FISHERY BULLETIN: VOL. 78, NO. 3 



either means or variances using one or a combina- 

 tion of the three sampling schemes? 



While our analyses are limited to data from 

 the Queen Charlotte Sound pilot survey and the 

 1977 survey, the questions repeatedly arise in 

 discussions of trawl surveys and thus are of 

 general interest. 



COMPARISONS OF RANDOM, 



STRATIFIED RANDOM, AND 



SYSTEMATIC SAMPLING 



Methodology 



Chapter 8 of Cochran ( 1964) discusses systemat- 

 ic sampling and presents methodology for choosing 

 among random, stratified random, and systematic 

 (every /jth) sampling. Similar discussions are 

 found in other sampling texts such as Hansen et 

 al. (1953) and Sukhatme and Sukhatme (1970). 

 The methodology used in comparing the three 

 sampling techniques assumes equal sampling 

 effort in each strata. If prior information indicated 

 that variance differs considerably among strata, 

 the optimal stratified random sampling scheme 

 would not be equal allocation of sampling effort 

 among strata. Unfortunately as shown by Abram- 

 son (1968), it can be difficult to obtain meaningful 

 information on within strata variance for trawl 

 surveys even if previous surveys have been made 

 in the area. The methodology also only considers 

 regularly spaced strata of uniform size. While 

 prior knowledge (catch records) made it possible to 

 design strata of unequal size on a large scale 

 basis, knowledge is insufficient to do so on the 

 scale considered in the analysis. The multispecies 

 aspects of the survey made it particularly difficult 

 to devise an optimal stratified random scheme. 



In this section we use Cochran's notation. How- 

 ever, instead of examining components of variance 

 for choosing among the three types of sampling as 

 Cochran did, we calculated the variances for each 

 type of sampling. Using the notation of Cochran, 

 let a population of k possible systematic samples 

 be represented by 



Member Systematic sample number 

 1 ... i ... k 



1 jii  Jh •■■ yk\ 



n 



yij 



yin 



yu 



yin 



ykj 



ykn 



where yij is the 7 th member of the jth systematic 

 sample. 



If the yifs are arranged as they actually occur 

 for a population of two systematic samples {k = 2) 

 with four (n = 4) members they appear as follows: 



Unit 12 3 4 5 6 7 8 



Variable y^ 3^21 y 12 3' 22 >'i3 3' 23 3'i4 >'24- 



In a systematic survey a number (i) is chosen 

 between 1 and k and then n members that are k 

 units apart are sampled. Under a scheme of 

 drawing one systematic sample, either units 1, 3, 

 5, and 7 or 2, 4, 6, and 8 would be observed. Under a 

 stratified random scheme one unit out of each of 

 four strata (units 1 and 2, 3 and 4, 5 and 6, and 

 7 and 8 ) would be chosen at random for observa- 

 tion. Under the corresponding random scheme 

 any four of the eight units would be chosen at 

 random to be observed. The example population 

 contains 2 possible systematic samples, 16 pos- 

 sible stratified random samples, and 70 possible 

 random samples. While more than 20'7f of the 

 possible random samples match with a stratified 

 random sample, <3% match with a systematic 

 sample. A systematic sample is much more struc- 

 tured or constrained than the other schemes. 

 Population variances of the means of random, 

 stratified random, and systematic sampling are 

 calculated as follows: 





(1) 



where V (yran) = variance of the mean calcu- 

 lated from random sampling 



(yran), 

 N=kXn, 



ran 



{y,j'yYIN-\, 



y, 1^^ 



660 



