440 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XV, 



No anthropologist would think of determining the correlation between height and weight 

 of a human tribe from data obtained by the use of apparatus that allowed him to weigh accu- 

 rately all individuals but allowed correct measurement of height of only the tallest 70 per cent, 

 the remaining 30 per cent being recorded merely as not taller than a certain amount. This is 

 precisely the case with certain of the alpha tests. The position is taken here that the alpha 

 tests are different techniques for the measurement of a rather general, but ill-defined, attribute 

 of the human being. It seems, therefore, that the comparison of techniques may be much more 

 precisely made if the absolute efficiency of each is determined within the limits of its effective 

 range, but stated as the value it would have if the technique in question were extended or 

 extendable to cover an unlimited range, with supplementary information as to the exact nature 

 and extent of the existing limitation. This is proposed as a needed improvement upon the 

 highly inaccurate and sometimes misleading statement of correlation coefficients calculated 

 uncritically from data containing relatively large proportions of unmeasured cases, which are 

 falsely treated as measured cases. 



The above proposal can be carried out in the following way: We calculate in the ordinary 

 manner a product-moment n, but ignoring completely all cases in the zero-class interval, or if 

 there is good reason to believe that scores of two or three points are accidental and nonsignificant 

 in a great many cases, even these may be ignored. The resulting n is substituted in the formula: 



r = - 



n 



Vi + (i-n 2 )J 



where J is a quantity depending on the percentage of cases ignored during the calculation of fi, 

 and may be determined from Sheppard's tables by the following formula: 



J- 



'4(1 +«) 



-fea(i + «)) J 



For the normal correlation surface this formula is identical with (2), p. 8 and J = -^— 1. 



Table 101 presents the correlation coefficients for each alpha and beta test, and for alpha 

 and beta total scores, calculated for ratings treated as outlined on page 22, and corrected for 

 length of range by formula (3). It is based upon the correlation tables, 102 to 118. 



Table 101. — Statistical constants. 



ALPHA TESTS (672 CASES). 



BETA TESTS (416 CASES). 



1 Total score (uncorrected). 



