RELATION OF VARIABLES. 131 



a small precipitation range should be materially less when based upon 

 both temperature and precipitation, than when based upon the latter 

 alone. Moreover, reduction of the standard deviation effected by 

 introducing the correction for temperature should be materially greater 

 for the slope method than for that of multiple linear correlation. In 

 eight instances the temperature ranges from 62.6° F. to 73.4° F., while 

 the precipitation varies only from 3.5 to 5.3 inches. The standard 

 deviation is reduced by 2.85, or from 4.94 to 2.09, for the slope method 

 by introducing the correction for temperature, while for the method 

 of multiple linear correlation, the standard deviation is reduced by 

 only 0.96, or from 3.53 to 2.57. These results are not only in agreement 

 with expectation but clearly indicate that part of the actual effect 

 of temperature on yield is attributed by the method of multiple 

 linear correlation to precipitation. To further test the truth of 

 this indication, each value of the wheat yield given in table 5 was 

 computed on the basis of the temperature relation ascertained by the 

 sloj^e method, and a linear correlation run between the residuals and 

 precipitation. As a result the precipitation-yield correlation coeffi- 

 cient is reduced from + 0.22 to + 0.015. 

 Reliability of results. 



Experience proves that, after all practicable efforts are made toward 

 controlling or correcting variations attributable to the independent 

 variables, some deviation between computed and observed values of 

 the dependent variable always remains. Although the magnitude 

 of these deviations is an index of the accuracy with which the empirical 

 relations describe the data at hand, one usually needs an estimate 

 of the reliability with which these empirical relations describe the 

 whole " universe" of which those data are a sample. Accordingly, the 

 complexity of these relations must be taken into account. This need 

 is particularly evident in the method of group averages, for, while 

 the empirical relations describe the data at hand with increasing pre- 

 cision as the number of groups approaches the number of observations, 

 reliability of this description decreases until, in the limiting case, it 

 is no ^greater than that afforded by the isolated observations them- 

 selves. It is evident, then, that in estimating reliability, the actual 

 number of observations should be reduced by an amount depending 

 on the total number of groups chosen to express the empirical relation, 

 and, for this purpose, each group in which a regression is used is equiva- 

 lent to two groups. An approximate rule is to add the number of 

 independent variables less 1 to the total number of observations and 

 to subtract from this sum the number of groups plus the number of 



