A SOMEWHAT detailed account of the mathematical theory of correlation and of the way in which it may be usefully applied to psychological measurements will be found in the later chapters of this Part. The object of the following introductory pages is to give the reader a general preliminary view of the method, free from mathematical complications, and to illustrate it by means of a simple example.

Correlation may be briefly denned as “tendency towards concomitant variation,” and a so-called correlation coefficient (or, again, correlation ratio) is simply a measure of such tendency, more or less adequate according to the circumstances of the case. J. S. Mill, in his “System of Logic,” distinguished a special scientific “Method of Concomitant Variations,” which he based upon the following principle:

“Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.”

The instances of this principle which Mill had in mind were mainly cases of approximately “complete” concomitance of variation, such as those usually met with in the domain of Physics. In such cases, the conditions of an experiment admit of a high degree of simplification, the phenomenon, or series of phenomena, under investigation can be isolated with tolerably complete success, and the “ irrelevant” factors can be reduced to a minimum. Under such conditions, when the degree of concomitance of the different corresponding measures of the two phenomena is found to be very high, the slight deviations from complete correspondence are put down to “errors of observation” or other unavoidable imperfections in the experimental method employed.

If the correspondence is one of simple proportionality, so that the graphical representation of it (one phenomenon being measured along the axis of x, the other along the axis of y) is a straight line, the correlation coefficient r will be unity. Example: the variation of the length of a metal rod with temperature.