STATISTICAL TREATMENTS 209 



Lines are commonly drawn through such variable data by the method 

 of least squares. The sum of the squares of all the deviations from the line 

 is made to be a minimum. In other w^ords, that line is found such that the 

 sum of the squared deviations is smallest. U y = a -\- hx, and if we know 

 a and h, the line is determined. In the least squares formula 



XyXx- — S ?c 2 y" 



"""^ NXx~-c^xy 



N S ?cy — S x S }' 

 h = 



Nix^-C^ xy 



where N is the number of points. These are the basic equations; varia- 

 tions may be used in special cases, as when the number of points is small. 



The correlation coefficient (r) was devised to express quantitatively the 

 relationship between two sets of variables. It has values ranging from 

 — 1 to to + 1 . If r is positive, larger values of x are also large with re- 

 spect to y. Taller men are heavier, in general. Zero correlation means 

 there is no consistent pattern between the x values and the y values. 

 Negative values of r mean that values large in respect to x are small in 

 respect to y. 



The correlation coefficient actually measures the relationship of x 

 and y in terms of their normal deviates (c), 



•^ CxCy 



r =^ 



n 



This expression is calculated by means of the following approximation: 



S(x — x/sa:')Cy — y/sy) _ S(x — xXy — y) 



r = 



n nSxSy 



The computation is easier on the electric calculator if the equation is 



rearranged to 



Xxy/n — xy 

 X = 



SxSy 



The correlation coefficient can provide valuable information but must 

 be used with discretion. It is too easy to attribute a cause-effect relation- 

 ship to a situation where r is large. For example, we might find a corre- 

 lation between the amount of water flowing in a river and the number of 

 automobiles sold. There is no conceivable way in which one of these 

 could cause the other even if they might be related through a third factor, 

 namely the season of the year. 



