ASTM E2334-03
(Practice)Standard Practice for Setting an Upper Confidence Bound For a Fraction or Number of Non-Conforming items, or a Rate of Occurrence for Non-conformities, Using Attribute Data, When There is a Zero Response in the Sample
Standard Practice for Setting an Upper Confidence Bound For a Fraction or Number of Non-Conforming items, or a Rate of Occurrence for Non-conformities, Using Attribute Data, When There is a Zero Response in the Sample
SCOPE
1.1 This practice presents methodology for the setting of an upper confidence bound regarding a unknown fraction or quantity non-conforming, or a rate of occurrence for nonconformities, in cases where the method of attributes is used and there is a zero response in a sample. Three cases are considered.
1.1.1 The sample is selected from a process or a very large population of discrete items, and the number of non-conforming items in the sample is zero.
1.1.2 A sample of items is selected at random from a finite lot of discrete items, and the number of non-conforming items in the sample is zero.
1.1.3 The sample is a portion of a continuum (time, space, volume, area etc.) and the number of non-conformities in the sample is zero.
1.2 Allowance is made for misclassification error in this standard, but only when misclassification rates are well understood or known and can be approximated numerically.
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An American National Standard
Designation: E 2334 – 03
Standard Practice for
Setting an Upper Confidence Bound For a Fraction or
Number of Non-Conforming items, or a Rate of Occurrence
for Non-conformities, Using Attribute Data, When There is a
Zero Response in the Sample
This standard is issued under the fixed designation E2334; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
1. Scope Probability and General Statistical Terms
ISO 3534-2 Statistics—Vocabulary and Symbols, Part 2:
1.1 This practice presents methodology for the setting of an
Statistical Quality Control
upper confidence bound regarding a unknown fraction or
quantity non-conforming, or a rate of occurrence for noncon-
NOTE 1—Samples discussed in this standard should meet the require-
formities, in cases where the method of attributes is used and ments (or approximately so) of a probability sample as defined in
Terminologies E1402 or E456.
there is a zero response in a sample. Three cases are consid-
ered.
3. Terminology
1.1.1 The sample is selected from a process or a very large
3.1 Definitions:
population of discrete items, and the number of non-
3.1.1 attributes, method of, n—measurement of quality by
conforming items in the sample is zero.
the method of attributes consists of noting the presence (or
1.1.2 Asample of items is selected at random from a finite
absence)ofsomecharacteristicorattributeineachoftheunits
lot of discrete items, and the number of non-conforming items
in the group under consideration, and counting how many of
in the sample is zero.
the units do (or do not) possess the quality attribute, or how
1.1.3 The sample is a portion of a continuum (time, space,
many such events occur in the unit, group or area. E 456
volume, area etc.) and the number of non-conformities in the
3.1.2 confidence bound, n—see confidence limit.
sample is zero.
3.1.3 confidence coeffıcient, n—the value, C, of the prob-
1.2 Allowance is made for misclassification error in this
ability associated with a confidence interval or statistical
standard, but only when misclassification rates are well under-
coverage interval. It is often expressed as a percentage.
stood or known and can be approximated numerically.
ISO 3534-1
2. Referenced Documents
3.1.4 confidence level, n—see confidence coeffıcient.
3.1.5 confidence limit, n—each of the limits, T and T,of
2.1 ASTM Standards:
1 2
the two sided confidence interval, or the limit T of the one
E141 Practice for Acceptance of Evidence Based on the
sided confidence interval. ISO 3534-1
Results of Probability Sampling
3.1.6 one sided confidence interval, n—when Tisafunction
E456 Terminology Relating to Quality and Statistics
of the observed values such that, u being a population
E1402 Terminology Relating to Sampling
parameter to be estimated, the probability P (T $ u)orthe
E1994 Practice for Use of Process Oriented AOQL and
probability P (T# u) is at least equal to C where C is a fixed
LTPD Sampling Plans
positive number less than 1. The interval from the smallest
2.2 ISO Standards:
value of u up to T or the interval from T to the largest possible
ISO 3534-1 Statistics—Vocabulary and Symbols, Part 1:
value of u is a one sided, C, confidence interval for u.
ISO 3534-1
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
3.1.7 confidence limit, n—each of the limits, T and T,of
1 2
StatisticsandisthedirectresponsibilityofSubcommitteeE11.30onDataAnalysis.
the two sided confidence interval, or the limit T of the one
Current edition approved Oct. 1, 2003. Published February 2004.
2 sided confidence interval. ISO 3534-1
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contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Standardsvolume information, refer to teh standard’s Document Summary page on
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NOTICE: This standard has either been superseded and replaced by a new version or withdrawn. Please
contact ASTM International (www.astm.org) for the latest information.
E2334–03
3.1.8 non-conformity, n—the non-fulfillment of a specified
l = a specific value of l for which a researcher will
requirement. ISO 3534-2
calculate a confidence coefficient for the statement, l
# l , when there is a zero response in the sample
3.1.8.1 Discussion—The term “defect” is also used in this
l = the upper confidence bound for the parameter l
context. u
u = the probability of classifying a conforming item as
3.1.9 non-conforming item, n—an item containing at least
non-conforming;oroffindinganonconformitywhere
one non-conformity. ISO 3534-2
none exists
3.1.9.1 Discussion—The term “defective item” is also used
u = the probability of classifying a non-conforming item
in this context.
as conforming; or of failing to find a non-conformity
3.1.10 population, n—the totality of items or units of where one should have been found
material under consideration. E 456
4. Significance and Use
3.1.11 sample, n—a group of items, observations or test
4.1 In Case 1, the sample is selected from a process or a
results, or portion of material taken from a large collection of
very large population of interest. The population is essentially
items or quantity of material, which serves to provide infor-
unlimited, and each item either has or has not the defined
mation that may be used as a basis for making a decision
attribute.The population (process) has an unknown fraction of
concerning the larger collection or quantity. E 456
items p (long run average process non-conforming) having the
3.1.12 probability sample, n—a sample of which the sam-
attribute. The sample is a group of n discrete items selected at
pling units have been selected by a chance process. At each
random from the process or population under consideration,
step of selection, a specified probability of selection can be
and the attribute is not exhibited in the sample. The objective
attached to each sampling unit available for selection.
istodetermineanupperconfidencebound,p ,fortheunknown
u
E 1402
fraction p whereby one can claim that p # p with some
u
3.1.13 item, n—anobjectorquantityofmaterialonwhicha
confidence coefficient (probability) C. The binomial distribu-
set of observations can be made. E 456
tion is the sampling distribution in this case.
4.2 In Case 2, a sample of n items is selected at random
3.1.13.1 Discussion—As used in this standard, “set” de-
from a finite lot of N items. Like Case 1, each item either has
notes a single variable (the defined attribute). The term
or has not the defined attribute, and the population has an
“sampling unit” is also used to denote an “item” (see Practice
unknownnumber, D,ofitemshavingtheattribute.Thesample
E141).
does not exhibit the attribute. The objective is to determine an
3.2 Definitions of Terms Specific to This Standard:
upper confidence bound, D , for the unknown number D,
u
3.2.1 zero response, n—in the method of attributes, the
whereby one can claim that D # D with some confidence
u
phrase used to denote that zero non-conforming items or zero
coefficient (probability) C. The hypergeometric distribution is
non-conformities were found (observed) in the item(s), unit,
the sampling distribution in this case.
group or area sampled.
4.3 In Case 3, there is a process, but the output is a
continuum, such as area (for example, a roll of paper or other
Symbols:
material, a field of crop), volume (for example, a volume of
A = the assurance index
liquidorgas),ortime(forexample,hours,days,quarterly,etc.)
C = confidence coefficient as a percent or as a probability
The sample size is defined as that portion of the “continuum”
value
sampled, and the defined attribute may occur any number of
C = the confidence coefficient calculated that a parameter
d
times over the sampled portion. There is an unknown average
meets a certain requirement, that is, that p# p , that
rateofoccurrence, l,forthedefinedattributeoverthesampled
D# D or that l# l , when there is a zero response
0 0
interval of the continuum that is of interest. The sample does
in the sample
not exhibit the attribute. For a roll of paper this might be
D = the number of non-conforming items in a finite
blemishes per 100 ft ; for a volume of liquid, microbes per
population containing N items
D = a specified value of D for which a researcher will cubic litre; for a field of crop, spores per acre; for a time
interval, calls per hour, customers per day or accidents per
calculateaconfidencecoefficientforthestatement, D
quarter.Therate, l,isproportionaltothesizeoftheintervalof
# D , when there is a zero response in the sample
D = the upper confidence bound for the parameter D interest. Thus, if l = 12 blemishes per 100 ft of paper, this is
u
N = the number of items in a finite population equivalent to 1.2 blemishes per 10 ft or 30 blemishes per 250
n = the sample size, that is, the number of items in a
ft .Itisimportanttokeepinmindthesizeoftheintervalinthe
sample
analysis and interpretation. The objective is to determine an
n = the sample size required
R upper confidence bound, l , for the unknown occurrence rate
u
p = a process fraction non-conforming
l, whereby one can claim that l# l with some confidence
u
p = a specified value of p for which a researcher will
coefficient (probability) C. The Poisson distribution is the
calculate a confidence coefficient, for the statement p
sampling distribution in this case.
# p , when there is a zero response in the sample
4.4 AvariationonCase3isthesituationwherethesampled
p = the upper confidence bound for the parameter p
u
“interval” is really a group of discrete items, and the defined
l = themeannumberofnon-conformities(orevents)over
attribute may occur any number of times within an item. This
some area of interest for a Poisson process
might be the case where the continuum is a process producing
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn. Please
contact ASTM International (www.astm.org) for the latest information.
E2334–03
discrete items such as metal parts, and the attribute is defined process remains in statistical control. The item having the
as a scratch. Any number of scratches could occur on any attribute is often referred to as a defective item or a non-
single item. In such a case the occurrence rate, l, might be conforming item or unit. The sample consists of n randomly
defined as scratches per 1000 parts or some similar metric. selected items from the population of interest. The n items are
4.5 In each case a sample of items or a portion of a inspectedforthedefinedattribute.Thesamplingdistributionis
continuum is examined for the presence of a defined attribute, the binomial with parameters p equal to the process (popula-
and the attribute is not observed (that is, a zero response).The tion) fraction non-conforming and n the sample size. When
objective is to determine an upper confidence bound for either zero non-conforming items are observed in the sample (the
an unknown proportion, p (Case 1), an unknown quantity, D event“all_zeros”),andtherearenomisclassificationerrors,the
(Case2),oranunknownrateofoccurrence, l(Case3).Inthis upper confidence bound, p , at confidence level C (0 < C <1),
u
standard, confidence means the probability that the unknown for the population proportion non-conforming is:
parameter is not more than the upper bound. More generally,
n
p 51 2 1 2 C (1)
=
u
these methods determine a relationship among sample size,
5.3.1.1 Forthecasewithmisclassificationerrors,whenzero
confidence and the upper confidence bound. They can be used
non-conforming items are observed in the sample (all_zeros),
todeterminethesamplesizerequiredtodemonstrateaspecific
the upper confidence bound, p , at confidence level C is:
p, D or l with some degree of confidence. They can also be u
n
used to determine the degree of confidence achieved in
12u 2 1 2 C
=
p 5 (2)
demonstrating a specified p, D or l. u
~12u 2u !
1 2
4.6 In this standard allowance is made for misclassification
5.3.1.2 Eq 2 reduces to Eq 1 when u = u = 0. To find the
1 2
errorbutonlywhenmisclassificationratesarewellunderstood
minimumsamplesizerequired(n )tostateaconfidencebound
R
or known, and can be approximated numerically.
of p at confidence C if zero non-conforming items are to be
u
4.7 It is possible to impose the language of classical
observed in the sample, solve Eq 2 for n. This is:
acceptancesamplingtheoryonthismethod.TermssuchasLot
ln~1 2 C!
Tolerance Percent Defective, Acceptable Quality Level, Con-
n 5 (3)
R
ln~~1 2 p ! ~12u ! 1 p u !
u 1 u 2
sumer Quality Level are not used in this standard. For more
information on these terms, see Practice E1994.
5.3.1.3 To find the confidence demonstrated (C ) in the
d
claim that an unknown fraction non-conforming p is no more
5. Procedure
thanaspecifiedvalue,say p ,whenzeronon-conformancesare
5.1 When a sample is inspected and a zero response is
observed in a sample of n items solve Eq 2 for C. This is:
exhibited with respect to a defined attribute, we refer to this
C 51 2 ~~1 2 p ! ~12u ! 1 p u ! (4)
d 0 1 0 2
event as “all_zeros.” Formulas for calculating the probability
5.3.2 Case 2—The item is a completely discrete object and
of “all_zeros” in a sample are based on the binomial, the
the attribute is either present or not within the item. Only one
hypergeometric and the Poisson probability distributions.
response is recorded per item (either go or no-go).The sample
When there is the possibility of misclassification error, adjust-
items originate from a finite lot or population of N items. The
ments to these distributions are used. This practice will clarify
sample consists of n randomly selected items from among the
wheneachdistributionisappropriateandhowmisclassification
N,withoutreplacement.Thepopulationproportiondefectiveis
error is incorporated. Three basic cases are considered as
p = D/N where the unknown D is the integer number of
described in Section 4. Formulas and examples for each case
non-conforming (defective) items among the N. The sampling
are given below. Mathematical notes are given in Appendix
distribution is the hypergeometric with parameters N, D and n.
X1.
When zero non-conforming items are observed in the sample
5.2 In some applications, the measurement method is
(all_zeros), and there are no misclassification erro
...
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