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FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



180 190 200 210 220 230 240 



STANDARD LENGTH IN MILLIMETERS 

 Figure 5. — Fecundity-length regression F=6A''. 



250 



formula Y=bX^. The use of this formula makes 

 the curves much easier to compare, since the Y- 

 intercept is taken as zero for all samples and the 

 slopes may be compared directly. 



When the curve is fitted by the least squares 

 method to the data, using the formula Y=a + bX^, 

 both the a and b values are determined by the data. 

 The sum of the plus deviations will equal the sum 

 of the minus deviations, while the sums of the 

 squares of the deviations will be at a minimum. 

 When the formula Y=bX^ is used, the T-intercept 

 is forced to be zero, and both of these conditions 

 cannot be met if the distribution of data is at all 

 skewed. A regression line about which the sura 

 of the plus deviations equals the sum of the minus 

 deviations can be obtained using a value obtained 

 2F 



from the formula b= 



2(X') 



A line about which 



the sum of the squares of the deviations is at a 



minimum can be obtained using b= 



ZiX^Y) 



Al- 



^{xy 



though there is little difference between the b val- 

 ues derived by these two methods ^ from the data 

 for the 116 sardines, the latter method has been 

 used because the standard errors of estimate and 



' Simpson apparently used the formulai) = z:f -^^ j -^iV which also gives a 6 

 value differing very little from the other two for the Ufi sardines. 



correlation coefficients based upon it should theo- 

 retically be more nearly comparable with those 

 obtained in conjunction with the least squares 

 methods in which the J^-intercept is determined 

 by the data. 



Figure 5 shows that samples SP-8 (Jan. 11) and 

 SP-9 (Jan. 24) almost coincide and that samples 

 SP-11 (Feb. 9),SP-12 (Feb. 21), and SP-13 (Feb. 

 27) show an apparent increase in fecundity as the 

 season progresses. Because of the restricted range 

 of lengths and the great variation in ovum count 

 at any given length, no significance is attached to 

 this apparent temporal fecundity increase. In 

 figure 6 are plotted the regression lines for each of 

 the 5 samples, fitted (least squares) by the formula 

 Y=a+bX (which differs most from the formula 

 Y=bX^). In table 4 each of the 10 possible com- 

 binations of pairs of b's is compared by t test for 

 the possibility of significantly different slopes for 

 both formulas. There are no pairs of regression 

 slopes that are significantly different when both 

 formulas are taken into consideration. The ap- 

 parent significant difi"ereiices in the slopes of the 

 pairs, SP-8 and SP-9, and SP-8 and SP-13, when 

 the formula Y=a+bX is used are primarily a re- 

 sult of the comparatively great variation in Y 

 values and the restricted range in X values. 



