1. No category

Sample Sizes for Attributes Inspection When
Strata are Heterogeneous
John L. Jaech and Marian Donoho
International Atomic Energy Agency
Vienna, Austria
ABSTRACT
that at least one of the M/yx defects that
are hypothesized to exist among the N
population (or stratum) items will be contained
in the sample of size n. This statement
implies that IAEA attributes inspection is
generally based on zero-acceptance number
sampling plans. Such plans require a minimum
amount of inspection which is an important
consideration when inspection resources are
limited.
The formula that is in common usage at the
International Atomic Energy Agency (IAEA) when
calculating sample sizes for inspection by
attributes is based on the assumption that the
stratum in question is homogeneous with respect
to element/isotope weight, i.e., all items in
the stratum have the same weight. In practice,
this assumption is generally invalid to some
extent. This paper examines the effect of
heterogeneity in item weights on the probability
of detection, and indicates how sample sizes
might be altered to account for the
heterogeneity. After considering the simplest
type of heterogeneity that occurs when the item
weights cluster tightly about a small number of
modal points, a more general pattern of
heterogeneity is examined. Modifications to the
sample sizes to account for this general pattern
are derived based on a specific diversion
strategy. A number of numerical results are
presented in the form of plots relating sample
size to mean item weight and to the scatter in
item weights.
An important feature of the sample size
formula is its simplicity. The IAEA inspectors
quickly commit the formula to memory through
repeated usage. The sample size calculation
itself is very simple with the aid of a pocket
calculator.
Notwithstanding the widespread application
of formula (1), it is not without potential
shortcomings. One question that has reoccurred
on frequent occasions is the impact of
heterogeneity in item weights on the validity of
the formula. Formula (1) is based
on the assumption that all item weights are
equal to x, although it is realized in
practice that this is never exactly true. One
purpose of stratification is to limit the
scatter in the item weights within each stratum
so that formula (1) yields the sample size that
will guarantee a detection probability of 1-|3.
INTRODUCTION
In calculating inspection sample sizes for
the detection of gross/partial defects by
attributes inspection, the formula in common
usage at the IAEA is of the form
where y = 1 for gross defects and is some
value less than one for partial defects. In
(1),
(3 is the nondetection probability, x
is the mean item weight (element or isotope),
M is the goal amount, N the number of items
in the stratum, and n the sample size.
In practice, limitations on the number of
strata that can be reasonably accommodated
result in strata that may well be quite
heterogeneous with respect to item weights.
This introduces concern about the validity of
applying the sample size formula (1), and about
how sample sizes should be calculated to provide
the desired detection probability if the degree
of heterogeneity is excessive. Of course, a
basic question to consider is just how much
heterogeneity can be tolerated without concern,
i.e., how much is "excessive".
Formula (1) has been applied virtually since
the beginning of IAEA inspections. It is based
on an approximation to the hypergeometric
probability density function and is the sample
size needed to assure with probability (1 - |3)
In addressing the question, a particular
type of heterogeneity is examined first with
respect to its effect on the sample
size detection probability question. The type
of heterogeneity in question is that in which
n/N = (1 - P
(1)
173
respect to permissible variation in item weights
within a stratum on the one hand, and problems
of implementation and data evaluation that are
associated with moderately large heterogeneity
in item weights on the other hand. In practice,
item weights within a stratum may vary by a
considerable amount. A particular pattern of
heterogeneity is considered here.
individual item weights may cluster tightly
around a small number of mean weights. This
would occur, e.g., when two or more homogeneous
strata that differ only in mean item weights are
combined into a single stratum in order to
reduce the number of strata. The obvious
solution to the sample size problem in this
instance is to compute the total sample size by
summing the sample sizes calculated separately
for each "sub-stratum". Under appropriate
assumptions, this is equivalent to simply
calculating the sample size for the combined
stratum. However, the detection probability is
not necessarily > (1-0), as will be
illustrated.
Characterize this specific pattern as one
in which item weights are clustered around m
modal points. In the limiting case, Nj items
have item weight xj; i = 1, 2, ..., m. In
essence, the stratum consists of m sub-strata.
Without loss of generality, set y = 1 in
formula (1), in which case the sample size in
substratum i would be
The major focus of the paper is to consider
a general pattern of heterogeneity in item
weights in which the heterogeneity is
characterized by the standard deviation in
stratum item weights. No further distributional
assumptions are made. A conservative approach is
to insert some value other than x in formula
(1), where this value exceeds x by some amount,
the amount depending on the value of the
standard deviation. This conservative approach
assumes inherently that the adversary will
choose to falsify the items with the largest
weights, thus minimizing the number of items
that need be falsified. The increased sample
size may be quite excessive, but the desired
detection probability of (1-P) should be
virtually guaranteed.
nj
and
£ nj
is the total sample size.
For small ij/M, say x^/M < 0.1 as is
often the case, the following approximation
holds:
(1 - P>*i/M) * - (Xi/MHnp
(3)
so that
Another approach is to assume a random
strategy for the adversary, one in which he
selects items to falsify at random from the
stratum, stopping when the sum of the selected
item weights equals or exceeds the goal amount.
One benefit of this approach from the
adversary's viewpoint is that the particular
items that are defected are completely
unpredictable to the inspector, and he must
therefore tailor his sampling plan to cover all
possibilities. It is also noted that this model
is certainly a reasonable one in a
non- adversarial situation in which the defects
are the results of mistakes in the records as
opposed to data falsifications. In this
situation, it is entirely reasonable to assume
that the defected items are selected at random.
(4)
Now, the stratum average is
m
(5)
= I
and it is noted that if
x
In the next section, the pattern of
heterogeneity in which individual item weights
cluster about a small number of mean weights is
considered. In the section to follow, the
effect of a general pattern of heterogeneity is
examined.
and N =
are used in formula (1), the sample size is
- pX/M>
CMJSTKK HETEKOGKNKITY
Whenever strata are created within some
component of a material balance, decisions must
be made regarding the definition of specific
criteria for stratification. The IAKA has
developed guidelines for stratification in
attempting to strike a compromise between the
excessively large number of strata that would
result in the event of rigid criteria with
- ififi
I
NjXi
(6)
which is identical to (4). Thus, the sample
size for the stratum is the same as the total
sample size for m pseudo- substrata.
A numerical example may enhance
174
understanding of this result. Table 1 gives
the frequency distribution for grams pluLonium
in containers of mixed oxide powders. The
frequencies are given in terms of Nj/N. for
P = 0.05, the sample size calculations are
also indicated in the table. The goal amount,
M = 8000 g Pu.
T a b l e
PATTKKN OK
In Table 1, a l l N^ itenis in substratum i
are assumed to have the same item weight. . In
generalizing the heterogeneity pattern, this
assumption is no longer made. Rather, the
heterogeneity is characterized by the standard
deviation in stratum item weights. No other
distributional assumptions are made, i.e., the
density function for item weights is not
specified.
1
Frequency Distribution of Grains
Plutonium in Containers of MOX Powder
As previously indicated, for this general
model a random strategy is assumed for the
adversary. Items to falsify (defects) are
selected at random from the stratum, this
process continuing until the goal amount is
achieved.
Nj/N
700-800
600- 700
500-600
400-500
300-400
200-300
100- 200
0-100
Pu.
750
650
550
450
350
250
150
50
0.28
0.30
0.10
0.02
0.02
0.20
0.06
0.02
1.00
0.2486
0.2160
0..1861
0..1551
0..1228
0.0893
0.0546
0.0185
0..0696
0..0648
0.0186
0.0031
0..0024
0..0179
0..0033
0.0004
0.1801
Under this assumption the number of defects
among the N items is a random variable,
denoted by Z. The distribution of 2 is
difficult to determine, but it may be inferred
by proceeding as follows. For a given value of
Z, denoted by zj, the probability that the sum
of the item weights exceeds M may be calculated
assuming, (by appealing to the Central Limit
Theorem) that this sum is normally distributed.
Let this probability be denoted by fz{- For
the next integer, the corresponding probability
is Pzi_l, and hence, the probability that
Z = Zi is qzi = pzi - Pzi.iThen, for given
zj, the nondetection probability is approximately
The stratum average item weight is 536 g
From (1),
n/N = (1 - pS36/8°°°) = 0.1819
as compared with n/N = 0.1801
pseudo-substrata.
NKTK.WC l-'NK IT Y
when forming the
(1 - n/N) l = a *
It is noted that although the total sample
size is essentially the same whether or not the
substrata are actually formed, the item weights
for the items sampled at random from the entire
stratum will not have the same frequency
distribution as indicated in the last column of
Table 1 except on an expectation basis. For
example, the expected number of items sampled
that have item weights between 700-800 g Pu is
0.0696N; the actual number sampled may be quite
different. For example, if N = 100, on the
average 7 items would be sampled from this
substratum. The actual number sampled in a
given realization of the sample could easily
range between 3 and 12.
The overall nondetection probability is then
z
v
I qzi a
(8)
and a is found by trial and error such that
the indicated sum is <_ P.
This process is illustrated.
data,
For the Table 1
x = 536 g Pu
The conclusion with respect to combining the
substrata into a single stratum and selecting
the n items at random from this single stratum
may be stated from another equivalent perceptive.
s = 220 g Pu
and also,
Conclusion:
(7)
If sampling is performed within
each substratum by repeated
application of formula (1), the
overall nondetection probability
probability is P independent
of how the defects may be
distributed. If sampling is
performed within the overall
stratum, then the expected nondetection probability is P; the
actual probability may be cither
greater than or less than p.
M = 8000 g Pu.
(It is noted that x and s, being population
values rather than sample values should be
replaced by M and a respectively. The
results are, of course, independent of the
notation, and since the accepted notation at the
IAEA for a stratum average weight is x, the
notation is preserved in this discussion.
for
175
= 13,
say.
p l3 = Prob( I
mean item weight, the sample size given by
application of equation (1) is actually slightly
larger than needed to achieve the desired value
of P. Only when the standard deviation
exceeds roughly 80 % of the mean need additional
items be inspected.
Xi > M) = Prob(t
/13s
= Prob(t > 1.301).
where t is N<0,1). From a table of the
standardized normal distribution function,
P13 = 0.097.
For
z
A simple example will serve to explain this
phenomenon. Consider the case where M/x = 4
and c = 0. In this case, all items in the
stratum weigh M/4 units (e.g., g Pu) and no
matter which items are included in the random
sample, exactly 4 would have to be falsified to
accumulate to the goal amount, H. Now suppose
that c is just slightly larger than 0, i.e.,
s it 0 but is small relative to x. Then the
required number of items needed to accumulate
to M units is either 4 or 5, and each value
of Z is equally likely. Thus, with
probability 0.5, Z = 4 and the required sample
size is the same as that required for s = c =
0. But also with probability 0.5, Z = 5, and
the required sample is smaller than that needed
for s = c = 0 , there being more defects in the
population. Thus, on an expectation basis, the
required sample size is actually smaller for
c > 0 by some amount that it is for c = 0. At
some point, however, at about c = s/x = 0.8,
the required sample size is greater than for
c = 0 because the standard deviation of Z
becomes so large relative to its mean. It is
noted that c ^ 0.8 corresponds to a stratum
with a considerable degree of heterogeneity, and
very few strata have coefficients of variation
that exceed 0.8.
=14,
i
P14 = Prob(t >
--JUX) = Prob(t > 0.603) = 0.273
vO4~ s
Since 0.097 is the probability that the
goal amount would be exceeded by the sum of the
weights of 13 randomly selected items, and
0.273 is the corresponding probability for 14
items, the probability that z = 14 is
Pl4
0.176
In this way the probability distribution of
the random variable Z is inferred. Some small
scale simulations were performed with good
results to verify the validity of this approach
to finding the distribution of Z.
Now, for a given value of Z - zj,
detection probability is approximately
SUMMARY AND CONCLUSIONS
non-
the
A computer program was written to permit
calculation of a large number of results. The
results are displayed in graphical form,
Figures (l)-(4). In the figures, "SQ" refers to
a significant quantity, the usual value for
M in the previous discussion. The parameter
c is the ratio s/x, i.e., is the coefficient
of variation. The parameter re is the ratio
of the sample size for given c to that given
by formula (1), i.e., at c = 0.
Two approaches were used to study the effect
of heterogeneity in item weights on required
inspection sample sizes for attributes
inspection. In the event that item weights are
clustered tightly around a number of modal
points, the overall sample size found by forming
pseudo-substrata as defined by the modal points
is essentially the same as that found for the
overall stratum when the stratum average item
weight is used in the sample size formula (1).
When sample items are selected randomly from the
entire stratum, the detection probability is
(1 - p) on an expectation basis, although in a
given realization of the sample, the detection
probability is a function of the diversion
strategy. If the sample items were to be drawn
at random from each of the pseudo-substrata by
applying formula (1) to each substratum m,
then the detection probability is essentially
(1 - P) independent of the diversion
strategy. Under the assumption that it is
appropriate to calculate the detection
probability on an expectation basis, it is valid
to apply formula (1) to the entire stratum in
spite of the degreee of heterogeneity when the
item weights exhibit the "cluster" pattern of
heterogeneity.
Rather surprisingly, perhaps, the analysis
shows that for values of the standard deviation
of item weights that are as large as 80 % of the
For a more general pattern of heterogeneity
characterized by the coefficient of variation,
c = s/x, and for a diversion strategy in which
- n/N)Zi
(9)
Since Z = zj with probability qz,, the
overall expected nondetection probability, P,
is equal to the sum
z
i
(10)
For a desired value of |3, equation (10)
may be solved by trial and error for the
fractional sample size, n/N.
176
weights. This conclusion holds as long as it is
judged satisfactory to characterize detection
probabilities on an expectation basis.
Otherwise, one can as one alternative form the
pseudo substrata indicated in the discussion on
the cluster pattern of heterogeneity and inspect
the appropriate number of items from each
substratum by repeated application of formula
(1). This obviously may greatly increase the
inspection planning implementation, and
evaluation effort. As another alternative,
given inspection resources, the conservative
approach of adjusting x upwards to characterize
the items with the large weights may be utilized.
Of course, it would be necessary in this latter
instance to randomly sample the items from the
entire stratum, and not just inspect those with
the largest item weights. The required sample
size with this conservative approach might well
exceed that given by formula (1) by a large
amount.
items are selected at random until the goal
amount is attained, an algorithm was developed
that gives the sample size needed to achieve the
required detection probability on an expectation
basis. The necessary computer software was
developed and calculations performed for a large
number of cases. The results indicate that even
for reasonably large values of s/x, i.e., as
large as 0.8, the sample size is actually
smaller for the heterogeneous stratum than for
the limiting case in which s = 0. Thus, formula
(1) tends to be conservative on the high side
for the very few instances in which s/x becomes
quite large, i.e., > 0.8. In such instances,
figures (l)-(4) may be used to calculate required
sample sizes.
The overall major
general, the commonly
(1) may be applied in
being concerned about
f
conclusion is that in
used sample size formula
most instances without
heterogeneity in item
«.l
•.4
•••
Fig.
Fig.
1
3
9Q/XBAH far •
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1.1 " ^_
Fig.
2
177
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