42 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS OHARACTKRISTICS. 



According to formulae (34*) II we find 



J/| = H 0,0 7 "J 2 + 0,77 9 2 i"— 0,0 7 92^- — 0,0 1 1 (i ; :; . 

 s v = — 0,0288 + 0,8366»; + 0,0288 IJ 2 — 0,0098 / :; . 



Hence it follows from (35) II that 



//, — m, —■ + 0,i 120 + L,03i c (x - m-) — 0,0 7:; i (x — m,)*— 0,oo~9(a: - w,) ;; , 

 x y -m, = — 0,o.39i l 0,G3i8(j/ — m 2 ) + 0,0121 (y — m 2 )* — 0,oo23(// — w 2 ) 3 . 



The first of these curves will be found to fit the data very well. The second 

 curve does not deviate very much from a straight line. Indeed, the regression of 

 x on y is very nearly linear. However, as is well seen from fig. 2, the yi-function 

 does not give the correlation very far beyond y = m 2 + 2a 2 . Indeed for y — .m z = 2 9 ia 2 

 the curve given by (32*) II will have an asymptote parallel to the .r-axis. Conse- 

 quentl} 7 it will not be advisable to use our second curve beyond y — m 2 — 2o 2 . But 

 on the negative side, the curve may be extended further. Here, however, the curve 

 of form (32*) will give a better fit. Its equation is 



0,781 0J) — 0,1 1 8 3 RAtj) — 0,32 1 R 2 M — 0, 109 R. (>:) — 0,205 2jR.(j/) — 0,009 u #, (* ) 

 V l ' 1 + 0,1 51 5 R t (lj) + 0,077 2 B t (ri) + 0,01 1 5 B t (lj) 



Taking for t] the values — 2.o, —1,5, — l,o, —0,5, — 0,o, +0,5, +l,o, +1,5, + 2,o, 

 + 2,6, +3,o and +3.5 the corresponding value of x. y — m x should be plotted against the 

 arguments y — m., = — 3,r>ooo, — 2, 7000, — l,80oo, — 0,9ooo, 0,oooo, l-0,90oo, +l,sooo, + 2, 7000, 

 + 3,cooo, +4.5000, +5.4000, +6 ; :5ooo. 



Finally we must add one more word about the curve of regression of x on y. 

 Really this regression miglit as well be assumed to be exactly linear. As it is a well 

 known fact that in such a case the straight line 



Xy — m i — — r(y — ?n 2 ) , 



gives the best fit to the data whatever may be the form of the correlation fnnction, we 

 can use this line instead of the curves derived from (32*) II and (35) II. Then we 

 have the advantage of being able to use the line within the whole domain of the 

 observations. Numerically the equation to the regression line will be 



x y — m 2 = 0, 5 8 9 8 (y — )«,). 



That it well fits the data will be seen from plate 2, where also for the sake 

 of comparison the other curves are plotted out. The curve of regression of y on x 

 will not be improved by applying a formula derived from (32*) II. Nor can it be 

 regarded as linear. 



