IENG 486 - Lecture 16
P, NP, C, & U Control Charts
(Attributes Charts)
4/7/2015
IENG 486: Statistical Quality & Process Control
1
Assignment:
 Reading:


Chapter 3.5
Chapter 7




Sections 7.1 – 7.2.2: pp. 288 – 304
Sections 7.3 – 7.3.2: pp. 308 - 321
Chapter 6.4: pp. 259 - 265
Chapter 9



Sections 9.1 – 9.1.5: pp. 399 - 410
Sections 9.2 – 9.2.4: pp. 419 - 425
Sections 9.3: pp. 428 - 430
 Assignment:


4/7/2015
CH7 # 6; 11; 27a,b; 31; 47
Access Excel Template for P, NP, C, & U Control Charts
IENG 486: Statistical Quality & Process Control
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Process for Statistical Control
Of Quality
 Removing
special causes
of variation


Statistical Quality Control and Improvement
Improving Process Capability and Performance
Continually Improve the System
Hypothesis
Tests
Ishikawa’s
Tools
Characterize Stable Process Capability
 Managing the
process with
control charts



Head Off Shifts in Location, Spread
Time
Process
Improvement
Process
Stabilization
Confidence in
“When to Act”
Identify Special Causes - Bad (Remove)
Identify Special Causes - Good (Incorporate)
Reduce Variability
Center the Process
LSL
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0
USL
IENG 486: Statistical Quality & Process Control
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Review
 Shewhart Control charts

Are like a sideways hypothesis test (2-sided!) from a
Normal distribution




When working with continuous variables, we use two
charts:



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UCL is like the right / upper critical region
CL is like the central location
LCL is like the left / lower critical region
X-bar for testing for change in location
R or s-chart for testing for change in spread
We check the charts using 4 Western Electric rules
IENG 486: Statistical Quality & Process Control
4
Continuous & Discrete
Distributions
Continuous

Discrete
Probability of a range of
outcomes is area under
PDF (integration)

Probability of a range of
outcomes is area under
PDF (sum of discrete
outcomes)
35.0 
2.5
30.4
(-3)
34.8
32.6
(-)
(-2)
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35.0 
2.5
37
()
39.2
(+)
43.6
41.4
(+3)
(+2)
30
32
34
36
()
IENG 486: Statistical Quality & Process Control
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40
42
5
Continuous & Attribute
Variables
Continuous Variables:

Take on a continuum of values.
 Ex.:

length, diameter, thickness
Modeled by the Normal Distribution
Attribute Variables:

Take on discrete values
 Ex.:

present/absent, conforming/non-conforming
Modeled by Binomial Distribution if classifying
inspection units into defectives
 (defective

inspection unit can have multiple defects)
Modeled by Poisson Distribution if counting defects
occurring within an inspection unit
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IENG 486: Statistical Quality & Process Control
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Discrete Variables Classes
 Defectives

The presence of a non-conformity ruins the entire unit – the
unit is defective

Example – fuses with disconnects
 Defects

The presence of one or more non-conformities may lower the
value of the unit, but does not render the entire unit defective

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Example – paneling with scratches
IENG 486: Statistical Quality & Process Control
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Binomial Distribution




Sequence of n trials
Outcome of each trial is “success” or “failure”
Probability of success = p
r.v. X - number of successes in n trials
X ~ Bin  n , p 
 So:
n x
n x
P  X  x     p 1  p 
 x
 Mean:   E  X   np
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where  n 
Variance: 
n!
 
 x  x ! n  x  !
2
 V  X   np  1  p 
IENG 486: Statistical Quality & Process Control
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Binomial Distribution
Example
 A lot of size 30 contains three defective fuses.

What is the probability that a sample of five fuses selected at
random contains exactly one defective fuse?
1
P [ X  1]

 5  3  
3 
   
 1 

1
30
30
 

 
5 1
 ( 5 )(. 1)(. 9 )
What is the probability that it contains one or more
defectives?
0
P [ X  1]  1  P [ X  0 ]
 5  3  
3 
 1    
 1 

0
30
30




 
 1  (1)(1)(. 9 )
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 . 328
4
50
5
 1  . 5905
IENG 486: Statistical Quality &
Process Control
 . 4095
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Poisson Distribution
 Let X be the number of times that a certain event
occurs per unit of length, area, volume, or time
X ~ Pois   
 So:
P  X  x 

e 
x
x!
where x = 0, 1, 2, …
 Mean:   E  X   
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Variance:  2  V  X   
IENG 486: Statistical Quality & Process Control
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Poisson Distribution
Example
 A sheet of 4’x8’ paneling (= 4608 in2) has 22
scratches.

What is the expected number of scratches if checking only
one square inch (randomly selected)?
λ1 

22
 . 00477
4608
What is the probability of finding at least two scratches in 25
25
in2?
λ 25 
λ
1
 25 ( λ1 )
 25 (. 00477 )
 . 119
i 1
P [ X  2 ]  1   P [ X  0 ]  P [ X  1] 
 e  .119 (. 119 ) 0 e  .119 (. 119 ) 1
 1  

0!
1!

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



 . 888 (1) . 888 (. 119 ) 
 1 


1
1


IENG 486: Statistical Quality & Process Control
 1  (. 888  . 106 )
 . 007
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Moving from Hypothesis
Testing to Control Charts
 Attribute control charts are also like a sideways
hypothesis test
Detects a shift in the process
 Heads-off costly errors by detecting trends –

if constant control
limits are used
2

UCL

2

2
0
2-Sided Hypothesis Test
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0
CL

2
Sideways Hypothesis
Test
LCL
Sample Number
Shewhart Control Chart
IENG 486: Statistical Quality & Process Control
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P-Charts
Tracks proportion defective in
Can
have a constant number of inspection units in the sample
 Sample Control Limits:

a sample of insp. units
Approximate 3σ limits are
found from trial samples:
UCL  p  3
p (1  p )
 Standard Control Limits:

Approximate 3σ limits
continue from standard:
UCL  p  3
n
n
CL  p
CL  p
LCL  p  3
p (1  p )
LCL  p  3
n
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p (1  p )
IENG 486: Statistical Quality & Process Control
p (1  p )
n
13
P-Charts (continued)
 More
commonly has variable number of inspection units
 Can’t use
run rules with variable control limits
 Mean Sample Size Limits:  Variable Width Limits:

Approximate 3σ limits are
found from sample mean:
UCL  p  3
p (1  p )

Approximate 3σ limits vary
with individual sample size:
UCL  p  3
ni
n
CL  p
CL  p
LCL  p  3
p (1  p )
LCL  p  3
n
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p (1  p )
IENG 486: Statistical Quality & Process Control
p (1  p )
ni
14
NP-Charts
Tracks number
of defectives in a sample of insp. units
Must
have a constant number of inspection units in each sample
Use of run rules is allowed if LCL > 0 - adds power !
 Sample Control Limits:
 Standard Control Limits:

Approximate 3σ limits are
found from trial samples:

Approximate 3σ limits
continue from standard:
UCL  n p  3 n p (1  p )
UCL  np  3 np (1  p )
CL  n p
CL  np
LCL  n p  3 n p (1  p )
LCL  np  3 np (1  p )
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IENG 486: Statistical Quality & Process Control
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C-Charts
Tracks number
of defects in a logical inspection unit
Must
have a constant size inspection unit containing the defects
Use of run rules is allowed if LCL > 0 - adds power !
 Sample Control Limits:

Approximate 3σ limits are
found from trial samples:
UCL  c  3 c
CL  c
LCL  c  3 c
 Standard Control Limits:UCL

Approximate 3σ limits
continue from standard:
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or
0
if
LCL
is
negative
or
0
if
LCL
is
negative
 c3 c
CL  c
LCL  c  3 c
IENG 486: Statistical Quality & Process Control
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U-Charts
 Number of defects occurring in variably sized inspection
 (Ex. Solder defects per 100 joints - 350 joints in board = 3.5 insp. units)
 Can’t use run rules with variable control limits, watch clustering!
unit
 Mean Sample Size Limits:  Variable Width Limits:

Approximate 3σ limits are
found from sample mean:
UCL  u  3
u

Approximate 3σ limits vary
with individual sample size:
UCL  u  3
ni
n
CL  u
CL  u
LCL  u  3
u
LCL  u  3
n
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u
IENG 486: Statistical Quality & Process Control
u
ni
17
Summary of Control Charts
Continuous Variable Charts Attribute Charts
Smaller changes detected faster
 Require smaller sample sizes
 Can be applied to attributes data as
well (by CLT)*

 Use
4/7/2015
Can cover several
defects with one chart
 Less costly inspection

of the control chart decision tree…
IENG 486: Statistical Quality & Process Control
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Control Chart Decision Tree
Defective Units
(possibly with multiple defects)
Binomial Distribution
Is the size of
the inspection
sample fixed?
No, varies
Yes,
constant
Discrete
Attribute
What is the
inspection
basis?
Individual Defects
Poisson Distribution
Is the size of
the inspection
unit fixed?
Kind of
inspection
variable?
Continuous
Variable
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Yes,
constant
No, varies
Which spread
method
preferred?
Range
Standard Deviation
IENG 486: Statistical Quality & Process Control
Use p-Chart
Use np-Chart
Use c-Chart
Use u-Chart
Use X-bar and
R-Chart
Use X-bar and
S-Chart
19
Attribute Chart Applications
 Attribute control charts apply to “service”
applications, too!



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Number of incorrect invoices per customer
Proportion of incorrect orders taken in a day
Number of return service calls to resolve problem
IENG 486: Statistical Quality & Process Control
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Questions & Issues
4/7/2015
IENG 486: Statistical Quality & Process Control
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