THE QUARTERLY REVIEW OF BIOLOGY 



(ab, of) = o for all tetrads obtained from 

 a table of correlations, then the abilities 

 involved can be analyzed into a (g, s) 

 system, and that if (ab, cd) =(= o, no (g, s) 

 system can be found; we shall require 

 some more complicated system. Spear- 

 man's procedure in any given case, then, 

 is to construct a table showing the inter- 

 correlations of abilities, as measured by 

 tests or otherwise, and to determine 

 whether or not the tetrad difference 

 vanishes. If it does, the abilities involved 

 can be analyzed into the form of equa- 

 tions (i). 



In practice, of course, the correlations 

 which we obtain will always be affected 

 by sampling errors, and accordingly our 

 tetrad differences will not in general be 

 zero even if they are derived from a (g, s) 

 system, but will tend to fluctuate around 

 zero. It therefore becomes important to 

 determine the permissible range of fluctua- 

 tion within which we may regard the 

 tetrad difference as sensibly zero. This 

 requires, clearly, determining the theoreti- 

 cal standard deviation of a tetrad differ- 

 ence in terms of the correlations on which 

 it is based. This can be done by the usual 

 procedure; and Spearman proceeds to make 

 comparisons of the observed values of his 

 tetrad differences with the theoretical 

 probable error (derived by the usual con- 

 vention that the p.e. = 0.6745 °0- Now 

 this assumes, among other things, that 

 the frequency distribution of tetrad dif- 

 ferences is approximately normal. This, 

 however (as was pointed out in a review 

 in Nature, August 6, 192.7, pp. 181-183) 

 is by no means even a plausible assumption. 

 We know that, for small samples and for 

 even moderately large correlations, the 

 frequency distribution of correlation coeffi- 

 cients is widely different from the normal. 

 Further, in a table of tetrad differences, 

 the individual differences have correla- 

 tion (over and above that due to their 



limited number) due to the fact that the 

 same correlation coefficient enters into 

 different tetrads. Thus the presence of a 

 common element will introduce correla- 

 tion between such quantities as (ab, cd) 

 and (ae, fg). What kind of frequency dis- 

 tribution might result is by no means 

 obvious; but to assume it Gaussian seems 

 clearly unjustified until we have at least 

 experimental evidence that it is approxi- 

 mately so. We should feel easier _ in our 

 minds if Professor Spearman had given a 

 frequency distribution of tetrad differences 

 for a set of correlations actually known to 

 be derived from a (g, s) system. 



Professor Spearman does, however, give 

 us something upon which to form an 

 idea of how tetrad differences may vary. 

 He has taken two tables of correlations of 

 mental tests, in one of which 14 tests were 

 applied to 37 persons, and has worked out 

 the frequency distribution of tetrad differ- 

 ences for the two cases. Upon the fre- 

 quency distributions so obtained he places 

 normal curves having standard deviations 

 equal to the theoretical and mean stand- 

 ard deviations of the tetrad differences of 

 his tables. Graphically, the fit is good; 

 but unfortunately he gives only a graphic 

 comparison. The reviewer in Nature, 

 cited above, stated that an actual goodness 

 of fit test indicated that the fit is really 

 very bad in one case, and distinctly not 

 good in the other. 



Hoping to obtain more light, I have 

 myself calculated for the correlations of 

 W. Brown, cited by Spearman (p. 147) the 

 frequency distribution of tetrad differ- 

 ences. In this case I find that a normal 

 curve will not give even an approximate 

 fit; and that even the Pearson Type VII 

 curve indicated by the values of the ob- I 

 served moments is an extremely bad fit 

 to the data. On the other hand, the 

 difficulty in the fit is that the tetrad differ- 

 ences are more closely concentrated about 



