clxxvi Tables for Statisticians and Biometricians [XXXI a h , LI a 



Then we find 



...... (xlvi), 



where E n+i , E n , etc. are the E's found from (xlv) above by substituting p for p. 

 Eliminating E n +i we find finally 



......... (xlvii), 



while (n-l)(l-rV)#n = (2w-3)rj5 + ^- ............ (xlviii), 



^n-l 



............... (xlix). 



In the case of a very small sample tire successive E's must be worked out up 

 to E n . For a larger sample 25 or upwards good results may be obtained by taking 

 E n = E n _!=E' in (xlviii) and solving the quadratic to find E' ; the value of E' is 

 then substituted in (xlvii) to find -fy. 



Illustration. In a sample of 25 cases only of parent and child the correlation 

 for a certain character was found to be '6. What is "the most reasonable" value to 

 give to the p of the sampled population ? 



If we distributed our ignorance equally the most likely value of p would be 

 (see pp. clxxiii clxxiv): 



P = -59182. 



But our experience is not that all values of p are equally likely ; the correlation of 

 parent and child has never been found to be negative, and the mean value of p as 

 deduced from long series of observations is known to be about 4- '-A6, and the range 

 is hardly more than 4- '40 to + '52, corresponding to a standard deviation of about 

 02. Hence 



which leads to ra = '000,507, say, and - ^ =82'1828, or it is the dominating 



m(n 1) 



term in the equation (xlvii). We find accordingly from (xlvii) 



-437,006 #'- -424,959 }} 



*" ~ 70-034,491 + 2-790,925 E' 



Again, (xlviii) becomes 22-171,776 E' 2 - 12'972 E' - 23 = 0, 



which gives E' = 1-352,2185, 



thus from (1) ijr = '00225, and p = p + ^ = '46225. 



