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Chapter 22
Quality Control
Copyright © 2012 Pearson Education.
Quality control today is a tool in the larger effort of continual
process improvement in both the manufacturing and service
industries. In this view, work is a process on which data can be
gathered and studied. This study can then lead to improvement in
the process.
The process improvement strategy includes understanding the
process, collecting the data, assessing the process, eliminating
causes of variation, studying the cause and effect of relationships
within the process, and implementing changes.
In this chapter we will study some tools used in the assessment of
the process data to investigate whether the process has changed,
either in its mean or in its variation.
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22.1 A Short History of Quality Control
• Walter A. Shewhart was a statistician and engineer
working at Bell Laboratories in the early part of the
twentieth century. He developed a time plot of a
measure of quality which became known as a control
chart, or a Shewhart chart, in his honor.
Shewhart defined the fundamental rule of statistical process
control (SPC) as:
• “Variation from common-cause systems should be left
to chance, but special causes of variation should be
identified and eliminated.”
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22.1 A Short History of Quality Control
Walter A. Shewhart noticed and defined two distinct sources of
variation in manufacturing.
1) Common-cause Variation: Variation due to random
fluctuations intrinsic to the process.
2) Special-cause Variation: Variation caused by issues in the
process that can be identified. When special-cause variation
is present, the process is said to be out of control.
The short-term focus of quality control is to detect and
correct the special causes of variation. The long-term
focus of continual quality improvement is to reduce the
amount of variation from the common causes.
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22.1 A Short History of Quality Control
Shewhart’s process was broken down into four distinct
stages:
1) Plan: Identify and describe
the process, and suggest
improvements
2) Do: Implement the changes
at a smaller scale.
3) Check: Examine the new
process to see if decreased
variation occurred.
4) Act: If applicable, implement
the improvements on a
large scale.
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22.1 A Short History of Quality Control
Control and Specification Limits
A control chart is used to determine if an in-control process might
change and slip “out of control.”
Control Limits are established to decide when to take action.
When points fall outside of the control limits, this signals that the
process may have changed and some action is needed.
There are two type of errors that can be made in using a
control chart:
• Type I: A false alarm; requiring an unnecessary adjustment to
the process
• Type II: A failure to see a change in the process.
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22.1 A Short History of Quality Control
Control and Specification Limits
Specification Limits are external limits placed on the process
based on quality concerns not directly related to how well the
parts are produced.
Example: Bolts have a specific diameter range they must fall in
in order to work.
Process Capability Studies are used to understand how well the
process is working given the specification limits and the actual
process parameters (mean and Variance).
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22.1 A Short History of Quality Control
Control and Specification Limits
The ratio C p measures the ratio of the entire specification range
(both sides of the mean) to 6 times the standard deviation of the
actual process variation.
USL  LSL
Cp 
6
USL denotes the upper specification limit, LSL denotes the lower
specification limit and σ denotes the standard deviation.
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22.1 A Short History of Quality Control
Control and Specification Limits
Because C p does not explicitly use the location of the mean it
tends to over estimate the potential of the process when the
mean is not centered and can be misleading. A measure known
as C pk is often used instead. When the process mean and
standard deviation are known, C pk is defined as:
C pk
Copyright © 2012 Pearson Education.
USL     LSL 
 min 
,

3

3



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22.1 A Short History of Quality Control
Example: Potato chips
A potato chip company claims that their bags are filled with 5.0 oz.
of chips. Historical data from the bag filing process shows that the
standard deviation of the bag weights are 0.1 oz and the mean is
5.02 oz. The quality control team has set the specification limits to
4.75 and 5.25 oz. Determine the values of C p and C pk .
C p  .833
C pk  min .766,0.9  .766
Copyright © 2012 Pearson Education.
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22.2 Control Charts for Individual
Observation (Run Charts)
Control Charts for individual observations (sometimes called run
charts) are created using measurements made on every product
produced to determine if the process has changed or has gone
out of control.
The average time between
alarms that the process has
changed is called the average
run length or ARL.
The average time between false
alarms for an in-control process
is called the in-control ARL.
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22.2 Control Charts for Individual
Obsevation (Run Charts)
The operating-characteristic (OC) curve describes the ability of
any chart in detecting a shift or change in the process.
Shifts are measured in terms
of standard deviations. The
OC curve gives the probability
of detecting a shift of a certain
size in the next observation. A
level shift is an abrupt change
while a gradual change is
called a trend.
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22.2 Control Charts for Individual
Obsevation (Run Charts)
The Western Electric Handbook (WECO, 1956) suggests several
rules for deciding whether a process is out of control:
1) One point outside of the 3σ control limits on either side
2) Two out of three consecutive points beyond the 2σ “warning
limits”
3) Four out of five consecutive points beyond 1σ from the mean
4) Eight (or seven) points in a row on the same side of the mean
5) Six points in a row increasing (or decreasing)
6) Fourteen points in a row alternatively increasing and
decreasing
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22.2 Control Charts for Individual
Observation (Run Charts)
Example: A producer of beverage containers wants to ensure that a liquid
at 90°C will lose no more than 4°C after 30 minutes. Historical data shows
the standard deviation to be 0.2°C. The quality control team has set 3.5°C
and 4.5°C as the specification limits. Construct a run chart for the data below
and determine if the process is out-of-control.
Container #
1
2
3
4
5
6
7
8
9
10
11
12
Temperature
loss (°C)
3.89
3.71
4.14
4.12
3.74
3.99
4.22
4.02
4.48
3.93
3.87
3.94
Copyright © 2012 Pearson Education.
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22.3 Control Charts for Measurements:
X and R Charts
X charts are control charts of sample means of periodic
samples. These plots have several advantages over run
charts.
• Less frequent testing is a simple advantage especially
when testing can take time or is destructive.
• Sample means have a distribution closer to the normal
distribution.
The control limits for the charts are as follows:
LCL    3
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
n
UCL    3

n
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22.3 Control Charts for Measurements:
X and R Charts
The Xchart below shows 25 five-item samples at regular
intervals with   300 and   33 .
LCL    3
UCL    3

n

n
 255.7
 344.3
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22.3 Control Charts for Measurements:
X and R Charts
R charts are control charts of the ranges of periodic samples.
The control limits for the charts are as follows:
LCL  R  3d3
R
d2
R
UCL  R  3d3
d2
Where, for k sample:
R1  ...  Rk
R
k
d2 and d3 are found for various sample sizes in Table R.
An alternative to the R chart is the S chart, which uses the
standard deviations of samples rather than their ranges.
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22.3 Control Charts for Measurements:
X and R Charts
The R chart below shows 25 samples of size 5 and R  81.32.
R
LCL  R  3d3
d2
R
UCL  R  3d3
d2
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22.3 Control Charts for Measurements:
X and R Charts
To choose an appropriate sample size, quality control monitors
examine the operating characteristic curves such as the
one below.
Both the process shift
and the sample size can
be used to find the
probability of a Type II
error for a next sample.
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22.4 Actions for Out-of-Control Processes
For an established manufacturing process there may be an outof-control action plan (OCAP) in place based on past
experience. For a new process for which no such plans exists
use the following basic steps to determine the problem.
• Use a team of all those involved in the process to determine the
possible causes. Use a cause-and-effect diagram to organize the
possible causes.
• Use a series of experiments to test the most probable causes.
• Once the most likely cause is established, correct the problem
in hopes of driving the process back into control.
• Restart the monitoring process.
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22.5 Control Charts for Attributes: p
Charts and c Charts
The most common type of categorical measurements used in
quality control are dichotomous variables (yes/no responses).
Example: A customer service phone call being rated by the
customer as either satisfactory or unsatisfactory
A p chart is a control chart of proportions used to monitor the
process for changes in proportions of defectives. The control
limits for the proportions are as follows:
pq
LCL  p  3
n
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pq
UCL  p  3
n
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22.5 Control Charts for Attributes: p
Charts and c Charts
As with charts there are a variety of alternative rules to WECO to
determine if a process is out of control. Minitab, for example,
uses the following four:
1) One point outside of the 3σ lines
2) Nine points in a row outside of 1σ on the same side
3) Six increasing or decreasing points in a row
4) Fourteen points in a row that alternate up and down
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22.5 Control Charts for Attributes: p
Charts and c Charts
A c chart is another option for monitoring dichotomous variables
used in the quality control process. Rather than using
proportions, the c chart uses the number of defective items in a
sample.
The Poisson model is used to determine the probabilities of a
certain number of defective products given a low expected value
of defects. A Poisson variable has a mean  and a standard
deviation  . Thus the control limits given the standard number
of defects c are:
LCL  c  c
UCL  c  c
If the LCL is less than 0 use 0 as the LCL.
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22.5 Control Charts for Attributes: p
Charts and c Charts
Example: Rope
Rope used for climbing is rated to be able to hold a certain
weight. The rope fails testing if it snaps when brought to a
certain tension. Historically only 1.5% of the tested ropes snap.
Suppose a sample of boxes containing 50 ropes are selected at
random and tested.
a) What is the standard deviation for the sample proportion from
a box of 50.
σ =0.017
b) What are the upper and lower 3σ control limits.
LCL=0
UCL=.071
c) How many ropes in a box would have to snap for the
process to be out of control.
4 ropes would have to snap.
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22.6 Philosophies of Quality Control
The three common ideas in quality control from statistical
thinking are:
• All work occurs in a system of interconnected processes.
• Variation exists in all processes.
• Understanding and reducing variation are keys to success.
Total quality management (TQM) and Six Sigma® are two
business improvement approaches.
TQM extends quality control to encompass the total organization.
Six Sigma® focuses more on the identification and elimination of
defects or special-cause variations in the business processes.
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What Have We Learned?
Understand the principles of quality control.
• A process is in control if the aspects being followed stay within
the control limits established for them.
• When a process drifts out of control, it is important to recognize
the fact so that adjustments can be made promptly. But we hope
to avoid making too many false alarms, so a balance must be
found.
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What Have We Learned?
Use run charts to track individual observations.
• A level shift flags a sudden change. A trend is a more gradual
change.
• The ability of a chart to detect shifts is described by its operating
characteristic (OC) curve.
• The Western Electric handbook specified several rules for
deciding when a process is out of control.
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What Have We Learned?
Use X , R, and S charts to track a process by plotting means
and ranges or standard deviations of small samples drawn from
the process.
• The control limits are often based on estimated standard
deviations and means.
• The S chart is more appropriate than the R chart when samples
are moderately large.
Understand the basic steps in creating an out-of-control action
plan (OCAP).
1) Organize and prioritize
3) Correct the cause.
possible causes.
4) Restart the monitoring process.
2) Perform experiments to
determine the cause.
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What Have We Learned?
Use p charts and c charts to track processes that have
categorical measurements.
• A p chart tracks proportions using control limits based on the
standard deviation of a proportion chart.
• A c chart tracks the number of defective items in a process.
Understand the basic philosophies of quality control, such as the
Total Quality Management (TQM) and Six Sigma®, and how they
are applied to improve business performance and quality.
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