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Fishery Bulletin 94(3). 1996 



var ( , s ,(c) = exp(0.350){^ 5 2 (1.961)-^ 50 (3.841)} 

 = 1.42{(6.39) 2 - 34.07} = 9.7. 



Thus the estimated standard error ofc is 3.1 (= (9.7) 172 ) 

 as compared with an expected value of 3.8 (from Equa- 

 tion 6, p = 1 ). For further examples of the performance 

 of the estimators on lognormal data, see Aitchison and 

 Brown (1957), Blackwood (1991), McConnaughey and 

 Conquest ( 1992) and Conquest et al. ( 1996). 



The next three examples are survey data that are 

 similar in appearance to the artificial data set in that 

 each contains a single large isolated catch. The first 

 data set is from a trawl survey in the southeastern 

 Bering Sea in 1991. In Table 1 is shown the survey 

 catch per unit of effort (CPUE) of male red king crab, 

 Paralithodes camtschatica, of legal size. The largest 

 CPUE is six times greater than the second largest 

 value and accounts for nearly 50% of the total sur- 

 vey catch. In Table 2 the estimates of the mean and 

 the standard errors are calculated as above. The pat- 

 tern of the estimates is similar to that for the artifi- 

 cial data. In particular, the estimate of the mean, c - 

 545.0, is much smaller than the sample mean (864.8) 

 and appears to be more precise. The reason that c is 

 so much lower than .v is that, based on the A-lognor- 

 mal model of the data, a CPUE as large or larger 

 than the biggest value (32,538) would have occurred 

 for approximately 1 in 2,200 tows during the 1991 

 crab survey, and c weighs the value accordingly. In 

 contrast, the sample mean gives each CPUE value 

 equal weight. 



In Table 1, CPUE data are given for petrale sole, 

 Eopsetta jordani, from a 1992 trawl survey off the 

 west coast of the United States. The largest catch is 

 65% of the total catch and is 12 times larger than 

 the next largest catch. Estimates of the mean and 

 standard errors are given in Table 2. 



The last example of this type of data set is from a 

 trawl survey off the east coast of the United States. 

 The data (Sissenwine, 1978) are the catch per tow in 

 1973 of Atlantic mackerel, Scomber scombrus. The 

 largest catch (5,182 kg) is more than 25 times greater 

 than the next largest ( 194 kg) and is 92% of the total 

 catch. This is the one example presented for which 

 lognormality of the nonzero values was rejected 

 (P=0 .02). Though the estimate c = 2.0 kg/tow is con- 

 siderably smaller than the sample mean ( x =26.2 kg/ 

 tow), it is much more consistent with previous and 

 subsequent survey indices (e.g. 1.6 kg/tow in 1972 

 and 2.5 kg/tow in 1974) than is the sample mean (see 

 Fig. 5 in Sissenwine, 1978). 



No dominating large catch 



The more usual type of survey data set is one that is 

 highly skewed but does not contain a relatively large 

 isolated value. A typical example of this sort of data is 

 seen in Figure 2 which shows the catch per tow of juve- 

 nile Arcto-Norwegian cod, Gadus morhua, collected 

 during a 1989 midwater trawl survey in the western 

 Barents Sea (Helle, 1994). The estimate of the mean 

 from c is 55.2 and from x is 49.7. Similarly, the esti- 

 mate d is greater than the sample variance (Table 2). 



Another example is from a 1989 zooplankton sur- 

 vey in the Barents Sea (Helle, 1994). Figure 3 is a 

 plot of the biomass per tow of copepods sampled with 

 a Juday plankton net. The frequency distribution is 

 similar to that in Figure 2, and, again, the estimates 

 c and d are larger than the ordinary sample esti- 

 mates (Table 2). 



The sample average and variance will be underes- 

 timates for most samples (i.e. be smaller than the 

 true values). This is due to the sampling distribu- 

 tion of x and s", which, for a highly skewed distri- 

 bution, will still be skewed to the right for small to 



