ASTM C970-87(1997)
(Practice)Standard Practice for Sampling Special Nuclear Materials in Multi-Container Lots
Standard Practice for Sampling Special Nuclear Materials in Multi-Container Lots
SCOPE
1.1 This practice provides an aid in designing a sampling and analysis plan for the purpose of minimizing random error in the measurement of the amount of nuclear material in a lot consisting of several containers. The problem addressed is the selection of the number of containers to be sampled, the number of samples to be taken from each sampled container, and the number of aliquot analyses to be performed on each sample.
1.2 This practice provides examples for application as well as the necessary development for understanding the statistics involved. The uniqueness of most situations does not allow presentation of step-by-step procedures for designing sampling plans. It is recommended that a statistician experienced in materials sampling be consulted when developing such plans.
1.3 The values stated in SI units are to be regarded as the standard.
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.
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Designation:C970–87 (Reapproved 1997)
Standard Practice for
Sampling Special Nuclear Materials in Multi-Container
Lots
This standard is issued under the fixed designation C970; 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 measurements, is partitioned into several component sums of
squares, each attributable to some meaningful cause (source of
1.1 This practice provides an aid in designing a sampling
variation).
and analysis plan for the purpose of minimizing random error
3.2 confidence interval—(a)an interval estimator used to
in the measurement of the amount of nuclear material in a lot
bound the value of a population parameter and to which a
consisting of several containers. The problem addressed is the
measure of confidence can be associated, and (b) the interval
selection of the number of containers to be sampled, the
estimate, based on a realization of a sample drawn from the
number of samples to be taken from each sampled container,
population of interest, that bounds the value of a population
and the number of aliquot analyses to be performed on each
parameter [with at least a stated confidence].
sample.
3.3 Estimation, Estimator, Estimate:
1.2 This practice provides examples for application as well
3.3.1 Estimation, in statistics, has a specific meaning, con-
as the necessary development for understanding the statistics
siderably different from the common interpretation of guess-
involved. The uniqueness of most situations does not allow
ing, playing a hunch, or grabbing out of the air. Instead,
presentationofstep-by-stepproceduresfordesigningsampling
estimation is the process of following certain statistical prin-
plans. It is recommended that a statistician experienced in
ciples to derive an approximation (estimate) to the unknown
materials sampling be consulted when developing such plans.
value of a population parameter. This estimate is based on the
1.3 The values stated in SI units are to be regarded as the
information available in a sample drawn from the population.
standard.
3.4 estimator—a function of a sample (X,X , . , X ) used
1 2 n
1.4 This standard does not purport to address all of the
to estimate a population parameter.
safety problems, if any, associated with its use. It is the
responsibility of the user of this standard to establish appro-
NOTE 1—An estimator is a random variable; therefore, not every
priate safety and health practices and determine the applica-
realization (x,x , . , x ) of the sample (X,X , . , X ) will lead to the
1 2 n 1 2 n
same value (realization) of the estimator. An estimator can be a function
bility of regulatory limitations prior to use.
that, when evaluated, results in a single value or results in an interval or
2. Referenced Documents region of values. In the former case the estimator is called a point
estimator, and in the latter case it is referred to as an interval estimator.
2.1 ASTM Standards:
3.5 estimate, (a: n)—aparticularvalueorvaluesrealizedby
E300 Practice for Sampling Industrial Chemicals
applying an estimator to a particular realization of a sample,
2.2 Other Standard:
that is, to a particular set of sample values (x,x , . , x ). (b:
NUREG/CR-0087, Considerations for Sampling Nuclear
1 2 n
v)—to use an estimator.
Materials for SNM Accounting Measurements
3.6 nesteddesign—oneofaparticularclassofexperimental
3. Terminology Definitions
designs, characterized by “nesting” of the sources of variation:
for each sampled value of a variable A, a given number of
3.1 analysis of variance—the body of statistical theory,
values of a second variable B is sampled; for each of these, a
methods, and practice in which the variation in a set of
given number of values of the next variable C is sampled, etc.
measurements, as measured by the sum of squares of the
The result is that each line of the “Expected Value of Mean
Square”columninananalysisofvariancetablecontainsallbut
This practice is under the jurisdiction of ASTM Committee C26 on Nuclear
one of the terms of the preceding line.
Fuel Cycle and is the direct responsibility of Subcommittee C26.08 on Quality
3.7 random variable— a variable that takes on any one of
Assurance and Reference Materials.
the values in its range according to a [fixed] probability
Current edition approved Jan. 30, 1987. Published March 1987. Originally
published as C970–82. Last previous edition C970–82.
distribution. (Synonyms: chance variable, stochastic variable,
Annual Book of ASTM Standards, Vol 15.05.
variate.)
AvailablefromNationalTechnicalInformationService,Springfield,VA22161.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
C970
n m r
3.8 standarddeviation(s.d.)—thepositivesquarerootofthe 1
¯
p¯ 5 X 5 X (2)
( ( (
ijk
nmr
variance.
i 51 j 51 k 51
3.9 variance—(a: population) the expected value of the
is an estimator of the true value p. The true variance of p¯ is
squareofthedifferencebetweenarandomvariableanditsown
then
expectedvalue;thatis,thesecondmomentaboutthemean.(b:
2 2 2
s ~N 2 n! s s
b s a
sample) The sum of squared deviations from the sample mean 2
s 5 1 1 (3)
p¯
n N 21 nm nmr
divided by one less than the number of values involved.
where:
4. Significance and Use
s = truevarianceamongtheNcontainersinthe
b
−1 2 −2 2
4.1 Plans for sampling and analysis of nuclear material are
givenlot,definedasN (p −N ((p ) ;
i i
s = true variance among samples taken from a
designed with two purposes in mind: the first is related to
s
material accountability and the second to material specifica- single container,
s = true variance of the laboratory analysis on
tions.
a
a homogeneous sample, and
4.2 Fortheaccountingofspecialnuclearmaterial,sampling
N 2 n
= finite population correction factor.
and analysis plans should be established to determine the
N 21
quantity of special nuclear material held in inventory, shipped
NOTE 2—If the ith container has g grams of material, then the true
between buyers and sellers, or discarded. Likewise, material
i
N N
average concentration is ( wp , where w = g /( g. However, the
1 i i i i 1 i
specification requires the determination of the quantity of
variance of the corresponding estimate can still be calculated as shown in
nuclear material present. Inevitably there is uncertainty asso-
thisguideline;thetruevariancewillbeonlyslightlylargerifthe g values
i
ciated with such measurements. This practice presents a tool
donotdiffertoomuch.Forexample,ifthes.d.ofthe g were20%ofthe
i
for developing sampling plans that control the random error
average g, it can be shown that the s.d. of p would be underestimated by
i
component of this uncertainty.
about 2% of the true standard deviation; for g’s having s.d.’s of 10% or
i
4.3 Precision and accuracy statements are highly desirable, 30% of their average, the underestimation is 0.5% or 4.5% respectively.
Note that a set of 25 weights g, uniformly spread from 3.3 to 6.7 kg, has
if not required, to qualify measurement methods. This practice i
a s.d. equal to 20% of the average (5 kg). (It is assumed that errors in the
relates to“ precision” that is generally a statement on the
estimation of net weights are insignificant compared to differences
random error component of uncertainty.
between containers, sampling variability, and analytical uncertainty, or
both.)
5. Designing the Sampling Plan—Measuring Random
2 2 2
5.3 Since the true variances s , s , and s are generally
b s a
Error
unknown,theymaybeestimatedusingappropriatedata.Those
5.1 The random error component of measurement uncer-
data can be historical data obtained from analyzing production
tainty is due to the various random errors involved in each
samples, as long as there have been no changes in the process
operation such as weighing, sampling, and analysis. The
with time. If such data are not available, as for example during
quantification of the random error is usually given in terms of
thestart-upofafacilityorafterachangeinprocessconditions,
thevarianceofthemeanofthemeasurements.Whenanalyzing
a designed experiment is required to obtain estimates of the
a lot of nuclear material to estimate the true concentration, p,
variances.
of a constituent such as uranium, the sample mean, p¯,isthe 2
5.4 An estimate s of the variance of the sample mean can
p¯
calculated estimator. The variance of p¯, s , is a measure of
p¯
be obtained from Eq 3, by inserting estimates of the variances
the random error associated with the measurement process.
appearing there. If a designed experiment is performed, the
This practice deals primarily with random error; measurement
estimates can be obtained from the mean squares.
process systematic error will be discussed briefly in 8.2.
It is shown in Appendix X1 that estimates of the variances
5.2 To estimate the true concentration, p, in a lot consisting
are as follows:
of N containers using a completely balanced nested design,
s 5 MS , (4)
randomly select n of the N containers; from each of the n a a
containers, randomly select m samples; perform r laboratory 1
s 5 MS 2 MS , (5)
~ !
s s a
analyses on each of the nm samples. (It is assumed that the r
amount of material withdrawn for samples is only a small
N 21
s 5 ~MS 2 MS !, (6)
b b s
fraction of the total quantity of material.) Let
Nmr
X 5 measuredconcentrationoftheconstituentinthe kthanalysis
where:
ijk
onthe jthsamplefromthe ithcontainer,or
MS,MS , and MS are the “mean squares” for analyses,
a b s
5 p 1 b 1 s 1 a . (1)
i ij ijk
where:
p = true concentration,
This topic can be found in many standard statistical texts, for example,
b = effect due to container i,
i Brownlee, K. A., Statistical Theory and Methodology in Science and Engineering,
th
s = effect due to the j sample from container i, and 2nded.,JohnWileyandSons,NewYork,1965;Bennett,C.A.,andFranklin,N.L.,
ij
th th
Statistical Analysis in Chemistry and the Chemical Industry, John Wiley and Sons,
a = effect due to the k analysis on the j sample from
ijk
New York, 1954; Mendenhall, William, Introduction to Linear Models and the
container i.
Design and Analysis of Experiments, Duxbury Press, Belmont, CA, 1968; and in
Then, if each container holds the same amount of material,
Jaech, J. L., “Statistical Methods in Nuclear Material Control,” (TID-26298,
(Note 2), the sample mean USAEC, 1973).
C970
containers and samples. The estimated variance of p¯ is ob- Z .Sincetheminimumcostisachievedwhentheconstraint
1−a/2
tained by replacing the true variances in Eq 3 by their is barely satisfied, we need to minimize cost subject to the
estimates: constraint
1 N 2 n 1 1
2 2 2 2 s 5 K (10)
p¯
s 5 s 1 s 1 s (7)
¯p b s a
n N 21 nm nmr
Finally, expressed in terms of the mean squares, this be-
where K is a constant, either specified directly or computed
comes
from D and a.
o
6.2.2 When the underlying variances are known from pre-
1 N 2 n 1
s 5 MS 1 MS (8)
p¯ b s.
vioushistory,theproblemofachievingaminimumcostwithin
nmr N Nmr
a stated confidence interval width reduces to finding a suitable
5.5 The variance of the sample mean, s , or its estimate, s
p¯
set of values for n, m, and r. In Appendix X2 it is shown that
p¯ , is used to calculate confidence limits for the quantity and
the optimum r and m are given by
concentration of nuclear materials. Therefore, it is desirable to
1 2
s c /
a s
reduce this variance and, in this way, reduce the random error.
r 5 (11)
S D
s c
s a
Obviously, this can be done by using large values of n, m, and
1 2
s c N 21 /
r (large number of samples and laboratory analyses). The cost
s b
m 5 (12)
S D
s c N
b s
and time required by that approach could be prohibitive.
Another approach is to improve the overall process such that
2 2 2
the basic variances s , s , s are reduced.
b s a
where:
5.6 Eq 8 gives an estimate of the variances for any given
c = marginal cost of choosing one additional container
p¯
b
n, m, and r and therefore can be used for comparing different
and preparing it for sampling,
sampling plans. An example of two sampling plans involving c = marginalcostofdrawinganadditionalsamplefroma
s
the same number of analyses but having different random
container and preparing it for analysis, and
errors is given in Appendix X3. c = marginal cost of an additional laboratory analysis.
a
Therefore,theoptimumvaluesfor rand mdonotdependon
5.7 When one has fixed resources within which the sam-
n, and in fact can be calculated immediately from the vari-
plingplanmustfunction,thequestionarisesashowtoallocate
ances, the “costs,” and N.
these resources to obtain the “best” sampling plan. Sections 6
6.2.3 Once mand raredeterminedandinsertedintoEq3, s
and 7 discuss this problem when “cost” is considered. “Cost”
p¯ is seen to be a monotonic decreasing function of n, so that
isusedgenericallyhere—itneednotbeamonetaryquantity;it
one need only make n large enough to achieve the required
could be time or something else.
bound on s (Note 3). Letting c =c =c =1.0 provides the
p¯ s a b
optimumvaluesof r, m,andnwhencostsareconsideredequal.
6. Determining Sample Sizes
In practice, the optimum values for m and r obtained this way
6.1 There are two common situations in which sampling
are unlikely to be integers. Unless these values are very close
plans must be developed for use in nuclear material measure-
tointegers,itisprudenttoconsiderbothbracketingvalues,that
ment when there are constraints on resources. In the first
is, if the optimum value for r is 1.4, try bothr=1 andr=2.
situation a constraint is imposed upon the “cost” of sampling
Thereasonisthatthefinalvalueofnwillgenerallybedifferent
and analysis. In this case, the problem is to find a plan that
and it is not clear beforehand which set of values of r, m, and
minimizesthevarianceofthesamplemean(minimizesrandom
nwillachievetherequiredvarianceatminimumcost.Itisalso
error) subject to the cost constraint. In the second situation, a
possibletousedifferentvaluesof m(or r,orboth)fordifferent
constraint is imposed upon the variance of the sample mean
containers or samples, or both, to obtain a non-integer “effec-
(upontherandomerror)andtheproblemistofindaplanwhich
tive” value of m (or r, or both). In this case, p¯ should be
minimizes cost subject to this constraint. Since this latter
replaced by a weighted average; s becomes more compli-
p¯
problem is the most frequently encountered, methods for its
cated; and the expected values of the mean squares also
solution will be given. The former problem, for which the
be
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