Title: Attribute Control Charts
1Attribute Control Charts
2Learning Objectives
 Defective vs Defect
 Binomial and Poisson Distribution
 p Chart
 np Chart
 c Chart
 u Chart
 Tests for Instability
3Shewhart Control Charts  Overview
4Defective and Defect
 Defective
 A unit of product that does not meet customers
requirement or specification.  Also known as a nonconforming unit.
 Example
 A base casting that fails porosity specification
is a defective.  A disc clamp that does not meet the parallelism
specification is a defective.
5Defective and Defect
 Defect
 A flaw or a single quality characteristic that
does not meet customers requirement or
specification.  Also known as a nonconformity.
 There can be one or more defects in a defective.
 Example
 A dent on a VCM pole that fails customers
specification is a defect.  A stain on a cover that fails customers
specification is a defect.
6Shewhart Control Charts for Attribute Data
 There are 4 types of Attribute Control Charts
Defectives
p
np
(Binomial Distribution)
Defects
u
c
(Poisson Distribution)
43
7Learning Objectives
?
 Defective vs Defect
 Binomial and Poisson Distribution
 p Chart
 np Chart
 c Chart
 u Chart
 Tests for Instability
Mean defective rate
Mean defect rate
8Types of Data and Distributions
 Discrete Data (Attribute)
 Binomial
 Poisson
 Continuous Data (Variable)
 Normal
 Exponential
 Weibull
 Lognormal
 t
 c2
 F
Discrete Distributions
Continuous Distributions
9Types of Distributions
Discrete Distributions
Continuous Distributions
10Discrete Distributions
 Binomial Distribution
 Useful for attribute data (or binary data)
 Result from inspection criteria which are binary
in nature, e.g. pass/fail, go/nogo,
accept/reject, etc.  Data generated from counting of defectives.
11Discrete Distributions
 Binomial Distribution
 If a process typically gives 10 reject rate (p
0.10), what is the chance of finding 0, 1, 2 or 3
defectives within a sample of 20 units (n 20)?
 Commonly used in Acceptance Sampling
12Binomial Distribution
 Commonly used in Acceptance Sampling, where p is
the probability of success (defective rate), n is
the number of trials (sample size), and x is the
number of successes (defectives found).
13Binomial Distribution
 Properties
 each trial has only 2 possible outcomes  success
or failure  probability of success p remains constant
throughout the n trials  the trials are statistically independent
 the mean and variance of a Binomial Distribution
are
14Discrete Distributions
Binomial Distribution
The location, dispersion and shape of a binomial
distribution are affected by the sample size (n)
and defective rate (p).
15James Bernoulli
Discrete Distributions
Binomial Distribution
16Discrete Distributions
 Poisson Distribution
 Useful for discrete data involving error rate,
defect rate (dpu, dpmo), particle count rate,
etc.  Data generated from counting of defects.
17Discrete Distributions
 Poisson Distribution
 If a process typically gives 4.0 defect rate (l
4 dpu), what is the chance of finding 0, 1, 2 or
3 defects per unit?
 Commonly used as an approximation of the binomial
distribution when  p lt 0.1 (10)
 n is large
18Poisson Distribution
 This distribution have been found to be relevant
for applications involving error rates, particle
count, chemical concentration, etc, where ? is
the mean number of events (or defect rate) within
a  given unit of time or space.
19Poisson Distribution
 Properties
 number of outcomes in a time interval (or space
region) is independent of the outcomes in another
time interval (or space region)  probability of an occurrence within a very short
time interval (or space region) is proportional
to the time interval (or space region)  probability of more than 1 outcome occurring
within a short time interval (or space region) is
negligible  the mean and variance for a Poisson Distribution
are
20Discrete Distributions
Poisson Distribution
The location, dispersion and shape of a Poisson
distribution are affected by the mean (l).
21Simeon D Poisson
Discrete Distributions
Poisson Distribution
22Summary of Approximation
Binomial
The smaller p and larger n the better
p lt 0.1
np gt 5, gt 10 p 0.5, lt 0.5
Poisson
? ? 15
The larger the better
Normal
23Learning Objectives
?
 Defective vs Defect
 Binomial and Poisson Distribution
 p Chart
 np Chart
 c Chart
 u Chart
 Tests for Instability
?
24p Chart
 Fraction NonConforming
 Reject Rate / Defective Rate
 Percent Fallout
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25p Chart
 Fraction nonconforming (p)
 Ratio of number of defectives (or nonconforming
items) in a population to the number of items in
that population.  Sample fraction nonconforming (p)
 Ratio of number of defectives (d) in a sample to
the sample size (n), i.e.
Is p a sample statistic?
26p Chart
 The underlying principles of the p chart are
based on the binomial distribution.  This means that if a process has a typical
fraction nonconforming, p, the mean and variance
of the distribution for ps are computed from the
binomial equation, giving
Xk number of defective unit in subgroup k
which has a total sample size of nk units
k number of subgroup, should be between 20 to
25 before constructing control limits.
27p Chart
 The p chart also assumes a symmetrical bellshape
distribution, with symmetrical control limits on
each side of the center line.  This implies that the binomial distribution is
approximately close to the shape of the normal
distribution, which can happen under certain
conditions of p and n  p ? 1/2 and n gt 10 implying np gt 5
 For other values of p, the general guideline is
to have np gt 10 to get a satisfactory
approximation of the normal to the binomial.
28p Chart
 Following Shewharts principle, the Center Line
and Control Limits of a p chart are
29p Chart
 If the sample size is not constant, then the
Control Limits of a p chart may be computed by
either method  a) Variable Control Limits


 where ni is the actual sample size of each
sampling i  b) Control Limits Based on Average Sample Size

 where n is the average (or typical) sample size
of all the samples
30p Chart  Average Sample Size
 When to Use Control Limits Based on Average
Sample Size instead of Variable Control Limits  Smallest subgroup size, nmin, is at least 30 of
the largest subgroup size, nmax.  Future sample sizes will not differ greatly from
those previously observed.  When using Control Limits Based on Average Sample
Size, the exact control limits of a point should
be determined and examined relative to that value
if  There is an unusually large variation in the size
of a particular sample  There is a point which is near the control
limits.
31Example 1 p Chart
 S/N Sampled Rejects
 1 50 12
 2 50 15
 3 50 8
 4 50 10
 5 50 4
 6 50 7
 7 50 16
 8 50 9
 9 50 14
 10 50 10
 11 50 5
 12 50 6
 13 50 17
 14 50 12
 15 50 22
 16 50 8
 17 50 10
 18 50 5
Frozen orange juice concentrate is packed in 6oz
cardboard cans. A metal bottom panel is attached
to the cardboard body. The cans are inspected
for possible leak. 20 samplings of different
sampling size were obtained. Verify if the
process is in control. The data are found in
AttributeSPC.MTW.
32Example 1 p Chart
 MiniTab Stat ? Control Charts ? P
33Example 1 p Chart
34Example 1 p Chart
Minitab allows different set of control charts to
be plotted on one chart MiniTab Stat ? Control
Charts ? P
35Example 1 p Chart
36Establish Trial Control Limits
 When to use it?
 New process, modified process, no historical data
available to calculate p  How to do it?
 Calculate p based on the preliminary 20 to 25
subgroups.  Calculate the trial control limits using the
formula mentioned in slide 21 or 22.  Sample values of p from the preliminary subgroups
to be plotted against the trial control limits.  Any points exceed the trial control limits should
be investigated.  If assignable causes for these points are
discovered, they should be discarded and new
trial control limits to be determined.
37np Chart
 If the sample size is constant, it is possible to
base a control chart on the number nonconforming
(np), rather than the fraction nonconforming (p).  The Center Line and Control Limits of an np chart
are
38Example 2 np Chart
 S/N Sampled Rejects
 1 50 12
 2 50 15
 3 50 8
 4 50 10
 5 50 4
 6 50 7
 7 50 16
 8 50 9
 9 50 14
 10 50 10
 11 50 5
 12 50 6
 13 50 17
 14 50 12
 15 50 22
 16 50 8
 17 50 10
 18 50 5
Frozen orange juice concentrate is packed in 6oz
cardboard cans. A metal bottom panel is attached
to the cardboard body. The cans are inspected
for possible leak. 20 samplings of 50
cans/sampling were obtained. Verify if the
process is in control. The data are found in
AttributeSPC.MTW.
39Example 2 np Chart
 MiniTab Stat ? Control Charts ? NP
40Example 2 np Chart
41p Chart vs np Chart
 For ease of recording, the np chart is preferred.
 The p chart offers the following advantages
 accommodation for variable sample size
 provides information about process capability
42Sample Size for p and np Charts
 Sample Size is determined based on the 2
criteria  Assumption to approximate Binomial Distribution
to a  Normal Distribution
 To ensure that the LCL is greater than zero.
For p ? 0.5
For p other values
43Learning Objectives
?
 Defective vs Defect
 Binomial and Poisson Distribution
 p Chart
 np Chart
 c Chart
 u Chart
 Tests for Instability
?
?
?
44c Chart
 Defects per Unit (DPU)
 Error Rate / Defect Rate
 Defects per Opportunity
2
0
3
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0
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7
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45c Chart
 Each specific point at which a specification is
not  satisfied results in a defect or nonconformity.
 The c chart is
 a control chart for the total number of defects
in an inspection unit  based on the normal distribution as an
approximation for the Poisson distribution, which
can happen when  c or ? ? 15
46c Chart
 Inspection Unit
 The area of opportunity for the occurrence of
nonconformities.  e.g. a HSA, a media, a PCBA
 This is an entity chosen for convenience of
recordkeeping.  It may constitute more than 1 unit of product.
 e.g. a HSA, both surfaces of a media, 10 pieces
of PCBA
47c Chart
 If the number of nonconformities (defects) per
inspection unit is denoted by c, then  The Center Line and Control Limits of a c chart
are
48u Chart
 In cases where the number of inspection units is
not constant, the u chart may be used instead,
with  If the average number of defects per inspection
unit is denoted by u, then
Where ci is the count of the number of defects in
number of inspection units, ai
49u Chart
 The Center Line and Control Limits of a u chart
are
50Example 3 c and u Charts
 S/N Units Defects
 1 5 10
 2 5 12
 3 5 8
 4 5 14
 5 5 10
 6 5 16
 7 5 11
 8 5 7
 9 5 10
 10 5 15
 11 5 9
 12 5 5
 13 5 7
 14 5 11
 15 5 12
 16 5 6
 17 5 8
 18 5 10
A personal computer manufacturer plans to
establish a control chart for nonconformities at
the final assembly line. The number of
nonconformities in 20 samples of 5 PCs are shown
here. Verify if the process is incontrol.
51Example 3 c and u Charts
 MiniTabs Stat ? Control Charts ? C
52Example 3 c and u Charts
 MiniTabs Stat ? Control Charts ? U
53Example 3 c and u Charts
54u (or c) Chart vs p (np) Chart
 The u (or c) chart offers the following
advantages  More informative as the type of nonconformity is
noted.  Facilitates Pareto analysis.
 Facilitates Cause Effect Analysis.
55Learning Objectives
?
 Defective vs Defect
 Binomial and Poisson Distribution
 p Chart
 np Chart
 c Chart
 u Chart
 Tests for Instability
?
?
?
?
?
56Selecting the Appropriate Chart
 c  Chart
 Measures the total number of defects in a
subgroup  The subgroup size can be 1 unit of product if we
expect to have a relatively large number of
defects/unit  Requires a constant subgroup size
 u  Chart
 Measures the number of defects/unit of product
(dpu)  The subgroup size can be constant or variable
 p  Chart
 Measures the proportion of defective units in a
subgroup  The subgroup size can be constant or variable
 np  Chart
 Measures the number of defective items in a
subgroup  Requires a constant subgroup size
57Exercise 1
 Strength of 5 test pieces sampled every
hour(XbarR)  Number of defectives in 100 parts(np)
 Number of solder defects in a printed circuit
board assembly(C)  Diameter of 40 units of products sampled every
day(XbarS)  Percent defective of a lot produced in every
30min period(p)  Surface defects of surface area of varying
sizes(u)  In a maintenance group dealing with repair work,
 the number of maintenance requests that
require  a second call to complete the repair every
week
58Test for Instability
Suitable for all charts
_
Suitable only for XChart
59Tests for Instability
 CAUTION Do not apply tests blindly
 Not every test is relevant for all charts
 Excessive number of tests ? Increased ?error
 Nature of application
60Variables vs Attributes Charts
 Attributes Control Charts facilitate monitoring
of more than 1 quality characteristics.  Variables Control Charts provide leading
indicators of trouble Attributes Control Charts
react after the process has actually produced bad
parts.  For a specified level of protection against
process drift, Variables Control Charts require a
smaller sample size.
61Learning Objectives
?
 Defective vs Defect
 Binomial and Poisson Distribution
 p Chart
 np Chart
 c Chart
 u Chart
 Tests for Instability
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62 End of Topic
 What Question
 Do You
 Have
63Reading Reference
 Introduction to Statistical Quality
Control,  Douglas C. Montgomery, John Wiley Sons,
 ISBN 0471303534