106 



MCEWEN AND MICHAEL. 



independent variable, in which event Httle gain in accuracy is made 

 by taking account of variabiUty within the group. But, when the 

 number of groups is small, this is not generally true. In such cases 

 variability within the group may be significantly decreased by intro- 

 ducing corrections based upon an assumed linear regression of the 

 dependent on the independent variable, e. g., a linear regression of 

 w on a; in a {w, x) group. By this means each value of the dependent 

 variable may be approximately reduced to what it would have been 

 had the independent variable remained at its constant average value, 

 e. g. each value of w in the (w, Xi) group may be reduced to a value 

 corresponding to x = Xi. If the central idea of group averages has 

 been made clear it will be obvious that the error introduced by an 

 assumption of linear regression tcithin the group is negligible. Accord- 

 ingly, after applying this correction, the outstanding variability is 

 legitimately attributed to " chance " and, after selecting the inde- 

 pendent variables, can be further reduced only by increasing the num- 

 ber of groups and the number of observations in each. 



For clearness, the analytical demonstration is given for the special 

 case of two independent variables and three groups of each, but the 

 same reasoning applies to the general case of any number of variables 

 and groups, as well as to the case in which regressions are run in some 

 of the groups and not in others. The notation is presented in table 3. 



TABLE 3. 

 General Notation Employed. 



