Introduction to Probability and Statistics for Engineers and Scientists

13.2Control Charts for Average Values: TheX-Control Chart 547

That is, if the process is in control throughout the production of subgroup√ i, then
n(Xi−μ)/σhas a standard normal distribution. Now it follows that a standard nor-
mal random variableZ will almost always be between−3 and+3. (Indeed,P{− 3 <
Z < 3 }=.9973.) Hence, if the process is in control throughout the production of the
items in subgroupi, then we would certainly expect that

− 3 <

√

n

Xi−μ

σ

< 3

or, equivalently, that

μ−

3 σ

√

n

<Xi<μ+

3 σ

√

n

The values

UCL≡μ+

3 σ

√

n

and

LCL≡μ−

3 σ

√

n

are called, respectively, theupperandlower control limits.
TheX-control chart, which is designed to detect a change in the average value of an
item produced, is obtained by plotting the successive subgroup averagesXiand declaring
that the process is out of control the first timeXidoes not fall between LCL and UCL
(see Figure 13.1).

EXAMPLE 13.2a A manufacturer produces steel shafts having diameters that should be
normally distributed with mean 3 mm and standard deviation .1 mm. Successive samples
of four shafts have yielded the following sample averages in millimeters.

Sample X Sample X

1 3.01 6 3.02

2 2.97 7 3.10

3 3.12 8 3.14

4 2.99 9 3.09

5 3.03 10 3.20

What conclusion should be drawn?