Least Square Method Solved Examples

Problem 1: Find the line of best fit for the following data points using the least squares method: (x,y) = (1,3), (2,4), (4,8), (6,10), (8,15).

Solution:

Here, we have x as the independent variable and y as the dependent variable. First, we calculate the means of x and y values denoted by X and Y respectively.

X = (1+2+4+6+8)/5 = 4.2

Y = (3+4+8+10+15)/5 = 8

xi	yi	X – xi	Y – yi	(X-xi)*(Y-yi)	(X – xi)2
1	3	3.2	5	16	10.24
2	4	2.2	4	8.8	4.84
4	8	0.2	0	0	0.04
6	10	-1.8	-2	3.6	3.24
8	15	-3.8	-7	26.6	14.44
Sum (Σ)		0	0	55	32.8

The slope of the line of best fit can be calculated from the formula as follows:

m = (Σ (X – x_i)*(Y – y_i)) /Σ(X – x_i)²

m = 55/32.8 = 1.68 (rounded upto 2 decimal places)

Now, the intercept will be calculated from the formula as follows:

c = Y – mX

c = 8 – 1.68*4.2 = 0.94

Thus, the equation of the line of best fit becomes, y = 1.68x + 0.94.

Problem 2: Find the line of best fit for the following data of heights and weights of students of a school using the least squares method:

• Height (in centimeters): [160, 162, 164, 166, 168]
• Weight (in kilograms): [52, 55, 57, 60, 61]

Solution:

Here, we denote Height as x (independent variable) and Weight as y (dependent variable). Now, we calculate the means of x and y values denoted by X and Y respectively.

X = (160 + 162 + 164 + 166 + 168 ) / 5 = 164

Y = (52 + 55 + 57 + 60 + 61) / 5 = 57

xi	yi	X – xi	Y – yi	(X-xi)*(Y-yi)	(X – xi)2
160	52	4	5	20	16
162	55	2	2	4	4
164	57	0	0	0	0
166	60	-2	-3	6	4
168	61	-4	-4	16	16
Sum ( Σ )		0	0	46	40

Now, the slope of the line of best fit can be calculated from the formula as follows:

m = (Σ (X – x_i)✕(Y – y_i)) / Σ(X – x_i)²

m = 46/40 = 1.15

Now, the intercept will be calculated from the formula as follows:

c = Y – mX

c = 57 – 1.15*164 = -131.6

Thus, the equation of the line of best fit becomes, y = 1.15x – 131.6

Least Square Method: In statistics, when we have data in the form of data points that can be represented on a cartesian plane by taking one of the variables as the independent variable represented as the x-coordinate and the other one as the dependent variable represented as the y-coordinate, it is called scatter data. This data might not be useful in making interpretations or predicting the values of the dependent variable for the independent variable where it is initially unknown. So, we try to get an equation of a line that fits best to the given data points with the help of the Least Square Method. 

In this article, we will learn the least square method, its formula, graph, and solved examples on it.