Least-Squares Method

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Widely used statistical technique employed to study trends in revenue, costs, production, and other data and to investigate the relationships among accounting and financial variables. It fits a straight line through a set of points in such a way that the sum of the squared distances from the data points to the line is minimized. The least-squares method involves the following steps:

(1) Define the distance from the data point from the line, denoted by u, as follows:

u = (y - y')

where y = observed value and y' = estimated value base on the line y' = a + bx (see the following figure).

(2) Minimize the sum of the squared distances:

Min Su² = S(y - y´)² = S(y - (a + bx))²

Using differential calculus yields the following equations, called Normal Equations:

Sy = na + bSx

Sxy = aSx + bSx²

Solving the equation for b and a yields:

To illustrate the computations of b and a, refer to the following data. All the sums required are computed and shown here:

Direct Labor- Factory Hours (x) Overhead (y) xy x²

9 hours $ 15 135 81

19 20 380 361

11 14 154 121

14 16 224 196

23 25 575 529

12 20 240 144

12 20 240 144

22 23 506 484

7 14 98 49

13 22 286 169

15 18 270 225

17 18 306 289

174 hours $225 3414 2792

From the table above:

Sx = 174 Sy = 225 Sxy = 3414 Sx² = 2792

x- = Sx/n = 174/12 = 14.5 y- = Sy/n = 225/12 = 18.75

Substituting these values into the formula for b first:

nSxy - (Sx)(Sy)

b = -------

nSx² - (Sx)²

(12)(3414) - (174)(225) 1818

= ----------- = --- = 0.5632

(12)(2792) - 174)² 3228

a = y- - bx-

= (18.75) - (0.5632)(14.5) = 18.75 - 8.1664 = 10.5836

Therefore, y´ = 10.5836 + 0.5632 x