152 Fi>;h(-rv Gullenn 88(I|, 1990 



And the variance of the total abundance is estimated by 



Var N = A-  War N. 



Appendix 2 



The centroid is simply the weighted bivariate mean of the sampled locations of the eggs or larvae. Its estimated 

 value depends on where the samples were taken and the size of the catch. If the samples are not evenly or ran- 

 domly spaced over the study area, then {X, Y) will be biased towards those areas with the highest sample density. 

 This bias can be avoided_by_either grouping the individual samples into strata (sectors) as done here, or by modify- 

 ing the computation of (X, Y) to account for unequal sample density as was done to estimate the mean and variance 

 for the Sette-Ahlstrom method (see Appendix I). 



The rotation of the ,Y and Y axes is specified by the slope of the principle axis, 6, (Sokal and Rohlf l'.)81): 



Cov{X,Y) 



Oi = . 



A, - Var(.Y) 



J.N,  (X,-X}  {Y,-Y) 

 where Cov(X,Y) = — 



'^ NA -1 



Ai = the first latent root of the variance-covariance matrix of A" and Y, 



= 0.5  [Var(A) + Var(r) + v^(Var(A) + Var(F))- - 4  (Var(A)  Var(y) - Cov(;(',y)2)], 



I.N,  (A,-X)2 

 Var (A') = ' ^ , and 



2! A' 



^N,  (F, -}•)■- 

 Yard") = 



I.N,\ - 1 



I j 



The ellipse about the centroid is specified by the standard deviations along the rotated axes. The standard 

 deviation in the direction of the major axis is equal to A] as defined above. The standard deviation in the direc- 

 tion of the minor axis is equal to X-,, which is the second latent root of the variance-covariance matrix of A and Y: 



I, = Var(A) + Var(}') - A,. 



Sokal and Rohlf (1981, box 15.5) give a formula for an ellipse of this shape but of a different size: 



Var(F)  (A -A)- - 2 Cov(A,y) • (Y-Y)  (A -A) + Var(A)  {Y-Y)- = C. 



By setting C to the appropriate value, this formula can define the ellipse as specified above and shown in Appen- 

 dix Figure 1. The value for C is derived from the formula for the point (A, 7) given in Sokal and Rohlf (1981, 

 box 15.5): 



A = A -h 1 and Y = Y + 6,  (A -A). 



A, -(1 + 6,2) 



