMPLE 13.6a When a certain manufacturing process is in control, it produces items
whose values are normally distributed with mean 10 and standard deviation 2. The fol-
lowing simulated data represent the values of 25 subgroup averages of size 5 from a normal
population with mean 11 and standard deviation 2. That is, these data represent the sub-
group averages after the process has gone out of control with its mean value increasing
from 10 to 11. Table 13.3 presents these 25 values along with the moving averages based
on span sizek=8 as well as the upper and lower control limits. The lower and upper
control limits fort>8 are 9.051318 and 10.94868.
As the reader can see, the first moving average to fall outside its control limits occurred
at time 11, with other such occurrences at times 12, 13, 14, 16, and 25. (It is interesting
to note that the usual control chart — that is, the moving average withk=1 — would
have declared the process out of control at time 7 sinceX 7 was so large. However, this is
the only point where this chart would have indicated a lack of control (see Figure 13.3).