VARIABILITY IN TRAWL CATCHES 



161 



Proof that, with a Poisson distribution, all sizes 

 of sampling units are equally efficient. — Consider 

 the ith stratum. Let it contain M sampling units. 

 Let the number of sampling units to be taken be 

 m. Let F be the estimate of the total fish in the 

 ith stratum. By (5): 



0> = 



■V 



M(M-m) 2 

 m 



(8) 



Consider sampling unit k' > k=l. Since the 

 area of the ith stratum is fixed, the total number 

 of sampling units of size k' is M/k' . For equal 

 efficiency, it is required a 2 =a' F 2 without increasing 

 the total area of the sample taken, so the number 

 of samples to be taken for k' > k is m/k'. By 

 equation 5: 



or 



V 



M/ k' (M/k' -m /k') 

 m/k' 



Oi 



(9) 



where <r/ 2 is the variance observed with the new 

 sampling unit, k' . 



The comparative efficiency of the new sampling 

 unit will be equation (8) divided by equation (9) : 



a F 2 /a F ' 2 =<r t 2 -i-a t ' 2 /k' . 



(10) 



The means observed may be considered pro- 

 portional to the size of the sampling units used, 

 provided sufficient samples are taken, since both 

 are sampling the same population density. So 

 x=x'/k', where x' is the mean observed with 

 sampling unit k'^>k. By definition of the dis- 



tribution, x=a 2 and x'=a t ' 2 =k'x. Substituting 

 in equation 10: 



<j F 2 /e F ' 2 = x+k'x/k'=\.0 



Proof that the smaller sampling unit is mort 

 efficient if heterogeneity is present. — With hetero- 

 geneity as defined above, the variance is greater 

 than the mean. As with the Poisson distribu- 

 tion, the means observed will be proportional to 

 the size of the sampling units used, so that x=x'/k'. 

 Because of heterogeneity, however, the variances 

 are no longer proportional to the size of the sampl- 

 ing unit as in the Poisson distribution where 

 <r, 2 =oi' 2 /k'. For the negative binomial distribu- 

 tion and some other types of heterogeneous distri- 

 butions, the relation between the mean and 

 variance is of the type a 2 =ax-\-hxr. We have, 

 therefore, a 2 <^a t ' 2 /k' and it is immediately obvious 

 from inspection of (10) that a F -/a F ' 2 will be less 

 than 1.0. 



It hardly needs to be pointed out that the 

 application of this result to problems of hetero- 

 geneous sampling does not mean one's difficulties 

 will be solved merely by taking the smallest pos- 

 sible sampling unit. Many practical factors will 

 set a lower limit to the size of the sample it is 

 desirable to take. In sampling with an otter 

 trawl, for example, one would approach a point 

 where more time would be consumed in lowering 

 and raising the trawl than in fishing along the 

 bottom. It must also be remembered that with 

 the smaller sampling unit, minor sources of error 

 may remain of the same magnitude and so be- 

 come of proportionally greater importance with 

 the smaller sampling unit. 



