The class of problem of which the figures given in the last table furnish an illustration is one of great importance in statistical work. Thus the variation of the height and weight of men with their age, the questions relating to the transmission of characteris tics from parents to children, the association of different diseases or defects and the results of combinations of treatment of diseases or defects, the interpretation of the results of variation in different features of the weather, for example rainfall and temperature, on crop yields and many other problems give rise to such dis tributions, while the limitation of the illustration to the case of three kinds of variation should not be taken to imply that larger numbers of elements may not be considered.
The practical application of the procedure of calculating co efficients of correlation is not confined to cases such as that in connection with which the nature of correlation was illustrated. It may also be applied to determine, in cases such as that of the consumption of sugar and of tea, illustrated earlier, the closeness with which the variations in the two series of figures coincide. In some cases it is of importance to compare, not two series the trends of which are, on the whole, similar, but two series of which the trends are different, though the variations from the general trend may prove to be similar in the two cases. Whether for diagrammatic representation, or for preliminary arrange ment of material for testing by the calculation of coefficients of correlation, it is desirable to isolate the important features as far as possible. The relation between economic prosperity and the
marriage rate is a problem that has attracted much attention, and in using as a measure of economic prosperity the value of the exports year by year, or of imports and exports together, the con nection which appears in some periods may be obscured in later periods by other conditions which cause one or other, or both, of the curves compared to diverge from the trend previously shown. To a considerable extent the effect of these disturbing elements in obscuring the significant relation may be evaded by using for comparison, not the absolute measures, e.g., of the value of trade and of the marriage rate, but the proportion in which the figures for a given year are in excess or defect in comparison with those of the average of a series of several years of which the year under consideration is the middle year, or the final year.
It may be noted that when two series of quantities, showing the variations in time of phenomena under examination, are tested for closeness of fit by the calculation of correlation coefficients, it is possible to test whether one of the two series differs from the other mainly in respect of the time at which variations are greatest or smallest. In comparing two series such as X4 y1 Y2 Y3 Y4 the sum of products used in calculating the correlation coef ficient was conceived of as x1 yi + x2 Y2 + X3 y3 + It may be found that such product-sums as