622 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol.xv, 



In the first place, it is evident from study of score distributions of each of the alpha and 

 beta tests that they are not coextensive as regards range of ability measured. The alpha 

 examination itself is not a homogeneous scale, but a composite of eight short scales, each much 

 more nearly homogeneous than the composite total alpha. Alpha must be treated as a com- 

 posite for two reasons: (1) Certain of its component tests are very much more difficult than 

 others, so that examinees of less than average intelligence do not in general register their 

 ability in all of the tests. (2) Owing to the fact that the total time allowed for the alpha 

 examination is rigidly apportioned among the eight tests, since all subjects begin and stop 

 each test according to a fixed schedule, the collection of the eight tests into an "examination" 

 is mainly a matter of administrative convenience, with relatively little implication of quan- 

 titative or qualitative equivalence. It follows from what has just been stated that subjects who 

 are able to score in all of the alpha tests obtain total scores that are not comparable with total 

 scores earned by individuals who failed in some of the tests. The reason for this difference is 

 that in obtaining an individual's total score by adding together the scores in each test, scores 

 of are treated arithmetically the same as other scores. There are psychological reasons, how- 

 ever, why as a score can not be regarded as a point on a linear scale, although other scores 

 can be so regarded. This distinction between the mathematical zero and "psychological" 

 zero is an important one, for failure to recognize it is a serious pitfall in the statistical analysis 

 of psychological data. Zero as a test score means some unmeasured amount of ability that 

 is less than the amount required to earn a score of one point. Thus, if a high degree of ability 

 is necessary to earn a score of one point in a certain test, it is obvious that the range of ability 

 covered by the zero-score is a very wide one, although the ranges corresponding to scores of 

 one point, two points, etc., may be very short. A score of zero, therefore, does not mean no 

 ability at all; it does not mean the point of discontinuance of the thing measured ; it means the 

 point of discontinuance of the instrument of measurement, the test. Zero is an arbitrary 

 origin on the measuring scale, corresponding in some tests to much higher grades of ability 

 than in others; above zero we count in positive numbers and below it we should count in 

 negative numbers. Why in practice the range of negative scores is not used is the necessary 

 consequence of the methods of testing intelligence, but this limitation constitutes no obstacle 

 to the use of negative scores in the statistical analysis of test data. 



The fact that zero points are disposed at different levels of intelligence for different tests 

 would be of no consequence if negative scores could be registered ; but since they can not be, 

 the individual who fails to earn a positive score and is marked zero is actually thereby given a 

 bonus varying in value directly with his stupidity. Since in alpha the zero points in most of 

 the tests are relatively high in the scale of intelligence, many individuals of low intelligence 

 would obtain negative total scores, if their intelligence could betray itself in appropriate 

 negative scores in each of the component tests. As it actually works out, these low-grade 

 individuals are improperly piled up, in the total score distribution, above the zero point. 

 Thus the skewness or, better, piling-up of cases in the lower class intervals of the alpha total 

 score distribution is accounted for. 



The situation is somewhat different with beta total scores. Here certain tests, besides 

 being limited at the lower end of the acale, are also limited at the top, so that high-grade indi- 

 viduals do not have the opportunity to earn all that they are capable of earning. The beta 

 tests were primarily designed to measure low-grade intelligence, and they are for the most 

 part inadequate measures of high-grade intelligence. The piling up of cases in terminal class 

 intervals, which we interpret as indicativB of limitation of range, is predominantly at the 

 upper end of three of the seven beta tests, with a resulting negative skewness of the total score 

 distribution. 



These peculiarities of the components of the alpha and beta examinations also account 

 for the very marked skewness of regression of either variable on the other, that is apparent 

 from the form of the regression curves calculated by Prof. Pearson. To be satisfactory a 

 regression equation for alpha or beta, and vice versa, must be serviceable for extrapolation. 



