The first three columns of the following table give ranks in the place of the original figures which indicated numbers of errors in measurements and percentage in scholarship. Where two or more individuals are entitled to the same rank, the figure used is the middle value of the ranks. Thus in the case of the educational measurements scores, two girls made 16.5 errors. There are but two pupils making better showings, and therefore Ruth and Helen would normally rank third and fourth, but since we have no evidence as to which should rank third and which fourth, each is given a rank of 3.5. Similarly it will be observed that Alexander, LaMonte, and Leo each obtained a percentage of 93 in scholarship, therefore the three boys named share equally the fourth, fifth, and sixth rank, each being given 5 as a rank; and the next highest pupil, Amelia with a percentage of 92, is given 7 as a rank.
| RANKING OF SIXTH-GRADE PUPILS | DIFFERENCES IN RANKINGS | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Name of Pupil | A Educational Measurements | B Teacher’s Ranking | C Scholarship Marks | A to B | A to C | B to C | |||
| d | d2 | d | d2 | d | d2 | ||||
| Adelaide | 12 | 19 | 18.5 | 7 | 49 | 6.5 | 42.25 | 0.5 | 0.2 |
| Ruth | 3.5 | 15 | 9 | 11.5 | 132.25 | 5.5 | 30.25 | 6 | 36 |
| Alexander | 9 | 7 | 5 | –2 | 4 | –4 | 16 | 2 | 4 |
| LaMonte | 14 | 6 | 5 | –8 | 64 | –9 | 81 | 1 | 1 |
| Earl | 28 | 18 | 24 | –10 | 100 | –4 | 16 | –6 | 36 |
| Joseph | 6 | 20 | 18.5 | 14 | 196 | 12.5 | 156.25 | 1.5 | 2.2 |
| Amedeo | 27 | 14 | 18.5 | –13 | 169 | 8.5 | 72.25 | 4.5 | 20.2 |
| Leo | 16 | 3 | 5 | –13 | 169 | –11 | 121 | –2 | 4 |
| William | 17 | 9 | 21 | –8 | 64 | 4 | 16 | –12 | 144 |
| Isabel | 8 | 21 | 25 | 13 | 169 | 17 | 289 | –4 | 16 |
| Ida | 13 | 4 | 3 | –9 | 81 | –10 | 100 | 1 | 1 |
| Hazel | 1 | 10 | 9 | 9 | 81 | 8 | 64 | 1 | 1 |
| Frederick | 23 | 26 | 16 | 3 | 9 | –7 | 49 | 10 | 100 |
| Charles | 20 | 13 | 18.5 | –7 | 49 | –1.5 | 2.25 | 5.5 | 30.2 |
| Edward | 11 | 1 | 2 | –10 | 100 | –9 | 81 | –1 | 1 |
| Benjamin | 22 | 24 | 26 | 2 | 4 | 4 | 16 | –2 | 4 |
| Bruce | 19 | 22 | 14 | 3 | 9 | –5 | 25 | 8 | 64 |
| Alden | 18 | 12 | 14 | –6 | 36 | –4 | 16 | –2 | 4 |
| George | 21 | 17 | 14 | –4 | 16 | 7 | 49 | 3 | 9 |
| Alice | 10 | 11 | 12 | 1 | 1 | 2 | 4 | –1 | 1 |
| Almira | 2 | 5 | 1 | 3 | 9 | –1 | 1 | 4 | 16 |
| Helen | 3.5 | 2 | 9 | –1.5 | 2.25 | 5.5 | 30.25 | –7 | 49 |
| Elizabeth | 24 | 23 | 27 | –1 | 1 | 3 | 9 | –4 | 16 |
| Amelia | 7 | 8 | 7 | 1 | 1 | 0 | 0 | 1 | 1 |
| Edwin | 5 | 16 | 11 | 11 | 121 | 6 | 36 | 5 | 25 |
| Robert | 25 | 28 | 28 | 3 | 9 | 3 | 9 | 0 | 0 |
| Edna | 15 | 27 | 23 | 12 | 144 | 8 | 64 | 4 | 16 |
| Samuel | 26 | 25 | 22 | –1 | 1 | –4 | 16 | 3 | 9 |
| Σd2 = 1790.5 | 1411.5 | 611.0 | |||||||
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 = (6Σd2)/n(n2 − 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 Σd2. n, the number of pupils is in this case 28, and therefore n(n2 − 1) is 28 (28 squared less 1) = 28 (784 − 1) = 28 × 783 = 21924. The substitution in the formula then goes as follows;
| ρ = 1 − | 6Σd2 | = 1 − | 6 × 1790.5 | = 1 − | 10743. | = 1 − .490 = .510 |
| n(n2 − 1) | 28 × 783 | 21924. |
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.
The coefficient found between the teacher’s estimates of intelligence and the results of educational measurements, .51, indicates a really useful degree of coördination. Unless a Mentimeter test shows a coefficient of coordination of .25 or more with the production records (or other reliable measure of true ability), it may be considered as having little value in helping to select and differentiate men for that particular line of work. If the coefficient is above .5, the test is quite useful, and the nearer the coefficient approaches 1.00 the more confidence one may place in the test as a means of selecting and classifying men in that particular field.
The sixth column of the table on page [329] gives the difference between the test results rankings and the scholarship marks rankings, and the seventh column gives the squares of these differences, the sum of these squares being given at the bottom of the seventh column as 1411.5. By substituting in the formula,