160 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



to give estimates of equal precision. If, for exam- 

 ple, we sample area A with sampling unit a l; and 

 then sample area A/2 with sampling unit a 2 

 obtaining an estimate equally precise, we say that 

 sampling unit a t is one-half as efficient as sampling 

 unit a 2 . 



Heterogeneity is defined as a nonrandom condi- 

 tion where the presence of an item at any given 

 point tends to increase the probability of another 

 item being in its immediate neighborhood. Het- 

 erogeneity is to be distinguished from another type 

 of nonrandomness in which items, through com- 

 petition or whatever cause, act to repel each other 

 so that they tend to fill the available space equi- 

 distant from one another. With heterogeneity, as 

 here denned, the variance is always greater than 

 the mean. 



The problem of sampling with different sizes of 

 sampling units has been examined by Beall (1939) 

 whose statement of the problem is followed closely 

 in equations 1 through 7 below. It is hoped that 

 the proofs which follow in equations 8 through 10 

 will be helpful in clarifying some of the confusion 

 which appears to exist concerning the problem. 



Consider N strata of equal size and a sampling 

 unit of area k. If the total area of the itli stratum 

 is A, the total number of sampling units it contains 

 is A/k=M. M will, of course, vary with the size 

 of the sampling unit. 



Let X be the total number of fish in the N 

 strata. Let u i} be the number of fish in the jth 

 sampling unit (j=l,2, ... M) in the ith stratum 

 (i=l,2, . . . N). Let u be the average number of 

 fish per sampling unit calculated over the entire 

 N strata. Then 



X=MNu (1) 



Within the ith stratum, let the mean value of 

 u tJ be u t and the variance 



ai 



1 M 



- s (u u -u t y 



M-i pi 



(2) 



Let m t denote any number of samples taken in the 

 ith stratum. Let the number of fish observed per 

 sampling unit be x n , x i2 . . . x imf with mean x t 

 and estimated variance 



M being equal for all strata, Beall (1939) states: 

 "The best linear estimate of X, that is, the esti- 

 mate with the minimum S. D. will be 



F=MJ}x t ." 



The standard deviation will be, when m t sampling 

 units have been drawn at random 



a F -- 



-MX 



Mi(M t —m t ) 



")} 



(4) 



where M ( is the total number of sampling units 

 contained in the ith stratum. 



Since the strata are all equal in area, M t —M and 

 m i =m, so that equation 4 may be written 



<r F =y 



(5) 



M( M-m) * a , 



MAT' is the total number of sampling units. Let 

 MN=M„. mN is the total number of samples 

 taken. Let mN—m . Substituting in equation 

 5 we have 



0> = 



■V 



{ M.-m t )M„ 1 ^ 2 



m 



(6) 



Since M and m are fixed, the accuracy of the 

 estimate of F is determined by the average vari- 

 ance within strata: 



N IT 2 



The sampling unit which gives the smallest value 

 to a 2 gives the greatest accuracy to F, the esti- 

 mate of X. 



With the number of strata and the fraction of 

 the area to be examined fixed, a„ may be supposed 

 to be affected by an increase in the size of the 

 sampling unit. 



Let a F =a F ' with sampling unit fc=l. Let 

 a„=a/' with sampling unit k > 1. When fc=l, 

 M=M', and m=m'. When k > 1, M=m", and 

 m=M". For each value of k, there will be a 

 value of a 2 and M"/M=m"/m=l/k. By ele- 

 mentary algebra, we obtain from equation 5 



i mi 



(3) 



^V^-W^- 



(7) 



(Tj? 



