THE QUARTERLY REVIEW OF BIOLOGY 



zero than the normal curve calls for; and 

 accordingly we can hardly conclude from 

 this that their true value is not zero. 



Comparison of frequency distributions of tetrad differences 

 for mental tests (Browti) and physical traits of 



maturity (Gates) 

 (Calculated from Spearman, pages 143 and 147) 





NUMBER 





OF 





TETRAD 



CLASS 

 INTERVALS 



DIFFER- 

 ENCES 





I 



V 





£ 







O - O.5 



II 9 



J 3 



O.5-I.5 



73 



46 



1.5- z. 5 



13 



40 



*■• 5" 3-5 



5 



2.6 



3-5- 4-5 





16 



4-5-5-5 





*3 



5.5- 6.5 





13 



6.5-7.5 





10 



7.5-8.5 





7 



8-5-9-5 





6 



9.5-10.5 





8 



10.5-11.5 





4 



11.5-11.5 





X 



over ix . 5 





6 



The class intervals arc in 

 terms of the theoretical mean 

 standard deviation of a tetrad 

 difference. This has the follow- 

 ing values: 

 Brown's correlations 0.0514 

 Gates' correlations 0.0333 

 The table gives only positive 

 values of tetrad differences; the 

 distribution is symmetrical, as 

 shown in the curve. 



The class o to 0.5 covers only 

 a half interval in the table; the 

 corresponding class in the figure 

 covers the full interval from 

 —0.5 to +0.5 



Spearman also presents an interesting 

 comparison in the distributions of tetrad 

 differences for a set of correlations of 

 bodily dimensions (McDonnell, Biometrika, 

 1901, 1) and of physical traits taken as 

 indicating degree of maturity (Gates, 

 Journal of Educational Research, 192.4, 

 p. 341). These distributions are of an 

 entirely different character from those of 

 mental tests previously mentioned. Using 

 as our standard of measurement the theo- 

 retical mean standard deviation of a tetrad 

 difference (derived by the formula given 

 by Spearman in his appendix) we find the 

 scatter of the mental tests is very much 

 less than that of the physical measure- 

 ments. A comparison of the two indi- 

 cates that the system of causes in one case 



must be radically different from that in 

 the other. For purposes of comparison,. 

 I offer the distributions for the tetrad' 

 differences of Brown and for those of] 

 Gates. It will be noted that the unit ofl 

 measurement in each case is the mean stand- , 

 ard deviation of a tetrad difference, de- 

 rived by Spearman's formula 16A, Ap-' 

 pendix, p. xi. Clearly, the two dis-l 

 tributions are not even remotely similar. 

 The comparison, of course, proves nothing 

 as to the causes in either case, except that; 

 they must be different in the two cases. 



We now turn to a consideration of Pro- 1 

 fessor Spearman's theory in more detail. 

 Clearly we should properly consider his i 

 mathematics at some length. We shall, 

 however, confine ourselves to a few general 

 questions, which seem to us fundamental. ; 

 For the rest, the mathematical proofs in- 

 volved seem formally correct enough; but 

 unfortunately, a mathematical proof is i 

 only as sound as its initial assumptions, 

 and it is by no means easy for the present 

 reviewer to determine precisely what 

 Professor Spearman has assumed. One 

 thing, however, appears reasonably 

 clear; Professor Spearman assumes 

 throughout that his fundamental rela- 

 tions are linear. We have, for example, 

 the equation 



m ax — r ag g x + r a3a s ax 



where we might expect 



max = f (gx, Sax) 





the determination of the form of the func- 

 tion being one of the problems to be solved. 

 To this Professor Spearman would doubt- 

 less reply that in fact we have assumed 

 nothing of the sort; we have shown that 

 when (ab, cd) is zero, we can always write 

 (1). This, as far as it goes, is true; but 

 does not the proof involved assume that 

 the correlations r a b, etc., result from lin- 

 ear relationships? Have we not assumed 





