Statistical
Process Control
Learning Objectives
1. Explain the purpose of a control chart
2. Explain the role of the central limit theorem in SPC
3. Build 𝑥-charts and 𝑅-charts
4. List the five steps involved in building control charts
5. Build 𝑝-charts and 𝑐-charts
6. Explain acceptance sampling
Statistical Process
Control

Application of statistical
techniques to ensure that
processes meet standards.

Provide a statistical signal
when assignable (special)
causes of variation are present

Eliminate assignable causes of
variation
Statistical Process
Control

Variability is inherent
in every process

Natural or common
causes

Special or assignable
causes
Variations
Natural Variations
Assignable Causes
 Common causes that affect all
processes
 Indicates the presence of changes
in the process
 Expected variations
 Variations can be traced to a
specific reason - misadjusted
equipment, machine wear, fatigue,
untrained workers, new raw
material
 Output measures follow a
probability distribution
 Process “in control” if the
distribution of output falls within
acceptable limits
 Eliminate bad causes, and
incorporate new causes
Natural Variations / Common
Causes
 Affect virtually all production
processes
 Expected amount of variation
 Output measures follow a
probability distribution
 If the distribution of outputs falls
within acceptable limits, the
process is said to be “in control”
Assignable Variations / Special
Causes
 Generally this indicates the presence of some
changes in the process
 Variations that can be traced to a specific reason
 Machine wear, misadjusted equipment,
fatigue, untrained workers, new raw material
 The objective is to discover when assignable
causes are present
 Eliminate the bad causes
 Incorporate the good causes
Samples
To measure the process, we take samples and analyze the sample
statistics following these steps
# #
Frequency
(a) Samples of the product,
say five boxes of cereal
taken off the filling
machine line, vary from
each other in weight
Each of these
represents one
sample of five
boxes of cereal
# # #
# # # #
# # # # # # #
#
Weight
# # # # # # # # #
Samples
To measure the process, we take samples and analyze the sample
statistics following these steps
Frequency
(b) After enough samples are
taken from a stable process,
they form a pattern called a
distribution
The solid line
represents the
distribution
Weight
Samples
To measure the process, we take samples and analyze the sample
statistics following these steps
(c) There are many types of distributions, including the normal
(bell-shaped) distribution, but distributions do differ in terms of
central tendency (mean), standard deviation or variance,
and shape
Variation
Shape
Frequency
Central tendency
Weight
Weight
Weight
Samples
To measure the process, we take samples and analyze the sample
statistics following these steps
Prediction
Frequency
(d) If only natural causes of
variation are present, the
output of a process forms a
distribution that is stable over
time and is predictable
Weight
Samples
To measure the process, we take samples and analyze the sample
statistics following these steps
Prediction
Frequency
(e) If assignable causes are
present, the process
output is not stable over
time and is not predicable
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Weight
Control Charts
Constructed from historical data, the purpose of control
charts is to help distinguish between natural variations and
variations due to assignable causes
Process Control
Frequency
Lower control limit
(a)In statistical control and
capable of producing
within control limits
Upper control limit
(b) In statistical control but
not capable of producing
within control limits
(c) Out of control
Size
(weight, length, speed, etc.)
Central Limit Theorem
Regardless of the distribution of the population, the distribution of
sample means drawn from the population will tend to follow a
normal curve
1) The mean of the sampling distribution will
be the same as the population mean m
2) The standard deviation of the sampling
distribution ( s x) will equal the population
standard deviation (s ) divided by the
square root of the sample size, n
x= = m
sx =
s
n
Population and Sampling Distributions
Distribution of sample means
always approaches a normal
distribution
Population
distributions
=
Mean of sample means = x
Beta
Standard
deviation of
the sample
means
Normal
Uniform
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-3s x -2s x -1s x
|
x=
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+1s x +2s x +3s x
95.45% fall within ± 2s x
99.73% of all x
fall within ± 3s x
=sx =
s
n
Sampling Distribution
Sampling distribution
of means
Process distribution
of means
Process distribution from
which the sample was drawn
was also normal, but it could
have been any distribution
𝑥=𝜇
(mean)
n = 100
n = 50
As the sample size
increases,
the sampling
distribution narrows
n = 25
Mean of process
Types of Control Charts
𝑥 −chart : Changes in
mean
Continuous
variables
𝑅 −chart: Changes in
dispersion
Categorical
variables
𝑝 −chart: Fraction,
proportion or
percentage defects
𝑐 −chart: Count
defects per unit output
Known
standard
deviation
Unknown
standard
deviation
Control Charts for Variables

Continuous random variables with real values

May be in whole or in fractional numbers
𝒙 −chart
R-chart
• Tracks changes in
central tendency (mean)
• Indicates a gain or loss
of dispersion
• Due to factors such as
tool wear, gradual
increase in temperature,
new materials
• Due to changes in worn
bearings, loose tool,
sloppiness of operators
Setting Chart Limits
For 𝑥 −charts when we know 𝝈
Lower control limit (LCL𝑥 )= 𝑥 − 𝑧𝜎𝑥
Upper control limit (UCL𝑥 )= 𝑥 + 𝑧𝜎𝑥
where
𝑥 = mean of the sample means or a target value
set for the process
𝑧 = number of normal standard deviations= 𝜎 𝑛
𝜎𝑥 = standard deviation of the sample means
𝜎 = population (process) standard deviation
𝑛 = sample size
Example: Setting Control Limits
 Randomly select and weigh nine (𝑛 = 9) boxes of cereals at
each hour
 Managers want to set control limits that include 99.73% of the
sample means (𝑧 = 3)
 Population standard deviation is known to be 1 oz (𝑠 = 1)
Average weight in 17 +13 +16 +18 +17 +16 +15 +17 +16
=
= 16.1 ounces
the FIRST sample
9
Example: Setting Control Limits
WEIGHT OF
SAMPLE
HOUR
(AVG. OF
9 BOXES)
1
WEIGHT OF
SAMPLE
WEIGHT OF
SAMPLE
HOUR
(AVG. OF
9 BOXES)
HOUR
(AVG. OF 9
BOXES)
16.1
5
16.5
9
16.3
2
16.8
6
16.4
10
14.8
3
15.5
7
15.2
11
14.2
4
16.5
8
16.4
12
17.3
12
é
Avg of 9 boxes
ê
å
Average mean of = ê x= = i=1
ê
12 samples
12
ê
ë
(
)
ù
ú
ú
ú
ú
û
x= = 16 ounces
n=9
z= 3
s = 1 ounce
Example: Setting Control Limits
12
é
Avg of 9 boxes
ê
å
Average mean = ê x= = i=1
of 12 samples ê
12
ê
ë
(
)
ù
ú
ú
ú
ú
û
x= = 16 ounces
n=9
z= 3
s = 1 ounce
æ 1 ö
æ 1ö
UCL x = x + zs x = 16 + 3 ç
÷ = 16 + 3 ç ÷ = 17 ounces
è3ø
è 9ø
=
æ 1 ö
æ 1ö
LCL x = x - zs x = 16 - 3 ç
÷ = 16 - 3 ç ÷ = 15 ounces
è3ø
è 9ø
=
Example: Setting Control Limits
Control Chart
for samples of
9 boxes
Variation due
to assignable
causes
Out of
control
17 = UCL
Variation due to
natural causes
16 = Mean
15 = LCL
| | | | | | | | | | | |
1 2 3 4 5 6 7 8 9 10 11 12
Sample number
Out of
control
Variation due
to assignable
causes
Setting Chart Limits
For 𝑥 −charts when we don’t know 𝝈
Lower control limit (LCL𝑥 )= 𝑥 − 𝐴2 𝑅
Upper control limit (UCL𝑥 )= 𝑥 + 𝐴2 𝑅
where
𝑥 = mean of the sample means
𝐴2 = control chart factor found in Table S6.1
n
𝑅=
R=
å R = average range of the samples
i
i=1
n
=
Control Chart Factors
Table S6.1
Factors for Computing Control Chart Limits (3 sigma)
SAMPLE SIZE,
n
MEAN FACTOR,
A2
UPPER RANGE,
D4
LOWER RANGE,
D3
2
1.880
3.268
0
3
1.023
2.574
0
4
.729
2.282
0
5
.577
2.115
0
6
.483
2.004
0
7
.419
1.924
0.076
8
.373
1.864
0.136
9
.337
1.816
0.184
10
.308
1.777
0.223
12
.266
1.716
0.284
Example: Setting Control Limits using
Table Values
Given:
Super Cola Example
Labeled as “net weight
12 ounces”
Process average = 12 ounces
Average range = .25 ounce
Sample size = 5
UCL x = x= + A2 R
= 12 + (.577)(.25)
UCL = 12.144
= 12 + .144
= 12.144 ounces
LCL x = x= - A2 R
= 12 - .144
= 11.856 ounces
Mean = 12
From Table
S6.1
LCL = 11.856
𝑅-chart

Shows sample ranges over time

Difference between smallest and largest values in the
sample

Monitors process variability

Independent from the process mean
Setting Chart Limits
For 𝑅 −charts
Lower control limit (LCL𝑅 )= 𝐷3 𝑅
Upper control limit (UCL𝑅 )= 𝐷4 𝑅
where
𝑈𝐶𝐿𝑅 = upper control limit for the range
𝐿𝐶𝐿𝑅 = lower control limit for the range
𝐷3 and 𝐷4 = values from Table S6.1
Control Chart Factors
Table S6.1
Factors for Computing Control Chart Limits (3 sigma)
SAMPLE SIZE,
n
MEAN FACTOR,
A2
UPPER RANGE,
D4
LOWER RANGE,
D3
2
1.880
3.268
0
3
1.023
2.574
0
4
.729
2.282
0
5
.577
2.115
0
6
.483
2.004
0
7
.419
1.924
0.076
8
.373
1.864
0.136
9
.337
1.816
0.184
10
.308
1.777
0.223
12
.266
1.716
0.284
Setting Control Limits
Given:
Average range = 5.3 pounds
Sample size = 5
From Table S6.1 D4 = 2.115, D3 = 0
UCL R = D4 R
= (2.115)(5.3)
= 11.2 pounds
UCL = 11.2
Mean = 5.3
LCL R = D3 R
= (0)(5.3)
= 0 pounds
LCL = 0
Sample Mean
Salmon
Fillets at
Darden
Restaurant
x Bar Chart
11.5 –
UCL = 11.524
11.0 –
=
x = – 10.959
10.5 –
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1
3
5
7
9
11
13
15
17
LCL = – 10.394
Sample Range
Range Chart
0.8 –
UCL = 0.6943
0.4 –
– = 0.2125
R
0.0 – |
1
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3
5
7
9
11
13
15
17
LCL = 0
Even though the process average is under control, the dispersion
of the process may not be. Operations managers use control
charts for ranges to monitor process variability and control
charts for averages to monitor process central tendency
Mean and Range Charts
The mean chart is sensitive to shifts in the process mean
(a)
These sampling
distributions result
in the charts below
(Sampling mean is
shifting upward, but
range is consistent)
UCL
( 𝑥-chart detects shift
in central tendency)
𝑥-chart
LCL
UCL
(𝑅-chart does not detect
change in mean)
𝑅-chart
LCL
Mean and Range Charts
The 𝑹-chart is sensitive to shifts in the process standard deviation
(b)
These sampling
distributions result
in the charts below
(Sampling mean is
constant, but dispersion
is increasing)
UCL
( 𝑥-chart indicates no
change in central tendency)
𝑥-chart
LCL
UCL
(𝑅-chart detects increase
in dispersion)
𝑅-chart
LCL
Steps in Creating Control Charts
Collect samples
Compute overall means (𝑥 and 𝑅)
Set appropriate control limits (𝑧 values)
Calculate UCL and LCL
Graph 𝑥 and 𝑅 charts
Investigate patterns
Identify and address assignable causes
Revalidate with new data
Setting Other Control Limits
Common z Values
Desired control
limits (%)
90.0
95.0
95.45
99.0
99.73
𝑧 −value (Standard deviation
required for desired level of
confidence)
1.65
1.96
2.00
2.58
3.00
Control Charts for Attributes

For variables that are categorical

Defective/nondefective, good/bad, yes/no, acceptable/
unacceptable

Measurement is typically counting defectives

Charts may measure
1. Percent defective (𝑝-chart) – has a sample size
2. Number of defects (𝑐-chart) – does not require sample size
information
Control Charts for Variables

Categorical variables – defective/non-defective, good/bad,
yes/no, acceptable/non-acceptable

Measurement is typically counting defects
𝒑 −chart
𝒄 −chart
• Percent defective
• Number of defects
• Requires a sample size
• Does not require sample
size information
Control Limits for p–charts
Population will be a binomial distribution, but applying the Central
Limit Theorem allows us to assume a normal distribution for the
sample statistics
UCL p = p+ zs p
LCL p = p- zs p
where
s p is estimated by
ŝ p =
(
)
p 1- p
n
p = mean fraction (percent) defective in the samples
z= number of standard deviations
s p = standard deviation of the sampling distribution
n = number of observations in each sample
Example: 𝑝–Charts for Data Entry
CEO wants to set control limit to include 99.73% of the random variation in the
data entry process. He examines 100 records entered and counts the number of
errors and the fraction defective in each samples. Samples of the work of 20
clerks are as follows:
SAMPLE
NUMBER
NUMBER OF
ERRORS
FRACTION
DEFECTIVE
SAMPLE
NUMBER
NUMBER OF
ERRORS
1
6
.06
11
6
.06
2
5
.05
12
1
.01
3
0
.00
13
8
.08
4
1
.01
14
7
.07
5
4
.04
15
5
.05
6
2
.02
16
4
.04
7
5
.05
17
11
.11
8
3
.03
18
3
.03
9
3
.03
19
0
.00
10
2
.02
20
4
.04
80
FRACTION
DEFECTIVE
Example: p-Chart for Data
Entry
SAMPLE
NUMBER
p=
1
2
NUMBER
OF
FRACTION
SAMPLE
Total
number
of
errors
ERRORS
DEFECTIVE
NUMBER
NUMBER
OF
80
ERRORS
=
6
.06
11
6
Total number
of records
examined
(100)(20)
5
.05
12
1
(.04)(1-.04)
3
0
.00(rounded13up from .0196)
8
ŝ p =
= .02
100
4
1
.01
14
7
5
FRACTION
DEFECTIVE
4
.04
15
5
6
UCL
= p+.02zŝ p = .04 +3(.02)
= .10 4
2
16
p
7
5
8
3
.03
9
3
10
2
17
LCL p = p-.05zŝ p = .04 -3(.02)
=0
= .04
.06
.01
.08
.07
.05
.04
11
.11
18
3
.03
.03
19
0
.00
.02
20
4
.04
80
Fraction defective
Example: p-Chart for Data
Entry
.11
.10
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
–
–
–
–
–
–
–
–
–
–
–
–
UCLp = 0.10
p = 0.04
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2
4
6
8
10
12
14
16
18
20
Sample number
LCLp = 0.00
Example: p-Chart for Data
Entry
Fraction defective
Possible assignable
causes present
.11
.10
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
–
–
–
–
–
–
–
–
–
–
–
–
UCLp = 0.10
p = 0.04
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2
4
6
8
10
12
14
16
18
20
Sample number
LCLp = 0.00
Control Limits for c-Charts
Population will be a Poisson distribution, but applying the Central
Limit Theorem allows us to assume a normal distribution for the
sample statistics
Lower control limit (LCL𝑐 )= 𝑐 − 3 𝑐
Upper control limit (UCL𝑐 )= 𝑐 + 3 𝑐
where
𝑐 = mean number of defects per unit
𝑐 = standard deviation of defects per unit
Example: c-Chart for Cab Company
The company receives complaints everyday, and over a 9-day
period, the owner received the following number of calls: 3, 0, 8, 9,
6, 7, 4, 9, 8 for a total of 54 complaints. It wants to compute 99.73%
control limits.
c = 54 complaints/9 days = 6 complaints/day
= 6+3 6
= 13.35
LCLc = c - 3 c
= 6-3 6
=0
Number defective
UCLc = c + 3 c
–
–
–
–
UCLc = 13.35
6 –
4 –
c= 6
14
12
10
8
2 –
0 – |
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1 2
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3
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4
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5
Day
|
6
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7
|
8
LCLc = 0
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9
Managerial Issues and Control Charts
Select points in
the processes
that need SPC
Major
Management
Decisions
Determine the
appropriate
charting
technique
Set clear
policies and
procedures
Which parts are
critical to success?
Which parts have a
tendency to become
out of control?
Variable chart
monitor weights or
dimensions
Attribute charts are
more of a yes-no or
go-no go gauge
What to do when
the process is out
of control
Patterns in Control Charts
Run test – A test to examine the points in a control chart
to see if nonrandom variation is present
 Identify abnormalities in a process
 Runs of 5 or 6 points above or below the target or
centerline suggest assignable causes may be present
 There are a variety of run tests
UCL
Target
LCL
Normal behavior.
Process is “in
control.”
One plot out.
Investigate for
cause. Process is
“out of control.”
Two plots very near
lower (or upper)
control. Investigate
for cause.
Run of 5 above (or
below) central line.
Investigate for
cause.
Trends in either
direction, 5 plots.
Investigate for cause
of progressive
change.
UCL
Target
LCL
Erratic behavior.
Investigate for
cause.
Acceptance Sampling
Form of quality testing used for incoming
materials or finished goods
Sampling
Inspection
• Take samples
at random
from a “lot” /
batches of
items
• Inspect each
of the items in
the sample
Decision
• Whether to
reject the lot
based on the
inspection
result
Both attributes and variables can be inspected
Acceptance Sampling
 Only screen “lots” – does not drive quality
improvement efforts
 Rejects “lots” can be:
 Returned to supplier
 To be 100% inspected to cull out all defects
 May be re-graded to a lower specification
Operating Characteristic Curve
 Shows the relationship between the
probability of accepting a lot and its
quality level
 The curve pertains to a specific
plan – a combination of n (sample
size) and c (acceptance level)
AQL
LTPD
Acceptance Quality Level (AQL)
 Poorest level of quality we are willing to accept
Lot Tolerance Percent Defective (LTPD)
 Quality level that we consider bad
P(Accept Whole Shipment)
The “Perfect” OC Curve
Keep whole
shipment
100 –
75 –
Return whole
shipment
50 –
25 –
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0 |– |
0 10 20
Cut-Off
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30 40
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50 60
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70 80
% Defective in Lot
|
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90 100
An OC Curve
 = 0.05 producer's risk for AQL
The steeper the
curve, the better
the plan
Probability of
Acceptance
 = 0.10
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0
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1
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2
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3
|
4
|
5
AQL
Consumer's
risk for LTPD
Good
lots
|
6
|
7
|
8
Percent
defective
LTPD
Indifference
zone
Bad lots
Operating Characteristics Curve
Producer
 Responsible of replacing all the defects in the
rejected lot
 Producer’s risk – mistake of having a good lot
rejected through sampling
Consumer
 Lots accepted are the responsibility of the consumer
 Consumer’s risk – mistake of accepting a bad lot
overlooked through sampling
Producer’s and Consumer’s Risks


Producer's risk ( )

Probability of rejecting a good lot

Probability of rejecting a lot when the fraction
defective is at or above the AQL
Consumer's risk ( )

Probability of accepting a bad lot

Probability of accepting a lot when fraction
defective is below the LTPD
OC Curves for Different
Sampling Plans
n = 50, c = 1
n = 100, c = 2
Average Outgoing Quality
The percentage defective in an average lot of goods
inspected through acceptance sampling
1. If a sampling plan replaces all defectives
2. If we know the true incoming percent defective for the
lot
We can compute the average outgoing quality
(AOQ) in percent defective
The maximum AOQ is the highest percent defective or the
lowest average quality and is called the average outgoing
quality limit (AOQL)
Average Outgoing Quality
AOQ =
(Pd)(Pa)(N – n)
N
where
Pd = true percent defective of the lot
Pa = probability of accepting the lot
N = number of items in the lot
n = number of items in the sample
Automated Inspection
 Modern technologies allow
virtually 100% inspection at
minimal costs
 Not suitable for all situations
Recap Learning Objectives
1. Explain the purpose of a control chart
• Provide statistical signal when assignable causes of variation are
present
2. Explain the role of the central limit theorem in SPC
• What does the theorem says? What does it provide?
3. Build 𝑥 −charts and 𝑅 −charts
• Know the formula, calculate the upper and lower limits, plot the
graph, investigate patterns
Recap Learning Objectives
4. List the five steps involved in building control charts
• Collect samples, calculate overall means, graph the means and
ranges, investigate patterns, collect additional samples
5. Build p-charts and c-charts
• Know the formula, calculate the upper and lower limit, plot the
graph, investigate patterns
6. Explain acceptance sampling
• What is an acceptable lot and what is not, what are the risks and
probability of accepting a bad lot