132 MCEWEN AND MICHAEL. 



regression coefficients. A similar procedure is followed in the method 

 of least squares when, in determining the " probable error" of a value 

 computed by a formula having vi unknown coefficients, the number 

 of observations is decreased by m. 



Applying this rule to determine the reliability of the estimated 

 yield of wheat, the reduced number of observations is 27 — 3 = 24 

 for the multiple linear correlation method; 27 — 5 = 22 for the group 

 method used when variability within the group is neglected, and also 

 for the sloije method; and 27 — 11 = 16 for the group method in 

 which regressions are used. The reliability for each of the four 

 methods, given by the probable error of a single observation, is 



J15L2 ^ ^ jgg Qg-,^^ Jl^ = ± 1.70, 



N 24 M 22 



0.6745 _, . ^ 



24 >( 22 



0.6745 ^ 1^ = ±1.63; and 0.6745 ^ 



123.4 _ ^ ^_g^_ 



22 V 16 



In case the results given by any one of the three group methods are 

 "smoothed" by any process, a simpler relation is obtained which 

 justifies a corresponding increase in the number of observations on 

 which to base estimates of reliability. For example, in the case when 

 regressions are used, the functional relation is given independently 

 in two ways; first, by the series of corresponding averages of de- 

 pendent and independent variable; and, second, by the regression 

 coefficients and corresponding values of the independent variable. 

 Accordingly, if the functional relations defined in these two ways be 

 averaged, subtraction of the number of regression coefficients from 

 the total number of observations would not be required to obtain the 

 reduced number for estimating reliability, and the probable error of 

 =1= 1.87 would be correspondingly reduced. The slope method amounts 

 to averaging these relations at the outset. 

 Meaning of end results. 



In order to visualize the results obtained by this method of suc- 

 cessive approximation, each functional relation may be represented 

 graphically. In the case when regtessions are computed by the usual 

 method, or estimated from the averages, the functional relation of say 

 w to a: is approximately represented by a series of straight lines, having 

 slopes equal to the regression coefficients and drawn through points 

 whose respective coordinates are Wi, Xi; W2, Xo, etc. As the number 



