APPENDIX 



705 



The algebraic details of subjecting a line to this minimal condition are 

 well known to those familiar with the method of least squares. The equa- 

 tion of the resulting line is 



where <r x is the standard deviation of the population with respect to the 

 ^-character, cr,, is the standard deviation with respect to the j-character, 

 and r is the correlation coefficient given by 



where the summation is extended to every two corresponding variates of 

 the population. 



Similarly, the regression of the ^-character with respect to the ^-charac- 

 ter is given by 



* = "%/ < 2 > 



It should be noted that (2) cannot be obtained from (i) by solving for 

 x in equation (i). 



Standard deviation of arrays. Suppose that the regression is truly linear, 



so that the means of the ^/-arrays fall on the linej = r^, and furthermore 



that the standard deviations of all parallel arrays are equal. Then the 

 standard deviation of any array must be given by 



n 

 where the summation extends to the entire population. 



+ 

 n 



= <r/ - 2 r V/ + rV/ 



= <r/(i-r>). (3) 



Hence the standard deviation of a j-array is obtained from the stand- 

 ard deviation <r y of the total population with respect to the ^-character by 

 multiplying o> by Vi r 2 . 



Since the first number of (3) is a sum of squares divided by n, the 

 second number must be positive. Hence 



\<r< i. 



This shows that our correlation coefficient must take values between + I 

 and i . 



