ASTM E2334-09(2013)
(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
SIGNIFICANCE AND USE
4.1 In Case 1, the sample is selected from a process or a very large population of interest. The population is essentially unlimited, and each item either has or has not the defined attribute. The population (process) has an unknown fraction of items p (long run average process non-conforming) having the attribute. The sample is a group of n discrete items selected at random from the process or population under consideration, and the attribute is not exhibited in the sample. The objective is to determine an upper confidence bound, pu, for the unknown fraction p whereby one can claim that p ≤ pu with some confidence coefficient (probability) C. The binomial distribution is the sampling distribution in this case.
4.2 In Case 2, a sample of n items is selected at random from a finite lot of N items. Like Case 1, each item either has or has not the defined attribute, and the population has an unknown number, D, of items having the attribute. The sample does not exhibit the attribute. The objective is to determine an upper confidence bound, Du, for the unknown number D, whereby one can claim that D ≤ Du with some confidence coefficient (probability) C. The hypergeometric distribution is the sampling distribution in this case.
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 material, a field of crop), volume (for example, a volume of liquid or gas), or time (for example, hours, days, quarterly, etc.) The sample size is defined as that portion of the “continuum” sampled, and the defined attribute may occur any number of times over the sampled portion. There is an unknown average rate of occurrence, λ, for the defined attribute over the sampled interval of the continuum that is of interest. The sample does not exhibit the attribute. For a roll of paper this might be blemishes per 100 ft2; for a volume of liquid, microbes per cubic litre; for a field of crop, spores per acre; for a time interval, cal...
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.
General Information
Relations
Buy Standard
Standards Content (Sample)
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation:E2334 −09(Reapproved 2013) An American National Standard
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 (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2.2 ISO Standards:
ISO 3534-1Statistics—Vocabulary and Symbols, Part 1:
1.1 This practice presents methodology for the setting of an
Probability and General Statistical Terms
upper confidence bound regarding a unknown fraction or
ISO 3534-2Statistics—Vocabulary and Symbols, Part 2:
quantity non-conforming, or a rate of occurrence for
Statistical Quality Control
nonconformities, in cases where the method of attributes is
used and there is a zero response in a sample. Three cases are
NOTE 1—Samples discussed in this standard should meet the require-
ments (or approximately so) of a probability sample as defined in
considered.
Terminologies E1402 or E456.
1.1.1 The sample is selected from a process or a very large
population of discrete items, and the number of non-
3. Terminology
conforming items in the sample is zero.
3.1 Definitions:
1.1.2 Asample of items is selected at random from a finite
3.1.1 Terminology E456 provides a more extensive list of
lot of discrete items, and the number of non-conforming items
terms in E11 standards.
in the sample is zero.
3.1.2 attributes, method of, n—measurement of quality by
1.1.3 The sample is a portion of a continuum (time, space,
the method of attributes consists of noting the presence (or
volume, area etc.) and the number of non-conformities in the
absence) of some characteristic or attribute in each of the units
sample is zero.
in the group under consideration, and counting how many of
1.2 Allowance is made for misclassification error in this
the units do (or do not) possess the quality attribute, or how
standard, but only when misclassification rates are well under-
many such events occur in the unit, group or area.
stood or known and can be approximated numerically.
3.1.3 confidence bound, n—see confidence limit.
3.1.4 confidence coeffıcient, n—the value, C, of the prob-
2. Referenced Documents
ability associated with a confidence interval or statistical
2.1 ASTM Standards:
coverage interval. It is often expressed as a percentage. ISO
E141Practice for Acceptance of Evidence Based on the
3534-1
Results of Probability Sampling
3.1.5 confidence interval, n—an interval estimate of a
E456Terminology Relating to Quality and Statistics
population parameter, calculated such that there is a given
E1402Guide for Sampling Design
long-run probability that the parameter is included in the
E1994Practice for Use of Process Oriented AOQL and
interval.
LTPD Sampling Plans
3.1.5.1 Discussion—A one-sided confidence interval is one
E2586Practice for Calculating and Using Basic Statistics
for which one of the limits is plus infinity, minus infinity, or a
natural fixed limit (such as zero).
3.1.6 confidence level, n—see confidence coeffıcient.
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
Statistics and is the direct responsibility of Subcommittee E11.30 on Statistical
3.1.7 confidence limit, n—the upper or lower limit of a
Quality Control.
confidence interval.
Current edition approved April 1, 2013. Published April 2013. Originally
approved in 2003. Last previous edition approved in 2009 as E2334–09. DOI:
3.1.8 item, n—an object or quantity of material on which a
10.1520/E2334-09R13.
set of observations can be made.
For referenced ASTM Standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Standardsvolume information, refer to thestandard’s Document Summary page on Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
the ASTM website. 4th Floor, New York, NY 10036, http://www.ansi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2334−09 (2013)
3.1.8.1 Discussion—As used in this standard, “set” denotes 3.3.14 λ —a specific value of λ for which a researcher will
a single variable (the defined attribute). The term “sampling calculate a confidence coefficient for the statement, λ≤λ ,
unit” is also used to denote an “item” (see Practice E141). when there is a zero response in the sample.
3.3.15 λ —the upper confidence bound for the parameter λ.
3.1.9 non-conforming item, n—an item containing at least
u
one non-conformity. ISO 3534-2
3.3.16 θ —the probability of classifying a conforming item
3.1.9.1 Discussion—The term “defective item” is also used as non-conforming; or of finding a nonconformity where none
in this context. exists.
3.3.17 θ —the probability of classifying a non-conforming
3.1.10 non-conformity, n—the non-fulfillment of a specified
item as conforming; or of failing to find a non-conformity
requirement. ISO 3534-2
where one should have been found.
3.1.10.1 Discussion—The term “defect” is also used in this
context.
4. Significance and Use
3.1.11 population, n—the totality of items or units of
4.1 In Case 1, the sample is selected from a process or a
material under consideration. E2586
very large population of interest. The population is essentially
3.1.12 sample, n—a group of observations or test results unlimited, and each item either has or has not the defined
taken from a larger collection of observations or test results , attribute.The population (process) has an unknown fraction of
whichservestoprovideinformationthatmaybeusedasabasis items p (long run average process non-conforming) having the
for making a decision concerning the larger collection. E2586 attribute. The sample is a group of n discrete items selected at
random from the process or population under consideration,
3.2 Definitions of Terms Specific to This Standard:
and the attribute is not exhibited in the sample. The objective
3.2.1 probability sample, n—a sample of which the sam-
istodetermineanupperconfidencebound,p ,fortheunknown
u
pling units have been selected by a chance process. At each
fraction p whereby one can claim that p ≤ p with some
u
step of selection, a specified probability of selection can be
confidence coefficient (probability) C. The binomial distribu-
attached to each sampling unit available for selection. E1402
tion is the sampling distribution in this case.
3.2.2 zero response, n—in the method of attributes, the
4.2 In Case 2, a sample of n items is selected at random
phrase used to denote that zero non-conforming items or zero
from a finite lot of N items. Like Case 1, each item either has
non-conformities were found (observed) in the item(s), unit,
or has not the defined attribute, and the population has an
group or area sampled.
unknownnumber, D,ofitemshavingtheattribute.Thesample
3.3 Symbols:
does not exhibit the attribute. The objective is to determine an
3.3.1 A—the assurance index, as a percent or a probability upper confidence bound, D , for the unknown number D,
u
value. whereby one can claim that D ≤ D with some confidence
u
coefficient (probability) C. The hypergeometric distribution is
3.3.2 C—confidence coefficient as a percent or as a prob-
the sampling distribution in this case.
ability value.
4.3 In Case 3, there is a process, but the output is a
3.3.3 C —the confidence coefficient calculated that a pa-
d
continuum, such as area (for example, a roll of paper or other
rameter meets a certain requirement, that is, that p ≤ p , that D
material, a field of crop), volume (for example, a volume of
≤ D orthatλ≤λ ,whenthereisazeroresponseinthesample.
0 0
liquidorgas),ortime(forexample,hours,days,quarterly,etc.)
3.3.4 D—the number of non-conforming items in a finite
The sample size is defined as that portion of the “continuum”
population containing N items.
sampled, and the defined attribute may occur any number of
times over the sampled portion. There is an unknown average
3.3.5 D —aspecifiedvalueof Dforwhicharesearcherwill
rateofoccurrence, λ,forthedefinedattributeoverthesampled
calculate a confidence coefficient for the statement, D ≤ D ,
interval of the continuum that is of interest. The sample does
when there is a zero response in the sample.
not exhibit the attribute. For a roll of paper this might be
3.3.6 D —the upper confidence bound for the parameter D. 2
u
blemishes per 100 ft ; for a volume of liquid, microbes per
3.3.7 N—the number of items in a finite population. cubic litre; for a field of crop, spores per acre; for a time
interval, calls per hour, customers per day or accidents per
3.3.8 n—the sample size, that is, the number of items in a
quarter.Therate, λ,isproportionaltothesizeoftheintervalof
sample.
interest. Thus, if λ = 12 blemishes per 100 ft of paper, this is
3.3.9 n —the sample size required.
R equivalent to 1.2 blemishes per 10 ft or 30 blemishes per 250
ft .Itisimportanttokeepinmindthesizeoftheintervalinthe
3.3.10 p—a process fraction non-conforming.
analysis and interpretation. The objective is to determine an
3.3.11 p —aspecifiedvalueof pforwhicharesearcherwill
upperconfidencebound, λ ,fortheunknownoccurrencerate λ,
u
calculate a confidence coefficient, for the statement p ≤ p ,
whereby one can claim that λ≤λ with some confidence
u
when there is a zero response in the sample.
coefficient (probability) C. The Poisson distribution is the
3.3.12 p —the upper confidence bound for the parameter p. sampling distribution in this case.
u
3.3.13 λ—the mean number of non-conformities (or events) 4.4 AvariationonCase3isthesituationwherethesampled
over some area of interest for a Poisson process. “interval” is really a group of discrete items, and the defined
E2334−09 (2013)
attribute may occur any number of times within an item. This 5.3.1 Case 1—The item is a completely discrete object and
might be the case where the continuum is a process producing the attribute is either present or not within the item. Only one
discrete items such as metal parts, and the attribute is defined response is recorded per item (either go or no-go).The sample
as a scratch. Any number of scratches could occur on any items originate from a process and hence the future population
single item. In such a case the occurrence rate, λ, might be of interest is potentially unlimited in extent so long as the
defined as scratches per 1000 parts or some similar metric. process remains in statistical control. The item having the
attribute is often referred to as a defective item or a non-
4.5 In each case a sample of items or a portion of a
conforming item or unit. The sample consists of n randomly
continuum is examined for the presence of a defined attribute,
selected items from the population of interest. The n items are
and the attribute is not observed (that is, a zero response). The
inspectedforthedefinedattribute.Thesamplingdistributionis
objective is to determine an upper confidence bound for either
the binomial with parameters p equal to the process (popula-
an unknown proportion, p (Case 1), an unknown quantity, D
tion) fraction non-conforming and n the sample size. When
(Case 2), or an unknown rate of occurrence, λ (Case 3). In this
zero non-conforming items are observed in the sample (the
standard, confidence means the probability that the unknown
event“all_zeros”),andtherearenomisclassificationerrors,the
parameter is not more than the upper bound. More generally,
upper confidence bound, p , at confidence level C (0 < C <1),
u
these methods determine a relationship among sample size,
for the population proportion non-conforming is:
confidence and the upper confidence bound. They can be used
n
todeterminethesamplesizerequiredtodemonstrateaspecific
p 51 2 =1 2 C (1)
u
p, D or λ with some degree of confidence. They can also be
5.3.1.1 Table 1contains the calculated upper confidence
used to determine the degree of confidence achieved in
bound for the process fraction non-conforming when x=0
demonstrating a specified p, D or λ.
non-conformingitemsappearinasampleofsize n.Confidence
4.6 In this standard allowance is made for misclassification
is100C%.Forexample,ifn=250objectsaresampledandthere
errorbutonlywhenmisclassificationratesarewellunderstood
are x=0 non-conforming objects in the sample, then the upper
or known, and can be approximated numerically.
95%confidenceboundfortheprocessfractionnon-conforming
is approximately 0.01191 or 1.191% non-conforming. Eq 1
4.7 It is possible to impose the language of classical
was applied.
acceptancesamplingtheoryonthismethod.TermssuchasLot
5.3.1.2 Forthecasewithmisclassificationerrors,whenzero
Tolerance Percent Defective, Acceptable Quality Level, Con-
non-conforming items are observed in the sample (all_zeros),
sumer Quality Level are not used in this standard. For more
the upper confidence bound, p , at confidence level C is:
u
information on these terms, see Practice E1994.
TABLE 1 Upper 100C% Confidence Bound, p , for the Process
u
5. Procedure
Fraction Non-Conforming, p, When Zero non-conforming Units
appear in a sample of Size, n
5.1 When a sample is inspected and a zero response is
n C=0.90 C=0.95 C=0.99
exhibited with respect to a defined attribute, we refer to this
5 0.369043 0.450720 0.601893
event as “all_zeros.” Formulas for calculating the probability
10 0.205672 0.258866 0.369043
of “all_zeros” in a sample are based on the binomial, the 15 0.142304 0.181036 0.264358
20 0.108749 0.139108 0.205672
hypergeometric and the Poisson probability distributions.
30 0.073881 0.095034 0.142304
When there is the possibility of misclassification error, adjust-
40 0.055939 0.072158 0.108749
ments to these distributions are used. This practice will clarify 50 0.045007 0.058155 0.087989
60 0.037649 0.048703 0.073881
wheneachdistributionisappropriateandhowmisclassification
70 0.032359 0.041893 0.063671
error is incorporated. Three basic cases are considered as
80 0.028372 0.036754 0.055939
90 0.025260 0.032738 0.049881
described in Section 4. Formulas and examples for each case
100 0.022763 0.029513 0.045007
are given below. Mathematical notes are given in Appendix
150 0.015233 0.019773 0.030235
X1.
175 0.013071 0.016973 0.025972
200 0.011447 0.014867 0.022763
5.2 In some applications, the measurement method is
225 0.010182 0.013226 0.020259
250 0.09168 0.011911 0.018252
known to be fallible to some extent resulting in a significant
275 0.008338 0.010834 0.016607
misclassification error. If experiments wi
...
Questions, Comments and Discussion
Ask us and Technical Secretary will try to provide an answer. You can facilitate discussion about the standard in here.