Download Control Charts for Fraction Nonconforming

CHAPTER (7)
Attributes Control Charts
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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.
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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.
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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.
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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
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  np (1  p)
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Design of Fraction Nonconforming
Chart
•
Three parameters must be specified
1. The sample size
2. The frequency of sampling
3. The width of the control limits
•
•
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Common to base chart on 100% inspection
of all process output over a period of time
Rational subgroups may also play role in
determining sampling frequency
Sample size
• If p is very small, we should choose n sufficiently
large to find at least one nonconforming unit
• Otherwise the presence of only one non-conforming
in the sample would indicate out-of-control
condition (example)
• To avoid this, choose n such that the probability of
finding at least one nonconforming per sample is at
least γ (example)
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Example
• p = 0.01 and n = 8
p(1  p)
0.01(1  0.01)
UCL  p  3
 0.01  3
 0.1155
n
8
• If there is one nonconforming in the sample, then
p =1/8=0.125
and we conclude that the process is out of control
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Sample size
• The sample size can be determined so that the
probability of finding at least one nonconforming
unit per sample is at least γ
• Example p = 0.01 and γ = 0.95
• Find n such that P(D ≥ 1) ≥ 0.95
• Using Poisson approximation of the binomial with
λ =np
• From cumulative Poisson table λ must exceed 3.00
 np ≥ 3  n ≥ 300
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Sample size
• 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).
•
UCL = pout
If  is the magnitude of a process shift, then n must satisfy:
p(1  p)
L
n
Therefore,
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2
L
n    p(1  p)

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)
n 
L2
p
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Interpretation of Points on the Control
Chart for Fraction Nonconforming
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Variable Sample Size
Variable-Width Control Limits
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Variable Sample Size
Control Limits Based on an Average Sample Size
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Variable Sample Size
The Standardized Control Chart
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The Opening Characteristic Function
and Average Run Length Calculations
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Control Charts for Nonconformities
(Defects)
Procedures with Constant Sample Size
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• 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.
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Control Charts for Nonconformities
(Defects)
Procedures with Variable Sample Size
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