Fishery Bulletin 91(1). 1993 



such as these are commonly expanded according to 

 area-swept procedures to produce estimates of popula- 

 tion size. Both nearshore and estuarine populations 

 were sampled, and we examined data collected during 

 four consecutive years. Survey design and methodol- 

 ogy are discussed in Armstrong & Gunderson (1985) 

 and Gunderson et al. ( 1990). 



The spatial dispersion of a population determines 

 the relationship between its mean abundance and vari- 

 ance, and this information may be used to select an 

 advantageous data transformation (Elliott 1977). 

 Strong linear relationships between the means and 

 standard deviations of our density data (r 2 =0.98 and 

 0.70, with P«0.001 and P«0.001 for the coastal and 

 estuarine areas, respectively) and graphical analysis 

 of log-transformed data suggested a logarithmic trans- 

 formation would be appropriate. In order to test this 

 assumption, the Kolmogorov test for normality was 

 applied to the density data for each cruise, both before 

 and after transformation (Table 1). These preliminary 

 analyses suggested that the density data were lognor- 

 mally distributed and, as such, that individuals in the 

 crab population were aggregated in space. However, 

 lognormal theory cannot be applied directly to any 



sample that contains a zero value, since the logarithm 

 of zero is undefined. Since our data exhibit only the 

 occasional zero catch, we used the common In (X+l) 

 transformation to normalize the data. An alternative 

 approach, when a significant fraction of the data con- 

 sists of zero catches, would be to use the A-distribution 

 (Pennington 1983 and 1986, Smith 1988), which is es- 

 sentially a lognormal distribution with a proportion 

 (A) of zeros. 



The lognormal distribution and parameter 

 estimation 



The lognormal distribution may be represented as a 

 Gaussian distribution of logarithmic data or, equiva- 

 lent^, as a right-skewed distribution of untransformed 

 data (Aitchison & Brown 1969). A brief review of the 

 density function and relevant parameters for the log- 

 normal distribution appear in the Appendix. There, 

 and throughout the text, we use the following notation 

 to distinguish between untransformed and transformed 

 scales and between population parameters and their 

 estimates: X represents untransformed density val- 

 ues, while X (the ordinary sample mean) and s~ x (the 



