VARIABILITY IN TRAWL CATCHES 



155 



the progeny of a single ancestor. The condition 

 of uniform distribution in time possibly is approxi- 

 mated, so that one may be inclined to favor the 

 negative binomial as the more likely, although an 

 essential of Kendall's model is the self -duplication 

 of random individuals at random times. 



A stronger argument in favor of the negative 

 binomial distribution is a relation among the 

 negative binomial, the logarithmic series, and the 

 Poisson distributions which has been pointed out 

 by Quenouille (1949). It is demonstrated that in 

 sampling from a logarithmic series, the probability 

 distribution of the number of individuals in 

 random samples is the negative binomial. Que- 

 nouille further demonstrated that if any two of 

 the three distributions hold, the third distribution 

 is implied (see appendix A). 



A logarithmic distribution of species and indi- 

 viduals is demonstrated to hold with reasonable 

 probability for the trawl data, and it is further 

 shown that the hypothesis of a negative binomial 

 distribution satisfactorily conforms to the dis- 

 tribution of numbers per tow, as theoretically 

 required, given a logarithmic series distribution 

 from which samples are drawn. 



The implied Poisson distribution can occur in 

 several ways, some of which are suggested by the 

 models considered above. The Poisson distribu- 

 tion of the number of species per tow suggests the 

 hypothesis that the various species are distributed 

 at random and move independently of one another. 

 This hypothesis does not require that the distribu- 

 tion of a species be random, since a species occurs 

 only once, no matter what its abundance in num- 

 bers, in the sample. The assumption of a Poisson 

 distribution of species appears sufficient to com- 

 plete the theory, especially when we have a certain 

 amount of evidence that this distribution actually 

 holds. Alternatively, we might suppose that the 

 occurrence of schools or concentrations of fish is 

 random. This may, in fact, be true but it cannot 

 be demonstrated at present because of lack of the 

 necessary observations. 



Application of the distribution theory to the census 

 problem 



The problem of variance. — The primary purpose 

 of the census cruises made on Georges Bank was 

 to estimate and define the abundance and dis- 

 tribution of the fish populations inhabiting it. 

 It is not the purpose of this paper to extend the 

 distribution theory to numerical estimates of 



populations, but it is pertinent to examine 

 certain aspects bearing on them. 



In populations with individuals distributed at 

 random and independently of each other, it is 

 well known that the variance is equal to the mean. 

 In a negative binomial distribution, the relation 

 between mean and variance is 



a x 2 =m-\-m 2 jk. 



(10) 



One sees at once that the departure of the variance 

 of the negative binomial from that of the Poisson 

 depends on the value of k; the variance, as well 

 as the distributions themselves, becoming identi- 

 cal as k tends to infinity. On the other hand, if 

 k is less than one, the variance is never less than 

 the square of the mean and may be of considerably 

 greater magnitude. 



The high variance associated with the mean 

 of a negative binomial distribution introduces 

 a most serious problem in estimation. Since k 

 is an intrinsic property of the population being 

 sampled, it is independent of the sampling method 

 and its estimate is subject only to sampling 

 variation. The mean, however, depends on the 

 size of the sampling unit, e. g., in the census 

 sampling, the size of the trawl _net and the 

 duration of the tow. 



Taking as a measure of the precision of sampling 

 the reciprocal of variance, the amount of informa- 

 tion of a mean based on n observations is n/<r z 2 . 

 It is clear that the amount of information can be 

 increased b}' increasing the number of ob- 

 servations, by decreasing the variance, or both. 



Examining the relation between mean and vari- 

 ance (equation 10), we see that any change in 

 sampling method which increases the mean will 

 increase the variance even more rapidly. It has 

 been suggested that the trawl data could be 

 "smoothed out" by using a smaller net and 

 increasing the length of the tow. Increasing the 

 length of the tow would tend to increase the 

 mean, while decreasing the size of the net would 

 tend to decrease the mean. If the two effects 

 were of such magnitude as to cancel each other, 

 the variance would, of course, remain unchanged. 



On the other hand, it will be seen that changes 

 in sampling which reduce the mean will also reduce 

 the variance, and for a given number of observa- 

 tions, will increase the amount of information 

 given about the mean. The obvious ways to 



