GM, _
CM,
exp [+0.50°/Vq]
’
(16)
exp [0.5(0°/ (V/V, )]
where o* is the true variance of the logarithms for aliquots of volume
V,. Note that this ratio is necessarily greater than 1. Michels makes
the point that this ratio can be substantially greater than 1 when o is
greater than 2 or 3 and V/V; is greater than 5.
He also makes use of
the Poisson distribution to give guidance on assessing the number of Pu
particles that must be present for Equation 16 to be near l.
In Table 4 we have computed the ratio of Am geometric means for the five
aliquot sizes (data from Figure 1).
In all but one case the ratio is
greater than 1. The largest ratio is 1.157 for GMj00 g/GM, g, i-e.,
GMio9g is about 16% larger than GM,g.
Essentially the same results
are obtained using the sample medians rather than geometric means.
Since the arithmetic mean is not systematically biased in this way by
aliquot size, it is the preferred estimate of central tendency for
comparing "average" results from studies that have used different aliquot
sizes.
Table 4.
Aliquot
Size (g)
10
25
50
100
Ratio of Observed Geometric Means of Am for Aliquots
of Different Sizes Using Data From Figure l
1
1.072
1,042
1.102
1.157
10
0.972
1.028
1.079
25
1.058
1.110
50
1.049
Michels restricts his discussion to the lognormal distribution.
However,
aliquot size is also applicable for other skewed distributions.
The
the phenomenon of increasing medians and geometric means with increasing
essential ingredients are that the distribution for each aliquot size be
unimodal and skewed toward high values. Also, this skewness must decrease
with increasing aliquot size. The geometric mean (or median) is always
less than the true mean (expected value) of a distribution that is
Skewed to the right.
As the aliquot sizes increases and the distribution
of aliquot concentrations becomes less skewed (more symmetric), the true
geometric mean and median must approach in value the true mean of the
distribution. This occurs since the mean and median are identical in
value for a completely symmetric distribution.
The effect of skewness on geometric means and medians is illustrated in
Figure 5 where we have plotted the density functions of lognormal distributions with parameters uw and o as estimated from the experimental
421