Introduction to Probability and Statistics for Engineers and Scientists

560 Chapter 13:Quality Control

If we letXidenote the number of defects in theith unit, then, since the variance of
a Poisson random variable is equal to its mean, when the process is in control

E[Xi]=λ, Var(Xi)=λ

Hence, when in control eachXishould with high probability be withinλ± 3

√
λ, and so
the upper and lower control limits are given by

UCL=λ+ 3

√

λ, LCL=λ− 3

√

λ

As before, when the control chart is started andλis unknown, a sample ofkunits should
be used to estimateλby

X=(X 1 +···+Xk)/k

This results in trial control limits

X+ 3

√

X and X− 3

√

X

If all theXi,i=1,...,kfall within these limits, then we suppose that the process is in
control withλ=X. If some fall outside, then these points are eliminated and we recompute
X, and so on.
In situations where the mean number of defects per item (or per day) is small, one
should combine items (days) and use as data the number of defects in a given number —
say,n— of items (or days). Since the sum of independent Poisson random variables
remains a Poisson random variable, the data values will be Poisson distributed with a
larger mean valueλ. Such combining of items is useful when the mean number of defects
per item is less than 25.
To obtain a feel for the advantage in combining items, suppose that the mean number
of defects per item is 4 when the process is under control; and suppose that something
occurs that results in this value changing from 4 to 6, that is, an increase of 1 standard
deviation occurs. Let us see how many items will be produced, on average, until the process
is declared out of control when the successive data consist of the number of defects inn
items.
Since the number of defects in a sample ofnitems is, when under control, Poisson
distributed with mean and variance equal to 4n, the control limits are 4n± 3

√
4 nor
4 n± 6

√
n. Now if the mean number of defects per item changes to 6, then a data value
will be Poisson with mean 6nand so the probability that it will fall outside the control
limits — call itp(n) — is given by

p(n)=P{Y> 4 n+ 6

√

n}+P{Y< 4 n− 6

√

n}