98 MCEWEN AND MICHAEL. 



number of observations been infinite. This is one way of making 

 predictions. 



Again, suppose observations show that the yield of wheat either 

 increases or decreases on the average with respect to time, or that, 

 within some definite period, there is a cycle or typical variation that 

 is repeated in approximately the same manner in each period. In such 

 cases information regarding the general trend or cycles affords a more 

 satisfactory basis for prediction, i. e., it results in prediction within 

 smaller limits for a given probability than is possible by the simpler 

 frequency method. But, suppose some other phenomenon is also 

 measured, for example, the rainfall during a given season of each year. 

 If it be observed that, in general, a large rainfall is followed by a 

 large yield of wheat, knowledge of the former could also be used to 

 improve prediction of the latter. Still another improvement would 

 be expected if the temperature during the growing season were also 

 measured, and so on. 



Prediction, as thus illustrated, implies a lag of the quantity pre- 

 dicted (dependent variable) behind the remaining quantities (inde- 

 pendent variables). Considering the case in general this lag may be 

 of any magnitude between any of the selected independent variables 

 and the dependent one, or all may vary simultaneously. But, the 

 problem of determining the empirical relations is the same, and, as 

 more factors are measured and as the number of observations increases, 

 approximation is had to the ideal of precise determination. However, 

 under the most favorable conditions, some deviations between 

 observed and computed values always remain, and these are called 

 accidental or "chance" variations. Even in laboratory experiments, 

 where the idea of artificial control over the independent variables is 

 dominant, it is often necessary, in order to obtain best results, to cor- 

 rect the dependent variable for unavoidable fluctuations due to vari- 

 ables beyond control. In any natural problem, however, the factors 

 involved are all necessarily variable and, as a rule, mutually corre- 

 lated, so that, in any given case, one is confronted with the difficulty 

 of selecting the most important factors, and the necessity of determin- 

 ing the approximate functional relation of the dependent variable to 

 each of the mutually correlated independent ones. 



If the functional relation between the variables is known to be 

 approximately linear, or can be made so by introducing suitable func- 

 tions, the usual method of multiple correlation may be applied. 

 Again, in case the form of the functions expressing the relation of the 

 dependent to each independent variable is known, the method of 



