Transcript Introduction to Statistical Quality Control, 4th Edition

```Chapter 6
Control Charts for Attributes
Introduction to Statistical Quality Control,
4th Edition
6-1. Introduction
• Data that can be classified into one of several
categories or classifications is known as attribute
data.
• Classifications such as conforming and
nonconforming are commonly used in quality
control.
• Another example of attributes data is the count of
defects.
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
• Fraction nonconforming is the ratio of the
number of nonconforming items in a
population to the total number of items in
that population.
• Control charts for fraction nonconforming
are based on the binomial distribution.
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Recall: A quality characteristic follows a binomial
distribution if:
1. All trials are independent.
2. Each outcome is either a “success” or “failure”.
3. The probability of success on any trial is given as
p. The probability of a failure is
1-p.
4. The probability of a success is constant.
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
• The binomial distribution with parameters n
 0 and 0 < p < 1, is given by
n x
nx
p(x)   p (1  p)
x
• The mean and variance of the binomial
distribution are
  np
  np(1  p)
2
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Development of the Fraction Nonconforming
Control Chart
Assume
• n = number of units of product selected at random.
• D = number of nonconforming units from the sample
• p= probability of selecting a nonconforming unit from the
sample.
• Then:
n x
P(D  x)   p (1  p) n  x
x
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Development of the Fraction Nonconforming
Control Chart
• The sample fraction nonconforming is given as
D
pˆ 
n
ˆ is a random variable with mean and
where p
variance
p(1  p)
2
p
 
n
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Standard Given
• If a standard value of p is given, then the control
limits for the fraction nonconforming are
p(1 p)
UCL  p  3
n
CL  p
p(1 p)
LCL  p  3
n
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
No Standard Given
• If no standard value of p is given, then the control
limits for the fraction nonconforming are
p (1  p )
n
UCL  p  3
CL  p
where
LCL  p  3
m
p
 Di
i 1
mn
p (1  p )
n
m

 pˆ i
i 1
m
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Trial Control Limits
• Control limits that are based on a preliminary set
of data can often be referred to as trial control
limits.
• The quality characteristic is plotted against the
trial limits, if any points plot out of control,
assignable causes should be investigated and
points removed.
• With removal of the points, the limits are then
recalculated.
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Example
• A process that produces bearing housings is
investigated. Ten samples of size 100 are selected.
Sample #
# Nonconf.
1
5
2
2
3
3
4
8
5
4
6
1
7
2
8
6
9
3
• Is this process operating in statistical control?
Introduction to Statistical Quality Control,
4th Edition
10
4
6-2. Control Charts for Fraction
Nonconforming
Example
n = 100, m = 10
Sample #
# Nonconf.
Fraction
Nonconf.
1
5
2
2
3
3
4
8
5
4
6
1
7
2
8
6
9
3
10
4
0.05
0.02
0.03
0.08
0.04
0.01
0.02
0.06
0.03
0.04
m
p
ˆi
p
i 1
m
 0.038
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Example
Control Limits are:
0.038(1  0.038)
UCL  0.038 3
 0.095
100
CL  0.038
0.038(1  0.038)
LCL  0.038 3
 0.02  0
100
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Example
P Chart for C1
Proportion
0.10
3.0SL=0.09536
0.05
P=0.03800
0.00
-3.0SL=0.000
0
1
2
3
4
5
6
7
8
9
10
Sampl e Number
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Design of the Fraction Nonconforming Control
Chart
• The sample size can be determined so that a shift of some
specified amount,  can be detected with a stated level of
probability (50% chance of detection). If  is the
magnitude of a process shift, then n must satisfy:
p(1  p)
L
n
Therefore,
2
 L
n    p(1  p)

Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Positive Lower Control Limit
• The sample size n, can be chosen so that the lower
control limit would be nonzero:
p(1  p)
LCL  p  L
0
n
and
(1  p) 2
n
L
p
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
Interpretation of Points on the Control Chart for
Fraction Nonconforming
• Care must be exercised in interpreting points that
plot below the lower control limit.
– They often do not indicate a real improvement in
process quality.
– They are frequently caused by errors in the inspection
process or improperly calibrated test and inspection
equipment.
Introduction to Statistical Quality Control,
4th Edition
6-2. Control Charts for Fraction
Nonconforming
The np control chart
• The actual number of nonconforming can also be
charted. Let n = sample size, p = proportion of
nonconforming. The control limits are:
UCL  np  3 np(1  p)
CL  np
LCL  np  3 np(1  p)
(if a standard, p, is not given, use p )
Introduction to Statistical Quality Control,
4th Edition
6-2.2 Variable Sample Size
• In some applications of the control chart for
the fraction nonconforming, the sample is a
100% inspection of the process output over
some period of time.
• Since different numbers of units could be
produced in each period, the control chart
would then have a variable sample size.
Introduction to Statistical Quality Control,
4th Edition
6-2.2 Variable Sample Size
Three Approaches for Control Charts with
Variable Sample Size
1. Variable Width Control Limits
2. Control Limits Based on Average Sample Size
3. Standardized Control Chart
Introduction to Statistical Quality Control,
4th Edition
6-2.2 Variable Sample Size
Variable Width Control Limits
• Determine control limits for each individual
sample that are based on the specific sample
size.
• The upper and lower control limits are
p(1  p)
p3
ni
Introduction to Statistical Quality Control,
4th Edition
6-2.2 Variable Sample Size
Control Limits Based on an Average Sample Size
• Control charts based on the average sample size
results in an approximate set of control limits.
• The average sample size is given by
m
n
•
 ni
i 1
m
The upper and lower control limits are
p(1  p)
p3
n
Introduction to Statistical Quality Control,
4th Edition
6-2.2 Variable Sample Size
The Standardized Control Chart
• The points plotted are in terms of standard
deviation units. The standardized control chart
– Centerline at 0
– UCL = 3
LCL = -3
– The points plotted are given by:
zi
pˆ i  p
p(1  p)
ni
Introduction to Statistical Quality Control,
4th Edition
6-2.4 The Operating-Characteristic
Function and Average Run
Length Calculations
The OC Function
• The number of nonconforming units, D, follows
a binomial distribution. Let p be a standard
value for the fraction nonconforming. The
probability of committing a Type II error is
  P(pˆ  UCL | p)  P(pˆ  LCL | p)
 P(D  nUCL | p)  P(D  nLCL | p)
Introduction to Statistical Quality Control,
4th Edition
6-2.4 The Operating-Characteristic
Function and Average Run
Length Calculations
Example
•
Consider a fraction nonconforming process
where samples of size 50 have been collected
and the upper and lower control limits are
0.3697 and 0.0303, respectively.It is important
to detect a shift in the true fraction
nonconforming to 0.30. What is the probability
of committing a Type II error, if the shift has
occurred?
Introduction to Statistical Quality Control,
4th Edition
6-2.4 The Operating-Characteristic
Function and Average Run
Length Calculations
Example
• For this example, n = 50, p = 0.30, UCL =
0.3697, and LCL = 0.0303. Therefore, from the
binomial distribution,
  P(D  nUCL | p)  P(D  nLCL | p)
 P(D  50(0.3697) | 0.30)  P(D  50(0.0303| 0.30)
 P(D  18.48 | 0.30)  P(D  1.515| 0.30)
 P(D  18 | 0.30)  P(D  1 | 0.30)
 0.8594 0
 0.8594
Introduction to Statistical Quality Control,
4th Edition
6-2.4 The Operating-Characteristic
Function and Average Run
Length Calculations
OC curve for the fraction nonconforming control chart
with p = 20, LCL = 0.0303 and UCL = 0.3697.
1.0
B
•
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
p
Introduction to Statistical Quality Control,
4th Edition
6-2.4 The Operating-Characteristic
Function and Average Run
Length Calculations
ARL
• The average run lengths for fraction
nonconforming control charts can be found as
before:
1
• The in-control ARL is ARL 0 

•
1
The out-of-control ARL is ARL1 
1 
Introduction to Statistical Quality Control,
4th Edition
6-3. Control Charts for
Nonconformities (Defects)
•
•
There are many instances where an item will
contain nonconformities but the item itself is not
classified as nonconforming.
It is often important to construct control charts
for the total number of nonconformities or the
average number of nonconformities for a given
“area of opportunity”. The inspection unit must
be the same for each unit.
Introduction to Statistical Quality Control,
4th Edition
6-3. Control Charts for
Nonconformities (Defects)
Poisson Distribution
•
•
The number of nonconformities in a given area can be
modeled by the Poisson distribution. Let c be the
parameter for a Poisson distribution, then the mean and
variance of the Poisson distribution are equal to the
value c.
The probability of obtaining x nonconformities on a
single inspection unit, when the average number of
nonconformities is some constant, c, is found using:
ecc x
p( x ) 
x!
Introduction to Statistical Quality Control,
4th Edition
6-3.1 Procedures with Constant
Sample Size
c-chart
• Standard Given:
UCL  c  3 c
CL  c
LCL  c  3 c
UCL  c  3 c
•
No Standard Given:
CL  c
LCL  c  3 c
Introduction to Statistical Quality Control,
4th Edition
6-3.1 Procedures with Constant
Sample Size
Choice of Sample Size: The u Chart
•
•
If we find c total nonconformities in a sample of n
inspection units, then the average number of
nonconformities per inspection unit is u = c/n.
The control limits for the average number of
nonconformities is
u
UCL  u  3
n
CL  u
LCL  u  3
u
n
Introduction to Statistical Quality Control,
4th Edition
6-3.2 Procedures with Variable
Sample Size
Three Approaches for Control Charts with
Variable Sample Size
1. Variable Width Control Limits
2. Control Limits Based on Average Sample Size
3. Standardized Control Chart
Introduction to Statistical Quality Control,
4th Edition
6-3.2 Procedures with Variable
Sample Size
Variable Width Control Limits
• Determine control limits for each individual
sample that are based on the specific sample
size.
• The upper and lower control limits are
u
u3
ni
Introduction to Statistical Quality Control,
4th Edition
6-3.2 Procedures with Variable
Sample Size
Control Limits Based on an Average Sample Size
• Control charts based on the average sample size
results in an approximate set of control limits.
• The average sample size is given by
m
n
•
 ni
i 1
m
The upper and lower control limits are
u
u3
n
Introduction to Statistical Quality Control,
4th Edition
6-3.2 Procedures with Variable
Sample Size
The Standardized Control Chart
• The points plotted are in terms of standard
deviation units. The standardized control chart
– Centerline at 0
– UCL = 3
LCL = -3
– The points plotted are given by:
ui  u
zi
u
ni
Introduction to Statistical Quality Control,
4th Edition
6-3.3 Demerit Systems
•
When several less severe or minor defects
can occur, we may need some system for
classifying nonconformities or defects
according to severity; or to weigh various
types of defects in some reasonable
manner.
Introduction to Statistical Quality Control,
4th Edition
6-3.3 Demerit Systems
Demerit Schemes
1.
2.
3.
4.
Class A Defects - very serious
Class B Defects - serious
Class C Defects - Moderately serious
Class D Defects - Minor
•
Let ciA, ciB, ciC, and ciD represent the number of
units in each of the four classes.
Introduction to Statistical Quality Control,
4th Edition
6-3.3 Demerit Systems
Demerit Schemes
• The following weights are fairly popular in
practice:
–
Class A-100, Class B - 50, Class C – 10, Class D - 1
di = 100ciA + 50ciB + 10ciC + ciD
di - the number of demerits in an inspection unit
Introduction to Statistical Quality Control,
4th Edition
6-3.3 Demerit Systems
Control Chart Development
• Number of demerits per unit:
D
ui 
n
where n = number of inspection units
n
D =  di
i 1
Introduction to Statistical Quality Control,
4th Edition
6-3.3 Demerit Systems
Control Chart Development
UCL  u  3ˆ u
CL  u
LCL  u  3ˆ u
where
and
u  100u A  50u B  10u C  u D
 100 u A  50 u B  10 u C  u D 
ˆ u  

n


2
2
2
Introduction to Statistical Quality Control,
4th Edition
1/ 2
6-3.4 The OperatingCharacteristic Function
•
•
The OC curve (and thus the P(Type II
Error)) can be obtained for the c- and uchart using the Poisson distribution.
For the c-chart:
  P( x  UCL| c)  P( X  LCL| c)
where x follows a Poisson distribution
with parameter c (where c is the true mean
number of defects).
Introduction to Statistical Quality Control,
4th Edition
6-3.4 The OperatingCharacteristic Function
•
For the u-chart:
  P(x  UCL | u)  P(x  LCL | u)
 P(c  nUCL | u)  P(c  nLCL| u)
Introduction to Statistical Quality Control,
4th Edition
6-3.5 Dealing with Low-Defect
Levels
•
•
•
When defect levels or count rates in a process
become very low, say under 1000 occurrences
per million, then there are long periods of time
between the occurrence of a nonconforming
unit.
Zero defects occur
Control charts (u and c) with statistic
consistently plotting at zero are uninformative.
Introduction to Statistical Quality Control,
4th Edition
6-3.5 Dealing with Low-Defect
Levels
Alternative
• Chart the time between successive occurrences
of the counts – or time between events control
charts.
• If defects or counts occur according to a Poisson
distribution, then the time between counts occur
according to an exponential distribution.
Introduction to Statistical Quality Control,
4th Edition
6-3.5 Dealing with Low-Defect
Levels
Consideration
•
Exponential distribution is skewed.
•
Corresponding control chart very asymmetric.
•
One possible solution is to transform the exponential
random variable to a Weibull random variable using x =
y1/3.6 (where y is an exponential random variable) – this
Weibull distribution is well-approximated by a normal.
•
Construct a control chart on x assuming that x follows a
normal distribution.
•
See Example 6-6, page 326.
Introduction to Statistical Quality Control,
4th Edition
6-4. Choice Between Attributes
and Variables Control Charts
•
•
•
•
•
Attributes data is easy to collect and several
characteristics may be collected per unit.
information about the process mean and variance is
obtained directly.
Variables control charts provide an indication of
impending trouble (corrective action may be taken
before any defectives are produced).
Attributes control charts will not react unless the process
has already changed (more nonconforming items may be
produced.
Introduction to Statistical Quality Control,
4th Edition
6-5. Guidelines for Implementing
Control Charts
1. Determine which process characteristics to
control.
2. Determine where the charts should be
implemented in the process.
3. Choose the proper type of control chart.
4. Take action to improve processes as the result of
SPC/control chart analysis.
5. Select data-collection systems and computer
software.
Introduction to Statistical Quality Control,
4th Edition
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