396 THE GRAMMAR OF SCIENCE 



points a, d, c, d, etc., so obtained form when joined the 

 polygon of regression. Now the mathematician shows us 

 how to draw a straight line which shall fit as closely as 

 possible such a system of points abode . . . i j k. He 

 does this by making the sum of the squares of the distances 

 of the points from the line as small as possible. The line 

 AB thus obtained is termed the line of regression, and 

 although we cannot prove here all the properties of this 

 line, they are of such interest that one or two must be 

 stated. In the first place the regression line will, as 

 a rule, very nearly pass through the mean points. In the 

 case of the stigmatic bands above, a and k are based on 

 the average of far too few capsules for us to lay much 

 weight upon them, and the other points will be seen to be 

 close to the line AB. When all the points ^ lie on the 

 line, the regression and correlation are said to be liitear, 

 otherwise they are termed skezv. The frequency given for 

 any group of capsules associated with a selected capsule 

 is termed an array. Thus 4, 9, 27, 52, 35, 23, 28, i 2 is 

 termed the array corresponding to 14. Now by the 

 method given on p. 387 we might find the standard 

 deviation of an array. Let us represent this by the letter 

 S, and let a represent the standard deviation of all poppy 

 capsules whatsoever. 



Now let us suppose there to be no regression whatever, 

 then the mean of the 5 array will be 5, of the 6 array 6, 

 of the 7 array 7, and so on ; the regression line accord- 

 ingly becomes the diagonal DE. In other words, when 

 the regression is zero, the slope of the regression line {i.e. 

 of the 45° line) is unity. On the other hand, if AB were 

 horizontal, the mean of each array would be 10.04 or the 

 mean of the general population, i.e. there would be perfect 

 regression, or no correlation at all. Hence we see 

 that the steepness between o and i of the line AB is a 

 measure of the amount of regression or of correlation 

 exhibited in the pairs of capsules. This steepness, or the 



1 Within the degree of exactness warranted by the probable errors of the 

 observations. Of course in practice the points will never lie exactly on a 

 straight line. 



