38 W1CKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



The equations for the regression of the means (x — m x expressed in a unit of 

 300 gr.; y — m 2 expressed in a unit of 80 gr.) 



(35) I. 



y x — m 2 = — 0,109 4 + 0,6 3 34 {x— mj + 0,03 31 (x — m,) 2 — 0,009 6 (a; — m, 



X y —m 1 = +0,1353 + 0,8353 (lj — 171.) — 0,0654 (y — W,)" — 0,0128 (y — m 2 



In the case of the second curve the range of applicability will be small, amounting 

 to less than 2a 2 on tlie positive side of the mean. The reason for this lies in the 

 large values for /?„, /? 03 and p' 13 . Of course this could be avoided by using formula 

 (32*). In this case we have 



»'i -'V 



0,6455?; i 0,0519 Rjir]) I 0,0753 B 2 {rj) f- 0,0029 R 5 (tj) I 0,0345 R 3 (rj) 



1 — 0,0804 Rj(rj) — 0,0046 R 4 (r t ) 



Plotting this curve in the xy plane we have to multiply by a v =1,6836. Taking for 

 y — m 2 the values ± 4,3140, ± 3,5955, ± 2,8764, ± 2, 1573, ± 1,4382, ± 0,7191 and 0,oooo as mea- 

 sured in the class range as unit, the corresponding values of // will be ± 3, o, ± 2,5, 

 ± 2,o, ± 1,5, ± 1,0, ±0,5 and 0,o and the tables for Ri{rj) can be used. 



In plate 1 all three curves will be seen. As the .4-function in approxima- 

 tion I is not valid below q = — 2,? the fractional curve will naturally not fit the 

 observations below that limit. 



The scedastic curves computed from (40) I are 



ol(x) = 1,13 6 5 -|- 0,i 3 >.)3 (y — m 2 ) + 0,1050 (y — w 2 ) 8 , 



°s(y) = 1,1436 + 0,0155 (X — mj + 0,0222 (x — mj 2 . 



As the material used consists of only 1223 individuals it is evident that the 

 points representing the observed square dispersion in the arrays must be much scat- 

 tered. For the same reason as in the case of the regression curves we find that 

 the curve for ol(x) is applicable only within a range of less than 2a 2 from the mean. 



The curves are seen from plate 1. 



It must be remarked that the mean errors do not affect the fit of the curves 

 to our given data. Their influence affects only the reliability of the curves and 

 the extent to which their special forms are really characteristic to the correlation in 

 general between weight of child and weight of placenta. 



Example 3. Case of non-linear regression. Correlation between weight of new- 

 born child and weight of placenta. Giiis. Same source as in example 2. 



The remarks and discussion in case of example 2 apply here also. The only 

 difference is that the range of applicability of formulae (34*) I here is somewhat 

 greater than in example 2. We will give only the numerical data. The mean er- 

 rors are nearly the same as in example 2. 



