clxxii Tables for Statisticians and Biometricians [XXXI a ~ A , LI 6 



may suppose the probability of p lying between p and p + dp to be <p (p) dp. Now 

 the chance that, for a given value of p, r lies between r and r + dr 



where y n is the ordinate at r of the r-frequency curve. This may be expressed in 



the form 



n-l n - 4 



^(1-^ (!-) ''/^tir, 



f 00 cfe 



where /n-i= I , r- sri (xxxiv). 



J o (cosh p?-) w - 



Accordingly the probability that r lies between r and r + dr and /a between p and 



p + dp 



n-l n - 4 



71 2. 2 \~~2~/i 2\~2~r ,7 .1 / \ j 



= (1 p ) (1 r*) l n _-idr<b(p)dp. 



7T 



Thus the most probable value of p will be that which makes 



n-l 



a maximum, or the most probable value of p will be that obtained from the equation 



, ^ 



t* { / ~ Ov *?--X\'li-v f ^ 



If we know nothing about the sampled population it is assumed by some that 

 it is reasonable to take <f> (p) = a constant, or 



n-l 

 Jr ((l-p 1 )"*" /n-l} =0 (Xxxvi). 



The value of p obtained from this equation has been termed the "most likely 

 value," and we will represent it by p. p is the value of p which makes the observed 

 value of r have a maximum probability, but it assumes that all values of p through- 

 out the range 1 to +1 are equally probable. It may, however, be doubted 

 whether when knowing nothing of the sampled population we suppose it one of 

 any of the innumerable possible populations, we are justified in supposing these 

 populations to be equally likely to have correlation coefficients of any magnitude. 

 Most workers in statistics are accustomed to seek for high correlations, and find 

 them with disappointing rarity. Unless we include the field of physics, which is 

 not the field wherein the applied statistician is accustomed to investigate, we can 

 only look upon the relation < (p) a constant as a convenient hypothesis for finding 

 a solution of (xxxv), to be used in default of a more accurate knowledge of < (p). 

 When we endeavour from past experience to give <(/>), even approximately, a 

 more reasonable value than a constant one, we shall speak of this value as the 

 "most reasonable value" to distinguish it from the "most likely value," p, and we 

 shall denote it by p. 



