154 



Fishery Bulletin 88(1), 1990 



a bivariate normal distribution. In tliis case the proportion of the population expected to be enclosed by the ellipse 

 may be estimated as follows. 



Sokal and Rohlf (1981) state that if 



Ai  A2 • (n-l)  2 



C = - -  i'„|2,„-2| 



(n - 2) 



then the ellipse is expected to enclose 100 • (1 - a)% of the observations. 



The a that corresponds to the ellipse used in this study can be determined by finding the value for a such that 



A, -A,- (n-l) • 2 



A, • A^ = C = ^ ^;; ^a[2.n-2\< 



which can be rewritten as 



o[2.»i-2] 



(w-2) 

 in - 1) • 2 



If /( is very lai-ge, then (n -2)/(n - 1) is close to 1 and 



^ o|2.a>| 



which corresponds to a = 0.6065. Thus the ellipse is expected to enclose approximately 40% of the observations. 



Appendix 3 



The usual stratified estimate of variance is 



Xa,,;" • YavN, 



hi 



YarN,, = 



A,, 



(Jessen 1978) 



which may be rewritten as 



1a,,,- ^ • VariV;,,. + Ia,,,,,^ • VariV, 



h L 



VkrNi, = ^ 



where ?'+ refers to sectors with at least 2 samples, and ('n refers to sectors with 1 or samples. 



There are no data to estimate Var N),, directly, so it is estimated with the average variance from sectors 

 where there are at least two samples. 



^A,,J  YavNu, 

 Var N|^^^ = — — r^^ , for all i„ 



Let 



This estimate of Var N/,,^^ is substituted back into the formula estimating Var N/,, giving 



I A,,/ • VariV,.^ 



TAi,,J  VariV;,,^ + 1a 



h,„ 



VarN,. = 



l/u. 



which simplifies to 



1A;,, 2 . VariV,,. 



Var N,, 



A,2 



4,2 



1 + 



Ai,. 



