ASTM E2334-09(2013)e1
(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.
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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.
ε NOTE—Section 3 was editorially corrected in August 2013.
1. Scope E2586Practice for Calculating and Using Basic Statistics
2.2 ISO Standards:
1.1 This practice presents methodology for the setting of an
ISO 3534-1Statistics—Vocabulary and Symbols, Part 1:
upper confidence bound regarding a unknown fraction or
Probability and General Statistical Terms
quantity non-conforming, or a rate of occurrence for
ISO 3534-2Statistics—Vocabulary and Symbols, Part 2:
nonconformities, in cases where the method of attributes is
Statistical Quality Control
used and there is a zero response in a sample. Three cases are
considered.
NOTE 1—Samples discussed in this standard should meet the require-
1.1.1 The sample is selected from a process or a very large
ments (or approximately so) of a probability sample as defined in
population of discrete items, and the number of non- Terminologies E1402 or E456.
conforming items in the sample is zero.
3. Terminology
1.1.2 Asample of items is selected at random from a finite
lot of discrete items, and the number of non-conforming items
3.1 Definitions:
in the sample is zero.
3.1.1 Terminology E456 provides a more extensive list of
1.1.3 The sample is a portion of a continuum (time, space,
terms in E11 standards.
volume, area etc.) and the number of non-conformities in the
3.1.2 attributes, method of, n—measurement of quality by
sample is zero.
the method of attributes consists of noting the presence (or
absence) of some characteristic or attribute in each of the units
1.2 Allowance is made for misclassification error in this
in the group under consideration, and counting how many of
standard, but only when misclassification rates are well under-
the units do (or do not) possess the quality attribute, or how
stood or known and can be approximated numerically.
many such events occur in the unit, group or area.
2. Referenced Documents
3.1.3 confidence bound, n—see confidence limit. E2586
2.1 ASTM Standards:
3.1.4 confidence coeffıcient, n—the value, C, of the prob-
E141Practice for Acceptance of Evidence Based on the
ability associated with a confidence interval or statistical
Results of Probability Sampling
coverage interval. It is often expressed as a percentage.
E456Terminology Relating to Quality and Statistics
ISO 3534-1, E2586
E1402Guide for Sampling Design
3.1.5 confidence interval, n—an interval estimate of a
E1994Practice for Use of Process Oriented AOQL and
population parameter, calculated such that there is a given
LTPD Sampling Plans
long-run probability that the parameter is included in the
interval. E2586
3.1.5.1 Discussion—A one-sided confidence interval is one
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
Statistics and is the direct responsibility of Subcommittee E11.30 on Statistical
for which one of the limits is plus infinity, minus infinity, or a
Quality Control.
natural fixed limit (such as zero).
Current edition approved April 1, 2013. Published April 2013. Originally
approved in 2003. Last previous edition approved in 2009 as E2334–09. DOI: 3.1.6 confidence level, n—see confidence coeffıcient. E2586
10.1520/E2334-09R13E01.
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
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E2334−09 (2013)
3.1.7 confidence limit, n—the upper or lower limit of a 3.3.13 λ—the mean number of non-conformities (or events)
confidence interval. E2586 over some area of interest for a Poisson process.
3.3.14 λ —a specific value of λ for which a researcher will
3.1.8 item, n—an object or quantity of material on which a
set of observations can be made. calculate a confidence coefficient for the statement, λ≤λ ,
when there is a zero response in the sample.
3.1.8.1 Discussion—As used in this standard, “set” denotes
a single variable (the defined attribute). The term “sampling
3.3.15 λ —the upper confidence bound for the parameter λ.
u
unit” is also used to denote an “item” (see Practice E141).
3.3.16 θ —the probability of classifying a conforming item
3.1.9 non-conforming item, n—an item containing at least
as non-conforming; or of finding a nonconformity where none
one non-conformity. ISO 3534-2
exists.
3.1.9.1 Discussion—The term “defective item” is also used
3.3.17 θ —the probability of classifying a non-conforming
in this context.
item as conforming; or of failing to find a non-conformity
3.1.10 non-conformity, n—the non-fulfillment of a specified
where one should have been found.
requirement. ISO 3534-2
4. Significance and Use
3.1.10.1 Discussion—The term “defect” is also used in this
context.
4.1 In Case 1, the sample is selected from a process or a
very large population of interest. The population is essentially
3.1.11 population, n—the totality of items or units of
unlimited, and each item either has or has not the defined
material under consideration. E2586
attribute.The population (process) has an unknown fraction of
3.1.12 sample, n—a group of observations or test results
items p (long run average process non-conforming) having the
taken from a larger collection of observations or test results,
attribute. The sample is a group of n discrete items selected at
whichservestoprovideinformationthatmaybeusedasabasis
random from the process or population under consideration,
for making a decision concerning the larger collection. E2586
and the attribute is not exhibited in the sample. The objective
3.2 Definitions of Terms Specific to This Standard:
istodetermineanupperconfidencebound,p ,fortheunknown
u
3.2.1 probability sample, n—a sample of which the sam-
fraction p whereby one can claim that p ≤ p with some
u
pling units have been selected by a chance process. At each
confidence coefficient (probability) C. The binomial distribu-
step of selection, a specified probability of selection can be
tion is the sampling distribution in this case.
attached to each sampling unit available for selection. E1402
4.2 In Case 2, a sample of n items is selected at random
3.2.2 zero response, n—in the method of attributes, the
from a finite lot of N items. Like Case 1, each item either has
phrase used to denote that zero non-conforming items or zero
or has not the defined attribute, and the population has an
non-conformities were found (observed) in the item(s), unit,
unknownnumber, D,ofitemshavingtheattribute.Thesample
group, or area sampled.
does not exhibit the attribute. The objective is to determine an
upper confidence bound, D , for the unknown number D,
3.3 Symbols:
u
whereby one can claim that D ≤ D with some confidence
3.3.1 A—the assurance index, as a percent or a probability
u
coefficient (probability) C. The hypergeometric distribution is
value.
the sampling distribution in this case.
3.3.2 C—confidence coefficient as a percent or as a prob-
4.3 In Case 3, there is a process, but the output is a
ability value.
continuum, such as area (for example, a roll of paper or other
3.3.3 C —the confidence coefficient calculated that a pa-
d
material, a field of crop), volume (for example, a volume of
rameter meets a certain requirement, that is, that p ≤ p , that D
liquidorgas),ortime(forexample,hours,days,quarterly,etc.)
≤ D orthatλ≤λ ,whenthereisazeroresponseinthesample.
0 0
The sample size is defined as that portion of the “continuum”
3.3.4 D—the number of non-conforming items in a finite
sampled, and the defined attribute may occur any number of
population containing N items.
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
blemishes per 100 ft ; for a volume of liquid, microbes per
3.3.6 D —the upper confidence bound for the parameter D.
u
cubic litre; for a field of crop, spores per acre; for a time
3.3.7 N—the number of items in a finite population.
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
equivalent to 1.2 blemishes per 10 ft or 30 blemishes per 250
3.3.9 n —the sample size required.
R
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
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E2334−09 (2013)
4.4 AvariationonCase3isthesituationwherethesampled 5.3 Formulas for upper confidence bounds in three cases:
“interval” is really a group of discrete items, and the defined 5.3.1 Case 1—The item is a completely discrete object and
attribute may occur any number of times within an item. This the attribute is either present or not within the item. Only one
might be the case where the continuum is a process producing response is recorded per item (either go or no-go).The sample
discrete items such as metal parts, and the attribute is defined items originate from a process and hence the future population
as a scratch. Any number of scratches could occur on any of interest is potentially unlimited in extent so long as the
single item. In such a case the occurrence rate, λ, might be process remains in statistical control. The item having the
defined as scratches per 1000 parts or some similar metric. attribute is often referred to as a defective item or a non-
conforming item or unit. The sample consists of n randomly
4.5 In each case a sample of items or a portion of a
selected items from the population of interest. The n items are
continuum is examined for the presence of a defined attribute,
inspectedforthedefinedattribute.Thesamplingdistributionis
and the attribute is not observed (that is, a zero response). The
the binomial with parameters p equal to the process (popula-
objective is to determine an upper confidence bound for either
tion) fraction non-conforming and n the sample size. When
an unknown proportion, p (Case 1), an unknown quantity, D
zero non-conforming items are observed in the sample (the
(Case 2), or an unknown rate of occurrence, λ (Case 3). In this
event“all_zeros”),andtherearenomisclassificationerrors,the
standard, confidence means the probability that the unknown
upper confidence bound, p , at confidence level C (0 < C <1),
u
parameter is not more than the upper bound. More generally,
for the population proportion non-conforming is:
these methods determine a relationship among sample size,
n
confidence and the upper confidence bound. They can be used
p 51 2 =1 2 C (1)
u
todeterminethesamplesizerequiredtodemonstrateaspecific
5.3.1.1 Table 1contains the calculated upper confidence
p, D or λ with some degree of confidence. They can also be
bound for the process fraction non-conforming when x=0
used to determine the degree of confidence achieved in
non-conformingitemsappearinasampleofsize n.Confidence
demonstrating a specified p, D or λ.
is100C%.Forexample,ifn=250objectsaresampledandthere
4.6 In this standard allowance is made for misclassification
are x=0 non-conforming objects in the sample, then the upper
errorbutonlywhenmisclassificationratesarewellunderstood
95%confidenceboundfortheprocessfractionnon-conforming
or known, and can be approximated numerically.
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
Tolerance Percent Defective, Acceptable Quality Level, Con-
sumer Quality Level are not used in this standard. For more
TABLE 1 Upper 100C% Confidence Bound, p , for the Process
u
information on these terms, see Practice E1994.
Fraction Non-Conforming, p, When Zero non-conforming Units
appear in a sample of Size, n
5. Procedure
n C=0.90 C=0.95 C=0.99
5 0.369043 0.450720 0.601893
5.1 When a sample is inspected and a zero response is
10 0.205672 0.258866 0.369043
exhibited with respect to a defined attribute, we refer to this
15 0.142304 0.181036 0.264358
event as “all_zeros.” Formulas for calculating the probability 20 0.108749 0.139108 0.205672
30 0.073881 0.095034 0.142304
of “all_zeros” in a sample are based on the binomial, the
40 0.055939 0.072158 0.108749
hypergeometric and the Poisson probability distributions.
50 0.045007 0.058155 0.087989
60 0.037649 0.048703 0.073881
When there is the possibility of misclassification error, adjust-
70 0.032359 0.041893 0.063671
ments to these distributions are used. This practice will clarify
80 0.028372 0.036754 0.055939
wheneachdistributionisappropriateandhowmisclassification
90 0.025260 0.032738 0.049881
error is incorporated. Three basic cases are considered as 100 0.022763 0.029513 0.045007
150 0.015233 0.019773 0.030235
described in Section 4. Formulas and examples for each case
175 0.013071 0.016973 0.025972
are given below. Mathematical notes are given in Appendix
200 0.011447 0.014867 0.022763
X1. 225 0.010182 0.013226 0.020259
250 0.09168 0.011911 0.018252
5.2 In some applications, the measurement method is
275 0.008338 0.010834 0.016607
300 0.007646 0.009936 0.015233
known to be fallible to some extent resulting in a significant
350 0.006557 0.008523 0
...
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