SQC (Chapter 2 NP)
SQC (Chapter 2 NP)
SQC (Chapter 2 NP)
quality characteristic.
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A) Control Chart for Fraction Nonconforming
(Control Chart for Proportion)
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The mean of D is np and the variance is np (1-p). The sample
proportion nonconforming is the ratio of the number of
nonconforming units in the sample, D, to the sample size n,
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The mean and variance of this estimator are
and
For p-chart:
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When the process fraction (proportion) p is not known, it
must be estimated from the available data.
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Hence for p-chart:
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Example: To establish the control chart, 30 samples of n = 50 cans each were
selected at half-hour intervals overa three-shift period in which the machine
was in continuous operation. The results are given in the following table
Sample Number of Sample Sample Number of Sample
Number Nonconfor Fraction Number Nonconfor Fraction
ming Cans, Nonconfor ming Cans, Nonconfor
Di ming, p̂ i Di ming, p̂ i
1 12 0.24 17 10 0.20
2 15 0.30 18 5 0.10
3 8 0.16 19 13 0.26
4 10 0.20 20 11 0.22
5 4 0.08 21 20 0.40
6 7 0.14 22 18 0.36
7 16 0.32 23 24 0.48
8 9 0.18 24 15 0.30
9 14 0.28 25 9 0.18
10 10 0.20 26 12 0.24
11 5 0.10 27 7 0.14
12 6 0.12 28 13 0.26
13 17 0.34 29 9 0.18
14 12 0.24 30 6 0.12
15 22 0.44 Total 347 p 0.2313
16 8 0.16
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B) np-chart:
If the sample size is constant for all samples, say n, then the
sampling distribution of the statistic, d = np number of
nonconforming/defective in the sample if 𝑝 is known
given by
if 𝑝 is unknown given by
Remarks:
if the sample size is varies from sample to sample, then np
chart would be quite uncomfortable to use because the
central lines as well as the control limits would vary from
sample to sample. In such a case p chart would be better to
use.
However, in case of constant sample size for all samples
any one of np or p charts may be used but, in practice, p
chart commonly used. In the other way, many non-
statistically trained personnel find the np chart easier to
interpret than the usual fraction nonconforming control
chart.
The p-chart and np chart tend to be used when n is large
since the 3-sigma rule is inspired by normal
distribution.
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Example: consider the data in the example above
for the fraction nonconforming Orange Juice
concentrate cans.
C) Variable Sample Size
In some applications of the control chart for fraction
nonconforming, the sample is a 100% inspection of 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. There are different approaches
to constructing and operating a control chart with a variable
sample size.
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Example. To illustrate this approach, consider the data given below for the
25 samples.
ii) Control Limits Based on an Average Sample Size
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Example: For the data in the above example, construct control
limits based on an average sample size.
Solution
iii) The Standardized Control Chart
The third approach to dealing with variable sample size is to use
a "standardized" control chart, where the points are plotted in
standard deviation units.
Such a control chart has the center 1ine at zero, and upper and
lower control limits of +3 and -3, respectively. The variable
plotted on the chart is
pi − p pi − p
Zi = or Zi = .
p(1−p) p(1−p)
ni ni
Ex: construct the standardized control chart for the above example. 23
D) Control Chart for defects per unit (c-chart)
Often rather than classifying an item being as defective
or non-defective, the number of defects in each unit
may be counted.
e−c cx
p(x)= ; x = 0, 1,2 ...., where x is the number of
x!
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If no standard is given, then c may be estimated as the
observed average number of nonconformities in a
preliminary sample of inspection units--say, c.
• UCL = c + 3 c ,
• CL = c and
• LCL = c - 3 c .
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Example: : Consider the number of nonconformities observed in 26
successive samples of 100 printed circuit boards data and construct the
control chart for c. Note that, for reasons of convenience, the inspection
unit is defined as 100 boards
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Solution:
total number of nonconformities 516
• c = = = 19.58
number of sample 26
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E) u-chart
If we find x total nonconformities in a sample of n inspection
units, then the average number of nonconformities per
inspection unit is u = x/n, where x is a Poisson random
variable.
The parameters of the control chart for the average number of
nonconformities per unit are as:
• UCL = u + 3 u/ n ,
• CL = u and
• UCL = u - 3 u/ n
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2.4. Process and Measurement System Capability
Analysis
Statistical techniques can be helpful throughout the product
cycle, including development activities prior to
manufacturing, in quantifying process variability, in
analyzing this variability relative to product requirements or
specifications, and in assisting development and
manufacturing in eliminating or greatly reducing this
variability. This general activity is called process
capability analysis
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Process capability refers to the uniformity of the process.
Obviously, the variability of critical-to- quality
characteristics in the process is a measure of the uniformity
of output.
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A process capability study usually measures functional
parameters or critical-to- quality characteristics on the
product, not the process itself.
When the analyst can directly observe the process and
can control or monitor the data-collection activity, the
study is a true process capability study, because by
controlling the data collection and knowing the time
sequence of the data, inferences can be made about the
stability of the process over time
Process capability analysis is a vital part of an overall quality-
improvement program
process capability analysis is a technique that has application in
many segments of the product cycle, including product and
process design, supply chain management, production or
manufacturing planning, and reducing the variability in a
manufacturing process,
Note that, the estimate of process capability may be in the
form of a probability distribution having a specified shape,
center (mean), and spread (standard deviation).
A) Process Capability Analysis Using Histogram
For the strength data shown above and using the figure, we find that
𝜎̂= 84th percentile - 50th percentile =298 -265 psi=33 psi. Note
that µ̂ = 265 psi and 𝜎̂ = 33 psi are not far from the sample
average ̅X = 264.06 and standard deviation s = 32.02.
C) Process Capability Ratio, using Cp
In order to manufacture within a specification, the difference
variation.
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USL −LSL
Cp =
6σ
where USL and LSL are the upper and lower specification limits,
respectively.
Clearly, any value of Cp < 1 means that the process variation is
greater than the specified tolerance band so the process is
incapable.
For increasing values of Cp the process becomes increasingly
capable.
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In a practical application, the process standard deviation
σ is almost always unknown and must be replaced by an
estimate of σ.
To estimate σ we typically use either the sample
R
standard deviation, s or .
d2,n
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Example:
USL −LSL 20 −6
Cp = = = 1.167.
6σ 6(2)
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Example: consider the container bursting-strength data in the above
example. Suppose that the lower specification limit on bursting
strength is 200 psi. We will use µ̂ = ̅X=264 and s =32 as estimates
of µ and 𝜎, respectively, and the resulting estimate of the one-sided
lower process-capability ratio is:
Cpk
The Cp index does not take into account where the process
mean is located relative to the specifications.
Cp simply measures the spread of the specifications relative
to the 6-sigma spread in the process.
This situation may be more accurately reflected by defining
a new process capability ratio that takes process centering
into account.
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Cpk takes in to account both the process variation and the
centering.
The Cpk can be used when there is only one specification
limit, upper or lower – a one-sided specification.
For upper and lower specification limits, there are two Cpk
values, Cpu and Cpl
These relate the difference between the process mean and
the upper and the lower specification limits respectively, to 3
(half the total process variation)
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𝑈𝑆𝐿 −𝜇
Cpu = ,upper specification only and
3𝜎
𝜇 − 𝐿𝑆𝐿
Cpl = , lower specification only.
3𝜎
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Note that Cp measures the potential capability of the
process, if centered;
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Example: Suppose that a stable process has upper and
lower specifications at USL = 62 and LSL = 38 with a
sample mean of 53 and sample standard deviation of 2.
Then compute the PCR for off-center process.
Solution:
Confidence interval for Cp
χ21−α χ2α
USL −LSL 2,n−1 USL− LSL 2,n−1
≤ Cp ≤
6σ n−1 6σ n−1
χ21−α χ2α
2,n−1 2,n−1
= Cp ≤ Cp ≤ Cp ,
n−1 n−1
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2 2 α α
where χ1−α
2,n−1
and χα
2,n−1
are the lower 2 and upper 2
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Solution:
χ21−α χ2α
2,n−1 2,n−1
The 95% CI for Cp = Cp ≤ Cp ≤ Cp
n−1 n−1
8.91 32.85
= 2.29 ≤ Cp ≤ 2.29
20−1 20−1
=1.57 ≤ Cp ≤3.01,
Cpk 1 1 ≤ Cpk
1−Z +
α 2 9nC2pk 2(n−1)
/
≤ Cpk 1 1
1+Z +
2
9nC2pk 2(n−1)
α
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Example: Consider the above example and construct the 95%
CI for Cpk by assuming the observed sample mean is 53.
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= Cpk 1 1 ≤ Cpk
1−Z +
𝑎 9nC2pk 2(n−1)
2
≤ Cpk 1 1
1+Z +
𝑎 9nC2pk 2(n−1)
2
1 1
= 1.714 1 − 1.96 + ≤ Cpk
9∗20∗1.714 2 2(20−1)
≤ 1.714 1 1
1 + 1.96 +
9∗20∗1.714 2 2(20−1)
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What happens if the process is not approximately
normally distributed?
The process capacities that we considered so far are based on
normality of the process distribution.
This poses a problem when the process distribution is not
normal.
Some of the remedies are
Transform the data so that they become approximately
normal
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Use or develop another set of process capacity, that apply to
non normal distributions. One statistic is called Cnpk (for non-
parametric Cpk). Its estimator is calculated by:
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