HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2010 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Chapter 17
Statistical Process Control
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
Objectives:
• To learn the meaning of a mean chart using standard
deviation.
• To learn the meaning of a mean chart using range.
• To find the center line of a mean chart.
• To find the upper and lower control limits.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
Definitions:
Process control relies on two assumptions:
• No matter what the specifications are for a product, the process that
produces the product will create output that has variation.
• Improving a process requires removing variation from it. Though it
would be ideal to remove all variation, this cannot be achieved. The
goal then is to move towards the ideal. This notion is known as
continuous improvement.
• Control chart – a graphical display of values of a process over
time. This chart has an upper and lower control limit. If a value is
above the upper control limit or below the lower control limit the
process is out of control. The control chart also contains a
centerline that represents the average value of the quality
characteristic corresponding to the in-control state.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
Definitions:
•
x Chart – a control chart that plots the means of small samples
taken from the process at regular intervals over time. There are two
ways to construct an x chart:
• If the process mean and standard deviation are known the
chart is constructed such that any sample average within ±3σ is
considered in control and any sample average outside of ±3σ is
considered out of control. It is defined by the formulas:
Centerline = 
Upper Control Limit UCL  =  
3
n
Lower Control Limit LCL  =  
3
n
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
Definitions:
• If the process mean and standard deviation are not known the
chart is constructed from factors in the table on the next slide.
The table provides 3σ factors that relate to the sample ranges.
It is defined by the formulas:
Upper Control Limit UCL  = x  AR
Lower Control Limit LCL  = x  AR
where
R  the mean of the sample ranges,
x  the mean of the sample means (grand mean), and
A  a factor that relates 3 to R.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
3σ Factors for Computing Control Charts Limits
Sample
Size
Mean Factor, A
Lower Range,
D3
Upper Range,
D4
2
1.880
0.000
3.267
3
1.023
0.000
2.574
4
0.729
0.000
2.282
5
0.577
0.000
2.114
6
0.483
0.000
2.004
7
0.419
0.076
1.924
8
0.373
0.136
1.864
9
0.337
0.184
1.816
10
0.308
0.223
1.777
15
0.223
0.348
1.652
20
0.180
0.414
1.586
25
0.153
0.459
1.541
Over 25
 1 
0.75 

 n
155 – 0.0015n
0.45 + 0.001n
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
Process Controls:
Suppose that a company produces bolts with an average width of 0.45
inches and a standard deviation of 0.11 inches. A total of 24 samples
of size 25 are taken. Determine the upper and lower control limits for
the process and determine if any of the sample values on the graph
are considered out of control.
Solution:
Centerline =   0.45
3  0.11
3
= 0.45 
= 0.516
n
25
3  0.11
3
Lower Control Limit LCL  =  
= 0.45 
= 0.384
n
25
Upper Control Limit UCL  =  
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
Process Controls:
Suppose that a company produces bolts with an average width of 0.45
inches and a standard deviation of 0.11 inches. A total of 24 samples
of size 25 are taken. Determine the upper and lower control limits for
the process and determine if any of the sample values on the graph
are considered out of control.
Solution:
Adding the Upper Control
Limit and Lower Control
Limit lines to the graph,
we can see that one of the
sample averages is
considered out of control.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
Process Controls:
Suppose that a company produces bolts but the true mean and
standard deviation are not known. A total of 24 samples of size 25 are
taken. The mean of the sample means (grand mean) is 0.4525 and
the mean of the sample ranges is 0.4029. Determine the upper and
lower control limits for the process and determine if any of the sample
values on the graph are considered out of control.
Solution:
Centerline  x  0.4525
Upper Control Limit UCL   x  AR  0.4525  0.153  0.4029   0.5141
Lower Control Limit LCL   x  AR  0.4525  0.153  0.4029   0.3909
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.3 Monitoring with an x Chart
Process Controls:
Suppose that a company produces bolts but the true mean and
standard deviation are not known. A total of 24 samples of size 25 are
taken. The mean of the sample means (grand mean) is 0.4525 and
the mean of the sample ranges is 0.4029. Determine the upper and
lower control limits for the process and determine if any of the sample
values on the graph are considered out of control.
Solution:
Adding the Upper Control
Limit and Lower Control
Limit lines to the graph,
we can see that one of the
sample averages is
considered out of control.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.4 Monitoring with an R Chart
Objectives:
• To learn the meaning of an R chart using standard deviation.
• To find the centerline of a R chart.
• To find the upper and lower control limits.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.4 Monitoring with an R Chart
Definitions:
• R Chart – a control chart that monitors the variation of the samples
of process.
• The chart uses the average of the sample range as the center
line and upper and lower level factors based on the sample
size. A table of these factors is given on the next slide.
Centerline  R
Upper Control Limit UCL  = RD4
Lower Control Limit LCL   RD3
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.4 Monitoring with an R Chart
3σ Factors for Computing Control Charts Limits
Sample
Size
Mean Factor, A
Lower Range,
D3
Upper Range,
D4
2
1.880
0.000
3.267
3
1.023
0.000
2.574
4
0.729
0.000
2.282
5
0.577
0.000
2.114
6
0.483
0.000
2.004
7
0.419
0.076
1.924
8
0.373
0.136
1.864
9
0.337
0.184
1.816
10
0.308
0.223
1.777
15
0.223
0.348
1.652
20
0.180
0.414
1.586
25
0.153
0.459
1.541
Over 25
 1 
0.75 

 n
155 – 0.0015n
0.45 + 0.001n
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.4 Monitoring with an R Chart
Process Controls:
Suppose that a company produces bolts but is concerned about the
variation in the process. A total of 24 samples of size 25 are taken.
The mean of the sample ranges is 0.4029. Determine the upper and
lower control limits for the process using an R Chart and determine if
any of the values of the sample ranges are considered out of control.
Solution:
Centerline  R  0.4029
Upper Control Limit UCL   RD4  0.4029 1.541  0.6209
Lower Control Limit LCL   RD3  0.4029  0.459   0.1849
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.4 Monitoring with an R Chart
Process Controls:
Suppose that a company produces bolts but is concerned about the
variation in the process. A total of 24 samples of size 25 are taken.
The mean of the sample ranges is 0.4029. Determine the upper and
lower control limits for the process using an R Chart and determine if
any of the values of the sample ranges are considered out of control.
Solution:
Adding the Upper Control
Limit and Lower Control
Limit lines to the graph,
we can see that all of the
sample ranges are
considered in control.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.5 Monitoring with a p Chart
Objectives:
• To learn the meaning of an p chart using standard deviation.
• To find the center line of a p chart.
• To find the upper and lower control limits.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.5 Monitoring with a p Chart
Definitions:
• p Chart – a control chart that monitors the proportions of a sample
that possess a certain attribute. There are two ways to construct a
p chart:
• If the true proportion of a population that possess a certain
attribute is known, the centerline, Upper Control Limits, and
Lower Control Limits are defined by the formulas:
Centerline  p
p 1  p 
Upper Control Limit UCL   p  3
Lower Control Limit LCL   p  3
where
n
p 1  p 
p = true population proportion and
n = the sample size.
n
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.5 Monitoring with a p Chart
Definitions:
• If the true proportion is not known the chart is constructed from
the sample proportion. It is defined by the formulas:
Centerline  p
Upper Control Limit UCL   p  3
Lower Control Limit LCL   p  3
where
p

p 1 p

n

p 1 p

n
total number of defects found in all samples
total number of items sampled
n = the sample size.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.5 Monitoring with a p Chart
Process Controls:
Silvia Garcia is monitoring the operation of a machine that makes
radar components. Historically, she expects 2 percent defectives and
some chance variation. After studying the process, Silvia decides to
construct a 3σ p chart. There were 10 samples that each had 200
observations.
Solution:
Centerline  p  0.02
Upper Control Limit UCL   p  3
Lower Control Limit LCL   p  3
p 1  p 
n
p 1  p 
n
 0.02  3
 0.02  3
0.02 1  0.02 
200
0.02 1  0.02 
200
 0.0497
 0.0097
Note: Since a proportion cannot be negative, the Lower Control Level
will be defined as 0.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.5 Monitoring with a p Chart
Process Controls:
Silvia Garcia is monitoring the operation of a machine that makes
radar components. Historically, she expects 2 percent defectives and
some chance variation. After studying the process , Silvia decides to
construct a 3σ p chart. There were 10 samples that each had 200
observations.
Solution:
Adding the Upper Control
Limit and Lower Control
Limit lines to the graph,
we can see that two of the
sample proportions are
considered out of control.
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.5 Monitoring with a p Chart
Process Controls:
Silvia Garcia is monitoring the operation of a machine that makes
radar components. She does not have historical data and does not
know the true proportion of defectives. The sample proportion of
defectives is 0.0445. After studying the process , Silvia decides to
construct a 3σ p chart. There were 10 samples that each had 200
observations.
Solution:
Centerline  p  0.0445
Upper Control Limit UCL   p  3
Lower Control Limit LCL   p  3

  0.0445  3
0.0445 1  0.0445 
  0.0445  3
0.0445 1  0.0445 
p 1 p
n

p 1 p
n
200
200
 0.0882
 0.0008
HAWKES LEARNING SYSTEMS
Statistical Process Control
math courseware specialists
Section 17.5 Monitoring with a p Chart
Process Controls:
Silvia Garcia is monitoring the operation of a machine that makes
radar components. She does not have historical data and does not
know the true proportion of defectives. The sample proportion of
defectives is 0.0445. After studying the process, Silvia decides to
construct a 3σ p chart. There were 10 samples that each had 200
observations.
Solution:
Adding the Upper Control
Limit and Lower Control
Limit lines to the graph,
we can see that none of the
sample proportions are
considered out of control.