CORRELATED VARIABILITY. 51 



2^ " ) anc * taking the positive root. Substi- 

 tute this value in the whole equation to the 4th power for 

 /(r), and in the first derivative of the same equation for /'(r) 

 (remembering that the first derivative of /(#) is obtained by 

 multiplying each term in f(x) by the exponent of x in that 

 term and diminishing the exponent of x by 1). The correc- 

 tion ^7-7^7 should be added to the value of r used in substi- 

 tuting. Repeat this process as often as the correction affects 

 the fourth place of decimals, and go to r 5 if necessary. 

 The probable error of r as thus determined is 



found as follows: First calculate the relations &= _= 



k rh 

 and /? 2 = ". . Also find 



and 



from Table TV. Moreover, 



1 



Then, 



Prob. error of r=-. i[j( o 



(a + b)(d + c) &* + 2(ad - be) fa 



which can be easily solved by substitution. In using the 

 foregoing formula, it must be noted that "a is the quadrant 

 in which the mean falls, so that h and k are both positive." 

 In other words, a + c > b + d and a + 6 > c + d. (Pearson, '00.) 



Example. The eye-colors of a certain set of people (see Bio- 

 metrika, II, 2 pp. 237-240) and of their great -grandparents were 

 found to be distributed as follows 



