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FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



reduce the mean are to reduce the length of the 

 tow and to reduce the size of the net within 

 practical limits. 



We have, for example, in depth zone II, 171 

 tows in which the mean catch of haddock was 130. 

 The value of k, calculated from the maximum- 

 liklihood equation, is 0.25 (0.2542). The expected 

 variance is 



130 2 



<7 2 =130- 



0.25 



or 67,730. Let us suppose that by decreasing the 

 length of the tow and the size of the net each by 

 one-half, we reduce the observed mean by 

 one-fourth so that it is now 130/4 or 32. The 

 expected variance is now <r I 2 =32+32 2 /0.25 or 

 4,128. The amount of information has been 

 increased by 67,730/4,128 or about 16 times. 

 Reduction in the size of the net and the length 

 of the tow would doubtless be accompanied by 

 other practical advantages, such as a smaller 

 crew to handle the net, greater ease in handling, 

 more data on the smaller catches, and the exe- 

 cution of a greater number of tows within a 

 given time period. 



It is to be pointed out that, in the final analysis, 

 we are interested in the precision of an estimate of 

 the total population in the area sampled. The 

 mean we observe in a series of samples with any 

 given sampling unit is, of course, the basis for this 

 estimate. This problem, with particular refer- 

 ence to a heterogeneous population, is examined in 

 appendix E. 



Ecological relations. — Since the numbers of each 

 species occurring in the trawl catches were re- 

 corded, we have in these data, in a sense, a third 

 dimension of observation: the relation of the 

 species to each other and to various environmental 

 conditions, not only in qualitative but also in 

 quantitative terms. While some of the prop- 

 erties of the index a of the logarithmic series have 

 been pointed out above, the primary purpose in 

 the present paper has been to examine the reality 

 of the logarithmic-series distribution of species 

 and individuals. 



Although the analysis of the indices of diversity 

 by bottom type and depth zone within and between 

 years does not reveal any significant differences in 

 richness of species, an entirely different result is 

 obtained if one uses Fisher's formula for variance 

 (7, appendix C) rather than that given by Ans- 



combe (8, appendix C). In our data Fisher's 

 formula gives a variance roughly one-tenth as 

 great as that given by the formula we have used. 

 As a result many significant differences appear 

 between bottom types, with an apparent tendency 

 for these differences to diminish with depth. 

 With Fisher's formula there still appear no signif- 

 icant differences between depth zones. As 

 Anscombe points out, the conditions under which 

 Fisher's formula is applicable are rather special- 

 ized. It is possible that closer examination of our 

 data may reveal that some of the specialized 

 conditions are met. 



An interesting relation which may be of prac- 

 tical use after further study of the data is given 

 by the formula 



S=alog e (l+iv7a) (11) 



When N is large compared to a, we may write 



S=a\o ge (Nla) (12) 



Making the necessary assumption that the 

 density of population within the total area is the 

 same as that within the area actually sampled, 

 we should expect to observe a total of S t species 

 if we sampled the entire area. Designating S, 

 as the number of species observed in the sample 

 area, A s say, the expected increase in species if we 

 sampled the entire area, A„ is 



S t -S 8 =alog e (A t /A s ) 



(13) 



from which we can readily estimate S t . Sub- 

 stituting S, and the sample estimate of a in 

 equation 12 we can then estimate N,, the total 

 number of individuals within the total area of all 

 species. Limits of reliability of estimate may be 

 set from the standard error of a. It is pointed 

 out here merely that the method may be of some 

 use when the area under consideration is not too 

 large as compared with the area sampled, when the 

 sampling is done within a fairly short period, and 

 when the assumption of uniform population 

 density can reasonably be made. Under these 

 conditions Fisher's formula for the variance of a 

 would be applicable. 



SUMMARY 



Analysis of the census-trawl data collected 

 during three summers on Georges Bank shows 

 that the hypothesis of random distribution of 



