24 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



transformed by arranging the terms in powers of ij by the aid of equations (7). Then 

 no tables are required for the computation. Some work will however be spared by 

 using a table. We give here a short table which is quite sufficient for the construc- 

 tion of the regression cnrves. Any required values of §' v for an assigned argument 

 r g may then be found graphically or by interpolation. 



? 



Ä,(?) 



»,(*) 



**(*) 





K, ? 





*.(*) 



*ifo) 



J's i) 



— 3,5 



+ 11,2500 



+ 32,3750 



+ 79,5025 



+ 148,9688 



+ 



123,5781 



— 461,2891 







2479.5538 



-3.0 



+ 8,0000 



+ 18,0000 



+ 30,0000 



+ 



18,0000 



— 



96,0000 



— 396,0000 



- 



516,0000 



2,5 



+ 5,2500 



+ 8,1250 



+ 4,5625 



— 



21,0938 



— 



75,5469 



— 62.3047 



-f 



373,0664 



-2,0 



+ 3,0000 



+ 2,0000 



— 5,0000 



— 



18,0000 



— 



11,0000 



+ 86,uooo 



+ 



249,0000 



— 1 ,5 



+ 1,2500 



— 1,1250 



— 5,4375 



— 



3,6563 



+ 



21,7031 



+ 54,4922 



— 



70,1835 



- 1,0 



- 0,0000 



— 2,0000 



— 2,0000 



+ 



0,0000 



+ 



16,0000 



— 20,0000 



— 



132,0000 



— 0,5 



— 0,7500 



— 1,3750 



+ 1.5625 



+ 



6,2813 



— 



4,6719 



— 40,0234 



+ 



12,0914 



± 0,o 



— 1,0000 



0,0000 



+ 3,0000 





0,0000 



— 



15,0000 



0,0000 



+ 



105,0000 



+ 0,5 



— 0,7500 



+ 1,3750 



+ 1,5625 



— 



0,2813 



— 



4,6719 



+ 40,0234 



+ 



12,6914 



+ 1,0 



— 0,0000 



+ 2,0000 



— 2,0000 



— 



0,0000 



+ 



16,0000 



+ 20,0000 



— 



132,0000 



+ 1,5 



+ 1,2500 



+ 1,1250 



— 5,4375 



+ 



3,6563 



+ 



21,7031 



— 54,4922 



— 



70,1835 



+ 2,0 



+ 3,0000 



— 2,0000 



— 5,0000 



+ 



18,0000 



— 



11,0000 



— 86,0000 



J- 



249,0000 



+ 2,5 



+ .""',2500 



— 8,1250 



+ 4,5625 



+ 



21,0938 



— 



75,5469 



+ 62,304 7 



+ 



373,0604 



+ 3,0 



+ 8,0000 



— 18,0000 



+ 30,0000 



— 



18,0000 



— 



96,0000 



+ 396,0000 



— 



516,0000 



+ 3,5 



+ 11,2500 



— 32,3750 



+ 79,5625 



— 



148,9688 



+ 



123,5781 



+ 461,2891 



— 



2479,5588 



The table has been confined to the interval ij = — 3,5 to t] = + 3,5. It will very 

 seldom be of any use to construct the regression curves further than to 3,5 times 

 the dispersion reckoned from the mean. Generally, when the correlation is mode- 

 rately skew, some 10000 observations should be required in order that the domain 

 of variation of more than the distance 3.5 a from the mean may contain 5 individu- 

 als. In addition the number of terms included in the formulae will generally not 

 be enough to give a sufficient degree of approximation when |>;|>3.5. 



However, when the correlation is not very considerably skew formulae I and 

 II may be written in other forms that are somewhat easier to handle numeric- 

 ally. If the denominator be inverted and transformed into a series in powers of 

 ^0 3-ft 3 (r/) + /?0 4-K,(?7) + etc. this series will be convergent so long as the denominator does 

 not become zero or exceed 2. 



The range within which this convergence takes place for different values of ^ 03 

 and p 0i will be the following: 



for |/? 0S | = 0,02; |/? J = 0,02 when 



» |/?osl = 0>°6; 1^04 1 = 0,oo » 



» |/*03 1 = 0,i o; |/^o4| = 0,oi » 



» I /^oa | == 0, 1 7 ; |/Am I = 0,0 3 » 



» IA.I-0,12; \p ot \-0,i* » 



|; | < 3,o 



ease 



I 



l'J<2,r, 



» 



r 



hl < 2,4 



» 



I! 



1 ', 1 < 2, i 



» 



II 



hl < 2 »o 



» 



I 



