Kendall and Picquelle Egg and larval distributions of Theragra chslcogrsmnw 



153 



Appendix Figure 1 



Ellipse with the center at (A'. Y ) and the major and 

 minor axes each two standard deviations long. 



The criterion that the leng:th of the major axis is 2  A] implies that 



A, = (A- -A')- + (Y-Y)- 



based on the Pythagorean theorem. By substituting in the formulas for A' and Y given above, this formula can 

 be rewritten as 



C 6r • C C 



solving for C: 



C 



A, = 



A.,. 



A2-(l+V) A2-(l + 6i2) A2 



Thus the equation for the ellipse is 

 Var(F) • {X-X)- 



Cov(A,i') • (}'-!')  (A -A') + Var(A) • (Y -Y)- = A, 



Ao, 



This ellipse is estimated by substituting the estimated values for the variances, covariance, means, and latent roots. 



No distributional assumptions ^ire needed to estimate the centroid, the slope of the principal axis, or the shape 

 of the ellipse. The centroid is the standard measure of central tendency and is valid as long as the stations repre- 

 sent a valid sample of the survey area. The slope of the principal axis is defined by the line that minimizes the 

 sum of the squared perpendicular distances between the data and the line. The size and shape of the ellipse are 

 set by the variance of the data about the centroid. The ellipse is simply a two-dimensional version of a mean with 

 standard error bars. 



In the univariate case, the mean and standard error provide information about the location of the data's center 

 of gravity and the amount of dispersion about that center, no matter what probability function the data come 

 from (Shuster 1982). However, if the data are from a normal distribution, a confidence interval may be computed 

 directly from the mean and standard error. If the data are from a bimodal or skewed distribution, the mean and 

 standard error are still valid, but viewing the mean with standard error bars can give the false impression that 

 the data are distributed symmetrically about the mean, when in fact the confidence interval would be asymmetrical 

 if the data are asymmetrical. 



Similarly, caution should be employed in interpreting the centroid and ellipse. No matter what probability 

 function describes the spatial distribution of the eggs and larvae, the centroid is the measure of central tendency 

 and the ellipse shows the orientation of the data and the amount of dispersion along the two axes. However, the 

 ellipse can only be considered a contour or an "equal frequency ellipse" about the centroid if the data are from 



