APPENDIX 703 



In what follows 



a- is to represent the standard deviation ; 

 n is to represent the number of variates ; 

 c is to represent the coefficient of variability ; 

 r is to represent the coefficient of correlation. 



1. E s 0.6745 " probable error in a single observation. 



E f 0.6745 " 



2. E M = := = = = probable error in the mean. 



v V// 



E M 0.6745 " 



3. E -7= = probable error in standard deviation. 



V 2 V 2 n 



4. E c = ' . i + 2( ) 2 = probable error in coefficient of varia- 



v 2 L V'/J bility 



0.6745 ^ 

 = . approximately, if C\s not greater than 10 per cent. 



V2# 



0.6745 ?"*) 



5. E r - = probable error in coefficient of correlation. 



6. E R = \l = probable error in the regression coefficient 



(Ta * 11 





SECTION VIII CORRELATION THEORY 



Definition. Two measurable characters of an individual, or of related 

 individuals, are said to be correlated if to a selected series of sizes of the one 

 there correspond sizes of the other whose mean values are functions of the 

 selected values. The word " sizes," here used, should be taken to mean 

 " numerical measure." 



For the sake of concreteness and simplicity, we may think of measuring 

 the correlation of sons with respect to their fathers. To render the above 

 definition in symbolic language and to develop the method of determining 

 the function mentioned in the definition are the first points in the application 

 of mathematics to the theory of correlation. For this purpose, let x and/ 

 represent variables such that y = <f>(x) gives the mean value oiy correspond- 

 ing to a selected x. Then the problem is to determine <(X). 



Suppose the following system of corresponding values results from 

 measurement: (x',y'}, (x",y"}, , (x<"\ _y (;/) ), where n is a very large 

 number indicating the population of fathers and corresponding sons. 

 These observations are said to form a total population or universe of 

 observations. As it will be more convenient to deal with the deviations 

 of the observations from their mean value than with the measurements 

 themselves, let ( / r 1 ,.) / 1 ), (^'Ja)' ' ' '> (* /) re P resent the deviations of 



