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TABLE A.-Age-grade table, Dedham, Mass., 1917-18, showing the distribution of pupils and the method employed in computing r and

Ages (relative items; lower numbers are substitutes for the exact ages).

P2.

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1 Abbreviations: S-subject items; R-relative items; T-total; f-frequency; 2=summation of; r-coefficient of correlation; psecond coefficient of regression.

TABLE B.-Age-grade table, Goshen, Ind., 1917-18. showing the distribution of pupils and the method employed in computing r and

Ages (relative items; lower numbers are substitutes for the exact ages).

P2.

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TABLE C.-Age-grade table, Donora, Pa., 1917-18, showing the distribution of pupils and the method employed in computing r and p2.

Ages (relative items; lower numbers are substitutes for the exact ages).

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THE COEFFICIENT OF CORRELATION AND THE COEFFICIENT OF REGRESSION FOR THREE TYPICAL AGE-GRADE DISTRIBUTIONS.

An inspection of figure 5 might lead one to believe that little relationship exists between the age and grade of pupils in certain cities. For instance, what is the relationship between the age and grade of pupils in Dedham, Mass., where the proportion of underage pupils is very large, and in Goshen, Ind., where only a negligible percentage of pupils is under age, but where a comparatively large percentage of pupils is of normal age? In the former city pupils enter school at the age of 5, while in the latter city they do not enter until the age of 6. It is hardly fair, therefore, to measure these cities by the standard proposed above. A better conception as to the relationship between the age and grade of pupils in these cities is gained from the coefficients of correlation and regression. An inspection of the age-grade tables (A and B) for these cities discloses a certain similarity, there being comparatively few ages represented in any grade or relatively few grades enrolling pupils of any given age. The "compactness" of these distributions is in striking contrast to the scattered" distribution of pupils in the age-grade table for Donora, Pa. Let us proceed to measure these distributions by the coefficients of correlation and regression.

The coefficient of correlation is usually expressed as follows:

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A shorter method of computing this coefficient is proposed by Ayres? as follows:

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This formula has been employed in computing r in Tables A, B, and C. The coefficient of correlation for Dedham is found to be 0.949 and that for Goshen 0.947, while that for Donora is only 0.906. The first two coefficients indicate practically the same relationship between the age and grade of pupils in Dedham and in Goshen. It is seen, therefore, that a very close and essentially equal corre

2 A shorter method of computing the coefficient of correlation. By Leonard P. Ayres. Journal of Educational Research, March, 1920.

lation between age and grade exists in Dedham and Goshen. The differences so apparent in figure 5 are undoubtedly due to the fact that children enter school earlier in Dedham than in Goshen, one school system being graded about as well as the other. In Donora the coefficient is comparatively low, indicating that pupils having a wide range of ages will be found in each grade.

If all the pupils in Goshen who are 6 years of age were in the first grade; those 7 years of age, in the second grade; those 8 years of age, in the third grade, etc., the correlation between age and grade would be perfect and positive and the coefficient (r) would equal 1. On the other hand, if all pupils entered school in the twelfth grade and were regularly demoted the correlation between age and grade would be perfect but negative, and the coefficient (r) would equal -1. If all pupils remain in the first grade the coefficient is indeterminate, 0/0. As age-grade distributions usually have high positive coefficients, it may be said that the coefficient for Donora (0.906) is relatively low.

The coefficient of regression tells a more significant story concerning the average rate of progress of pupils through the grades. This coefficient is found by dividing the numerator of the Ayres formula for r by the quantity under the second radical.3

The coefficient of regression for Dedham is 0.922; for Goshen, 0.883; and for Donora, 0.795. These coefficients show what fractional part of a grade the average pupil advances for each additional year of age. Again the coefficients for Dedham and Goshen are essentially equal, indicating that the average pupil makes about nine-tenths of a grade a year. In Donora the corresponding rate of progress is less than eight-tenths of a grade a year.

3 Coefficients of regression. By Leonard P. Ayres. Journal of Educational Research, May, 1920.

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