Appendix 193 



all of the n observations of x, the sum of the corresponding 



(Y-yT , 



— ^r s will be a minimum, i.e. 



That —A X-L be a minimum. 



N 

 But substituting the value mX for y', the condition becomes: 



(1) That S(Y ~ mX)2 be a minimum. 



N 



(2) Expanding (1) we have S ( y2 ~ 2mXY+m 2 X 2 ) which ^ 



m S Y 2 9™ SxY , m ,2X 2 



N N N 



2 Y 2 

 But, by definition is <Ty 2 



V vj 



Similarly — — = a x 2 an d (3) 



becomes (4) <jy 2 — 2 m 5^X + m 2 <7x 2 



V 

 Call this equation tt. Now if m should increase by an 



infinitesimally small quantity, e, a new equation would result: 



(5) Y.' = (j y 2 -2(m+e)^M + (m+e)«cr x 2 



V 



But for the original equation (4) ^ to be a minimum it 



V 

 must be less than (5) ^r and if e is such a small quantity 



that its square can be disregarded (5) becomes 

 (6) Z = a y 2 — 2m^X — 2e?^X + m 8 <J 8 + 2mea x » 

 If e is an infinitesimally small quantity then for all practical 



purposes X = Y_ and ~ = o. Subtract (4) from (6) and 



N N N 



the result is (7 ) V '~ V = — 2e 5^X + 2e(m ax 8 ), therefore: 



N N 



Sxr 



(8) 2e(max«-^p , )= o 



For (8) to be true a sufficient condition is that the coefficient 

 of the constant 2e be zero, i. e. : 



mtf-^- Y = o or »«#*.£*£ or (9) m = |2? 

 N N N <Jx 2 



The expression (9) gives us the slope of the line. But, since 



the coefficient desired was one expressed in terms of the two 



standard deviations, we may multiply both sides by 



