[252] The Census Bureau figures have been subject to a good deal of criticism, and I therefore refrain from trying to draw precise conclusions from them.
[253] The figures showing the number of banks reporting from each State, together with the number of reports rejected, will be found on pp. 47-49 of his monograph. The figures above are combinations of figures from his various tables. These tables are so carefully indexed in Dean Kinley's monograph that detailed page references are unnecessary here.
[254] Cf. our discussion of this topic in the statistical chapter, infra.
[255] Loc. cit., pp. 153-154.
[256] Discussions in Economics and Statistics, I, 204. Quoted by Kinley, loc. cit., 152.
[257] The coefficient of correlation has been developed by the biologists, chiefly Karl Pearson, but has been applied to problems in many fields, especially economics, sociology, psychology, and education. A good source is Yule's Introduction to the Theory of Statistics. Professor H. L. Moore has made extensive use of the method in his Laws of Wages, and his Economic Cycles.
Connected with the coefficient of correlation, usually, is a figure for "probable error," which depends, primarily, on the square root of the number of observations. When the probable error is low, and the coefficient of correlation high (as .8), it is commonly supposed that a very high degree of causal connection is established. I shall not go into detail in discussion of the method. My personal judgment is that it is overrated, that "spurious" correlations, leading to quite erroneous conclusions, have frequently resulted from it, and that the labor involved in calculating coefficients of correlation is frequently too great for the results obtained. I should never be disposed to accept conclusions based on a "correlation coefficient" unless there were other converging evidence to support it. In effect we have, in the coefficient of correlation, nothing more than a refinement of the method of comparing two curves on a graph. The curves tell the story, in a general way, whereas the coefficient of correlation sums up all the comcomitant variations (and disagreements) in one figure. The eye does not readily compare the degree of relation between two curves with the degree of relation between two others. When it is desired to know which, of several relationships, is closest, the graphic method, or the method of comparing series of figures, burdens the attention. The coefficient of correlation condenses the information to such a degree as to make comparison easy. It is, then, merely a refinement of familiar statistical methods. Used wisely, guided by sound theory, it aids in presenting facts. It enables us to state quantitatively things we already know qualitatively. But there is no magic in it! As I have mentioned both Mr. Silberling and Professor Moore in this connection, it is proper to say that both of them are fully alive to the dangers and limitations of the method, and that Professor Moore emphasises strongly the need for sound a priori testing of hypotheses before submitting them to the test of correlation. One danger, that of getting a high correlation merely because both of the variables compared are growing rapidly, has been avoided by Mr. Silberling by the use of successive percentage deviations, instead of absolute figures. For reasons explained by Mr. Silberling in a footnote, he uses, instead of the "probable error," a statement of the number of observations. Thus, "r = .78 (46)" means that the coefficient of correlation is .78, and that there are 46 observations for each of the two variables compared.
[258] They get into clearings, however, two days after.
[259] Professor Kemmerer, also. See his index of variation of trade, op. cit., pp. 130-131.
[260] It is unfortunate that weekly figures from railways do not exist in such number, or for roads of sufficient importance, to justify correlations of the weekly figures with clearings.