RELATION OF VARIABLES. 127 



5. Supplementary Considerations. 



For clearness in presenting the central idea of this method of suc- 

 cessive approximation, and for convenience in the analytic demonstra- 

 tions and concrete illustrations, certain matters were barely men- 

 tioned that require further consideration. They are: first, manner of 

 grouping; second, basis of mathematical reasoning; third, reliability 

 of results; and fourth, meaning of end results. 

 Manner of grouping. 



It is desirable so to group the values of each independent variable as 

 to equally distribute the effect of "chance" on the corresponding 

 average of the dependent variable. Among other things, this effect 

 of chance increases in any particular group as the number of entries 

 in that group decreases, whence, in order to distribute it equally, each 

 group should contain approximately the same number of entries. But, 

 this effect of chance in any particular group also depends upon the 

 range in value of the independent variable within that group, and, for 

 this reason, the entries should be so grouped as to make this range 

 the same in each group. In practice, however, it is usually impossible 

 to group the data so as to realize, even approximately, both of these 

 conditions. Thus, in the illustrative wheat problem (see table 5), 

 although the first condition is completely realized by dividing the 

 twenty-seven entries into three groups of nine entries each with respect 

 to both temperature and precipitation, the second condition is not, 

 the temperature range being 3.8° F. in group 1, 2.7° F. in group 2, and 

 6.4° F. in group 3, and the precipitation range being 2.5 inches in 

 group 4, 2.1 inches in group 5, and 3.5 inches in group 6. When 

 regressions are used it is probably best to meet the first requirement 

 at the expense of the second, as is here done, but when variability 

 within the group is neglected, this is not so e"vident. No general rule 

 can be stated, and choice of the number of groups and number of 

 entries in each is a matter of judgment, just as in fitting a parabolic 

 function 



w = f/o + aiT + Chx^ + • • . • -\-hiy + 62.V" + • ■ ■ • 



to empirical data, the number of terms to retain is a matter of judg- 

 ment. In either case the final result is fixed as soon as the choice is 

 made. 



After deciding upon the number of groups and number of entries 

 in each, it is not uncommon to find that the last value of the inde- 



