Correlated Averages. 197 



And to determine A we have 



19, --08 



A 2 (-36x-19--08 2 ) 



A = A^ 



-•08, -36 



= A 2, 0620 (a result which may be verified by observing that 

 the other two principal minors afford, as they ought, the same 

 equation for A). Whence A = 16*129. Substituting this 

 value in the values of the coefficients above found, we have 

 for the sought expression 



5-806^ 2 + 3-064v 3 2 + 5*806tf 3 2 -2 x 1-290^ 



-2 x 4-194iv?3-2 x l-290ayi' 3 . 



Thus we see that the dispersion of x x corresponding to 



assigned values of a? 2 , oc z has for modulus The 



most probable deviation of one organ, e. g. the cubit, corre- 

 sponding to assigned deviations of the two other organs is 

 found by differentiating with respect to m % the expression above 

 written and equating to zero. Thus, if .i'/, x£ be the assigned 

 deviations of stature and height of knee, and f 2 t ne most 

 probable corresponding deviation of cubit, 3*064| 3 = 1*290^!' 



+ 1-29(W- ? 2 = 1 ' 29 3^ ig3 ' ) ='«W+*fl. 



I have verified this deduction by actually observing the 

 value of cubit-deviation which on an average corresponds to 

 assigned values of the other deviations. For this purpose I 

 employ a table such as Table I. at p. 192, with an additional 

 column for height of knee. This material, consisting of un- 

 manipulated observations on the stature, cubit, and knee-height 

 of three hundred men, was kindly furnished to me by Mr. 

 Galton. Out of these three hundred triplets I pick out 

 ninety-six which have the stature and knee-height above the 

 respective means of those organs ; and proceed in a manner 

 analogous to the simpler calculation discussed at p. 193. The 

 columns for stature and cubit will be nearly the same as those 

 of Table II. in that passage ; but not quite the same. For 

 negative values are admissible there for the deviation of 

 stature which is there the dependent variable, but not here, 

 where both stature and knee-height are treated as independent 

 variables. On the other hand, negative deviations of cubit 

 are admissible here, but not there. The rationale of this dis- 

 tinction is sufficiently explained by the remark at p. 194 (top). 



Here is a specimen of the calculation, the columns for 



