FISHERY BULLETIN: VOL. 78, NO. 3 



While this argument is correct, it does not mean 

 that the estimate of total numbers of fish over the 

 entire strata is more precise with decreasing 

 sample element size. The effect of reducing sample 

 element size on the variance of the estimate of the 

 total number of fish in a stratum under the condi- 

 tion of a fixed sample size is considered below. 

 Using the definitions of A, a, n from above, 

 then A^ = total possible number of samples in a 

 stratum where the sample element size equals a 

 (i.e.,N — A/a). The variance of the total number of 

 fish in a stratum from n samples of the standard 

 element size ( V^^ ) is 



V, 



iVr 



V, 



nbi 



or 



V, = 



a^n 



m + 



m' 



(Cochran 1964). The variance (V^g^ ^^ the total 

 number of fish in a stratum for the sample ele- 

 ment size reduced by 1/6 is 



V, =— Y 



nbf 



mm 



\ 



1^6^ \m , m^ A" T , m^l 



The difference in variance between the different 

 sample element sizes is 



relative increase = 





1 



i-f 



The purpose of many surveys is to produce total 

 biomass estimates. These total biomass estimates 

 are made by expanding a density estimate (usual- 

 ly in the form of a catch per unit effort measure) 

 (Gunderson and Sample 1980) by the total area 

 (Cochran 1964). Since measurement of the area 

 involved can be made with relatively little error 

 compared with the density estimate, we ignore 

 error in area measurements in the following 

 discussion. The precision of an estimate will vary 

 inversely to its standard error. An index of preci- 

 sion (Pj) is: 



Pr 



dISE. 



(9) 



This index is the inverse of the coefficient of 

 variation and is used here because it varies 

 directly rather than inversely with precision. The 

 density of a population is equal to the mean of the 

 negative binomial distribution divided by the 

 sample element size (S): 



d = mlS 



(10) 



Vd = v,^ 



V, 



^- 6m + — 2- m+-— 



a-n L k J a^n L kJ 



where m — the mean of a number of tows of sam- 

 ple element size ( S ) , 

 S = a constant sample element size with 

 no variance. 



The variance of m is 



2 



an 



[m(6-l)]. 



(8) 



Although there is actually an increase in overall 

 variance by reducing sample element size with a 

 constant sample size, it will be relatively small 

 compared with the overall variance when m is 

 large in value and/or k small: 



V,, = {m + m^lk)ln. 



(11) 



Therefore the variance of the density estimate is 



Vj = {m + m^ lk)lnS'' 



(12) 



and from Equation (10) 



670 



