148 



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 

 NUMBER OF HADDOCK 



Figure 2. — Number of tows, number of haddock, and the fitted binomial distribution. 



Fisher (1941) considered the simplest frequency 

 distribution which allows for some variation of m 

 to be the Eulerian distribution with frequency 

 element: 



df=-7T — tt-. p- k m k - 1 e- m/p dm (1) 



(k l) ! 



When m varies in this way, the frequency of 

 occurrence of x units in the sample is: 



i 



i 



o (fc-1)! 



v^tj^k—\^—mlVp~ m 



m 1 



dm 



leading to the integral of the Eulerian type: 

 Qc+x— 1)1 p x 



(2) 



(3) 



xl(k—l)l (l+p) k+z 



which is identical with the standard form of the 

 negative binomial. 



In the derivation of the negative binomial 

 distribution, it is important to consider that while 

 the Poisson distribution is that expected for 

 homogeneous material, the Poisson no longer 

 holds if (1) there is variation in the size of the 

 samples or (2) if the material is heterogeneous. 



The variation (in excess of that expected for a 

 Poisson distribution) observed in plankton and 

 trawl sampling has been attributed in the past 

 to factors falling under (1), e. g., variations in the 

 volume of water sampled. It is clear that vari- 

 ability may be ascribed equally to heterogeneity 

 in the population sampled, i. e., "true" sampling 

 variation. This heterogeneity, furthermore, does 

 not exclude the possibility of a stable 5 mathemati- 

 cal distribution amenable to statistical treatment. 



DISTRIBUTION OF SPECIES AND NUMBERS OF 

 FISH 



The number of fish of each species was recorded 

 for each tow made over the period 1948-50. The 

 logarithmic series having been developed by R. A. 

 Fisher (1943) to account for peculiarities in the 

 distribution of species and numbers of individuals 

 of each, it appeared desirable to determine 

 whether the species occurring in the trawl samples 

 were so distributed. 



1 By "stable" we mean a distribution whose general characteristics do not 

 change between successive sets of samples. Heterogeneity, while a condition 

 for a negative binomial distribution, does not necessarily imply the negative 

 binomial distribution as the only possibility. 



