Example Method of Least Squares

The given example explains how to find the equation of a straight line or a 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 for the following data. Also find the 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 from equation (2). Thus 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 the above equation (see column 4 in the table above).