ASTM E141-91(2003)e1
(Practice)Standard Practice for Acceptance of Evidence Based on the Results of Probability Sampling
Standard Practice for Acceptance of Evidence Based on the Results of Probability Sampling
SIGNIFICANCE AND USE
This practice is designed to permit users of sample survey data to judge the trustworthiness of results from such surveys. Section 5 gives extended definitions of the concepts basic to survey sampling and the user should verify that such concepts were indeed used and understood by those who conducted the survey. What was the frame? How large (exactly) was the quantity N? How was the parameter θ estimated and its standard error calculated? If replicate subsamples were not used, why not?
Adequate answers should be given for all questions. There are many acceptable answers to the last question. If the sample design was relatively simple, such as simple random or stratified, then good estimates of sampling variance are easily available. If a more complex design was used then methods such as discussed in [1] may be acceptable. Replicate subsamples is the most straightforward way to estimate sampling variances when the survey design is complex.
Once the survey procedures that were used satisfy Section 5, consult Section 4 to see if any increase in sample size is needed. The calculations for making it are objectively described in Section 4.
Refer to Section 6 to guide in the interpretation of the uncertainty in the reported value of the parameter estimate, θ, i.e. the value of its standard error, se(θ). The quantity se(θ) should be reviewed to verify that the risks it entails are commensurate with the size of the sample.
SCOPE
1.1 This practice presents rules for accepting or rejecting evidence based on a sample. Statistical evidence for this practice is in the form of an estimate of a proportion, an average, a total, or other numerical characteristic of a finite population or lot. It is an estimate of the result which would have been obtained by investigating the entire lot or population under the same rules and with the same care as was used for the sample.
1.2 One purpose of this practice is to describe straightforward sample selection and data calculation procedures so that courts, commissions, etc. will be able to verify whether such procedures have been applied. The methods may not give least uncertainty at least cost, they should however furnish a reasonable estimate with calculable uncertainty.
1.3 This practice is primarily intended for one-of-a-kind studies. Repetitive surveys allow estimates of sampling uncertainties to be pooled; the emphasis of this practice is on estimation of sampling uncertainty from the sample itself. The parameter of interest for this practice is effectively a constant. Thus, the principal inference is a simple point estimate to be used as if it were the unknown constant, rather than, for example, a forecast or prediction interval or distribution devised to match a random quantity of interest.
This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.
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
An American National Standard
´1
Designation:E141–91(Reapproved 2003)
Standard Practice for
Acceptance of Evidence Based on the Results of Probability
Sampling
This standard is issued under the fixed designation E141; 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.
´ NOTE—Editorial changes were made throughout in November 2003.
1. Scope 2. Referenced Documents
1.1 This practice presents rules for accepting or rejecting 2.1 ASTM Standards:
,
evidence based on a sample. Statistical evidence for this E105 Practice for Probability Sampling of Materials
practice is in the form of an estimate of a proportion, an E122 Practice for Calculating Sample Size to Estimate,
average, a total, or other numerical characteristic of a finite With Specified Precision, the Average for a Characteristic
population or lot. It is an estimate of the result which would of a Lot or Process
havebeenobtainedbyinvestigatingtheentirelotorpopulation E178 Practice for Dealing With Outlying Observations
underthesamerulesandwiththesamecareaswasusedforthe E456 Terminology Relating to Quality and Statistics
sample.
NOTE 1—Practice E105 provides a statement of principles for guidance
1.2 One purpose of this practice is to describe straightfor-
ofASTMtechnicalcommitteesandothersinthepreparationofasampling
ward sample selection and data calculation procedures so that
planforaspecificmaterial.PracticeE122aidsindecidingontherequired
courts, commissions, etc. will be able to verify whether such
sample size. Practice E178 helps insure better behaved estimates. Termi-
nology E456 provides definitions of statistical terms used in this standard.
procedures have been applied.The methods may not give least
uncertainty at least cost, they should however furnish a
3. Terminology
reasonable estimate with calculable uncertainty.
3.1 Definitions:
1.3 This practice is primarily intended for one-of-a-kind
3.1.1 Equal Complete Coverage Result, n—the numerical
studies. Repetitive surveys allow estimates of sampling uncer-
characteristic(u)ofinterestcalculatedfromobservationsmade
tainties to be pooled; the emphasis of this practice is on
by drawing randomly from the frame, all of the sampling units
estimation of sampling uncertainty from the sample itself. The
covered by the frame.
parameter of interest for this practice is effectively a constant.
3.1.1.1 Discussion—Locatingtheunitsandevaluatingthem
Thus, the principal inference is a simple point estimate to be
are supposed to be done in exactly the same way and at the
used as if it were the unknown constant, rather than, for
same time as was done for the sample. The quantity itself is
example, a forecast or prediction interval or distribution
denoted u.Theequalcompletecoverageresultisneveractually
devised to match a random quantity of interest.
calculated. Its purpose is to serve as the objectively defined
1.4 This standard does not purport to address all of the
concrete goal of the investigation. The quantity u may be the
safety concerns, if any, associated with its use. It is the
¯
population mean,(Y), total (Y), median (M), the proportion
responsibility of the user of this standard to establish appro-
(P), or any other such quantity.
priate safety and health practices and determine the applica-
3.1.2 frame, n—a list, compiled for sampling purposes,
bility of regulatory limitations prior to use.
which designates all of the sampling units (items or groups) of
a population or universe to be considered in a specific study.
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling /
Statistics. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved Oct. 1, 2003. Published November 2003. Originally contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
approved in 1959. Last previous edition approved in 1991 as E141–91. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/E0141-91R03E01. the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
´1
E141–91 (2003)
3.1.2.1 Discussion—The list may cover a specific shipment size is needed. The calculations for making it are objectively
or lot, all households in a county, a state, or country; for described in Section 4.
example,anypopulationofinterest.Everysamplingunitinthe
4.4 Refer to Section 6 to guide in the interpretation of the
frame (1) has a unique serial number, which may be preas- uncertainty in the reported value of the parameter estimate, u,
signed or determined by some definite rule, (2) has an
i.e. the value of its standard error, se(u). The quantity se(u)
address—a complete and clear instruction (or rules for its should be reviewed to verify that the risks it entails are
formulation) as to where and when to make the observation or
commensurate with the size of the sample.
evaluation, (3) is based on physically concrete clerical mate-
rials such as directories, dials of clocks or of meters, ledgers,
5. Descriptive Terms and Procedures
maps, aerial photographs, etc., referred to in the addresses.
5.1 Probability Sampling Plans—include instructions for
3.1.3 sample, n—agroupofitems,observations,testresults,
using either:
or portions of material, taken from a larger collection of such
5.1.1 carefully prepared tables of random number,
items; it provides information for decisions concerning the
5.1.2 computer algorithms, carefully programmed and run
larger collection.
on a large computer, to generate pseudo-random numbers or,
3.1.3.1 Discussion—Aparticular sample is identified by the
5.1.3 certifiably honest physical devices, such as coin flips,
setofserialnumbersfromtherandomizationdeviceandbythe
toselectthesampleunitssothatinferencesmaybedrawnfrom
addresses on the frame generated by those serial numbers.
the test results and decisions may be made with risks correctly
3.1.4 sampling unit, n—an item, test specimen or portion of
calculated by probability theory.
material that is to be subjected to evaluation as part of the
5.1.4 Such plans are defined and their relative advantages
sampling plan.
discussed in Refs. (1), (2), and (3).
3.1.4.1 Discussion—If it is not feasible to select test speci-
5.2 Replicate Subsamples—a number of disjoint samples,
mens or laboratory samples individually, the sampling unit
each one separately drawn from the frame in accord with the
may be a group of items, for example, a row, an entire case of
same probability sampling plan. When appropriate, separate
items,oraprescribedarea(asintheexaminationofafinishing
laboratoriesshouldeachworkonseparatereplicatesubsamples
process).
and teams of investigators should be assigned to separate
3.1.4.2 By a more expensive method of measurement (fu-
replicate subsamples.This approach insures that the calculated
ture time, more elaborate frame) it may be possible to define a
standard error will not be a systematic underestimate. Such
quantity, u8, as a target parameter or ideal goal of an investi-
subsamples were called interpenetrating in Ref. (4) where
gation. Criticism that holds u to be an inappropriate goal
many of their basic properties were described. See Ref. (2) for
should demonstrate that the numerical difference between u
further theory and applications.
and u8 is substantial. Measurements may be imprecise but so
5.2.1 Discussion—For some types of material a sample
long as measurement errors are not too biased, a large size of
selected with uniform spacing along the frame (systematic
the lot or population, N, insures that u and u8 are essentially
sample) has increased precision over a selection made with
equal.
randomlyvaryingspacings(simplerandomsample).Examples
4. Significance and Use include sampling mineral ore or grain from a conveyor belt or
sampling from a list of households along a street. If the
4.1 This practice is designed to permit users of sample
systematicsampleisobtainedbyasinglerandomstarttheplan
survey data to judge the trustworthiness of results from such
is then a probability sampling plan, but it does not permit
surveys. Section 5 gives extended definitions of the concepts
calculatingthestandarderrorasrequiredbythispractice.After
basic to survey sampling and the user should verify that such
dividing the sample size by an integer k (such as k =4or k
concepts were indeed used and understood by those who
=10) and using a random start for each of k replicate
conducted the survey. What was the frame? How large (ex-
subsamples, some of the increased precision of systematic
actly) was the quantity N? How was the parameter u estimated
sampling (and a standard error on k−1 degrees of freedom)
and its standard error calculated? If replicate subsamples were
can be achieved.
not used, why not?
5.2.2 Audit Subsample—a small subsample of the survey
4.2 Adequate answers should be given for all questions.
sample(asfewas10observationsmaybeadequate)forreview
There are many acceptable answers to the last question. If the
of all procedures from use of the random numbers through
sampledesignwasrelativelysimple,suchassimplerandomor
locating and measurement, to editing, coding, data entry and
stratified, then good estimates of sampling variance are easily
tabulation. Selection of the audit subsample may be done by
available. If a more complex design was used then methods
puttingthensampleobservationsinorderastheyarecollected,
such as discussed in Ref. (1) may be acceptable. Replicate
calculating the nearest integer to n , or some other conve-
=
subsamples is the most straightforward way to estimate sam-
nient integer, and taking this number to be the spacing for
pling variances when the survey design is complex.
systematicselectionoftheauditsubsample.Thereviewshould
4.3 Once the survey procedures that were used satisfy
uncover any gross departures from prescribed practices or any
Section 5, consult Section 4 to see if any increase in sample
conceptual misunderstandings in the definitions. If the audit
subsample is large enough (say 30 observations or more) the
regression of audited values on initial observations may be
The boldface numbers in parentheses refer to a list of references at the end of
this standard. used to calibrate the estimate. This technique is the method of
´1
E141–91 (2003)
two-phase sampling as discussed in Ref. (1). Helpful discus-
se~p!5 p~1 2 p!/~n 21! (4)
=
sion of an audit appears in Ref. (2).
5.2.3 Estimate—a quantity calculated on the n sample
(b) when u=the population total (u=Y) then
observations in the same way as the equal complete coverage
u5 Ny¯ and se ~u! 5 Nse ~y¯! (5)
result u would have been calculated from the entire set of N
possible observations of the population; the symbol u denotes
If a simple random sample of size n=200 has a=25 items
the estimate. (In calculating u, replicate subsample member-
with the attribute then the conclusion is u=0.125 and se
ship is ignored.)
(u)=0.023 on 199 degrees of freedom.
5.2.3.1 Discussion—An estimate has a sampling distribu-
5.2.4.4 Example 4—If uisaparameterotherthanameanor
tion induced from the randomness in sample selection. The
if the sample design is complex, then replicate subsamples
equal complete coverage result is effectively a constant while
should be used in the sample design. Denote the k separate
any estimate is only the value from one particular sample.
estimatesas u,i=1,2,., kanddenoteby utheestimatebased
i
Thus, there is a mean value of the sampling distribution and
onthewholesample.Theaverageofthe u willbecloseto,but
i
there is also a standard deviation of the sampling distribution.
in general not equal to u. The standard error of u is calculated
5.2.4 Standard Error—the quantity computed from the
as:
observations as an estimate of the sampling standard deviation
k
of the estimate; se (u) denotes the standard error. 2
se u 5 u 2u /k k 21 (6)
~ ! ~ ! ~ !
Π( i
i 51
5.2.4.1 Example1—When uisthepopulationaverageofthe
N quantities and a simple random sample of size n was drawn,
where u is the average of the u . The standard error is
then the sample average y becomes the usual estimate u, i
based on k−1 degrees of freedom.
where
The following estimates of percent “drug-in-suit” sales of
n
prescriptiondrugswerebasedon20replicatesubsamples;each
u5 y¯ 5 y /n. (1)
( i
i 51
followed a stratified cluster sampling design. The separate
The quantities y , y , ., y denote the observations. The estimateswere:6.8,7.1,8.4,9.5,8.6,4.1,3.7,3.2,3.8,5.8,8.8,
1 2 n
standard error is calculated as: 5.0, 7.9, 8.8, 8.4, 8.1, 6.0, 6.3, 4.5, 5.8. The value of u was
6.74% and se(u)=0.43% on 19 degrees of freedom. Notice
n
se u! 5 se y¯! 5 y 2 y¯! /n~n 21!. (2) that u =6.58 does not equal u=6.74. This is because u is a
~ ~ ~
Π( i
i 51
ratio of two overall averages while u is the average of 20
1 3
There are n −1 degrees of freedom in this standard error.
ratios. For an example with k=2, average ⁄3 and ⁄5 and
When the observations are: compare to (1+3)/(3+5).
81.6, 78.7, 79.7, 78.3, 80.9, 79.5, 79.8, 80.3, 79.5, 80.7
5.2.5 Procedures—must be described in written form and
should cover the following matters; (1) parties interested in
then y=79.90and se(y¯)=0.32.Asthisexampleillustrates,
collecting data should agree on the importance of knowing u
formula (2) is correct when k replaces n and subsample
and its definition including measurement methods, (2) the
estimates are used in place of observations.
frame shall be carefully and explicitly constructed; N shall be
5.2.4.2 Example 2 on the Finite Population Correction
well established, (3) random numbers (or a certifiably honest
(fpc)—Multiplying se (y¯) by 1 2 n/N is always correct
=
physical random device) shall dictate selection of the sample.
when the goal of the survey is to estimate the finite population
There will be no substitution of one sampling unit for another.
mean (u= Y). Using the previous data and if N=50, then
The method of sample selection shall permit calculation of a
se(y¯) becomes se(y¯)=0.28 after applying the fpc. If random
standard error of the estimate (4) the use of replicate sub-
measurement error exists in the observations, then u8 based on
samples is recommended (see 5.2); an audit subsample should
a reference measurement method may be a more appropriate
be selected and processed and any departures from prescribed
survey goal than u (see section 4.1.4.1). If so, then se(y¯) would
...
This document is not anASTM standard and is intended only to provide the user of anASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
An American National Standard
e1
Designation:E141–91(Reapproved 1997) Designation:E141–91(Reapproved 2003)
Standard Practice for
Acceptance of Evidence Based on the Results of Probability
Sampling
This standard is issued under the fixed designation E141; 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.
e NOTE—Editorial changes were made throughout in November 2003.
1. Scope
1.1 This practice presents rules for accepting or rejecting evidence based on a sample. Statistical evidence for this practice is
in the form of an estimate of a proportion, an average, a total, or other numerical characteristic of a finite population or lot. It is
an estimate of the result which would have been obtained by investigating the entire lot or population under the same rules and
with the same care as was used for the sample.
1.2 One purpose of this practice is to describe straightforward sample selection and data calculation procedures so that courts,
commissions, etc. will be able to verify whether such procedures have been applied. The methods may not give least uncertainty
at least cost, they should however furnish a reasonable estimate with calculable uncertainty.
1.3 This practice is primarily intended for one-of-a-kind studies. Repetitive surveys allow estimates of sampling uncertainties
tobepooled;theemphasisofthispracticeisonestimationofsamplinguncertaintyfromthesampleitself.Theparameterofinterest
for this practice is effectively a constant. Thus, the principal inference is a simple point estimate to be used as if it were the
unknown constant, rather than, for example, a forecast or prediction interval or distribution devised to match a random quantity
of interest.
1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory
limitations prior to use.
2. Referenced Documents
2.1 ASTM Standards:
E105 Practice for Probability Sampling of Materials
E122 Practice for Choice of Sample Size to Estimate a Measure of Quality for a Lot or Process
E178 Practice for Dealing with Outlying Observations
E456 Terminology for Statistical Methods
NOTE 1—Practice E105 provides a statement of principles for guidance of ASTM technical committees and others in the preparation of a sampling
planforaspecificmaterial.PracticeE122aidsindecidingontherequiredsamplesize.PracticeE178helpsinsurebetterbehavedestimates.Terminology
E456 provides definitions of statistical terms used in this standard.
3. Terminology
3.1 Definitions:
3.1.1 Equal Complete Coverage Result, n— the numerical characteristic (u) of interest calculated from observations made by
drawing randomly from the frame, all of the sampling units covered by the frame.
3.1.1.1 Discussion—Locating the units and evaluating them are supposed to be done in exactly the same way and at the same
time as was done for the sample.The quantity itself is denoted u.The equal complete coverage result is never actually calculated.
Its purpose is to serve as the objectively defined concrete goal of the investigation. The quantity u may be the population mean,
¯
(Y), total (Y), median (M), the proportion (P), or any other such quantity.
3.1.2 frame, n—a list, compiled for sampling purposes, which designates all of the sampling units (items or groups) of a
population or universe to be considered in a specific study.
3.1.2.1 Discussion—The list may cover a specific shipment or lot, all households in a county, a state, or country; for example,
This practice is under the jurisdiction of ASTM Committee E-11E11 on Quality and Statistical Methods and is the direct responsibility of Subcommittee E11.10 on
Sampling and Data Analysis. Sampling.
Current edition approved August 15, 1991. Published November 1991. Originally published as E141–59 T. Last previous edition E141–69 (1975).
Annual Book of ASTM Standards, Vol 14.02.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
e1
E141–91 (2003)
any population of interest. Every sampling unit in the frame (1) has a unique serial number, which may be preassigned or
determined by some definite rule, (2) has an address—a complete and clear instruction (or rules for its formulation) as to where
and when to make the observation or evaluation, (3) is based on physically concrete clerical materials such as directories, dials
of clocks or of meters, ledgers, maps, aerial photographs, etc., referred to in the addresses.
3.1.3 sample,n—agroupofitems,observations,testresults,orportionsofmaterial,takenfromalargercollectionofsuchitems;
it provides information for decisions concerning the larger collection.
3.1.3.1 Discussion—A particular sample is identified by the set of serial numbers from the randomization device and by the
addresses on the frame generated by those serial numbers.
3.1.4 sampling unit, n—anitem,testspecimenorportionofmaterialthatistobesubjectedtoevaluationaspartofthesampling
plan.
3.1.4.1 Discussion—If it is not feasible to select test specimens or laboratory samples individually, the sampling unit may be
a group of items, for example, a row, an entire case of items, or a prescribed area (as in the examination of a finishing process).
3.1.4.2 Byamoreexpensivemethodofmeasurement(futuretime,moreelaborateframe)itmaybepossibletodefineaquantity,
u8, as a target parameter or ideal goal of an investigation. Criticism that holds u to be an inappropriate goal should demonstrate
that the numerical difference between u and u8 is substantial. Measurements may be imprecise but so long as measurement errors
are not too biased, a large size of the lot or population, N, insures that u and u8 are essentially equal.
4. Significance and Use
4.1 This practice is designed to permit users of sample survey data to judge the trustworthiness of results from such surveys.
Section 5 gives extended definitions of the concepts basic to survey sampling and the user should verify that such concepts were
indeed used and understood by those who conducted the survey. What was the frame? How large (exactly) was the quantity N?
How was the parameter u estimated and its standard error calculated? If replicate subsamples were not used, why not?
4.2 Adequate answers should be given for all questions. There are many acceptable answers to the last question. If the sample
design was relatively simple, such as simple random or stratified, then good estimates of sampling variance are easily available.
If a more complex design was used then methods such as discussed in [1] may be acceptable. Replicate subsamples is the most
straightforward way to estimate sampling variances when the survey design is complex.
4.3 Once the survey procedures that were used satisfy Section 5, consult Section 4 to see if any increase in sample size is
needed. The calculations for making it are objectively described in Section 4.
4.4 Refer to Section 6 to guide in the interpretation of the uncertainty in the reported value of the parameter estimate, u, i.e.
thevalueofitsstandarderror,se(u).Thequantityse(u)shouldbereviewedtoverifythattherisksitentailsarecommensuratewith
the size of the sample.
5. Descriptive Terms and Procedures
5.1 Probability Sampling Plans—include instructions for using either:
5.1.1 carefully prepared tables of random number,
5.1.2 computer algorithms, carefully programmed and run on a large computer, to generate pseudo-random numbers or,
5.1.3 certifiably honest physical devices, such as coin flips, to select the sample units so that inferences may be drawn from the
test results and decisions may be made with risks correctly calculated by probability theory.
5.1.4 Such plans are defined and their relative advantages discussed in [1], [2] and [6].
5.2 Replicate Subsamples—a number of disjoint samples, each one separately drawn from the frame in accord with the same
probability sampling plan. When appropriate, separate laboratories should each work on separate replicate subsamples and teams
of investigators should be assigned to separate replicate subsamples. This approach insures that the calculated standard error will
not be a systematic underestimate. Such subsamples were called interpenetrating in [5] where many of their basic properties were
described. See [2] for further theory and applications.
5.2.1 Discussion—For some types of material a sample selected with uniform spacing along the frame (systematic sample) has
increased precision over a selection made with randomly varying spacings (simple random sample). Examples include sampling
mineraloreorgrainfromaconveyorbeltorsamplingfromalistofhouseholdsalongastreet.Ifthesystematicsampleisobtained
byasinglerandomstarttheplanisthenaprobabilitysamplingplan,butitdoesnotpermitcalculatingthestandarderrorasrequired
by this practice. After dividing the sample size by an integer k (such as k =4or k =10) and using a random start for each of k
replicate subsamples, some of the increased precision of systematic sampling (and a standard error on k−1 degrees of freedom)
can be achieved.
5.2.2 Audit Subsample—a small subsample of the survey sample (as few as 10 observations may be adequate) for review of all
procedures from use of the random numbers through locating and measurement, to editing, coding, data entry and tabulation.
Selection of the audit subsample may be done by putting the n sample observations in order as they are collected, calculating the
nearest integer to n , or some other convenient integer, and taking this number to be the spacing for systematic selection of the
=
audit subsample. The review should uncover any gross departures from prescribed practices or any conceptual misunderstandings
in the definitions. If the audit subsample is large enough (say 30 observations or more) the regression of audited values on initial
observationsmaybeusedtocalibratetheestimate.Thistechniqueisthemethodoftwo-phasesamplingasdiscussedin[1].Helpful
discussion of an audit appears in [2].
e1
E141–91 (2003)
5.2.3 Estimate—a quantity calculated on the n sample observations in the same way as the equal complete coverage result u
would have been calculated from the entire set of N possible observations of the population; the symbol u denotes the estimate.
(In calculating u, replicate subsample membership is ignored.)
5.2.3.1 Discussion—An estimate has a sampling distribution induced from the randomness in sample selection. The equal
complete coverage result is effectively a constant while any estimate is only the value from one particular sample. Thus, there is
a mean value of the sampling distribution and there is also a standard deviation of the sampling distribution.
5.2.4 Standard Error—the quantity computed from the observations as an estimate of the sampling standard deviation of the
estimate; se (u) denotes the standard error.
5.2.4.1 Example 1—When u is the population average of the N quantities and a simple random sample of size n was drawn,
then the sample average y becomes the usual estimate u, where
n
u5 y¯ 5 y /n. (1)
(
i
i51
n
u5 y¯ 5 y /n. (1)
( i
i 51
The quantities y , y , ., y denote the observations. The standard error is calculated as:
1 2 n
n
se ~u!5se ~y¯! 5 ~y 2 y¯! /n~n 21!. (2)
Œ(
i
i51
n
se ~ u! 5 se ~y¯! 5 ~y 2 y¯! /n~n 21!. (2)
Π(
i
i 51
There are n −1 degrees of freedom in this standard error. When the observations are:
81.6, 78.7, 79.7, 78.3, 80.9, 79.5, 79.8, 80.3, 79.5, 80.7
then y=79.90 and se(y¯)=0.32.As this example illustrates, formula (2) is correct when k replaces n and subsample estimates
are used in place of observations.
5.2.4.2 Example 2 on the Finite Population Correction (fpc)—Multiplyingse(y¯)by 1 2 n/Nisalwayscorrectwhenthegoal
=
of the survey is to estimate the finite population mean (u= Y). Using the previous data and if N=50, then se(y¯) becomes
se(y¯)=0.28 after applying the fpc. If random measurement error exists in the observations, then u8 based on a reference
measurementmethodmaybeamoreappropriatesurveygoalthan u(seesection4.1.4.1).Ifso,then se(y¯)wouldbefurtheradjusted
upward by an amount somewhat less than the downward adjustment of the fpc. Both of these adjustments are often numerically
so small that these adjustments may be omitted—leaving se(y¯) of (2) as a slight overestimate.
5.2.4.3 Example 3—If the quantity of interest is (a) a proportion or (b) a total and the sample is simple random then the above
formulas are still applicable. A proportion is the mean of zeroes and ones, while the total is a constant times the mean. Thus:
(a) when u is taken to be the population proportion (u= P) then;
u5 p 5 (y/n 5 a/n (3)
i
where:
a is the number of units in the sample with the attribute, and
se~p!5 =p~1 2 p!/~n 21! (4)
(b) when u=the population total (u=Y) then
u5 Ny¯ and se ~u! 5 Nse ~y¯! (5)
Ifasimplerandomsampleofsize n=200has a=25itemswiththeattributethentheconclusionis u=0.125and se(u)=0.023
on 199 degrees of freedom.
5.2.4.4 Example 4.If u is a parameter other than a mean or if the sample design is complex, then replicate subsamples should
be used in the sample design. Denote the k separate estimates as u,i=1, 2, ., k and denote by u the estimate based on the whole
i
sample. The average of the uwill be close to, but in general not equal to u. The standard error of u is calculated as:
i
k
se ~u! 5 ~u 2u! /k~k 21! (6)
Π(
i
i 51
k
se ~u! 5 ~u 2u! /k~k 21! (6)
Π(
i
i 51
where u is the average of the u. The standard error is based on k−1 degrees of freedom.
i
The following estimates of percent “drug-in-suit” sales of prescription drugs were based on 20 replicate subsamples; each
followed a stratified cluster sampling design. The separate estimates were: 6.8, 7.1, 8.4, 9.5, 8.6, 4.1, 3.7, 3.2, 3.8, 5.8, 8.8, 5.0,
e1
E141–91 (2003)
7.9,8.8,8.4,8.1,6.0,6.3,4.5,5.8.Thevalueof uwas6.74%andse(u)=0.43%on19degreesoffreedom.Noticethat u =6.58
does not equal u=6.74. This is because u is a ratio of two overall averages while u is the average of 20 ratios. For an example
1 3
with k=2, average ⁄3 and ⁄5 and compare to
...
Questions, Comments and Discussion
Ask us and Technical Secretary will try to provide an answer. You can facilitate discussion about the standard in here.