192 Appendix 



axes at the mean of the systems, — the point whose co-ordin- 

 ates from absolute zero are average x, average y. Plot the 

 cases from this zero point in terms of their deviations from 

 their averages. Each case is then represented by a point, whose 

 distance from the vertical axis parallel to the horizontal axis is 

 the deviation of the case from its x average and whose distance 

 from the horizontal axis parallel to the vertical axis is the 

 deviation of the case from the y average. In other words the 

 coordinates of the points representing the n cases would be 



XjYj, X 2 Y 2 , X a Y 8 X„Y n . In Diagram III a typical point is 



marked P and the X and Y for this point shown graphically. 



3. The problem is then to "fit" a straight line to this scatter 

 diagram of points which will describe the relationship between 

 the X and Y series in the n cases studied. The best fitting 

 straight line will be the one from which the average of the 

 squares of the deviations of the n points will be a minimum. 

 But the single point from which the average of the squares of 

 the deviations of all the points is a minimum is the point 

 whose coordinates are average x r average y, or the zero point 

 in the diagram as we have constructed it. This may there- 

 fore be assumed as a point on the best fitting line. (For the 

 student who wishes a mathematical proof that this point lies 

 on the best fitting line the proofs of Moore and Yule, are cited. 

 Since this is proven in other derivations of the correlation co- 

 efficient, it was deemed expedient to assume it 'here in the inter- 

 ests of brevity and clearness.) 



4. Since the line passes through the zero point and has no 

 intercept on either axis its equation will have the general form 

 y equals mx, m being some constant which will express its slope. 



5. For the series of observations plotted along the horizon- 

 tal axis as X , X 2 , X 8 ....X„, there will be a similar series of 

 points on the line. In the terms of the equation of the line the 



abscissae of these points will be y' v y' 2 y' 3 y' n . These points 



will have the coordinates X^'g, X 2 y' 2 , X 3 y' 3 X n y'n. (See 



Key of Diagram III.) The equation may therefore be written 

 y' equals mX. 



6. The vertical deviation of each point from the line will 

 then be Y—y\ Y 2 — y' 2 , etc. (On Diagram III Yp— y'p=the 

 line FP\) The problem is then to determine m so that, for 



