Example Method of Least Squares

The given example explains you that how to find the equation of straight line or least square line by using the method of least square, which is very useful in statistics as well as in mathematics.

Example:

Fit a least square line to the following data. Also find trend values and show that \sum \left( {Y - \widehat Y} \right) = 0.

X

1

2

3

4

5

Y

2

5

3

8

7

Solution:


X

Y

XY

{X^2}

\widehat Y = 1.1 + 1.3X

Y - \widehat Y

1

2

2

1

2.4

-0.4

2

5

10

4

3.7

+1.3

3

3

9

9

5.0

-2

4

8

32

16

6.3

1.7

5

7

35

25

7.6

-0.6

\sum X = 15

\sum Y = 25

\sum XY = 88

\sum {X^2} = 55

Trend Values

\sum \left( {Y - \widehat Y} \right) = 0

The equation of least square line Y = a + bX
Normal Equation for ‘a \sum Y = na + b\sum X{\text{ }}25 = 5a + 15b ---- (1)
Normal Equation for ‘b \sum XY = a\sum X + b\sum {X^2}{\text{ }}88 = 15a + 55b ----(2)

Eliminate a from equation (1) and (2), multiply equation (2) by 3 and subtract form equation (2), we get the values of a and b.

Here    a = 1.1 and b = 1.3, the equation of least square line becomes Y = 1.1 + 1.3X.

For the trends values, put the values of X in above equation, see the above table column 4.

Comments

comments