\ X X I '"*, LP'*] Introduction <-i\\ 



Clearly /9,-/9,-l = 0, as for every "two lump" Jn-.ju.-ncy (sec J'hil. 7 



Vol. 2 hi. A, p. I:; 



Table XXXI rt provides for various values of p the constants for cases of sniiipl< > 

 of two. It'will be seen that the mean e.,m Ration in Kain|, / markedly 1cm 

 than the correlation in the parent population. 



Illustration (v). 200 pairs of pairs of Fathers and Sons were taken at. random 

 from a population containing 806f pairs for correlation of stature, each pair of pairs 

 being returned to the population before the next drawing. 



The sampled population followed closely a normal distribution with correlation 

 = *5189 '0160. Of the 200 pairs of pairs 132 gave a positive, 68 a negative corre- 

 lation. The value of p found by formula (ix) from these numbers is '4818, with a 

 probable error of '0622. From the known value of p, the numbers of positive and 

 negative correlations should be 135 to 65, not 132 to 68. Had we used the 400 

 fathers of the 200 pairs of pairs to form a correlation table the probable error of the 

 resulting coefficient of correlation would have been '0246, showing how much more 

 accurate the product-moment method is. The present method is, however, far less 

 laborious. 



(ii) Samples of Three, n = 3. 



Here the solution is given by complete elliptic integrals, i.e. if 



n 



d<j> 



then r-|^0>)-(l-y)JiO)] ........................ (xi). 



1 1 p a 

 Again, /^' = 1 + - f- \og e (l-p z ) ........................ (xii), 



* P 



leading to o- r 2 =l-f 2 +^ ~~f*\oge(l - P 2 ) .................. (xii**). 



* P 

 Further, 



r l p , 

 = -jj -- , a form easier for computing 



P P 



(xiii), 



.(xiv). 



* By aid of Tippett's Random Numbers, Tracts for Computer, No. xv. Cambridge University Press. 



t The original population contained 1072 pairs of Father and Son, but as some of theso were on the 

 same card only 806 cards were used to avoid giving each card two numbers. The correlation -. r )189 was 

 calculated on 1000 pairs. I have to thank Dr A. E. R. Church for providing the 200 samples on which 

 this Illustration is based. 



