704 



APPENDIX 



the observations from their mean value. These deviations may be con- 

 veniently represented with respect to coordinate axes (Fig. 8). 



By the range along the ;r-axis we shall mean such an interval that ordi- 

 nates drawn at the extremities of the interval include between them the 

 total population. Thus, in Fig. 8, the range is taken from a to b. This 

 range may well be divided into some number, say s, of equal parts, each 

 of width AJT, by ordinates at the points of division. If we let x{, x, , 

 x s ' be the abscissas of the feet of the ordinates through the middle points 

 of the s classes, we shall call these the marks of the classes of /'s. The 

 values of y which belong to a given class of x are said to form a j-array. 



A'- 



X 



X 



A' 



x 



x 



X 



X 



r 



FIG. 8 



Let the crosses (x) in Fig. 8 represent the means of the j's in each of 

 the .r-arrays. If correlation exists, these means do not lie at random over 

 the field, but arrange themselves more or less in the form of a smooth 

 curve called the " curve of regression." This curve is a crude picture of the 

 function which defines the correlation of the ^-character relative to the 

 ^-character. Experience has shown that, in many sets of measurements, 

 this line is approximately a straight line. For this reason, and for sim- 

 plicity, the line subjected to the condition that the sum of the squares of the 

 deviations (measured parallel to the _y-axis and weighted with number of 

 points in array) of the means from it shall be a minimum, is called the 

 "line of regression." When the means lie exactly on the line the degres- 

 sion is said to be " truly linear." 



