Transcript DevStat8e_16_04

16
Quality Control
Methods
Copyright © Cengage Learning. All rights reserved.
16.4 Control Charts for Attributes
Copyright © Cengage Learning. All rights reserved.
Control Charts for Attributes
The term attribute data is used in the quality control
literature to describe two situations:
1. Each item produced is either defective or nondefective
(conforms to specifications or does not).
2. A single item may have one or more defects, and the
number of defects is determined.
In the former case, a control chart is based on the binomial
distribution; in the latter case, the Poisson distribution is the
basis for a chart.
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The p Chart for Fraction
Defective
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The p Chart for Fraction Defective
Suppose that when a process is in control, the probability
that any particular item is defective is p (equivalently, p is
the long-run proportion of defective items for an in-control
process) and that different items are independent of one
another with respect to their conditions.
Consider a sample of n items obtained at a particular time,
and let X be the number of defectives and = X/n. Because
X has a binomial distribution, E(X) = np and
V(X) = np(1 – p), so
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The p Chart for Fraction Defective
Also, if np  10 and n(1 – p)  10. has approximately a
normal distribution. In the case of known p (or a chart
based on target value), the control limits are
If each sample consists of n items, the number of defective
items in the ith sample is xi, and = xi /n, then
are plotted on the control chart.
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The p Chart for Fraction Defective
Usually the value of p must be estimated from the data.
Suppose that k samples from what is believed to be an
in-control process are available, and let
The estimate p is then used in place of p in the
aforementioned control limits.
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The p Chart for Fraction Defective
The p chart for the fraction of defective items has its center
line at height p and control limits
If LCL is negative, it is replaced by 0.
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Example 6
A sample of 100 cups from a particular dinnerware pattern
was selected on each of 25 successive days, and each
was examined for defects.
The resulting numbers of unacceptable cups and
corresponding sample proportions are as follows:
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Example 6
cont’d
Assuming that the process was in control during this
period, let’s establish control limits and construct a p chart.
Since,
= 1.52, p = 1.52/25 = .0608 and
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Example 6
cont’d
The LCL is therefore set at 0. The chart pictured in Figure
16.5 shows that all points are within the control limits. This
is consistent with an in-control process.
Control chart for fraction-defective data of Example 6
Figure 16.5
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The c Chart for Number of
Defectives
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The c Chart for Number of Defectives
We now consider situations in which the observation at
each time point is the number of defects in a unit of some
sort.
The unit may consist of a single item (e.g., one automobile)
or a group of items (e.g., blemishes on a set of four tires).
In the second case, the group size is assumed to be the
same at each time point.
The control chart for number of defectives is based on the
Poisson probability distribution.
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The c Chart for Number of Defectives
Recall that if Y is a Poisson random variable with
parameter , then
E(Y) = 
V(Y) = 
Y =
Also, Y has approximately a normal distribution when  is
large (  10 will suffice for most purposes).
Furthermore, if Y1, Y1, . . . , Yn, are independent Poisson
variables with parameters 1, 2, . . . , n it can be shown
that Y1 + . . . + Yn has a Poisson distribution with parameter
1 + . . . + n.
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The c Chart for Number of Defectives
In particular, if 1 = . . . = n =  (the distribution of the
number of defects per item is the same for each item), then
the Poisson parameter is n.
Let  denote the Poisson parameter for the number of
defects in a unit (it is the expected number of defects per
unit). In the case of known  (or a chart based on a target
value),
LCL =  – 3
UCL =  + 3
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The c Chart for Number of Defectives
With xi denoting the total number of defects in the ith unit
(i = 1, 2, 3, . . .), then points at heights x1, x2, x3 , . . . are
plotted on the chart.
Usually the value of  must be estimated from the data.
Since E(Xi) = , it is natural to use the estimate  = x
(based on x1 , x2, . . . , , xk).
The c chart for the number of defectives in a unit has
center line at x and
LCL = x – 3
UCL = x + 3
If LCL is negative, it is replaced by 0.
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Example 7
A company manufactures metal panels that are baked after
first being coated with a slurry of powdered ceramic. Flaws
sometimes appear in the finish of these panels, and the
company wishes to establish a control chart for the number
of flaws.
The number of flaws in each of the 24 panels sampled at
regular time intervals are as follows:
7 10 9 12 13
13 9 21 10 6
6 13 7 5 11 8 10
8 3 12 7 11 14 10
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Example 7
with xi = 235 and
are
cont’d
= x = 235/24 = 9.79. The control limits
LCL = 9.79 – 3
= .40
UCL = 9.30 + 3
= 19.18
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Example 7
cont’d
The control chart is in Figure 16.6. The point corresponding
to the fifteenth panel lies above the UCL.
Control chart for number of flaws data of Example 7
Figure 16.6
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Example 7
cont’d
Upon investigation, the slurry used on that panel was
discovered to be of unusually low viscosity (an assignable
cause).
Eliminating that observation gives x = 214/23 = 9.30 and
new control limits
LCL = 9.30 – 3
= .15
UCL = 9.30 + 3
= 18.45
The remaining 23 observations all lie between these limits,
indicating an in-control process.
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Control Charts Based on
Transformed Data
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Control Charts Based on Transformed Data
The use of 3-sigma control limits is presumed to result in
P (statistic < LCL)  P (statistic > UCL)  .0013
when the process is in control.
However, when p is small, the normal approximation to the
distribution of = X/n will often not be very accurate in the
extreme tails.
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Control Charts Based on Transformed Data
Table 16.3 gives evidence of this behavior for selected
values of p and n (the value of p is used to calculate the
control limits).
In-Control Probabilities for a p Chart
Table 16.3
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Control Charts Based on Transformed Data
Control chart for number of flaws data of Example 7
Figure 16.6
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Control Charts Based on Transformed Data
In many cases, the probability that a single point falls
outside the control limits is very different from the nominal
probability of .0026.
This problem can be remedied by applying a transformation
to the data.
Let h(X) denote a function applied to transform the binomial
variable X. Then h() should be chosen so that h(X) has
approximately a normal distribution and this approximation
is accurate in the tails.
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Control Charts Based on Transformed Data
A recommended transformation is based on the arcsin (i.e.,
sin–1) function:
Y = h(X ) = sin–1
Then Y is approximately normal with mean value sin–1
and variance 1/(4n); note that the variance is independent
of p. Let yi = sin–1
.
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Control Charts Based on Transformed Data
Then points on the control chart are at heights y1 , y2, . . . .
For known n, the control limits are
When p is not known, sin–1
is replaced by y.
Similar comments apply to the Poisson distribution when 
is Y = h(X) = 2 , small. The suggested transformation is,
which has mean value 2
and variance 1.
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Control Charts Based on Transformed Data
Resulting control limits are 2
y  3 otherwise.
 3 when  is known and
The book Statistical Methods for Quality Improvement
listed in the chapter bibliography discusses these issues in
greater detail.
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