====================================================================== |10TH |9TH |8TH |7TH |6TH |5TH |4TH |3D |2D |1ST ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+----- 1st tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 2d tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 3d tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 4th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 5th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 6th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 7th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 8th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 9th tenth |10 |10 |10 |10 |10 |10 |10 |10 |10 |10 10th tenth|10 |10 |10 |10 |10 |10 |10 |10 |10 |10 ====================================================================== The significance of any coefficient of correlation depends upon the group of individuals for which it is determined. A correlation of .40 between computation and problem-solving in eighth-grade pupils of 14 years would mean a much closer real relation than a correlation of .40 in all 14-year-olds, and a very, very much closer relation than a correlation of .40 for all children 8 to 15. Unless the individuals concerned are very elaborately tested on several days, the correlations obtained are "attenuated" toward 0 by the "accidental" errors in the original measurements. This effect was not known until 1904; consequently the correlations in the earlier studies of arithmetic are all too low. In general, the correlation between ability in any one important feature of computation and ability in any other important feature of computation is high. If we make enough tests to measure each individual exactly in:-- (_A_) Subtraction with integers and decimals, (_B_) Multiplication with integers and decimals, (_C_) Division with integers and decimals, (_D_) Multiplication and division with common fractions, and (_E_) Computing with percents, we shall probably find the intercorrelations for a thousand 14-year-olds to be near .90. Addition of integers (_F_) will, however, correlate less closely with any of the above, being apparently dependent on simpler and more isolated abilities. The correlation between problem-solving (_G_) and computation will be very much less, probably not over .60.