The coefficient of coördination, being an index number to show the closeness with which two rankings correspond, is dependent upon the differences between the rankings of the various individuals in the two measures being compared. The formula used is ρ = (6Σd^2)/n(n^2 − 1), where ρ stands for the coefficient of coordination, d stands for the difference between an individual’s rank in the two measures, and n stands for the number of individuals ranked in the two traits. The capital sigma, Σ, stands for the sum of whatever follows it, in this case the squares of the differences between the two rankings.
We may now employ the formula to find the coefficient of coördination between rank in educational measurements and rank in the teacher’s judgment as to intelligence. The difference between the ranks in column A and column B of the above table is given in the fourth column. Adelaide had a 12 in column A and a 19 in column B, so the difference (7) appears in the fourth column and its square (49) in the fifth column. Similarly the difference between Ruth’s 3.5 and her 15 is 11.5, the square of which is 132.25. Finding the squares of all the differences between rank in A and rank in B, and adding these squares together at the bottom of the table gives 1790.5, which may now be substituted in the formula for Σd^2. n, the number of pupils is in this case 28, and therefore n(n^2 − 1) is 28 (28 squared less 1) = 28 (784 − 1) = 28 × 783 = 21924. The substitution in the formula then goes as follows;
ρ = 1 − (6Σd^2)/(n(n^2 − 1)) = 1 − (6 × 1790.5)/(28 × 783) = 1 − 10743./21924. = 1 − .490 = .510
The coefficient of coordination between rank in the educational measurements and rank in the teacher’s estimate of intelligence for the sixth grade class is .51, which suggests the question of how to interpret a coefficient after it is found.
A coefficient of 1.00 would mean perfect coördination and would only be found when there were no differences whatever between the two rankings considered. Such a perfect relationship will probably never be found, except by some freak of chance, for even when a group of persons is retested with the same test there is almost certain to be some change in their relative standings. A coefficient of 0.00 would indicate no relation whatever between the two rankings, while a coefficient of –1.00 would mean perfect correlation of a negative sort, the person getting highest in one measure getting lowest in the other, the person scoring next to the highest in one scoring next to the lowest in the other, and so on. Perfect negative correlation is as infrequent as perfect positive correlation.