RELATION OF VARIABLES. 107 



The following equations, except for the terms involving R, are 

 derived as before: 



A = A^- + -^ {nud + «i6/} - ^ {R^(y - y,)i + R,-^{y - Y,)i 



+ i?6S(7/ - y6)i} (16) 



B = B' + ^ {n2,d + «26/} - ~ {R^iy - 74)2 + i?5S(?/ - 75)2 

 A 2 N2 



+ i?6S(7/ - y,),] (17) 

 C = C'- + ^ {t?34(/ + fh,f} - ^ {R^(y - 74)3 + Ro2{y - y,h 



+ i?62(;/ - yc)3l (18) 



D - D^' + — {71410 + 7?43C} - -Ir {Rj:ix - Xi)4 + R^{X - X2)4 

 ■l\ i A 4 



+ i?3S(.r - X3)4} (19) 



E = E^ + — {n,ia + n,,c} - ~ {R{L{x - x^s + R^{x - x.Ja 

 •i^ 5 A 5 



+ i?32(.T-X3)5} (20) 



F = F^ + ^ {^eia + 7?63c} - ^ {i?.i2(.r - x^e + R^^U - x^)^ 



■^ > 6 A 6 



+ i?32(.f - X3)6} (21) 

 where, 



?ii4 = number of observations of w common to group 1 and 4 

 7116 = " " " " " " " "1 and 6 



and so on, and where 



B - A = a, B - C = c, E - D = (^, and E - F = /. 



In order to correct for the position of w with respect to y in groups 

 1, 2, and 3, and for w with respect to .r in groups 4, 5, and 6, the terms 

 invoKang the regressions R are added. They are readily derived. 

 Consider, for example, the values of w common to groups 1 and 4 



(equation 16): is the correction to A^ on the assumption that 



A^i 



each value of ^v corresponds to the average value of y, i. e., to y^. 

 But, when the variability in y is taken account of, the difference 



