Prof. F. Y. Edgeworthon Correlated Averages. 191 



value of a l9 say £ v which most frequently corresponds to an 

 assigned value of x 2l say x 2 (or vice versa). From the equa- 



,. dR n t 

 tion -— =0 we have 

 ax x 



It should be observed that for the purpose of this calcula- 

 tion it is not necessary, as Mr. Galton has done, to pick out 

 the values of x Y corresponding to each value of x 2 . It is 

 sufficient to take the sum, or the mean, of all or some of the 

 positive, exclusive of negative, values [or negative, exclusive 

 of positive] and the sum or mean of the corresponding values, 

 not exclusive of negative [or positive] ; and to equate 



or 



Sf] _ &x 2 _ Sf x m s# 2 '. 



n ^ n ' n ' n ' 



omitting perhaps the extreme observations, with respect to 

 which the law of error is liable to break down. 



For example, let it be required to find the coefficient of 

 correlation between the stature and left cubit of adult males, 

 from the data utilized by Mr. Galton in Table II. of his paper 

 on "Co-Relations " (Proc. Roy. Soc. 1888, p. 138) /without the 

 trouble of the detailed selection, the elaborate "depouillement" 

 which the construction of his table requires. Take the cubit 

 as the independent variable, x 2 of the last paragraph, and 

 write down all the deviations of the cubit (from its mean 

 value) which are above zero and short of the extremity. 

 There are ninety-three such instances — between 18'5 and 

 19'5 inches — among the materials which Mr. Galton has 

 employed in constructing his Table II. (op. cit.). The sum 

 of these deviations is 1762*75 inches; the sum of the ninety- 

 three concurrent deviations of stature is 6422 inches. Each 

 of these has to be divided by the corresponding quartile ; 

 •56 inch in the case of the cubit, 1*75 inch in the case of the 

 statures. Thus 



^ Sg 1 = 1762-75 . 6422 = ^ 



- S# 2 '56 " 1*75 ~~ 



Which is the value for the coefficient found by Mr. Galton. 



In working this example I have taken the figures from the 

 compartments of Mr. Galton's Table II. ; for instance, reckon- 

 ing that there are 55 cubit-deviations ,: 18'5 and under 19*0/'' 



