Introduction to Probability and Statistics for Engineers and Scientists

546 Chapter 13:Quality Control

isolation but rather take into account the values of other subgroups. Three different control
charts of this type — known as moving-average, exponential weighted moving-average,
and cumulative sum control charts — are presented in Section 13.6.

13.2 Control Charts for Average Values: TheX-Control Chart

Suppose that when the process is in control the successive items produced have measurable
characteristics that are independent, normal random variables with meanμand variance
σ^2. However, due to special circumstances, suppose that the process may go out of control
and start producing items having a different distribution. We would like to be able to
recognize when this occurs so as to stop the process, find out what is wrong, and fix it.
LetX 1 ,X 2 ,...denote the measurable characteristics of the successive items produced.
To determine when the process goes out of control, we start by breaking the data up into
subgroups of some fixed size — call itn. The value ofnis chosen so as to yield uniformity
within subgroups. That is,nmay be chosen so that all data items within a subgroup were
produced on the same day, or on the same shift, or using the same settings, and so on.
In other words, the value ofnis chosen so that it is reasonable that a shift in distribution
would occur between and not within subgroups. Typical values ofnare 4, 5, or 6.
LetXi,i=1, 2,...denote the average of theith subgroup. That is,

X 1 =

X 1 +···+Xn

n

X 2 =

Xn+ 1 +···+X 2 n

n

X 3 =

X 2 n+ 1 +···+X 3 n

n

and so on. Since, when in control, each of theXihave meanμand varianceσ^2 , it follows
that

E(Xi)=μ

Var(Xi)=

σ^2

n

and so

Xi−μ

√

σ^2

n

∼N(0, 1)