METHODS OF STATISTICAL ANALYSIS. 15 



Expressing this relationship in terms of regression, we have (remember- 

 ing that <r w = 12.867 and <r A =245.12). 



(h-h) =r wh - A (w-w) =0.958 ^ (w-w) 



or 



(h-h)*= 18.250 (w-w) 



In a form somewhat more convenient for practical work, i.e., in that 

 of the characteristic equation, the relationship is 



7 /T &h \ &h 



n ( h T . . -j/i I 17* , _ in 



It/ \lli I w fi W I I ' t0/j 



Noting that numerically w =73.8 kilograms and h = 1876.6 calories, we 

 have 



19 



h = (1876.6 -0.958 -^ 73.8) + 0.958 =i w 



12.867 12.867 



which gives 



h = 529.7+18.3 w 



Such equations predict the average value of the y character 

 associated with a given grade of the x character, or the average value 

 of the x character associated with a given value of the y character. 

 For examples, the values of h predicted by the equation are the average 

 values of a series of individuals of given stature, body-weight, or any 

 other physical or physiological character used as a basis of prediction. 

 They represent the most probable heat-production of an individual deter- 

 mination providing that the distribution of variation in heat-production 

 is symmetrical about its mean and the relationship between the char- 

 acter from which prediction is made and heat-production be capable 

 of expression by a linear equation. 



In the following pages the straight lines due to such equations are 

 frequently represented on a diagram showing by the position of a dot 

 the value of both the x and the y character of all the individuals. 



Such scatter diagrams bring out clearly the fact that the predicted 

 measure is an average and can be taken to represent only the most 

 probable value of the individual case. It is necessary, therefore, to 

 consider the amount of deviation which may be expected to occur 

 about the predicted mean. 



The standard deviation of the predicted character, say h, for the 

 individuals of any group is 



where x denotes stature, body-weight, age or any other character 

 with respect to which the individuals may be classified in the investi- 

 gation of metabolism. 



