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