Attribute Control Charts

1
Attribute Control Charts
2
Learning Objectives

• Defective vs Defect
• Binomial and Poisson Distribution
• p Chart
• np Chart
• c Chart
• u Chart
• Tests for Instability

3
Shewhart Control Charts - Overview
4
Defective and Defect

• Defective
• A unit of product that does not meet customers
  requirement or specification.
• Also known as a non-conforming 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.

5
Defective and Defect

• Defect
• A flaw or a single quality characteristic that
  does not meet customers requirement or
  specification.
• Also known as a non-conformity.
• 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.

6
Shewhart Control Charts for Attribute Data

• There are 4 types of Attribute Control Charts

Defectives
p
np
(Binomial Distribution)
Defects
u
c
(Poisson Distribution)
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7
Learning 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
8
Types of Data and Distributions

• Discrete Data (Attribute)
• Binomial
• Poisson
• Continuous Data (Variable)
• Normal
• Exponential
• Weibull
• Lognormal
• t
• c2
• F

Discrete Distributions
Continuous Distributions
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Types of Distributions
Discrete Distributions
Continuous Distributions
10
Discrete 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.

11
Discrete 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

12
Binomial 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).

13
Binomial 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

14
Discrete Distributions
Binomial Distribution
The location, dispersion and shape of a binomial
distribution are affected by the sample size (n)
and defective rate (p).
15
James Bernoulli
Discrete Distributions
Binomial Distribution
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Discrete Distributions

• Poisson Distribution
• Useful for discrete data involving error rate,
  defect rate (dpu, dpmo), particle count rate,
  etc.
• Data generated from counting of defects.

17
Discrete 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

18
Poisson 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.

19
Poisson 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

20
Discrete Distributions
Poisson Distribution
The location, dispersion and shape of a Poisson
distribution are affected by the mean (l).
21
Simeon D Poisson
Discrete Distributions
Poisson Distribution
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Summary 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
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Learning Objectives
?

• Defective vs Defect
• Binomial and Poisson Distribution
• p Chart
• np Chart
• c Chart
• u Chart
• Tests for Instability

?
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p Chart

• Fraction Non-Conforming
• Reject Rate / Defective Rate
• Percent Fallout

0
.
5
1
0
.
4
3
.
0
S
L

0
.
3
8
8
0
n
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3
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p
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P

0
.
2
1
4
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r
0
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2
P
0
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1
-
3
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L

0
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0
4
0
0
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0
2
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1
0
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S
a
m
p
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e

N
u
m
b
e
r
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p Chart

• Fraction non-conforming (p)
• Ratio of number of defectives (or non-conforming
  items) in a population to the number of items in
  that population.
• Sample fraction non-conforming (p)
• Ratio of number of defectives (d) in a sample to
  the sample size (n), i.e.

Is p a sample statistic?
26
p Chart

• The underlying principles of the p chart are
  based on the binomial distribution.
• This means that if a process has a typical
  fraction non-conforming, 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.
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p Chart

• The p chart also assumes a symmetrical bell-shape
  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.

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p Chart

• Following Shewharts principle, the Center Line
  and Control Limits of a p chart are

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p 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

30
p 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.

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Example 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 6-oz
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.
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Example 1 p Chart

• MiniTab Stat ? Control Charts ? P

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Example 1 p Chart
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Example 1 p Chart
Minitab allows different set of control charts to
be plotted on one chart MiniTab Stat ? Control
Charts ? P
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Example 1 p Chart
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Establish 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.

37
np 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

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Example 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 6-oz
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.
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Example 2 np Chart

• MiniTab Stat ? Control Charts ? NP

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Example 2 np Chart
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p 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

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Sample 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
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Learning Objectives
?

• Defective vs Defect
• Binomial and Poisson Distribution
• p Chart
• np Chart
• c Chart
• u Chart
• Tests for Instability

?
?
?
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c Chart

• Defects per Unit (DPU)
• Error Rate / Defect Rate
• Defects per Opportunity

2
0
3
.
0
S
L

1
8
.
9
7
t
n
u
o
C

1
0
e
C

9
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6
5
0
l
p
m
a
S
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3
.
0
S
L

0
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3
3
0
7
0
2
0
1
0
0
S
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p
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N
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b
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c 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

46
c 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
  record-keeping.
• It may constitute more than 1 unit of product.
• e.g. a HSA, both surfaces of a media, 10 pieces
  of PCBA

47
c 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

48
u 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
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u Chart

• The Center Line and Control Limits of a u chart
  are

50
Example 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 in-control.
51
Example 3 c and u Charts

• MiniTabs Stat ? Control Charts ? C

52
Example 3 c and u Charts

• MiniTabs Stat ? Control Charts ? U

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Example 3 c and u Charts
54
u (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.

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Learning Objectives
?

• Defective vs Defect
• Binomial and Poisson Distribution
• p Chart
• np Chart
• c Chart
• u Chart
• Tests for Instability

?
?
?
?
?
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Selecting 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

57
Exercise 1

• Strength of 5 test pieces sampled every
  hour(Xbar-R)
• 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(Xbar-S)
• Percent defective of a lot produced in every
  30-min 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

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Test for Instability
Suitable for all charts
_
Suitable only for X-Chart
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Tests 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

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Variables 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.

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Learning Objectives
?

• Defective vs Defect
• Binomial and Poisson Distribution
• p Chart
• np Chart
• c Chart
• u Chart
• Tests for Instability

?
?
?
?
?
?
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• End of Topic
• What Question
• Do You
• Have

63
Reading Reference

• Introduction to Statistical Quality
  Control,
• Douglas C. Montgomery, John Wiley Sons,
  
• ISBN 0-471-30353-4