cqe经典复习资料6
Chapter 6 Sampling Plans 83
CHAPTER 6
SAMPLING PLANS
1.0 Introduction
2.0 Methods for Checking Product
3.0 Acceptance Sampling Plans
4.0 The Operating Characteristic Curve
5.0 The Average Outgoing Quality Curve
6.0 Probability Nomographs
7.0 Sampling Plan Construction
8.0 Glossary of Terms
“No wise fish would go anywhere without a porpoise.”
The Mock Turtle
Chapter 6 Sampling Plans 85
SAMPLING PLANS
1.0 INTRODUCTION
Sampling plans are hypothesis tests regarding product that has been submitted for an
appraisal and subsequent acceptance or rejection. The product may be grouped into lots or
may be single pieces from a continuous operation. A sample is selected and checked for
various characteristics. For products grouped into lots, the entire lot is accepted or rejected.
The decision is based on the specified criteria and the amount of defects or defective units
found in the sample. Accepting or rejecting a lot is analogous to not rejecting or rejecting the
null hypothesis in a hypothesis test. In the case of continuous production, a decision may be
made to continue sampling or to check subsequent product 100%.
Sampling at the end of a manufacturing process provides a check on the adequacy of the
quality control procedures of the manufacturing department. If the process has been
controlled satisfactorily, the product would be accepted and passed on to the next
organization or customer. If the process or quality controls have broken down, the sampling
procedures will prevent defective products from going any farther. The manufacturing
department, as part of the process or quality control program, may also use sampling
techniques.
As processes become more refined and the process capabilities are known, the need for
inspection becomes less important. The Inspection organization or end of the line appraisal
function has three objectives that will be achieved in part through sampling techniques.
•Report the quality level of the manufacturing department to management. This is
the primary objective of an inspection function.
•Provide adequate safeguards against the shipment of defective products.
•Assure that the manufacturing department has performed its quality functions
properly.
2.0 METHODS FOR CHECKING PRODUCT
Selecting product for appraising quality characteristics can be done by a number of different
methods. The six methods listed below are widely used.
•No checking
•100% checking
•Constant percentage sampling
•Random spot checking
•Audit sampling (no acceptance and rejection criteria)
•Acceptance sampling based on probability
No checking may be warranted when the process capability is known and the probability of
defective product is very small. A periodic audit to verify that conditions have not changed is
a recommended practice when products are not checked on a routine basis. In some
cases, incoming materials from various suppliers may not be inspected because the
supplier has demonstrated outstanding quality capabilities.
QReview 86
When the process capability and the product quality level is not known, no checking usually
results in increased costs for reworking defective product. When defective products are
unknowingly shipped to the next using organization, subsequent operations may have to be
halted to make corrections. When the risks involved are known, this technique will result in
significant savings and increased product velocity. When the risks are not known, this
technique may cause significant losses and problems to the company.
At the other extreme, product may be inspected 100%. In certain circumstances, 100% or
even 200% checking may be necessary, particularly where lives are involved. In most
routine processes, looking at each item is expensive, not always 100% effective and not
necessary to assure product quality. One hundred percent checking is a sorting operation to
separate good product from defective product. In addition, one hundred percent checking
cannot be used when a destructive test is made. As the number of quality characteristics
being checked increases, the effectiveness of the inspector decreases.
The unscientific sampling technique, known as the constant percentage sample, is a very
popular procedure. This sounds like a logical procedure to many people. Why not make it
easy and take a 10% sample from the lot? The problem with this method is that the sample
taken from small lots may be too small and the sample taken from large lots may be too
large. The inspection accuracy is not achieved for small lots and too much time and effort
may be spent on large lots. Also, the sampling risks involved are not known. After a certain
point, a larger sample will not yield more information. If the sample size is of sufficient size to
determine the quality level and a decision can be reached as to accept or reject the lot, then
further sampling would not be warranted.
Random spot-checking may sometimes be used when a process is in statistical control. The
random check is used to verify that the process is in control and to report the product quality
level. The sampling risks are not known, so this method will not guarantee that the outgoing
quality will be at an acceptable level. This type of sampling may be used when a supplier has
been certified as providing excellent quality products over some length of time or the process
capability is so good that other methods of inspection are not necessary.
Audit sampling is sampling that is done on a routine basis, but acceptance criterion is not
specified. A quality report is issued and the manufacturing organization will determine what
action is to be taken if the material is not acceptable. Audit sampling is used where the
manufacturing quality controls are known to be working correctly. The process capability
must be known and the chance of defective products arriving at the inspection point must be
very small.
Acceptance sampling based on probability is the most widely used sampling technique
throughout industry. Many sampling plans are tabled and published and can be used with
little training. The Dodge-Romig Sampling Inspection Tables are an example of published
tables. Some applications require special unique sampling plans, so an understanding of
how a sampling plan is developed is important. In acceptance sampling, the risks of making
a wrong decision are known. When inspection is performed by attributes, (product is
classified as good or defective) four types of acceptance sampling plans may be used, with
lot by lot single sampling plans being the most popular. This is because they are easier to
administer and implement than the other plans and they are very effective.
Chapter 6 Sampling Plans 87
3.0 ACCEPTANCE SAMPLING PLANS
1) Lot by Lot - Single Sampling
2) Lot by Lot - Double Sampling
3) Continuous Sampling
4) Sequential Sampling
3.1 Lot by Lot - Single Sampling
•A lot size (N) of product is delivered to the quality check or inspection position.
•A sample size (n) is selected randomly from the lot.
•If the number of defects or defectives in the sample exceed the acceptance number (c
or AN), the entire lot is rejected.
•If the number of defects or defectives in the sample do not exceed the acceptance
number, the entire lot is accepted.
•Rejected lots are usually detailed 100% for the requirements that caused the rejection.
In some cases the lot may be scrapped.
•Accepted lots and screened rejected lots are sent to their destination. The rejected
lots may be submitted for re-inspection.
3.2 Lot by Lot - Double Sampling
•A lot size (N) of product is delivered to the quality check or inspection position.
•Two sample sizes (n1, n2) and two acceptance numbers (c1, c2 or AN1, AN2) are
specified.
•A first sample of size n1 is taken.
•If the number of defects or defectives in the first sample exceed c2 , the lot is rejected
and a second sample is not taken.
•If the number of defects or defectives in the first sample do not exceed c1, the lot is
accepted and a second sample is not taken.
•If the number of defects or defectives in the first sample are more than c1 but less
than or equal to c2, a second sample n2 is selected and inspected.
•If a second sample is inspected:
a) and defects or defectives in combined first and second sample do not exceed c2,
the lot is accepted.
b) and defects or defectives in combined samples exceed c2, the lot is rejected.
i) Rejected lots are detailed or scrapped.
ii) Accepted lots and detailed rejected lots are sent to their destination.
QReview 88
3.3 Continuous Sampling
•Continuous sampling is used where product flow is continuous and not feasible to be
formed into lots.
•Two parameters are specified in a continuous sampling plan. The first is the frequency
of checking f and the second is the clearing number i. The frequency f is expressed as
1/10, 1/20, 1/X, etc. and i is a number such as 20 or 50.
•When inspection begins, the product is checked 100% until i parts are found to be
defect free. At this time, one out of X shall be inspected. If f = 1/10, then one out of 10
parts will be checked. The sampling will continue until a defect is found. When a
defect is found, 100% inspection shall resume and the cycle starts over. When i parts
are found to be defect free, the sample 1/X shall start again.
•In most cases, the inspector will not perform the 100% inspection. The inspector will
mark the last sampled part and the manufacturing department will perform the 100%
inspection or detailing operation.
3.4 Sequential Sampling
•The inspector will select one part from the lot and check for the specified
requirements.
•The part is classified as good or defective.
•A chart like the one shown below is specified for various sequential sampling plans.
The required quality levels determine the acceptance, rejection, and continue sampling
regions on the chart. The chart shows the inspector what decision to make after each
sample is inspected. The lot will either be accepted rejected or another sample will be
taken. This procedure is done on a lot by lot basis. The advantage of this type of
sampling plan is that a decision could be made based on a relatively small sample.
•Rejected lots are detailed 100% (usually by the manufacturing department). Accepted
and screened rejected lots are sent to their destination.
Reject
Continue Sampling
Accept
0 1 2 3 4 5 6 7 8 9 10
Cumulative defective units
Cumulative number of samples
Chapter 6 Sampling Plans 89
4.0 THE OPERATING CHARACTERISTIC CURVE (OC CURVE)
The Operating characteristic curve is a picture of a sampling plan. Each sampling plan has a
unique OC curve. The sample size and acceptance number define the OC curve and
determine its shape. The acceptance number is the maximum allowable defects or defective
parts in a sample for the lot to be accepted. The OC curve shows the probability of
acceptance for various values of incoming quality.
0% 1% 2% 3% 4% 5% 6% 7%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p - Quality of lots submitted to inspection (AIQ)
Approximately two
times the AOQL
Rejectable
Quality Level (RQL)
Probability of Acceptance
The Producer's Risk ()
Acceptable Quality
Level (AQL)
The Consumer’s
Risk ()
An OC curve is developed by determining the probability of acceptance for several values of
incoming quality. Incoming quality is denoted by p. The probability of acceptance is the
probability that the number of defects or defective units in the sample is equal to or less than
the acceptance number of the sampling plan. The AQL is the acceptable quality level and the
RQL is rejectable quality level. If the units on the abscissa are in terms of percent defective,
the RQL is called the LTPD or lot tolerance percent defective. The producer’s risk () is the
probability of rejecting a lot of AQL quality. The consumer’s risk () is the probability of
accepting a lot of RQL quality.
There are three probability distributions that may be used to find the probability of
acceptance. These distributions were covered in the Basic Probability chapter and are
reviewed here.
•The hypergeometric distribution
•The binomial distribution
•The Poisson distribution
Although the hypergeometric may be used when the lot sizes are small, the binomial and
Poisson are by far the most popular distributions to use when constructing sampling plans.
QReview 90
4.1 Hypergeometric Distribution
The hypergeometric distribution is used to calculate the probability of acceptance of a
sampling plan when the lot is relatively small. It can be defined as the true basic
probability distribution of attribute data but the calculations could become quite
cumbersome for large lot sizes.
The probability of exactly x defective parts in a sample n:

P x
x
n
n x
N n
n N
( ) 

4.2 Binomial Distribution
The binomial distribution is used when the lot is very large. For large lots, the nonreplacement
of the sampled product does not affect the probabilities. The
hypergeometric takes into consideration that each sample taken affects the probability
associated with the next sample. This is called sampling without replacement. The
binomial assumes that the probabilities associated with all samples are equal. This is
sometimes referred to as sampling with replacement although the parts are not
physically replaced. The binomial is used extensively in the construction of sampling
plans. The sampling plans in the Dodge-Romig Sampling Tables were derived from the
binomial distribution.
The probability of exactly x defective parts in a sample n:
P x p p x
n ( ) x (1)nx
The symbol p represents the value of incoming quality expressed as a decimal.
(1% = .01, 2% = .02, etc.)
4.3 Poisson Distribution
The Poisson distribution is used for sampling plans involving the number of defects or
defects per unit rather than the number of defective parts. It is also used to approximate
the binomial probabilities involving the number of defective parts when the sample (n) is
large and p is very small. When n is large and p is small, the Poisson distribution formula
may be used to approximate the binomial. Using the Poisson to calculate probabilities
associated with various sampling plans is relatively simple because the Poisson tables
can be used. The Thorndike chart, which will be discussed later, is a valuable aid in the
construction of sampling plans using the Poisson distribution.
The probability of exactly x defects or defective parts in a sample n:
P x
e np
x
np x
( )
( )
!


The letter e represents the value of the base of the natural logarithm system. It is a
constant value (e = 2.71828).
4.4 An OC Curve Using the Binomial Distribution
Chapter 6 Sampling Plans 91
For any value of the AIQ, the probability of acceptance is the probability of c or fewer
defective parts where c is the acceptance number for the sampling plan. The
probability of acceptance is usually expressed as a decimal rather than as a
percentage. It is represented by the symbol Pa. The letter n represents the sample size.
For a sampling plan with an acceptance number of 0, Pa = P(0). For a sampling plan
with an acceptance number of 1, Pa = P(0) + P(1).
Pa = P(0) + …+ P(c)
The probability of acceptance is calculated for a sampling plan with n = 30 and c = 1.
p = AIQ P(0) P(1) Pa
.01 30 
0 (.01)0 (.99)30 = .740 30 
1 (.01)1 (.99) 29
= .224 .964
.02 30 
0 (.02)0 (.98) 30
= .545 30 
1 (.02)1 (.98) 29
= .334 .879
.03 30 
0 (.03)0 (.97) 30
= .401 30 
1 (.03)1 (.97) 29
= .372 .773
.04 30 
0 (.04)0 (.96) 30 = .294 30 
1 (.04)1 (.96) 29 = .367 .661
.05 30 
0 (.05)0 (.95) 30
= .215 30 
1 (.05)1 (.95) 29
= .339 .554
.06 30 
0 (.06)0 (.94) 30 = .156 30 
1 (.06)1 (.94) 29
= .299 .455
.07 30 
0 (.07)0 (.93) 30
= .113 30 
1 (.07)1 (.93) 29
= .256 .369
.08 30 
0 (.08)0 (.92) 30
= .082 30 
1 (.08)1 (.92) 29 = .214 .296
.09 30 
0 (.09)0 (.91) 30
= .059 30 
1 (.09)1 (.91) 29 = .175 .234
.10 30 
0 (.10)0 (.90) 30
= .042 30 
1 (.10)1 (.90) 29 = .141 .183
.11 30 
0 (.11)0 (.89) 30
= .030 30 
1 (.11)1 (.89) 29 = .112 .142
.12 30 
0 (.12)0 (.88) 30
= .022 30 
1 (.12)1 (.88) 29 = .088 .110
The Operating Characteristic Curve for n = 30 and c = 1 using the Binomial Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
Probability of Acceptance
Average Incoming Quality (AIQ)
Fraction Defective in Decimal Form
QReview 92
4.5 An OC Curve Using the Poisson Distribution
The Poisson distribution is used to compute the probability of acceptance for defects per
unit sampling plans. It may also be used to approximate binomial probabilities and
compute the probability of acceptance for fraction defective sampling plans.
For the sampling plan n = 30 and c = 1, c may be in terms of number of defects or in
terms of number of defective parts. If c is in terms number of defects, the AIQ or
abscissa on the OC curve is in terms of defects per unit. The acceptance number for the
sampling plan n = 30 and c = 1 may either be 1 defect or 1 defective part.
The Poisson formula, P x
e np
x
np x
( )
( )
!


, is used to compute the probabilities of
acceptance.
p = AIQ m= np P(0) P(1) Pa
.01 .30 .741 .222 .963
.02 .60 .549 .329 .878
.03 .90 .407 .366 .772
.04 1.2 .301 .361 .663
.05 1.5 .223 .335 .558
.06 1.8 .165 .298 .463
.07 2.1 .122 .257 .380
.08 2.4 .091 .218 .308
.09 2.7 .067 .181 .249
.10 3.0 .050 .149 .199
.11 3.3 .037 .122 .159
.12 3.6 .027 .098 .126
For all practical purposes, the probabilities of acceptance are the same as those
obtained with the binomial formula. There are some minor differences. For this example,
the differences increase slightly as the curve approaches the tail.
The Operating Characteristic Curve for n = 30 and c = 1 using the Poisson Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
Probability of Acceptance
Average Incoming Quality (AIQ)
Defects per Unit or Fraction Defective in Decimal Form
Chapter 6 Sampling Plans 93
5.0 THE AVERAGE OUTGOING QUALITY CURVE
For every acceptance sampling plan, the outgoing quality will be somewhat better than the
incoming quality because a certain percent of the lots will be rejected and detailed. The
Average Outgoing Quality (AOQ) curve shows the relationship between incoming and
outgoing quality. The AOQ and OC curve, when used together, describe the characteristics
of the sampling plan and the risks involved.
AOQ p P
N n
N a ( )( )
( )
The letter p is the incoming quality level (AIQ) and Pa is the probability of acceptance. The
abscissa for the AOQ curve is the same as for the OC curve. The Average Outgoing Quality
Limit (AOQL) is, on average, the maximum value of the AOQ. When the lot (N) is very large,
the expression (N - n)/N approaches 1 and may be omitted. For this example, the lot size is
5000 pieces. Therefore, (N - n)/N = (5000 - 30)/5000 = .994 1 and is dropped out. For this
AQL curve, the Probabilities of acceptance (Pa) for the sampling plan n = 30 and c = 1 are
based on the binomial distribution.
p = AIQ Pa AOQ = (p)( Pa)
.01 .964 .010
.02 .879 .018
.03 .773 .023
.04 .661 .026
.05 .554 .028
.06 .455 .027
.07 .369 .026
.08 .296 .024
.09 .234 .021
.10 .184 .018
.11 .143 .016
.12 .110 .013
AOQ Curve
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
Average Incoming Quality (AIQ)
Average Outgoing Quality (AOQ)
AOQL = .028
The AOQL is approximately .028 at an incoming quality level of .05. The AOQL is the
QReview 94
outgoing quality level at the crest of the curve.
6.0 PROBABILITY NOMOGRAPHS
A Nomograph is a paper slide rule that helps to simplify certain computations. Many of the
same computations may be made more elegantly on a computer. Nomographs were
popular before there were computers. Since computers are not allowed at the CQE exam,
the nomographs may come in handy.
6.1 Binomial Nomograph
Ken Larson of the AT&T Company developed the binomial nomograph. The nomograph
greatly simplifies and reduces the computational burden involved with solving binomial
problems. It permits direct solution of some problems not otherwise directly solvable
except by approximation or computer. The nomograph contains the cumulative binomial
probabilities P(0) + … + P(c), and may be used to determine sample sizes and
acceptance numbers for acceptance sampling plans. The probability of accepting a lot
is the probability of c or fewer defective parts. The nomograph is used for fraction
defective sampling plans.
The values for the operating characteristic (OC) curve are obtained directly from the
nomograph.
6.2 Thorndike Chart
The Thorndike chart was developed by Frances Thorndike of Bell Laboratories in 1926.
It is a nomograph of the cumulative Poisson probability distribution. Like the binomial
nomograph, it may also be used to determine sample sizes and acceptance numbers for
sampling plan applications. The Thorndike chart is used for the following sampling plans:
•Defects per unit sampling plans.
•Approximation to the binomial probabilities for fraction defective sampling plans.
The abscissa on the Thorndike chart is np. The ordinate is the probability of c or fewer
occurrences. The curved lines in the body of the chart represent the cumulative number
of occurrences or successes that are of interest. The curved lines also represent the
acceptance numbers in a sampling plan. The Thorndike chart may be used as an
alternative to the Poisson tables when determining cumulative probabilities.
7.0 SAMPLING PLAN CONSTRUCTION
Sampling plans may be developed to meet certain criteria and to insure that the specified
outgoing quality levels are met. In the construction of a lot by lot single sampling plan, four
parameters must be determined prior to determining the sample size and acceptance
number.
The parameters are:
•The Acceptable Quality Level (AQL)
•Alpha (), The Producers Risk
•The Rejectable Quality Level (RQL)
Chapter 6 Sampling Plans 95
•Beta (), The Consumers Risk
The objective is to find a sample size and acceptance number whose OC curve meets the
above parameters. The sampling plan will have a 1 - chance of being accepted when the
incoming quality is at the AQL level and a chance of being accepted when the incoming
quality is at the RQL level. An easy way to find the sample size and acceptance number is to
use a the binomial nomograph or the cumulative Poisson nomograph called the Thorndike
chart. Copies of the binomial nomograph and Thorndike chart are included in the appendix.
They will be provided separately if you have the computer based version of QReview. They
are also included in various textbooks.
Sampling plans will be constructed using both the binomial nomograph and the Thorndike
chart. The AQL, RQL, , and must be specified. They may be determined by your
customer, special studies, or past experience. The most common values to use for and 
are .05 and .10 respectively. The following values are used for the binomial nomograph and
Thorndike chart examples:
AQL = 2% (.02), =.05, RQL = 8% (.08), = .10
7.1 Sampling Plan Construction Using the Binomial Nomograph
The AIQ is on the left scale and the probability of acceptance is on the right scale. The
semi vertical lines on the nomograph represent the sample sizes and the semi horizontal
lines are the acceptance numbers. Draw a line from the AQL (.02) to its probability of
acceptance (.95) and a line from the RQL (.08) to its probability of acceptance (.10). The
intersection will yield the sample size and acceptance number. Do not interpolate: Find
the closest sample size and acceptance number to the intersection point. Both the
sample size and acceptance numbers must be integers.
.02
.08
.10
.95
n =100
c = 4
Probability of c or fewer occurrences in n trials (P)
Probability of occurrence in a single trial (p)
Number of trials or sample size (n)
Number of occurrences (c)
(Probability of acceptance)
(Average Incoming Quality - AIQ)
QReview 96
The sample size is 100 and the acceptance number is 4.
Chapter 6 Sampling Plans 97
7.2 Sampling Plan Construction Using the Thorndike Chart
The Thorndike chart can also be used to find the sample size and acceptance number
for the plan. To assist in the task, a tool called an L will be used. The L may be modified
for any value of . The inside horizontal scale on the L must be lined up with and is
to coincide with 1 - on the vertical axis. The left or vertical side of the L coincides with
the probability of acceptance on the Thorndike chart.
R is the ratio of the RQL to the AQL (R = RQL/AQL). R is computed and marked with
an arrow as shown on the diagram.
The L is moved across the chart until one of the acceptance number curves goes
through both the value on the inside vertical scale and the R value on the inside
horizontal scale. The best fitting acceptance number curve is the one to select. The
curve represents the acceptance number (c) for the plan. The curve c = 4 goes through
both the points = .05 and R = 4 (R = .08/.02 = 4).
R = 4
1.0
0.95
0.1
c = 4
2 8
Value of pn
0.1
Probability of c or fewer defects
0.99999
0.00001
30
.05
.10
The sample size is determined as follows: At the corner of the L where the value is 1,
drop a straight line to the pn scale on the abscissa of the Thorndike chart. The value of
pn is 2. The value of p at this point is the AQL (AQL = .02). If pn =2 and p = .02 then n =
2/.02 = 100. Also a straight line can be dropped from R to the pn scale. The value of pn
is 8. The value of p at this point is the RQL (RQL = .08). If pn = 8 and
p = .08 then n = 8/.08 = 100. The same answer is obtained at either point.
The sample size is 100 and the acceptance number is 4.
Both the binomial nomograph and the Thorndike chart give the same sample size and
acceptance number. In some cases, there may be minor variations between the two
methods.
QReview 98
8.0 GLOSSARY OF TERMS
•Acceptance Number: The maximum allowable defects or defectives in a sample for
the lot to be accepted. (Acceptance number = AN or c)
•AIQ - Average Incoming Quality: This is the average quality level going into the
inspection point. The inspection data and subsequent report reflects this number. The
AIQ is the abscissa on the OC and AOQ curves.
•AOQ - Average Outgoing Quality: The average quality level leaving the inspection
point after rejection and acceptance of a number of lots. If rejected lots are not checked
100% and defective units removed or replaced with good units, the AOQ will be the same
as the AIQ.
•AOQ Curve - Average Outgoing Quality Curve: The curve or graph that shows the
Average Outgoing Quality for various values of incoming quality.
•AOQL - Average Outgoing Quality Limit: The maximum value of the AOQ. If the
sampling procedures are followed, the outgoing quality will, on average, not be worse
than the AOQL.
•AQL - Acceptable Quality Level: The quality level for which there is a high probability
of accepting the lot. The AQL is also defined as the maximum fraction defective or
defects per unit that can be considered satisfactory as a process average.
•Attribute data: Although measurements may be taken, they are not recorded on the
data sheet. The product is classified as good or defective. (Discrete data)
•Consumers risk (b): The probability of accepting a lot with a high number of defective
units. is usually set at .05 to .15 with .10 as the common value. can also be defined
as the probability of accepting a lot of RQL or LTPD quality.
•Lot: A collection of individual pieces from a common source, possessing a common set
of quality characteristics and submitted as a group for acceptance at one time.
(Lot size = N)
•LTPD - Lot Tolerance Percent Defective: This is the value of incoming quality where it
is desirable to reject most lots. The quality level is unacceptable. This is the RQL
expressed as a percent defective.
•OC Curve - Operating Characteristic Curve: The curve or graph shows the
probability of a lot being accepted for various values of incoming quality.
•Power of Test (1 - b): This is the probability of rejecting a lot when the incoming
quality is at the RQL level.
•Process average: The normal or stable quality level of a product or process for a
specified period of time. The quality level may be expressed as a fraction defective,
percent defective, or defects per unit.
Chapter 6 Sampling Plans 99
•Producers risk (a): The probability of rejecting a good lot. This is usually set at .01 to
.10 for most sampling plans. The symbol can also be defined as the probability of
rejecting a lot of AQL quality.
•Random sample: A sample selected in such a manner that any piece of product in the
lot has an equal chance of being chosen.
•RQL - Rejectable Quality Level: The generic term for the incoming quality level for
which there is a low probability of accepting the lot. The quality level is substandard.
•Sample: A subset of a lot selected to be inspected. (Sample size = n)
•Sampling plan: The procedure that specifies the number of units to be selected from a
lot or batch for appraisal. The sample size and acceptance number describes each
unique sampling plan.
•Variables data: Actual measurements are taken and recorded. (Continuous data)
CHAPTER 6
SAMPLING PLANS
1.0 Introduction
2.0 Methods for Checking Product
3.0 Acceptance Sampling Plans
4.0 The Operating Characteristic Curve
5.0 The Average Outgoing Quality Curve
6.0 Probability Nomographs
7.0 Sampling Plan Construction
8.0 Glossary of Terms
“No wise fish would go anywhere without a porpoise.”
The Mock Turtle
Chapter 6 Sampling Plans 85
SAMPLING PLANS
1.0 INTRODUCTION
Sampling plans are hypothesis tests regarding product that has been submitted for an
appraisal and subsequent acceptance or rejection. The product may be grouped into lots or
may be single pieces from a continuous operation. A sample is selected and checked for
various characteristics. For products grouped into lots, the entire lot is accepted or rejected.
The decision is based on the specified criteria and the amount of defects or defective units
found in the sample. Accepting or rejecting a lot is analogous to not rejecting or rejecting the
null hypothesis in a hypothesis test. In the case of continuous production, a decision may be
made to continue sampling or to check subsequent product 100%.
Sampling at the end of a manufacturing process provides a check on the adequacy of the
quality control procedures of the manufacturing department. If the process has been
controlled satisfactorily, the product would be accepted and passed on to the next
organization or customer. If the process or quality controls have broken down, the sampling
procedures will prevent defective products from going any farther. The manufacturing
department, as part of the process or quality control program, may also use sampling
techniques.
As processes become more refined and the process capabilities are known, the need for
inspection becomes less important. The Inspection organization or end of the line appraisal
function has three objectives that will be achieved in part through sampling techniques.
•Report the quality level of the manufacturing department to management. This is
the primary objective of an inspection function.
•Provide adequate safeguards against the shipment of defective products.
•Assure that the manufacturing department has performed its quality functions
properly.
2.0 METHODS FOR CHECKING PRODUCT
Selecting product for appraising quality characteristics can be done by a number of different
methods. The six methods listed below are widely used.
•No checking
•100% checking
•Constant percentage sampling
•Random spot checking
•Audit sampling (no acceptance and rejection criteria)
•Acceptance sampling based on probability
No checking may be warranted when the process capability is known and the probability of
defective product is very small. A periodic audit to verify that conditions have not changed is
a recommended practice when products are not checked on a routine basis. In some
cases, incoming materials from various suppliers may not be inspected because the
supplier has demonstrated outstanding quality capabilities.
QReview 86
When the process capability and the product quality level is not known, no checking usually
results in increased costs for reworking defective product. When defective products are
unknowingly shipped to the next using organization, subsequent operations may have to be
halted to make corrections. When the risks involved are known, this technique will result in
significant savings and increased product velocity. When the risks are not known, this
technique may cause significant losses and problems to the company.
At the other extreme, product may be inspected 100%. In certain circumstances, 100% or
even 200% checking may be necessary, particularly where lives are involved. In most
routine processes, looking at each item is expensive, not always 100% effective and not
necessary to assure product quality. One hundred percent checking is a sorting operation to
separate good product from defective product. In addition, one hundred percent checking
cannot be used when a destructive test is made. As the number of quality characteristics
being checked increases, the effectiveness of the inspector decreases.
The unscientific sampling technique, known as the constant percentage sample, is a very
popular procedure. This sounds like a logical procedure to many people. Why not make it
easy and take a 10% sample from the lot? The problem with this method is that the sample
taken from small lots may be too small and the sample taken from large lots may be too
large. The inspection accuracy is not achieved for small lots and too much time and effort
may be spent on large lots. Also, the sampling risks involved are not known. After a certain
point, a larger sample will not yield more information. If the sample size is of sufficient size to
determine the quality level and a decision can be reached as to accept or reject the lot, then
further sampling would not be warranted.
Random spot-checking may sometimes be used when a process is in statistical control. The
random check is used to verify that the process is in control and to report the product quality
level. The sampling risks are not known, so this method will not guarantee that the outgoing
quality will be at an acceptable level. This type of sampling may be used when a supplier has
been certified as providing excellent quality products over some length of time or the process
capability is so good that other methods of inspection are not necessary.
Audit sampling is sampling that is done on a routine basis, but acceptance criterion is not
specified. A quality report is issued and the manufacturing organization will determine what
action is to be taken if the material is not acceptable. Audit sampling is used where the
manufacturing quality controls are known to be working correctly. The process capability
must be known and the chance of defective products arriving at the inspection point must be
very small.
Acceptance sampling based on probability is the most widely used sampling technique
throughout industry. Many sampling plans are tabled and published and can be used with
little training. The Dodge-Romig Sampling Inspection Tables are an example of published
tables. Some applications require special unique sampling plans, so an understanding of
how a sampling plan is developed is important. In acceptance sampling, the risks of making
a wrong decision are known. When inspection is performed by attributes, (product is
classified as good or defective) four types of acceptance sampling plans may be used, with
lot by lot single sampling plans being the most popular. This is because they are easier to
administer and implement than the other plans and they are very effective.
Chapter 6 Sampling Plans 87
3.0 ACCEPTANCE SAMPLING PLANS
1) Lot by Lot - Single Sampling
2) Lot by Lot - Double Sampling
3) Continuous Sampling
4) Sequential Sampling
3.1 Lot by Lot - Single Sampling
•A lot size (N) of product is delivered to the quality check or inspection position.
•A sample size (n) is selected randomly from the lot.
•If the number of defects or defectives in the sample exceed the acceptance number (c
or AN), the entire lot is rejected.
•If the number of defects or defectives in the sample do not exceed the acceptance
number, the entire lot is accepted.
•Rejected lots are usually detailed 100% for the requirements that caused the rejection.
In some cases the lot may be scrapped.
•Accepted lots and screened rejected lots are sent to their destination. The rejected
lots may be submitted for re-inspection.
3.2 Lot by Lot - Double Sampling
•A lot size (N) of product is delivered to the quality check or inspection position.
•Two sample sizes (n1, n2) and two acceptance numbers (c1, c2 or AN1, AN2) are
specified.
•A first sample of size n1 is taken.
•If the number of defects or defectives in the first sample exceed c2 , the lot is rejected
and a second sample is not taken.
•If the number of defects or defectives in the first sample do not exceed c1, the lot is
accepted and a second sample is not taken.
•If the number of defects or defectives in the first sample are more than c1 but less
than or equal to c2, a second sample n2 is selected and inspected.
•If a second sample is inspected:
a) and defects or defectives in combined first and second sample do not exceed c2,
the lot is accepted.
b) and defects or defectives in combined samples exceed c2, the lot is rejected.
i) Rejected lots are detailed or scrapped.
ii) Accepted lots and detailed rejected lots are sent to their destination.
QReview 88
3.3 Continuous Sampling
•Continuous sampling is used where product flow is continuous and not feasible to be
formed into lots.
•Two parameters are specified in a continuous sampling plan. The first is the frequency
of checking f and the second is the clearing number i. The frequency f is expressed as
1/10, 1/20, 1/X, etc. and i is a number such as 20 or 50.
•When inspection begins, the product is checked 100% until i parts are found to be
defect free. At this time, one out of X shall be inspected. If f = 1/10, then one out of 10
parts will be checked. The sampling will continue until a defect is found. When a
defect is found, 100% inspection shall resume and the cycle starts over. When i parts
are found to be defect free, the sample 1/X shall start again.
•In most cases, the inspector will not perform the 100% inspection. The inspector will
mark the last sampled part and the manufacturing department will perform the 100%
inspection or detailing operation.
3.4 Sequential Sampling
•The inspector will select one part from the lot and check for the specified
requirements.
•The part is classified as good or defective.
•A chart like the one shown below is specified for various sequential sampling plans.
The required quality levels determine the acceptance, rejection, and continue sampling
regions on the chart. The chart shows the inspector what decision to make after each
sample is inspected. The lot will either be accepted rejected or another sample will be
taken. This procedure is done on a lot by lot basis. The advantage of this type of
sampling plan is that a decision could be made based on a relatively small sample.
•Rejected lots are detailed 100% (usually by the manufacturing department). Accepted
and screened rejected lots are sent to their destination.
Reject
Continue Sampling
Accept
0 1 2 3 4 5 6 7 8 9 10
Cumulative defective units
Cumulative number of samples
Chapter 6 Sampling Plans 89
4.0 THE OPERATING CHARACTERISTIC CURVE (OC CURVE)
The Operating characteristic curve is a picture of a sampling plan. Each sampling plan has a
unique OC curve. The sample size and acceptance number define the OC curve and
determine its shape. The acceptance number is the maximum allowable defects or defective
parts in a sample for the lot to be accepted. The OC curve shows the probability of
acceptance for various values of incoming quality.
0% 1% 2% 3% 4% 5% 6% 7%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p - Quality of lots submitted to inspection (AIQ)
Approximately two
times the AOQL
Rejectable
Quality Level (RQL)
Probability of Acceptance
The Producer's Risk ()
Acceptable Quality
Level (AQL)
The Consumer’s
Risk ()
An OC curve is developed by determining the probability of acceptance for several values of
incoming quality. Incoming quality is denoted by p. The probability of acceptance is the
probability that the number of defects or defective units in the sample is equal to or less than
the acceptance number of the sampling plan. The AQL is the acceptable quality level and the
RQL is rejectable quality level. If the units on the abscissa are in terms of percent defective,
the RQL is called the LTPD or lot tolerance percent defective. The producer’s risk () is the
probability of rejecting a lot of AQL quality. The consumer’s risk () is the probability of
accepting a lot of RQL quality.
There are three probability distributions that may be used to find the probability of
acceptance. These distributions were covered in the Basic Probability chapter and are
reviewed here.
•The hypergeometric distribution
•The binomial distribution
•The Poisson distribution
Although the hypergeometric may be used when the lot sizes are small, the binomial and
Poisson are by far the most popular distributions to use when constructing sampling plans.
QReview 90
4.1 Hypergeometric Distribution
The hypergeometric distribution is used to calculate the probability of acceptance of a
sampling plan when the lot is relatively small. It can be defined as the true basic
probability distribution of attribute data but the calculations could become quite
cumbersome for large lot sizes.
The probability of exactly x defective parts in a sample n:

P x
x
n
n x
N n
n N
( ) 

4.2 Binomial Distribution
The binomial distribution is used when the lot is very large. For large lots, the nonreplacement
of the sampled product does not affect the probabilities. The
hypergeometric takes into consideration that each sample taken affects the probability
associated with the next sample. This is called sampling without replacement. The
binomial assumes that the probabilities associated with all samples are equal. This is
sometimes referred to as sampling with replacement although the parts are not
physically replaced. The binomial is used extensively in the construction of sampling
plans. The sampling plans in the Dodge-Romig Sampling Tables were derived from the
binomial distribution.
The probability of exactly x defective parts in a sample n:
P x p p x
n ( ) x (1)nx
The symbol p represents the value of incoming quality expressed as a decimal.
(1% = .01, 2% = .02, etc.)
4.3 Poisson Distribution
The Poisson distribution is used for sampling plans involving the number of defects or
defects per unit rather than the number of defective parts. It is also used to approximate
the binomial probabilities involving the number of defective parts when the sample (n) is
large and p is very small. When n is large and p is small, the Poisson distribution formula
may be used to approximate the binomial. Using the Poisson to calculate probabilities
associated with various sampling plans is relatively simple because the Poisson tables
can be used. The Thorndike chart, which will be discussed later, is a valuable aid in the
construction of sampling plans using the Poisson distribution.
The probability of exactly x defects or defective parts in a sample n:
P x
e np
x
np x
( )
( )
!


The letter e represents the value of the base of the natural logarithm system. It is a
constant value (e = 2.71828).
4.4 An OC Curve Using the Binomial Distribution
Chapter 6 Sampling Plans 91
For any value of the AIQ, the probability of acceptance is the probability of c or fewer
defective parts where c is the acceptance number for the sampling plan. The
probability of acceptance is usually expressed as a decimal rather than as a
percentage. It is represented by the symbol Pa. The letter n represents the sample size.
For a sampling plan with an acceptance number of 0, Pa = P(0). For a sampling plan
with an acceptance number of 1, Pa = P(0) + P(1).
Pa = P(0) + …+ P(c)
The probability of acceptance is calculated for a sampling plan with n = 30 and c = 1.
p = AIQ P(0) P(1) Pa
.01 30 
0 (.01)0 (.99)30 = .740 30 
1 (.01)1 (.99) 29
= .224 .964
.02 30 
0 (.02)0 (.98) 30
= .545 30 
1 (.02)1 (.98) 29
= .334 .879
.03 30 
0 (.03)0 (.97) 30
= .401 30 
1 (.03)1 (.97) 29
= .372 .773
.04 30 
0 (.04)0 (.96) 30 = .294 30 
1 (.04)1 (.96) 29 = .367 .661
.05 30 
0 (.05)0 (.95) 30
= .215 30 
1 (.05)1 (.95) 29
= .339 .554
.06 30 
0 (.06)0 (.94) 30 = .156 30 
1 (.06)1 (.94) 29
= .299 .455
.07 30 
0 (.07)0 (.93) 30
= .113 30 
1 (.07)1 (.93) 29
= .256 .369
.08 30 
0 (.08)0 (.92) 30
= .082 30 
1 (.08)1 (.92) 29 = .214 .296
.09 30 
0 (.09)0 (.91) 30
= .059 30 
1 (.09)1 (.91) 29 = .175 .234
.10 30 
0 (.10)0 (.90) 30
= .042 30 
1 (.10)1 (.90) 29 = .141 .183
.11 30 
0 (.11)0 (.89) 30
= .030 30 
1 (.11)1 (.89) 29 = .112 .142
.12 30 
0 (.12)0 (.88) 30
= .022 30 
1 (.12)1 (.88) 29 = .088 .110
The Operating Characteristic Curve for n = 30 and c = 1 using the Binomial Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
Probability of Acceptance
Average Incoming Quality (AIQ)
Fraction Defective in Decimal Form
QReview 92
4.5 An OC Curve Using the Poisson Distribution
The Poisson distribution is used to compute the probability of acceptance for defects per
unit sampling plans. It may also be used to approximate binomial probabilities and
compute the probability of acceptance for fraction defective sampling plans.
For the sampling plan n = 30 and c = 1, c may be in terms of number of defects or in
terms of number of defective parts. If c is in terms number of defects, the AIQ or
abscissa on the OC curve is in terms of defects per unit. The acceptance number for the
sampling plan n = 30 and c = 1 may either be 1 defect or 1 defective part.
The Poisson formula, P x
e np
x
np x
( )
( )
!


, is used to compute the probabilities of
acceptance.
p = AIQ m= np P(0) P(1) Pa
.01 .30 .741 .222 .963
.02 .60 .549 .329 .878
.03 .90 .407 .366 .772
.04 1.2 .301 .361 .663
.05 1.5 .223 .335 .558
.06 1.8 .165 .298 .463
.07 2.1 .122 .257 .380
.08 2.4 .091 .218 .308
.09 2.7 .067 .181 .249
.10 3.0 .050 .149 .199
.11 3.3 .037 .122 .159
.12 3.6 .027 .098 .126
For all practical purposes, the probabilities of acceptance are the same as those
obtained with the binomial formula. There are some minor differences. For this example,
the differences increase slightly as the curve approaches the tail.
The Operating Characteristic Curve for n = 30 and c = 1 using the Poisson Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
Probability of Acceptance
Average Incoming Quality (AIQ)
Defects per Unit or Fraction Defective in Decimal Form
Chapter 6 Sampling Plans 93
5.0 THE AVERAGE OUTGOING QUALITY CURVE
For every acceptance sampling plan, the outgoing quality will be somewhat better than the
incoming quality because a certain percent of the lots will be rejected and detailed. The
Average Outgoing Quality (AOQ) curve shows the relationship between incoming and
outgoing quality. The AOQ and OC curve, when used together, describe the characteristics
of the sampling plan and the risks involved.
AOQ p P
N n
N a ( )( )
( )
The letter p is the incoming quality level (AIQ) and Pa is the probability of acceptance. The
abscissa for the AOQ curve is the same as for the OC curve. The Average Outgoing Quality
Limit (AOQL) is, on average, the maximum value of the AOQ. When the lot (N) is very large,
the expression (N - n)/N approaches 1 and may be omitted. For this example, the lot size is
5000 pieces. Therefore, (N - n)/N = (5000 - 30)/5000 = .994 1 and is dropped out. For this
AQL curve, the Probabilities of acceptance (Pa) for the sampling plan n = 30 and c = 1 are
based on the binomial distribution.
p = AIQ Pa AOQ = (p)( Pa)
.01 .964 .010
.02 .879 .018
.03 .773 .023
.04 .661 .026
.05 .554 .028
.06 .455 .027
.07 .369 .026
.08 .296 .024
.09 .234 .021
.10 .184 .018
.11 .143 .016
.12 .110 .013
AOQ Curve
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
Average Incoming Quality (AIQ)
Average Outgoing Quality (AOQ)
AOQL = .028
The AOQL is approximately .028 at an incoming quality level of .05. The AOQL is the
QReview 94
outgoing quality level at the crest of the curve.
6.0 PROBABILITY NOMOGRAPHS
A Nomograph is a paper slide rule that helps to simplify certain computations. Many of the
same computations may be made more elegantly on a computer. Nomographs were
popular before there were computers. Since computers are not allowed at the CQE exam,
the nomographs may come in handy.
6.1 Binomial Nomograph
Ken Larson of the AT&T Company developed the binomial nomograph. The nomograph
greatly simplifies and reduces the computational burden involved with solving binomial
problems. It permits direct solution of some problems not otherwise directly solvable
except by approximation or computer. The nomograph contains the cumulative binomial
probabilities P(0) + … + P(c), and may be used to determine sample sizes and
acceptance numbers for acceptance sampling plans. The probability of accepting a lot
is the probability of c or fewer defective parts. The nomograph is used for fraction
defective sampling plans.
The values for the operating characteristic (OC) curve are obtained directly from the
nomograph.
6.2 Thorndike Chart
The Thorndike chart was developed by Frances Thorndike of Bell Laboratories in 1926.
It is a nomograph of the cumulative Poisson probability distribution. Like the binomial
nomograph, it may also be used to determine sample sizes and acceptance numbers for
sampling plan applications. The Thorndike chart is used for the following sampling plans:
•Defects per unit sampling plans.
•Approximation to the binomial probabilities for fraction defective sampling plans.
The abscissa on the Thorndike chart is np. The ordinate is the probability of c or fewer
occurrences. The curved lines in the body of the chart represent the cumulative number
of occurrences or successes that are of interest. The curved lines also represent the
acceptance numbers in a sampling plan. The Thorndike chart may be used as an
alternative to the Poisson tables when determining cumulative probabilities.
7.0 SAMPLING PLAN CONSTRUCTION
Sampling plans may be developed to meet certain criteria and to insure that the specified
outgoing quality levels are met. In the construction of a lot by lot single sampling plan, four
parameters must be determined prior to determining the sample size and acceptance
number.
The parameters are:
•The Acceptable Quality Level (AQL)
•Alpha (), The Producers Risk
•The Rejectable Quality Level (RQL)
Chapter 6 Sampling Plans 95
•Beta (), The Consumers Risk
The objective is to find a sample size and acceptance number whose OC curve meets the
above parameters. The sampling plan will have a 1 - chance of being accepted when the
incoming quality is at the AQL level and a chance of being accepted when the incoming
quality is at the RQL level. An easy way to find the sample size and acceptance number is to
use a the binomial nomograph or the cumulative Poisson nomograph called the Thorndike
chart. Copies of the binomial nomograph and Thorndike chart are included in the appendix.
They will be provided separately if you have the computer based version of QReview. They
are also included in various textbooks.
Sampling plans will be constructed using both the binomial nomograph and the Thorndike
chart. The AQL, RQL, , and must be specified. They may be determined by your
customer, special studies, or past experience. The most common values to use for and 
are .05 and .10 respectively. The following values are used for the binomial nomograph and
Thorndike chart examples:
AQL = 2% (.02), =.05, RQL = 8% (.08), = .10
7.1 Sampling Plan Construction Using the Binomial Nomograph
The AIQ is on the left scale and the probability of acceptance is on the right scale. The
semi vertical lines on the nomograph represent the sample sizes and the semi horizontal
lines are the acceptance numbers. Draw a line from the AQL (.02) to its probability of
acceptance (.95) and a line from the RQL (.08) to its probability of acceptance (.10). The
intersection will yield the sample size and acceptance number. Do not interpolate: Find
the closest sample size and acceptance number to the intersection point. Both the
sample size and acceptance numbers must be integers.
.02
.08
.10
.95
n =100
c = 4
Probability of c or fewer occurrences in n trials (P)
Probability of occurrence in a single trial (p)
Number of trials or sample size (n)
Number of occurrences (c)
(Probability of acceptance)
(Average Incoming Quality - AIQ)
QReview 96
The sample size is 100 and the acceptance number is 4.
Chapter 6 Sampling Plans 97
7.2 Sampling Plan Construction Using the Thorndike Chart
The Thorndike chart can also be used to find the sample size and acceptance number
for the plan. To assist in the task, a tool called an L will be used. The L may be modified
for any value of . The inside horizontal scale on the L must be lined up with and is
to coincide with 1 - on the vertical axis. The left or vertical side of the L coincides with
the probability of acceptance on the Thorndike chart.
R is the ratio of the RQL to the AQL (R = RQL/AQL). R is computed and marked with
an arrow as shown on the diagram.
The L is moved across the chart until one of the acceptance number curves goes
through both the value on the inside vertical scale and the R value on the inside
horizontal scale. The best fitting acceptance number curve is the one to select. The
curve represents the acceptance number (c) for the plan. The curve c = 4 goes through
both the points = .05 and R = 4 (R = .08/.02 = 4).
R = 4
1.0
0.95
0.1
c = 4
2 8
Value of pn
0.1
Probability of c or fewer defects
0.99999
0.00001
30
.05
.10
The sample size is determined as follows: At the corner of the L where the value is 1,
drop a straight line to the pn scale on the abscissa of the Thorndike chart. The value of
pn is 2. The value of p at this point is the AQL (AQL = .02). If pn =2 and p = .02 then n =
2/.02 = 100. Also a straight line can be dropped from R to the pn scale. The value of pn
is 8. The value of p at this point is the RQL (RQL = .08). If pn = 8 and
p = .08 then n = 8/.08 = 100. The same answer is obtained at either point.
The sample size is 100 and the acceptance number is 4.
Both the binomial nomograph and the Thorndike chart give the same sample size and
acceptance number. In some cases, there may be minor variations between the two
methods.
QReview 98
8.0 GLOSSARY OF TERMS
•Acceptance Number: The maximum allowable defects or defectives in a sample for
the lot to be accepted. (Acceptance number = AN or c)
•AIQ - Average Incoming Quality: This is the average quality level going into the
inspection point. The inspection data and subsequent report reflects this number. The
AIQ is the abscissa on the OC and AOQ curves.
•AOQ - Average Outgoing Quality: The average quality level leaving the inspection
point after rejection and acceptance of a number of lots. If rejected lots are not checked
100% and defective units removed or replaced with good units, the AOQ will be the same
as the AIQ.
•AOQ Curve - Average Outgoing Quality Curve: The curve or graph that shows the
Average Outgoing Quality for various values of incoming quality.
•AOQL - Average Outgoing Quality Limit: The maximum value of the AOQ. If the
sampling procedures are followed, the outgoing quality will, on average, not be worse
than the AOQL.
•AQL - Acceptable Quality Level: The quality level for which there is a high probability
of accepting the lot. The AQL is also defined as the maximum fraction defective or
defects per unit that can be considered satisfactory as a process average.
•Attribute data: Although measurements may be taken, they are not recorded on the
data sheet. The product is classified as good or defective. (Discrete data)
•Consumers risk (b): The probability of accepting a lot with a high number of defective
units. is usually set at .05 to .15 with .10 as the common value. can also be defined
as the probability of accepting a lot of RQL or LTPD quality.
•Lot: A collection of individual pieces from a common source, possessing a common set
of quality characteristics and submitted as a group for acceptance at one time.
(Lot size = N)
•LTPD - Lot Tolerance Percent Defective: This is the value of incoming quality where it
is desirable to reject most lots. The quality level is unacceptable. This is the RQL
expressed as a percent defective.
•OC Curve - Operating Characteristic Curve: The curve or graph shows the
probability of a lot being accepted for various values of incoming quality.
•Power of Test (1 - b): This is the probability of rejecting a lot when the incoming
quality is at the RQL level.
•Process average: The normal or stable quality level of a product or process for a
specified period of time. The quality level may be expressed as a fraction defective,
percent defective, or defects per unit.
Chapter 6 Sampling Plans 99
•Producers risk (a): The probability of rejecting a good lot. This is usually set at .01 to
.10 for most sampling plans. The symbol can also be defined as the probability of
rejecting a lot of AQL quality.
•Random sample: A sample selected in such a manner that any piece of product in the
lot has an equal chance of being chosen.
•RQL - Rejectable Quality Level: The generic term for the incoming quality level for
which there is a low probability of accepting the lot. The quality level is substandard.
•Sample: A subset of a lot selected to be inspected. (Sample size = n)
•Sampling plan: The procedure that specifies the number of units to be selected from a
lot or batch for appraisal. The sample size and acceptance number describes each
unique sampling plan.
•Variables data: Actual measurements are taken and recorded. (Continuous data)
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