CHAPTER 27 • Statistical Process Control

• x Charts for Process Monitoring 27-11
PROCESS-MONITORING CONDITIONS
Measure a quantitative variable x that has a Normal distribution. The process has
been operating in control for a long period, so that we know the process mean and
the process standard deviation that describe the distribution of x as long as the
process remains in control.
In practice, we must of course estimate the process mean and standard deviation
from past data on the process. Under the process-monitoring conditions, we
have very many observations and the process has remained in control. The law
of large numbers tells us that estimates from past data will be very close to the
truth about the process. That is, at the process-monitoring stage we can act as if
we know the true values of and . Note carefully that and describe
the center and spread of the variable x only as long as the process remains in
control. A special cause may at any time disturb the process and change
the mean, the standard deviation, or both.
To make control charts, begin by taking small samples from the process at
regular intervals. For example, we might measure 4 or 5 consecutive parts or time
the responses to 4 or 5 consecutive customer calls. There is an important idea
here: the observations in a sample are so close together that we can assume that the
process is stable during this short period of time. Variation within the same sample
gives us a benchmark for the common cause variation in the process. The process
standard deviation refers to the standard deviation within the time period spanned by
one sample. If the process remains in control, the same describes the standard
deviation of observations across any time period. Control charts help us decide
whether this is the case.
We start with the x chart based on plotting the means of the successive samples.
Here is the outline:
1. Take samples of size n from the process at regular intervals. Plot the means
x of these samples against the order in which the samples were taken.
2. We know that the sampling distribution of x under the process-monitoring
conditions is Normal with mean and standard deviation s/ 1n (see text
page 294). Draw a solid center line on the chart at height .
3. The 99.7 part of the 68–95–99.7 rule for Normal distributions (text page
77) says that, as long as the process remains in control, 99.7% of the values
of x will fall between m 3s/ 1n and m 3s/ 1n. Draw dashed control
limits on the chart at these heights. The control limits mark off the range of
variation in sample means that we expect to see when the process remains
in control.
If the process remains in control and the process mean and standard deviation
do not change, we will rarely observe an x outside the control limits. Such an x is
therefore a signal that the process has been disturbed.
x _ chart
center line
control limits