360 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



400 600 800 1000 1500 2000 



TOTAL LENGTH mm. 

 Figure 1.- — Relations between length of pectoral fin and 

 total length. Open circles and fine line represent Costa 

 Rioan data. Solid circles and heavy line represent 

 Hawaiian data. 



lines depicted in the figures were in every case fitted 

 to the original data and not to the class means. 



As may be seen from figure 1, the pectoral fins 

 of Hawaiian ycllowfin tima, over the size range 

 considered, are on the average longer than those 

 of Costa Rican fish, and the difl^erence increases as 

 the size of fish increases. No elaborate statistical 

 analysis is required to show that these samples 

 caimot be considered as arising from the same 

 population. If inspection of the figure itself is not 

 sufficiently convincing, a very simple test suffices 

 to show that the probability of the two samples 

 arising by random sampling from a single popu- 

 lation is very small, regardless of whether or not 

 the growth law on the basis of which the regres- 

 sions were calculated is exactly correct. Under 

 the hypothesis that the Costa Rican sample was 



drawn from the same population as the Hawaiian 

 sample, we should expect the points for Costa 

 Rican fish to be half the time above and half the 

 time below the corresponding values predicted 

 from the Hawaiian sample. For each size class, 

 the Costa Rican value falls below the value which 

 would be expected on the basis of the Hawaiian 

 sample. The probability of this occurring by 

 chance alone for all 10 Costa Rican points is {]i) '" or 

 1 chance in 1024; it is, then, most imlikely. 



In figure 2 are plotted values of logarithm of 

 length of second dorsal fin against logarithm of 

 total length. This transformation yields a linear 

 regression for the Costa Rican sample, the fish 

 in which are from 54 cm. to 157 cm. in total length. 

 Similarly, the Hawaiian data for fish 62 cm. and 

 over in total length are rather well fitted by a 

 linear regression, as shown in the figure (we have 

 no Hawaiian specimens between 54 cm. and 62 

 cm.). We have also plotted in the figure the 

 second-degree polynomial that fits the Hawaiian 

 data for all sizes of fish in our sample. It is 

 obvious, whichever regression we employ for the 

 Hawaiian fish, that the second dorsal fins of yellow- 

 fin tuna from waters of the Hawaiian Islands grow, 

 relative to tot al length , faster than those of yellow- 

 fin tuna from waters ofl^ Costa Rica. The differ- 

 ence in fin lengths is small at smaller sizes of fish, 

 but increases with size of fish until among large 

 fish the difference is very striking. 



As may be seen from figure 3, the same situation 

 obtains for the length of anal fin relative to total 

 length. As has been reported for Costa Rican fish 

 and African fish, the variability of fin lengths of 

 second dorsal and anal fins, even on a logarithmic 

 scale, is not entirely independent of size of fish, 

 but tends to be greater at larger sizes. For this 

 reason the values of s for the corresponding equa- 

 tions in table 2 and on page 359 are average values, 

 and wdl be a little too small at large fish sizes and 

 too large at small sizes. 



Comparison of the linear regressions of figures 2 

 and 3 majj^ be made by means of analysis of 

 covariance (Kendall 1946, p. 237 et seq.); or, 

 without reference to regression equations, we may 

 simply compare the mean values of the several 

 size classes and , foUowmg the same sort of reason- 

 ing as above in the case of the pectoral fin, arrive 

 at the conclusion that the probability of the sam- 

 ples being drawn from a single population is very 

 small. 



