Since the abilities which together constitute arithmetical ability are thus specialized, the individual who is the best of a thousand of his age or grade in respect to, say, adding integers, may occupy different stations, perhaps from 1st to 600th, in multiplying with integers, placing the decimal point in division with decimals, solving novel problems, copying figures, etc., etc. Such specialization is in part due to his having had, relatively to the others in the thousand, more or better training in certain of these abilities than in others, and to various circumstances of life which have caused him to have, relatively to the others in the thousand, greater interest in certain of these achievements than in others. The specialization is not wholly due thereto, however. Certain inborn characteristics of an individual predispose him to different degrees of superiority or inferiority to other men in different features of arithmetic. We measure the extent to which ability of one sort goes with or fails to go with ability of some other sort by the coefficient of correlation between the two. If every individual keeps the same rank in the second ability--if the individual who is the best of the thousand in one is the best of the group in the other, and so on down the list--the correlation is 1.00. In proportion as the ranks of individuals vary in the two abilities the coefficient drops from 1.00, a coefficient of 0 meaning that the best individual in ability A is no more likely to be in first place in ability B than to be in any other rank. The meanings of coefficients of correlation of .90, .70, .50, and 0 are shown by Tables 15, 16, 17 and 18.[26] [26] Unless he has a thorough understanding of the underlying theory, the student should be very cautious in making inferences from coefficients of correlation. TABLE 15 DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .90 ====================================================================== |10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+----- 1st tenth | | | | | .1 | .4 | 1.8 | 6.6 |22.4 |68.7 2d tenth | | | .1 | .4 | 1.4 | 4.7 |11.5 |23.5 |36.0 |22.4 3d tenth | | .1 | .5 | 2.1 | 5.8 |12.8 |21.1 |27.4 |23.5 | 6.6 4th tenth | | .4 | 2.1 | 6.4 |12.8 |20.1 |23.8 |21.2 |11.5 | 1.8 5th tenth | .1 | 1.4 | 5.8 |12.8 |19.3 |22.6 |20.1 |12.8 | 4.7 | .4 6th tenth | .4 | 4.7 |12.8 |20.1 |22.6 |19.3 |12.8 | 5.8 | 1.4 | .1 7th tenth | 1.8 |11.5 |21.2 |23.8 |20.1 |12.8 | 6.4 | 2.1 | .4 | 8th tenth | 6.6 |23.5 |27.4 |21.1 |12.8 | 5.8 | 2.1 | .5 | .1 | 9th tenth |22.4 |36.0 |23.5 |11.5 | 4.7 | 1.4 | .4 | .1 | | 10th tenth|68.7 |22.4 | 6.6 | 1.8 | .4 | .1 | | | | ====================================================================== TABLE 16