BIOMETRICS 159 



Height of Fathers. Mean Height of Sons. 

 Inches. Inches. 



69 . . . . 69 



Fathers below the average height have sons of a mean 

 height greater than that of the fathers (the left-hand 

 column) ; fathers above the average height have sons of a 

 mean height less than that of the fathers (the right-hand 

 column), while fathers of average height (69 inches) have 

 sons of average height. This proves a distinct correlation 

 between the heights of father and son. But we want 

 further to measure the amount of this correlation. In 

 order to do this we construct from the given data a 

 diagram of correlation in the following way : We plot off 

 on a horizontal line the heights of the fathers in equal 

 distances from 62 to 76 inches, do the same on a vertical 

 line for the heights of the sons, and complete the square. 

 We draw the diagonal CD of the square thus constructed, 

 which goes through all the points where the heights of 

 fathers and sons are the same — i.e., where the horizontal 

 and vertical lines indicating equal heights cut each other. 

 If we now mark the mean of each array of sons on their 

 respective horizontal lines in such wise that each mean 

 comes to lie on the vertical which indicates the corre- 

 sponding class of fathers to which that array of sons 

 belongs — i.e., point a, indicating the mean 65-5 inches of 

 sons belonging to fathers of class 62 inches, lies on the 

 horizontal line 65-5 inches and on the vertical 62 inches ; 

 point h lies on the horizontal 66 inches and on the vertical 

 63 inches, etc. — we can draw a straight line AB going as 

 nearly as possible through all the points thus indicated. 

 Both lines, CD and AB, cut each other in O, which, as can 

 be seen, lies on the horizontal and vertical lines indicating 

 the mean of the population P =^ 69 inches. We have 

 now in the slope of the line AB a measure of the correlation 

 between the heights of the fathers and that of the sons 

 plotted on the lines CD and AB respectively. If AB 

 coincides with CD, the points a, b, c, d, etc., would all fall 



