If the data are truly lognormal, a minimum variance unbiased estimate of the
mean is obtained by multiplying the GM by a factor that is always greater than
1 (Aitchison and Brown, 1969, page 45, eq. 5.40). However, this estimator is
biased if the data are not truly lognormal, which may often be the case.
Other estimators of central tendency include weighted means, and means that
take into account a correlation that may exist between data values (discussed
by De Wijs, 1951, 1953).
These papers by De Wijs are concerned with ore assay
values rather than radionuclides.
However, both kinds of variables tend to
follow lognormal-type distributions, and the objective of estimating "average"
values is common to both fields of application. The applicability of De Wijs'
References
methods to radionuclide inventory studies needs to be evaluated.
symmetric
for
primarily
on the study of various estimators of central tendency
distributions include Andrews et aZ. (1972) and Huber (1972).
(1975, pp. 419-420) briefly related aspects of the problem.
We may ask:
Gilbert et al.
Which is the more meaningful average for estimating, say, the
inventory of 239Pu in surface soil?
The average concentration is clearly
related to the estimates of inventory obtained at safety-shot sites by Gilbert
et al., 1975 (see Appendix A), since
Area x Average Pu Concentration = Estimated Inventory
(m?)
(nCi/m*)
(nCi)
was used to estimate the inventory in each stratum.
Hence, if the GM were
used rather than the AM (the latter was used by Gilbert et al., 1975), a
smaller estimate of inventory would result.
The magnitude of the difference
in estimates depends on the degree of skewness in the data.
An example where
the estimate of inventory changes by a factor of two depending on whether the
AM or GM is used is given in Figure 6.
In this figure, we plot the 239,240 py
concentrations (nCi/g dry) obtained from 47 surface (0-5 cm) soil samples
collected at random locations within stratum 6 in the GZ area of Area 13
(Project 57) (see Figure 4). Notice that 46 of the 47 observations fall
between 0 and 22 nCi/g, but one observation is between 304 and 305 nCi/g. The
effect of this one extreme point on the median, GM, and AM is indicated in the
figure.
If the outlying datum is deleted from consideration (which is not
recommended since the datum does not appear to be a gross error), the AM drops
from 14 to about 7 nCi/g, whereas the GM and median decline only slightly.
Hence, if Gilbert et al. (1975) had deleted this one observation, their estimate
of *3%py inventory for stratum 6 soil would have been about 8 rather than 19
curies. This in turn would have reduced the total estimated inventory for all
six strata from 44 to 36 curies.!! The apparent precision of the estimate
would also have improved due to the elimination of this single datum.
1lThe estimated total inventory for 23°%Pu in Area 13 (Project 57) is roughly
24 Ci if the GM is used in place of the AM and the datum 305 nCi/g is not
deleted.
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