RELATION OF VARIABLES. Ill 



value computed, without taking into account the correlation between 

 the independent variables. These twelve equations (29) to (40), to- 

 gether with the four equivalents "B — A=a, B — C = c, "E — D=d, 

 and E — F = / determine the six required averages and the six 

 required regression coefficients, and may be most conveniently solved 

 by a process of successive approximation similar to that alreadj^ 

 presented for the simpler case in which variability within the group 

 is neglected. The details of this process are given in table 4. 



As in the simpler case (see p. 105), if the process of successive approxi- 

 mation, indicated by hues 1, 2, 3, 4, and 5, 7, 9, and 11 results in 

 convergence to definite limiting values, these values will satisfy 

 equations (29) to (40). In the third approximation, the procedure 

 indicated by lines 6, 8, 10, and 12, involving first differences, affords 

 a numerical check on the computation of AR'i\ AR'i\ etc., and AA"% 

 AB'", etc., of lines 5, 7, 9 and 11, and beginning with the fourth approxi- 

 mation, second differences (lines 14, 16, 18, and 20), afford a check 

 against the first differences. For a numerical illustration see page 122. 



If the reader has followed the reasoning thus far, he will doubtless 

 feel that, although the regression method is formally complete, the 

 labor involved in its application would; in many instances, be so great 

 as to make its use impracticable. To meet this objection, a slope 

 method has been devised which takes account of the variability with- 

 in the group in nearly as accurate a manner, but one that eliminates 

 half of the equations, namely, those similar to (23) to (28). The 

 basis of this method is the fact that the slope of a chord of a simple 

 curve is approximately equal to that of the tangent at the point 

 midway between the extremities of the chord. Accordingly, the slope 



•O A 



= Si-2 of the chord whose extremities are (xi, A) and (X2, B) 



X2 — Xi 



is approximately that of the tangent at the point whose abscissa is 

 ^— 1_^, Similarly, for the point midway between (X2, B) and (X3, C), 



rj g 



the slope of the tangent is approximately = /S2-3, and so on. 



X3 — X2 



But the slopes at the points (xi. A), (x., B), and (X3, C) are required. 

 That at (X2, B) is readily obtained by utilizing the rate of change in 

 slope as a means of interpolating between (Si_2 and S2-3. Thus 



