CORRELATION 97 



were 'no correlation between the heights of sons and 

 those of their fathers the mode of every array of sons 

 would be the same, and this value would be identical 

 with the mode of the heights of all the sons taken 

 at once. The actual result is found to be intermediate 

 between these two possible extremes. Thus we see 

 that sons tend to be like their fathers in respect of 

 stature, but not exactly like, and if the example given 

 were a real one the fundamental fact of a positive 

 resemblance or correlation between the statures of 

 fathers and sons would at once be clearly established. 



The way in which a numerical value is attached to 

 this correlation can be shown graphically. 



In the diagram opposite, the dots indicate the values 

 of the modes of the several arrays of sons as read off on 

 the vertical scale to the left of the figure, the heights 

 of the corresponding classes of fathers being read off 

 on the horizontal scale. It will be seen that this series 

 of dots lies nearly in a straight line which is inclined 

 at a certain angle to the horizontal. 



Now if there were perfect correlation between the 

 heights of fathers and sons, or if the modal value of 

 each array of sons were identical with the height of 

 the corresponding class of fathers, the inclination of 

 the line obtained in the above manner would be one of 

 45 degrees, as in the case of the line CD which passes 

 through the points at which the values as read off 

 in the vertical and horizontal scales are identical. If, 

 on the other hand, there were no correlation the line 

 would be horizontal, as EF. 



The value taken to represent the amount of correla- 



7 



