no. 3.] PSYCHOLOGICAL EXAMINING IN THE UNITED STATES ARMY. 623 



This requirement is not consistent with the fact that regression parabolae commonly do not 

 give admissible results as extrapolation formulae. On the other hand, a linear regression 

 equation, if satisfactorily descriptive of observation may generally be trusted to give good 

 extrapolated results for short distances at least beyond the ends of the range of observation. 

 Thus, when the regression cubic for alpha on beta (fig. 1) fails completely as an extrapolation 

 formula, Prof. Pearson considers a straight line as quite satisfactory even at rather great dis- 

 tances from the last observation point to which the line is fitted. 



Since the skew relationship between alpha and beta total scores can be traced to the 

 varying limitations of range of the separate alpha and beta tests, the problem of combination 

 of tests can be dealt with most directly by starting from the interrelations of the separate tests, 

 provided these form a linear system within the portions of the total range of intelligence common 

 to the tests taken in all possible pairs. 



Without making precise tests of the linearity of regression of each test on every other, 

 we have considered that but slight errors would be made if we treated our system of variables 

 as a linear one. Group X (see Chap. I, sec. 2, page 555) provides a set of data upon which 

 to base the plan of analysis. For this sample at least we are able to drop the terms alpha and 

 beta and deal with 16 variables (8 alpha tests, 7 beta tests, and Stanford-Binet mental age) 

 instead of 3, gaining an approximately linear system at the expense of the increase in number 

 of variables. We can immediately distribute the 1,047 cases of the experimental sample 

 upon a scale of the sort we have already suggested for the common denominator for the various 

 types of examination, viz, a scale of ratings obtained by summing scores in all tests; provided, 

 however, that we treat zero soores, not impartially with other scores, but consider them rather 

 as defects in the record, and that we supply values to substitute for them by means of multiple 

 regression equations based on all the tests in whioh points are earned. Such a distribution 

 might be reasonably regarded as descriptive of the intelligence of our sample measured by a 

 nearly linear scale, i. e., a scale equal subdivisions of which correspond to approximately equal 

 ranges of intelligence. 



We can not here enter into a lengthy philosophical discussion of the significance of different 

 types of frequency distribution for the linearity of the scales used in obtaining them, but one 

 point needs to be considered. Equality of the successive segments of a scale may be defined 

 in various ways. If we transpose some of the elements, say centimeters of a meter rule, we will 

 obtain exactly the same measurements as before, whether we measure the length of bone, the 

 height of plants, or the thickness of cakes of ice. This is true because the centimeter as a unit 

 of measurement is definable quite independently of the laws of growth of bones or plants or 

 of the physical conditions requisite to the freezing of ice. But the "unit" of measurement in 

 a psychological test presents a fundamentally different situation. As an element in a quanti- 

 tative scale of intelligence, or accuracy, or what not, it is definable solely in terms of the variable 

 measured. Therefore, for an answer to the question whether a given scale of intelligence is 

 linear, we lack, so to speak, a fixed point of reference. We may assume that intelligence in 

 the unselected population is distributed normally, but we shall never get beyond assumption 

 until we have a scale the "units" of which are equal, and demonstrably so, from other premises 

 than our original assumption. On the other hand, since the frequency distribution of variates, 

 the quantitative aspects of which depend upon a large number of partially independent and 

 relatively small factors, is precisely the Gaussian normal curve, and since most psychological 

 tests are anything but measures of specific, isolated types of reaction, but rather measures of 

 reactions determined by a great number of partially or completely independent and, considered 

 singly, almost insignificant factors, we might argue that in the case where an approximately nor- 

 mal distribution arises the "units " of the scale used are actually practically equal. The validity 

 of this argument rests, of course, upon the correctness of our supposition in regard to the 

 number, magnitude, and independence of the factors which determine the reactions measured. 

 This is, in fact, the assumption from which we proceed in treating our system of 16 variables as 

 a linear one. We consider that the frequency distributions of 13 out of the 16 variables suffi- 



