648 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol.xv, 



The above constants are in terms of class intervals. A class interval is taken as 25 points 

 of the combined scale. When it is considered that the range of the combined scale is approxi- 

 mately 600 points, and that our experimental sample is actually distributed over 20 of these 

 25-point class intervals, it will be apparent at once that the means and standard deviations 

 given above, especially the former, differ only negligibly, and that means and standard deviations 

 of subgroups of the principal sample are probably reliable to a degree commensurate with 

 interpretable differences. What significance is to be attached to the values of the third and 

 fourth moments and to the /3's is not clear. We believe, however, that the approximate sym- 

 etry of all four of the distributions warrants the conclusion that one of the objects aimed at 

 from the beginning, viz, the distribution of intelligence upon a linear or nearly linear scale has 

 been reached. If this has been attained, then the means and standard deviations of the sub- 

 groups of the principal sample which it may be desirable to compare are in less danger of 

 misinterpretation than they would be if the combined distribution had been based upon a scale 

 much foreshortened at the lower end as is the scale of alpha total scores. 



The theoretical mean combined scale score of our experimental sample as obtained by 

 summing means for all alpha and beta tests and mental age (treated as so many points, on the 

 basis of one point per quarter year) is 172.56, or, in terms of class intervals of 25 points, 13.901. 

 Since our combined scale is only an hypothetical one, we may as well drop the use of the original 

 points and adopt any convenient class interval as the unit of the scale. Thus, calling the interval 

 from — 150 to — 126, 1, — 125 to — 101, 2, etc., we have a scale of, theoretically, 24 subdivisions 

 or units. In terms of these class intervals or units the standard deviation of our experimental 

 sample is theoretically 3.245, as calculated from the standard deviations and intercorrelations 

 of all component variables. The close agreement between these expected values and the ones 

 actually obtained from the various combinations of alpha and beta total score distributions of 

 the sample that we have tried appears to be justification of the validity of our transformation 

 tables in addition to that given by the approximate agreement among the constants from the 

 four different trials of the method. 



An apology should be offered for the absence of probable errors and other constants relating 

 to the reliability of our results. Failure to provide these important auxiliaries of statistical 

 analysis has been due to a combination of circumstances, most important of which was the lack 

 of both time and methods for their calculation. The pressure of time has required the focusing 

 of attention upon the fundamental steps necessary to reach the goal, and it has been necessary 

 to be content with the assumption that the size of the experimental sample was sufficient to 

 justify results from a "common-sense" point of view, notwithstanding the failure to provide 

 precise determinations of their validity. 



No claim is made for completeness in treatment of the problem stated at the beginning of 

 this chapter, but it appears that in a rough way the more important difficulties of analysis 

 peculiar to the statistics of psychological data have been overcome, if only in a way that empha- 

 sizes more sharply than ever the need for more precise methods. 



Section 2. — Interrelation of alalia and beta tests. 



The following regression equations are by-products of the study outlined in the preceding 

 section. They are added because they tell a condensed story of the interdependence of the 

 tests, and because they are of practical value in predicting probable scores on tests in which 

 individuals may have for various reasons failed to score. 



All of these equations were calculated directly from the correlation determinant given in 

 table 155, the alpha-alpha section, and by the formula given on page — . Some of the longer 

 equations are given in two forms: In the usual form, X r = a + b 2 X 2 + b 3 X 3 + . . . b D X a ; 

 and, secondly, in the form 



Xj _ _ IRu ^ Xj Rj3 _ Xj _ Rin ^ Xa 



