Handbook of Civil Engineering Calculations

FIGURE 36. Regression line, or line of best fit.

2. Set up the criterion for the regression line
   Draw the arbitrary straight line in Fig. 36, and consider the vertical deviation e of a point
   in the scatter diagram from this arbitrary line. The regression line is taken as the line for
   which the sum of the squares of the deviations is minimum.
   Let 7 denote the ordinate of a point in the scatter diagram and YR the corresponding or-
   dinate on the regression line. By definition, Se^2 = 2(7- YR)^2 = minimum.


3. Write the equation of the regression line
   Let n denote the number of points in the scatter diagram, and let YR = a + bXbe the equa-
   tion of the regression line, where X denotes the year number as measured from some con-
   venient datum. To find the regression line, parameters a and b must be evaluated.
   Since Se^2 is to have a minimum value, express the partial derivatives of 2e^2 with re-
   spect to a and b, and set these both equal to zero. Then derive the following simultaneous
   equations containing the unknown quantities a and b:

2Y = an+bZX

^XY = aZX + b^X^2

4. Simplify the calculations
   Select the median date (the end of 19CC) as a datum. This selection causes the term ^X to
   vanish.


5. Determine the values of SX^2 and SXY
   Prepare a tabulation such as Table 33. Use the data for each year in question.


6. Solve for parameters a and b
   Substitute in the equations in step 3, and solve for a and b. Thus 2118 = 5a; 402 = 1OZ?;
   a -423.6; b -40.2.

Year

Arbitrary line

Annual

sales,

in

units

of $

1000