ASTM D4854-95

Standard Guide for Estimating the Magnitude of Variability from Expected Sources in Sampling Plans

SCOPE
1.1 This guide serves as an aid to subcommittees in writing specifications and sampling procedures.
1.2 The guide explains how to estimate the contributions of the variability of lot sampling units, laboratory sampling units, and specimens to the variation of the test result of a sampling plan.
1.3 The guide explains how to combine the estimates of the variability from the three sources to obtain an estimate of the variability of the sampling plan results.
1.4 The guide is applicable to all sampling plans that produce variables data (Note 1). It is not applicable to plans that produce attribute data, since such plans do not take specimens in stages, but require that specimens be taken at random from all of the individual items in the lot. Note 0This guide is applicable to all sampling plans that produce variables data regardless of the kind of frequency distribution of these data, because no estimates are made of any probabilities.
1.5 This guide includes the following topics:Topic TitleSectionNumberScope1Referenced Documents2Terminology3Significance and Use4Sampling Plans Producing Variables Data5Reducing Variability of Sampling Results6Keywords 7Analysis of Data Using ANOVAAnnex A1A Numerical ExampleAnnex A2

General Information

• Status: Historical
• Publication Date: 31-Dec-2000
• Technical Committee: D13 - Textiles

Relations

• Replaced By: ASTM D4854-95(2001) - Standard Guide for Estimating the Magnitude of Variability from Expected Sources in Sampling Plans (Withdrawn 2009)
• Effective Date: 01-Jan-2001

Standards Content (Sample)

NOTICE: This standard has either been superseded and replaced by a new version or discontinued.
Contact ASTM International (www.astm.org) for the latest information.
Designation: D 4854 – 95
Standard Guide for
Estimating the Magnitude of Variability from Expected
Sources in Sampling Plans
This standard is issued under the fixed designation D 4854; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
1. Scope E 456 Terminology Relating to Quality and Statistics
2.2 ASTM Adjuncts:
1.1 This guide serves as an aid to subcommittees in writing
TEX-PAC
specifications and sampling procedures.
1.2 The guide explains how to estimate the contributions of
NOTE 2—Tex-Pac is a group of PC programs on floppy disks, available
the variability of lot sampling units, laboratory sampling units,
through ASTM Headquarters, 100 Barr Harbor Drive, Conshohocken, PA
19428, USA. The calculations described in the annexes of this guide,
and specimens to the variation of the test result of a sampling
including the cost comparisons of various sampling plans, can be
plan.
conducted using one of these programs.
1.3 The guide explains how to combine the estimates of the
variability from the three sources to obtain an estimate of the
3. Terminology
variability of the sampling plan results.
3.1 Definitions:
1.4 The guide is applicable to all sampling plans that
3.1.1 analysis of variance (ANOVA), n—a procedure for
produce variables data (Note 1). It is not applicable to plans
dividing the total variation of a set of data into two or more
that produce attribute data, since such plans do not take
parts, one of which estimates the error due to selecting and
specimens in stages, but require that specimens be taken at
testing specimens and the other part(s) possible sources of
random from all of the individual items in the lot.
additional variation.
NOTE 1—This guide is applicable to all sampling plans that produce
3.1.2 attribute data, n—observed values or determinations
variables data regardless of the kind of frequency distribution of these
which indicate the presence or absence of specific character-
data, because no estimates are made of any probabilities.
istics.
1.5 This guide includes the following topics:
3.1.3 component of variance, n—a part of a total variance
Section identified with a specific source of variability.
Topic Title
Number
3.1.4 degrees of freedom, n—for a set, the number of values
Scope 1
that can be assigned arbitrarily and still get the same value for
Referenced Documents 2
Terminology 3 each of one or more statistics calculated from the set of data.
Significance and Use 4
3.1.4.1 Discussion— For example, if only an average is
Sampling Plans Producing Variables Data 5
specified for a set of five observations, there are four degrees of
Reducing Variability of Sampling Results 6
Keywords 7
freedom since the same average can be obtained with any
Analysis of Data Using ANOVA Annex A1
values substituted for four of the observations as long as the
A Numerical Example Annex A2
fifth value is set to give the correct total. If both the average
2. Referenced Documents and standard deviation have been specified, there are only three
degrees of freedom left.
2.1 ASTM Standards:
3.1.5 determination value, n—the numerical quantity calcu-
D 123 Terminology Relating to Textiles
lated by means of the test method equation from the measure-
D 2904 Practice for Interlaboratory Testing of a Textile Test
ment values obtained as directed in a test method. (Syn.
Method that Produces Normally Distributed Data
determination) (See also observation.)
D 4271 Practice for Writing Statements on Sampling in Test
3.1.6 laboratory sample, n—a portion of material taken to
Methods for Textiles
represent the lot sample, or the original material, and used in
D 4467 Practice for Interlaboratory Testing of a Textile Test
the laboratory as a source of test specimens.
Method that Produces Non-Normally Distributed Data
3.1.7 lot sample, n—one or more shipping units taken to
represent an acceptance sampling lot and used as a source of
This guide is under the jurisdiction of ASTM Committee D-13 on Textiles and
laboratory samples.
is the direct responsibility of Subcommittee D13.93 on Statistics.
Current edition approved May 15, 1995. Published July 1995. Originally
published as D 4854 – 88. Last previous edition D 4854 – 91. Annual Book of ASTM Standards, Vol 14.02.
2 5
Annual Book of ASTM Standards, Vol 07.01. PC programs on floppy disks are available through ASTM. For a 3 ⁄2 inch disk
Annual Book of ASTM Standards, Vol 07.02. request PCN:12-429040-18, for a 5 ⁄4 inch disk request PCN:12-429041-18.
Copyright © ASTM, 100 Barr Harbor Drive, West Conshohocken, PA 19428-2959, United States.
D 4854
3.1.8 mean square—in analysis of variance, a contraction of D 123 or Terminology E 456, or appropriate textbooks on
the expression “mean of the squared deviations from the statistics.
appropriate average(s)” where the divisor of each sum of
4. Significance and Use
squares is the appropriate degrees of freedom.
4.1 This guide is useful in estimating the variation due to lot
3.1.9 observation, n—(1) the process of determining the
sampling units, laboratory sampling units, and specimen selec-
presence or absence of attributes or making measurements of a
tion and testing during the sampling and testing of a lot of
variable, (2) a result of the process of determining the presence
material.
or absence of an attribute or making a measurement of a
4.2 Estimates of variation from the several sources will
variable. (Compare measurement value, determination value,
make it possible to write sampling plans which balance the cost
and test result.)
of sampling and testing with the desired precision of the plan.
3.1.10 precision, n—the degree of agreement within a set of
4.3 This guide is useful in: (1) designing process controls
observations or test results obtained as directed in a method.
and (2) developing sampling plans as parts of product specifi-
3.1.10.1 Discussion—The term “precision,” delimited in
cations.
various ways, is used to describe different aspects of precision.
4.4 This guide can be used for designing new sampling
This usage was chosen in preference to the use of “repeatabil-
plans or for improving old plans.
ity” and “reproducibility” which have been assigned conflict-
4.5 This guide is concerned with the process of sampling.
ing meanings by various authors and standardizing bodies.
This is unlike Practice D 2904 or Practice D 4467 which are
3.1.11 random sampling, n—the process of selecting units
concerned with the process of testing.
for a sample of size n in such a manner that all combinations
4.6 Studies based on this guide are applicable only to the
of n units under consideration have an equal or ascertainable
material(s) on which the studies are made. If the conclusions
chance of being selected as the sample. (Syn. simple random
are to be used for a specification, then separate studies should
sampling and sampling at random.)
be made on three or more kinds of materials of the type on
3.1.12 sample, n—(1) a portion of a lot of material which is
which the test method may be used and which produce test
taken for testing or record purposes; (2) a group of specimens
results covering the range of interest.
used, or observations made, which provide information that
can be used for making statistical inferences about the popu- 5. Sampling Plans Producing Variables Data
lation(s) from which they were drawn. (See also lot sample,
5.1 For the results of using this guide to be completely valid,
laboratory sample, and specimen.)
it is necessary that all of the sampling units at every stage be
3.1.13 sampling plan, n—a procedure for obtaining a
taken randomly. It is not always practical to achieve complete
sample.
randomness, but every reasonable effort should be made to do
3.1.14 sampling plan result, n—the number obtained for use so.
in judging the acceptability of a lot when applying a sampling 5.2 In sampling plans which produce variables data, there
plan. are three stages in which variation can occur. For a schematic
representation of these three stages see Fig. 1 (see also Practice
3.1.15 sampling unit, n—an identifiable, discrete unit or
D 4271):
subunit of material that could be taken as part of a sample.
5.2.1 Lot Sample—Variation among the averages of the
3.1.16 specimen, n—a specific portion of a material or
sampling units within a lot sample is due to differences
laboratory sample upon which a test is performed or which is
between such items as cases, cartons, and bolts, variation
taken for that purpose. (Syn. test specimen.)
among laboratory samples plus test method error and differ-
3.1.17 sum of squares—in analysis of variance, a contrac-
ences among specimens. To estimate variation due to lot
tion of the expression “sum of the squared deviations from the
sampling units alone, proceed as directed in 5.3 and 5.4.
appropriate average(s)” where the average(s) of interest may be
5.2.2 Laboratory Sample—Within the lot sampling units,
the average(s) of a specific subset(s) of data or of the entire set
variation among the averages of the laboratory sampling units
of data.
is due to differences among such items as cones within cases,
3.1.18 test result, n—a value obtained by applying a test
garments within cartons, and swatches within bolts, plus test
method, expressed either as a single determination or a
method error and differences among specimens. To estimate
specified combination of a number of determinations.
variation due to laboratory sampling units alone, proceed as
3.1.19 variables data, n—measurements which vary and
directed in 5.3 and 5.4.
may take any of a specified set of numerical values.
5.2.3 Specimens—Variation among determination values on
3.1.20 variance, s , n—of a population, a measure of the
specimens is due to the test method error and the differences
dispersion of members of the population expressed as a
among specimens within laboratory sampling units such as
function of the sum of the squared deviations from the
cones, garments, and swatches. Usually it is not feasible to
population mean.
separate these two errors. To estimate the variation among
3.1.21 variance, s , n—of a sample, a measure of the
specimens proceed as directed in 5.3 and 5.4.
dispersion of variates observed in a sample expressed as a
5.3 If a sampling plan has already been put into operation,
function of the squared deviations from the sample average.
or if a new plan is proposed, put it into operation, and collect
3.1.22 For definitions of textile terms, refer to Terminology the resulting data. In the case of either an old plan or a new
D 123. For definitions of statistical terms, refer to Terminology plan, obtain at least two sampling units at each of the stages of
D 4854
each of the stages, using the equations for mean squares
composition in Table A1.1 or Table A1.2. Details of how to
make these calculations are shown in Annex A1.
NOTE 3—There is disagreement among statisticians on if and when to
pool sums of squares and degrees of freedom. This guide recommends
pooling under certain circumstances. When and how to pool is discussed
in A1.2.1, A1.2.2, A1.2.3, and A1.3.1.
6. Reducing Variability of Sampling Results
6.1 Variability of Sampling Results—Calculate the esti-
mated variance of the sampling plan result (average of all
specimen determinations), v, for several sampling plans, using
Eq 1:
v 5 L/n 1 T/mn 1 E/mnk (1)
where:
v 5 estimated variance of sampling plan results,
L 5 mean squared deviation due to variation among lot
sampling units,
n 5 number of sampling units in the lot samples,
T 5 mean squared deviation due to laboratory samples,
m 5 number of laboratory sampling units from one lot
sampling unit,
E 5 mean squared deviation due to testing specimens, and
k 5 number of specimens per laboratory sampling unit.
FIG. 1 Sampling Plan—Three Stages
6.1.1 The values of L, T, and E are obtained by the use of
analysis of variance and estimation of the components of
sampling. Sample at least two lots and make an ANOVA for
variance as directed in 5.3 and 5.4, and explained in the
each lot as directed in Annex A1. Continue collecting data for
annexes.
successive lots and make a new ANOVA of the data for each
6.2 Sampling Plan Choice—Other things being equal, from
lot. Tabulate the resulting sums of squares, degrees of freedom,
those sampling plans examined as directed in 6.1, choose the
and mean squares in a format like that of Table A2.3. Calculate
plan which has an acceptable variability with an acceptable
the totals for the sums of squares and for the degrees of
cost. Once the sizes of L, T, and E have been determined, both
freedom to date and calculate the combined mean squares for
the anticipated variability and cost of obtaining a sampling
the lots sampled to date. Continue until the results become
result for any desired combination of m, n, and k may be
stable, that is, until the estimates of the mean squares change
calculated. See Annex A2 and Table A2.4.
very little with additional use of the sampling plan.
5.4 After the estimates of the mean squares have stabilized,
7. Keywords
do any desired pooling of sums of squares and degrees of
freedom (see Note 3). Calculate the components of variance for 7.1 sampling plans; statistics; variability
ANNEXES
(Mandatory Information)
A1. ANALYSIS OF DATA USING ANOVA
A1.1 Sampling Stages—Data taken as directed in 5.3 will sampling units serve as test specimens, the ANOVA takes the
be in three, two, or one stage as follows: form of lot sampling units with one stage of subsampling
(specimens within a unit of the lot sample). See A1.3.
A1.1.1 Three-Stage Sampling—For a sampling plan having
A1.1.3 One-Stage Sampling—For a sampling plan in which
distinct sampling units in the lot sample, laboratory samples,
the lot sampling units serve as specimens, there are no other
and specimens, the ANOVA takes the form of lot sampling
sources of variability than specimens to estimate. See A1.4.
units with two stages of subsampling (laboratory sampling
units within lot samples and specimens within laboratory
sampling units). See A1.2. A1.2 ANOVA for Three-Stage Sampling—(For a numerical
A1.1.2 Two-Stage Sampling—For a sampling plan having example, see Annex A2.) For a sampling plan having distinct
distinct sampling units in the lot sample, but the laboratory lot sampling units, laboratory samples, and specimens, make
D 4854
the following calculations: 5 0, and pool the mean squares of all of the sources of
variation to give
...