150 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 



NUMBER OF OCEAN PERCH 



Figure 4. — 'Number of tows, number of ocean perch, and the fitted binomial distribution. 



Williams (Fisher, Corbet t, and Williams, 1943) 

 has suggested that the index a of the logarithmic 

 series be called the "index of diversity," since it 

 is a measure of the richness in species from sample 

 to sample. Williams (1944) has given a summary 

 of the properties of the index of diversity, which 

 (slightly condensed) is as follows: 



(1) If several samples are taken from the same popula- 

 tion, not only will they have the same index of diversity, 

 but when two or more of them are added together the 

 combined sample will still have the same index. 



(2) Different populations may have, by chance, the 

 same index of diversity, and samples from them will have 

 the same index. When these samples are added together, 

 however, the alpha of the combined sample will be larger 

 than that of either of the two or more original samples. 



(3) If two populations are entirely apart, then the 

 index of one sample from each combined will be the sum 

 of the two indices for each sample separately. If there 

 is some overlap, alpha will be larger than either but smaller 

 than the combined total. 



(4) If two samples, not of very small size, are taken 

 from the same population, the larger sample being p times 

 the smaller, then the number of species in the larger 

 sample will be or log e p more than in the smaller; it follows 

 that the index a is the number of species added to a 

 sample by multiplying its size by e. 



(5) In the same population, x approaches 1.0 as the size 

 of the sample increases, and ni approaches a (where ni is 

 the number of singleton species). 



From these properties of the index of diversity, 

 it is apparent that a useful analytical device is 

 provided for examining the characteristics of pop- 

 ulations, provided the logarithmic distribution is 

 applicable. 



SOME APPLICATIONS OF THE LOGARITHMIC 

 SERIES TO THE TRAWL DATA 



The trawl data collected from 1948 to 1950 on 

 Georges Bank were classified by depth zone and 

 bottom type, and the index of diversity and its 



