Transcript Executive Overview

Knowing What to Do
Knowing How to Do It
Getting Better Every Day
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Acceptance Sampling
I
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What you will learn
 The purpose of Sampling
 How to draw a statistically valid Sample
 How to Develop a Sampling Plan
 How to construct an O-C curve for your sampling
plan
 How to use (and understand) ANSI/ASQ Z1.4
 How to use ANSI/ASQ Z1.9
 Assessing Inspection Economics
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What is Sampling
Sampling refers to the practice of evaluating
(inspecting) a portion -the sample - of a lot – the
population – for the purpose of inferring information
about the lot.
Statistically speaking, the properties of the sample
distribution are used to infer the properties of the
population (lot) distribution.
An accept/reject decision is normally made based on
the results of the sample
Sampling is an Audit practice
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Why Sample?
 Economy
 Less inspection labor
 Less time
 Less handling damage
 Provides check on process control
 Fewer errors ???
 i.e. inspection accuracy
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What does Sampling not do?
 Does not provide detailed information of lot quality
 Does not provide judgment of fitness for use (of
rejected items)
 Does not guarantee elimination of defectives – any
AQL permits defectives
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Sampling Caveats
 Size of sample is more important than percentage of lot
 Only random samples are statistically valid
 Access to samples does not guarantee randomness
 Acceptance sampling can place focus on wrong place
 Supplier should provide evidence of quality
 Focus should be on process control
 Misuse of sampling plans can be costly and misleading.
 No such thing as a single representative sample
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Representative Sample?
There is no such thing as a single
representative sample
Why?
 Draw repeated samples of 5 from a normally
distributed population.
 Record the X-bar (mean) and s (std.dev) for each
sample
 What is the result?
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Distribution of Means
The Distribution of Means obeys normal distribution – regardless of
distribution of parent population.
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Standard Error of the Mean
Central Limit Theorem
The relationship of the standard deviation of sample
means to the standard deviation of the population
Note: For a uniform distribution, Underestimates error by 25% with
n=2, but only by 5% with n=6
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The Random Sample
At any one time, each of the remaining items in the
population has an equal chance of being the next
item selected
One method is to use a table of Random Numbers
(handout from Grant & Leavenworth)
 Enter the table Randomly ( like pin-the-tail-on-thedonkey)
 Proceed in a predetermined direction – up, down, across
 Discard numbers which cannot be applied to the sample
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Random Number Table
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Source: Statistical Quality Control by Grant &
Leavenworth
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Stratified Sampling
 Random samples are selected from a “homogeneous lot”.
Often, the parts may not be homogeneous because they were
produced on different machines, by different operators, in
different plants, etc.
 With stratified sampling, random samples are drawn from
each “group” of processes that are different from other groups.
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Selecting the Sample
 Wrong way to select sample
 Judgement: often leads to Bias
 Convenience
 Right ways to select sample
 Randomly
 Systematically: e.g. every nth unit; risk of bias occurs
when selection routine matches a process pattern
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The O-C Curve
Operating Characteristic Curve
Ideal O-C Curve
Pa
Percent Defective
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The Typical O-C Curve
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Sampling Terms
 AQL – Acceptable Quality Level: The worst quality
level that can be considered acceptable.
 Acceptance Number: the largest number of defective
units permitted in the sample to accept a lot – usually
designated as “Ac” or “c”
 AOQ – Average Outgoing Quality: The expected
quality of outgoing product, after sampling, for a
given value of percent defective in the incoming
product. AOQ = p * Pa
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Sampling Terms (cont.)
 AOQL – Average Outgoing Quality Level: For a
given O-C curve, the maximum value of AOQ.
 Rejection Number – smallest number of defective
units in the sample which will cause the lot to be
rejected – usually designated as “Re”
 Sample Size – number of items in sample – usually
designated by “n”
 Lot Size – number of items in the lot (population) –
usually designated by “N”
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Sampling Risks
 Producers Risk – α: calling the population bad
when it is good; also called Type I error
 Consumers Risk – β: calling the population good
when it is bad; also called Type II error
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Sampling Risks (cont)
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Acceptance Sampling
II
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Constructing the O-C curve
We will do the following O-C curves
 Use Hyper-geometric and Poisson for each of the
following
•
•
•
•
N=60, n=6, Ac = 2
N=200, n=20, Ac = 2
N=1000, n=100, Ac = 2
N=1000, n=6, Ac = 2
Let’s do k (Ac, c - # of successes) = 0 first
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Hyper-geometric
The number of distinct combination of “n” items
taken “r” at a time is
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Hyper-geometric (cont)
= (DCk
NqCn-k)
/ NCn
Construct the following Table
p
D=Np P(k=0) P(k=1) P(k=2) P(k ≤ 2)
0%
1%
2%
3%
etc.
A Hyper-geometric calculator can be found at www.stattrek.com
Note: The Hyper-geometric distribution applies when the population, N, is
small compared to the sample size, however, it can always be used.
Sampling is done without replacement.
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Hypergeometric Calculator
N=
n=
p
0%
1%
2%
3%
4%
5%
6%
7%
100
10
D=Np
K
0
1
2
3
4
5
6
7
D=Defects in Pop.
Nq=N-Np
100
99
98
97
96
95
94
93
P(k=0)
0
1
0.9
0.809091
0.726531
0.651631
0.583752
0.522305
0.46674
P(k=1)
1
0.1
0.181818
0.247681
0.2996
0.339391
0.368686
0.38895
P(k=2)
2
0.009091
0.025046
0.045961
0.070219
0.096458
0.123549
P(k ≤ 2)
1
1
1
0.999258
0.997192
0.993362
0.987449
0.97924
total
successes
in Popl.
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Hypergeometric Calculator
Example: p=0.02, k=0, N=100,
n=10
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Hypergeometric Calculator
Example: p=0.02, k=0, N=100, n=10
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Hypergeometric Calculator
Example: p=0.02, k=0, N=100, n=10
P (k=0) = 0.809091
P (k=1) = 0.181818
P (k=2) = 0.009091
----------------------P(k≤2) = 1.0
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From QCI-CQE Primer 2005, pVI-9
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Poisson
Construct the following Table, using the Poisson Cumulative Table
p
np
P (k ≤ 2)
0%
1%
2%
3%
4%
etc.
Compare. When is Poisson a good approximation
Use the Poisson when n/N˂0.1 and np ˂5.
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Poisson Calculator
Example: p=0.02, n=10, c=0
X=k, the number of successes in the sample, i.e. “c”
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Poisson Calculator
Example: p=0.02, n=10, c=0
Mean = np
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Poisson Calculator
Example: p=0.02, n=10, c=0
TRUE for cumulative, i.e. Σk; FALSE for probability mass function, i.e.p(x=k)
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From QCI-CQE Primer 2005, pVI-8
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From QCI-CQE Primer 2005, pVI-8
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From QCI-CQE Primer 2005, pVI-9
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O-C Curve & AOQ
Determine the O-C curve.
 Prepare the following Table using the Poisson distribution
p
Pa
AOQ = p * Pa
0%
1%
2%
3%
etc
Graph the results: Pa and AOQ vs p.
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OC Curve & AOQ (2)
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OC Curve & AOQ (3)
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Acceptance Sampling
III
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Questions
1. What if this AOQ is not adequate?
2. What if you would like to add a 2nd sample when
the first sample fails?
Example
 OC curve after 1st Sample:
p=0.02, n=30, N=500, c (Ac)=0, Re=2
 OC curve after 2nd Sample (of 30 more):
p=0.02, n=60, N=500, c (Ac)= 1, Re=2
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Hypergeometric Multiple Sampling
p
D=Np
N=
500
500
500
500
n=
30
60
60
60
Nq=N-Np
K
P(k=0)
P(k=0)
P(k ≤ 1)
P(k=1)
0
0
1
1
0.00
0
500
1
0.01
5
495
0.73
0.53
0.36
0.89
0.02
10
490
0.54
0.28
0.38
0.66
0.03
15
485
0.39
0.14
0.30
0.44
0.04
20
480
0.28
0.07
0.21
0.28
0.05
25
475
0.20
0.04
0.14
0.17
0.06
30
470
0.15
0.02
0.08
0.10
0.07
35
465
0.11
0.01
0.05
0.06
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Hypergeometric Multiple Sampling
Hypergeometric Multiple Sample
Prob of Acceptance
N=500, n=30, c=0
N=500, n=60, c=1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Lot defective
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ANSI/ASQC Z1.4-1993
Mil-Std 105
 Sampling for Attributes; 95 page Document
 Pa’s from 83% to 99%
 Information necessary: N, AQL, Inspection Level
 How to Use
 Code Letters
 Single, Double, Multiple Plans
 Switching Rules
 Obtain: n, Ac, Re,
 O-C Curves
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ANSI/ASQC Z1.4-1993
Exercises
 N=475, AQL = 0.1%, Single Plan, Normal
 What is Code Letter
 What is Sample Size,
 What is Ac, Re
 Repeat for Tightened Inspection
 Repeat for Reduced Inspection
Note: 0.1% is 1000 ppm
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Z1.4 Code Letters
I-Reduced, II-Normal, III-tightened |||| For N=475, Normal, code letter is “H”
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Z1.4 Single Plan – Normal Insp.
Table II-A
n=125,
New code Letter “K”
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Z1.4 O-C Curve for Code Letter “K”
Table X-K
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Z1.4 Switching Rules
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ANSI/ASQC Z1.4-1993
What happens when AQL = . 1% isn’t
good enough
AQL = 0.1% => 1000 ppm




Is Z1.4 Adequate?
How would you decide?
If not, what would you do?
Construct O-C curve for n=1000, c=0 (Poisson). Use
100ppm < p < 5000 ppm (see slides 38 & 39)
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ANSI/ASQC Z1.9-1993
Mil-Std 414
Sampling for Variables; 110 page Document
Four Sections in the document
 Section A: General description of Plans
 Section B: Plans used when variability is unknown
(Std. deviation method is used)
 Section C: Plans used when variability is unknown
(range method is used)
 Section D: Plans used when the variability is known.
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ANSI/ASQC Z1.9-1993
Mil-Std 414
 Information necessary: N, AQL, Inspection Level
 How to Use
 Code Letters
 Single or Double Limit, Std. Dev or Range Method Plans
 Switching Rules
 Obtain: Code Letter, n, Accept/Reject criteria,
critical statistic (k)
 O-C Curves
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ANSI/ASQC Z1.9-1993
Exercise (From QCI, CQE Primer, pVI-37)
The specified max. temp for operation of a device is
209F. A lot of 40 is submitted for inspection. Use
Normal (Level II) with AQL = 0.75%. The Std.
Dev. is unknown.
Use Std. Dev. Method, variation unknown
 Find Code Letter, Sample Size, k
 Should lot be accepted or rejected
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Z1.9 Code Letters
For N=40, AQL=0.75 |||||| Use AQL=1.0 & Code Letter “D”
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Z1.9 – Finding Decision Criteria
Std. Dev method – Table B-1
 For Code Letter “D”, n=5 & AQL=1, k=1.52
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ANSI/ASQC Z1.9-1993
What is “k”
“k” is a critical statistic (term used in hypothesis testing).
It defines the maximum area of the distribution which can be
above the USL.
When Qcalc > k, there is less of distribution above Qcalc than above
“k” and lot is accepted. (Compare to “Z” table)
Increasing (USL - X-bar) increases Pa
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ANSI/ASQC Z1.9-1993
Exercise Solution
The five reading are 197F, 188F, 184F, 205F, 201F.
 X-bar (mean) = 195F
 S (Std. Dev) = 8.8F
 Qcalc = (USL – X-bar)/s = 1.59
 Because Qcalc = 1.59 is greater than k=1.52, lot is
accepted
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Z1.9 – OC Curve for “D”
Table A-3 (p9)
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ANSI/ASQC Z1.9-1993
Another Exercise
 Same information as before
 AQL = 0.1
 Find Code Letter, n, k
 Accept or Reject Lot?
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Solution – 2nd Exercise
New code letter is “E”, n=7, & k=2.22
The seven reading are 197F, 188F, 184F, 205F, 201F,
193F & 197F.
 X-bar (mean) = 195F
 S (std. Dev) = 7.3F
 Qcalc = (USL – X-bar)/s = 1.91
 Because Qcalc = 1.91 is less than k=2.22, lot is
rejected
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Inspection Economics
 Average Total Inspection: The average number
of devices inspected per lot by the defined sampling
plan
ATI = n Pa + N(1- Pa)
which assumes each rejected lot is 100% inspected.
 Average Fraction Inspected:
AFI = ATI/N
 Average Outgoing Quality:
AOQ = AQL (1 – AFI)
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Inspection Economics
Exercise (from Grant & Leavenworth, p395)
 AQL = 0.5%, N=1000
 Which sampling plan would have least ATI.
 n = 100, c = 0
 n = 170, c = 1
 n = 240, c = 2
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Inspection Economics
Exercise Solution
N
1000
1000
1000
n
100
170
240
c
0
1
2
Pa
0.59
0.8
0.92
n Pa
59
136
220.8
N(1- Pa)
410
200
80
ATI
460
336
300.8
AFI
0.460
0.336
0.301
AOQ
0.0027 0.00332
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Inspection Economics
Comparison of Cost Alternatives
 No Inspection
NpD
 100% Inspection
NC
 Sampling
nC + (N-n)pDPa + (N-n)(1-Pa)C
D = Cost if defective passes; C = Inspection cost/item
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Inspection Economics
Sample Size Break-Even Point
nBE = D/C
D = Cost if defective passes; C = Inspection cost/item
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Resources
 American Society for Quality
 Quality Press
 www.asq.org
 ASQ/NC A&T partnership quality courses
 CQIA, CMI, CQT, CQA, CQMgr, CQE, CSSBB
 Quality Progress Magazine
 And others
 Web-Sites
 www.stattrek.com – excellent basic stat site
 http://mathworld.wolfram.com/ - greaqt math and stat site
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