Transcript P, NP, C, & U Control Charts

ENGM 720 - Lecture 08
p, np, c, & u Control Charts
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Outline
 Assignment
 Discrete
Distributions and Probability of Outcomes
• Examples of discrete distributions
 Hypothesis
Testing to Control Charts
 p- & np-Charts
 c- & u-Charts
 Summary of Control Chart Options
• Using the Control Chart Decision Chart
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Assignment:

Reading:
•
•

Chapter 6
•
Finish reading
Chapter 7
•
•
•
Sections 7.1 and 7.2 through p.313
Sections 7.3 through p.325
Sections 7.3.2 and 7.5
Assignments:
•
•
•
Obtain the Control Chart Factors table from Materials Page
Access Excel Template for X-bar, R, & S Control Charts:
•
•
Download Assignment 5 for practice
Use the data on the HW5 Excel sheet to do the charting, verify the
control limits by hand calculations
Access Excel Template for P, NP, C, & U Control Charts
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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
• Hypothesis
Tests
• Ishikawa’s
Continually Improve the System
Characterize Stable Process Capability
Tools

Managing the
process with
control charts
Head Off Shifts in Location, Spread
• Process
Improvement
• Process
Stabilization
• Confidence in
Time
Identify Special Causes - Bad (Remove)
Identify Special Causes - Good (Incorporate)
Reduce Variability
“When to Act”
Center the Process
LSL
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0
USL
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Review

Shewhart Control charts
•
•
•
Are like a sideways hypothesis test (2-sided!) from a
Normal distribution
• UCL is like the right / upper critical region
• CL is like the central location
• LCL is like the left / lower critical region
When working with continuous variables, we use two
charts:
• 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
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Continuous & Discrete Distributions
 Continuous
• Probability of a range of
 Discrete
• Probability of a range of
outcomes is area under
PDF (integration)
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
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()
38
40
6
42
Discrete Distribution Example
 Sum
of two six-sided dice:
• Outcomes range from 2 to 12.
• Count the possible ways to obtain each individual sum - forms a
histogram
• What is the most frequently occurring sum that you could roll?
• Most likely outcome is a sum of 7 (there are 6 ways to obtain it)
• What is the probability of obtaining the most likely sum in a
single roll of the dice?
• 6  36 = .167
• What is the probability of obtaining a sum greater than 2 and
less than 11?
• 32  36 = .889
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Continuous & Attribute Variables
 Continuous
Variables:
• Take on a continuum of values.
• Ex.: length, diameter, thickness, temperature …
• 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
• Modeled by Poisson Distribution if counting defects
occurring within an inspection unit
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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
• Example – paneling with scratches
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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 
 n x
n x
P  X  x     p 1  p 
 x

So:

Mean:   E  X   np
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where  n 
n!
 x   x! n  x !


 
2
Variance:   V  X   np 1  p 
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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?
P[ X  1]
•
 5  3 
   
 1  30 
1
3

1  
 30 
51
 .328
 (5)(.1)(.9) 4
What is the probability that it contains one or more
defectives?
P[ X  1]  1  P[ X  0]
 5  3 
 1    
 0  30 
0
3

1



 30 
50
 1 (1)(1)(.9)5
 1 .5905
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 .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:
e  x
P  X  x 
x!
where x = 0, 1, 2, …

Mean:   E  X   
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Variance:  2  V  X   
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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)?
22
 .00477
λ1 
4608
What is the probability of finding at least two scratches in 25
in2?
25
 .119
λ25 
λ1  25( λ1 )  25(.00477)

i 1
P[ X  2]  1  P[ X  0]  P[ X  1]
 e .119 (.119) 0 e .119 (.119)1 
 .888(1) .888(.119) 

  1  
  1  (.888  .106)
 1  

1
1


0
!
1
!


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 .007
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

2

2
0
2-Sided Hypothesis Test
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UCL
0
CL

2
Sideways Hypothesis
Test
LCL
Sample Number
Shewhart Control Chart
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P-Charts
Tracks proportion
Can

defective in a sample of insp. units
have a constant number of inspection units in the sample
Sample Control Limits:
• Approximate 3σ limits are
found from trial samples:
UCL  p  3
p(1  p)
n

Standard Control Limits:
• Approximate 3σ limits
continue from standard:
UCL  p  3
CL  p
CL  p
p(1  p)
LCL  p  3
n
LCL  p  3
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p(1  p)
n
p(1  p)
n
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P-Charts (continued)
More
commonly has variable number of inspection units
Can’t

use run rules with variable control limits
Mean Sample Size Limits:
• Approximate 3σ limits are
found from sample mean:
UCL  p  3
p(1  p )
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Variable Width Limits:
• Approximate 3σ limits vary
with individual sample size:
UCL  p  3
n
p(1  p)
ni
CL  p
CL  p
LCL  p  3

p(1  p )
LCL  p  3
n
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p(1  p)
ni
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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  np  3 np(1  p)
UCL  np  3 np(1  p)
CL  np
CL  np
LCL  np  3 np(1  p)
LCL  np  3 np(1  p)
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C-Charts
Tracks the
count 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:
• Approximate 3σ limits
continue from standard:
UCL  c  3 c
CL  c
LCL  c  3 c
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or 0 if LCL is negative
or 0 if LCL is negative
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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!

Mean Sample Size Limits:
• Approximate 3σ limits are
found from sample mean:
UCL  u  3
u

Variable Width Limits:
• Approximate 3σ limits vary
with individual sample size:
UCL  u  3
n
CL  u
LCL  u  3
n
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u
ni
CL  u
u
unit
LCL  u  3
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u
ni
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Steps for Trial Control Limits




Start with 20 to 25 samples
Use all data to calculate initial control limits
Plot each sample in time-order on chart.
Check for out of control sample points
•
•
If one (or more) found, then:
1. Investigate the process;
2. Remove the special cause; and
3. Remove the special cause point and recalculate
control limits.
If can’t find special cause - drop point & recalculate
anyway
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Summary of Control Charts
 Use

of the control chart decision table.
(Continuous) Variables Charts
• Smaller changes detected
faster
• Apply to attributes data as well
(by CLT)*
• Require smaller sample sizes
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
Attributes Charts
• Can cover several
defects with one chart
• Less costly inspection
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Control Chart Decision Table
Defective Units
(possibly with multiple defects)
Binomial Distribution
Is the size of
the inspection
sample fixed?
No, varies
Use p-Chart
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?
Use np-Chart
Yes,
constant
Use c-Chart
No, varies
Continuous
Variable
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Which spread
method
preferred?
Use u-Chart
Range
Use X-bar and
R-Chart
Standard Deviation
ENGM 720: Statistical Process Control
Use X-bar and
S-Chart
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Control Chart Sensitizing Rules


Western Electric Rules:
1.
One point plots outside the three-sigma limits;
2.
Eight consecutive points plot on one side of the center line
(run rule!);
3.
Two out of three consecutive points plot beyond two-sigma
warning limits on the same side of the center line (run rule!);
3.
Four out of five consecutive points plot beyond one-sigma
warning limits on the same side of the center line (run rule!).
If chart shows lack of control, investigate for special cause
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Attribute Chart Applications

Attribute control charts apply to “service”
applications, too.
• Number of incorrect invoices per customer
• Proportion of incorrect orders taken in a day
• Number of return service calls to resolve problem
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Questions & Issues
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