Sec. 7.3 STANDARD DEVIATION ABOUT TREND LINE 209 



about 15 units greater than the Y for X = 7; hence some of any ob- 

 served difference between the F for X = 7 and the Y for X = 10 can 

 be accounted for and need not be considered as sampling error. 



Graphically the preceding remarks mean that if Y and X can be 

 considered to be linearly related the standard deviation of the F, : 

 should be calculated from the trend line rather than from the horizon- 

 tal line: Y = y. This means that the quantity 2(F — F) 2 should be 

 employed in this calculation instead of 2(F — y) 2 . However, the divi- 

 sor in this calculation will not be (n — 1) as it is for Sy, above. 



The divisor needed in the computation of the standard deviation 

 about the trend line is (n — 2). The reason for this cannot be given 

 conclusively without mathematical analysis which is not appropriate 

 to this book; but it can be rationalized in the following manner. 

 Suppose that a sample of 5 observations on X and F simultaneously 

 were as follows: 



X: 1 2 3 4 5 x = 3 

 F: 54687 y = 6 



It is readily determined that F = 0.8CLY + 3.6; hence the following 

 table can be set up for purposes of illustration : 



(Yi 



What are the deviations from the trend line for X = 4 and X = 5, 

 respectively? The fact that S(F,- — Y { ) = will be found to account 

 for one of these deviations. The fact that b = 0.80 will allow the de- 

 termination of the second unknown deviation. 



Let the unknown deviations F 4 — F 4 and F 5 — F 5 corresponding 

 to X = 4 and X = 5 be denoted by v and w, respectively. It follows 

 that 



S(Fi - Yd = 0.6 + (-1.2) + + v + w = 0, 



which reduces easily to 



(7.31) v + w = 0.6. 



The slope of a straight line can be computed by determining the 

 amount by which F changes for any chosen change in X, and by taking 

 the ratio of the former to the latter. For example, if it is determined 

 by measurement on the graph or by substitution into a mathematical 

 formula that for the interval from X = 1 to X = 5 the height of the 



