i 



210 LINEAR REGRESSION AND CORRELATION Ch. 7 



straight line above the horizontal axis increases from 10 to 30, the 

 slope of this line is measured by 20/4 = 5. Hence, in the situation of 

 the preceding paragraph the slope is given by (F 5 — Y 4 )/(X 5 — X 4 ). 

 But (X 5 — X 4 ) = 1, and the slope is known to be 0.80; hence (F 5 — F 4 ) 

 -J- 1 = 0.80. In order to transform this equation into one involving v 

 and w consider the following two equations: 



Y$ — F 4 = F 5 — F 4 



f 5 - F 4 = 0.80. 



When the left and right members of the second equation are sub- 

 tracted from the corresponding members of the first equation, it is 

 found that 



(F 5 - F 5 ) - (F 4 - F 4 ) = Y 5 -Y 4 - 0.80; 



but Y 5 — Y 5 = w, F 4 — F 4 = v, and Y 5 — Y4 = — 1 ; therefore, 



(7.32) v -w = 1.8. 



When equations 7.31 and 7.32 are solved simultaneously it is found 

 that v = 1.2 and w = —0.6. Hence, two of the deviations from the 

 trend line can be calculated from the size of b and from the fact that 

 2(F — F) = 0. Although there are five actual deviations from the 

 linear trend line, only three (any three) of them actually should be 

 considered chance deviations from the regression line determined from 

 the sample. Hence, in the present problem, n — 2 = 5 — 2 = 3 will 

 be used as the divisor of 2(Fj — F z ) 2 in the computation of the stand- 

 ard deviation about the linear trend line. 



This divisor, n — 2, is generally called the number of degrees of free- 

 dom for the estimated standard deviation about the linear trend line, 

 just as the number, n — 1, is the number of degrees of freedom for the 

 estimated standard deviation, sy, about the mean. 



With the above discussion as a background, the formula for the 

 estimated standard deviation of the Y { about the trend line becomes 



(7.33) s y . x = Vs(7 t - - Yd 2 /(n - 2) , 



wherein the symbol, s y . x , is read "s sub y dot x." 



For the data used as illustration in this section, 2(F; — F;) 2 = 3.60, 

 n — 2 = 3; hence s y . x = V 3.60/3 = 1.10. This is a measure of the 

 variation among the F-measurements which remains unexplained even 

 after the linear trend with X has been taken into account. When the 

 trend with X is ignored, sy = V 10/4 = 1.58; so s y . x is 0.48 of a unit 

 smaller than sy. In other words, the variability of the F 2 - (as measured 



