Least Square Method Practice Questions

The Least Squares Method is a mathematical technique used to find the best-fitting curve or line through a set of data points by minimizing the sum of the squares of the differences between observed values and predicted values.

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

Practice Questions on Least Square Method

Question 1: Find the equation of the best-fit line for the data:

Question 2: Fit a straight line using least squares for:

Question 3: For the data:

(a) Find the least squares line.
(b) Comment on how well it fits the data.

Question 4: The following data represents the year and population (in thousands):

Fit a straight line and estimate the population in year 6.