Linear Regression Formula

Linear regression is defined as a data technique that determines the relationship between two variables by applying a linear equation to the given data. Here, one variable is supposed to be independent, while the other is supposed to be dependent. It is used with the help of a linear regression equation, which is similar to the slope-intercept form. To measure the strength of the relationship between two variables, we use a correlation coefficient which has a value range of -1 to +1. The slope of the regression line is calculated by putting the independent variable equal to zero in the equation and then solving for the dependent variable.

Linear regression formula

y = a + bx

Where,

• y is the dependent variable that lies along the y-axis,
• a is the y-intercept,
• b is the slope of regression line,
• x is the independent variable that lies along the x-axis,

The intercept value, a, and slope of the line, b, are evaluated using the formulas given below:

Where,

• x denotes the values of independent data set,
• y denotes the values of dependent data set.

Sample Problems

Problem 1: Find the linear regression equation for the given data:

x	y
3	8
9	6
5	4
3	2

Solution:

Calculate the intercept and slope value.

x	y	x2	xy
3	9	9	27
5	3	25	15
8	6	64	48
4	2	16	8
∑x = 20	∑y = 20	∑x2 = 114	∑xy = 98

Using the formula we get,

= (20 (114) – 20 (98)) / (4 (114) – 400)

= 320/56

= 5.71

= (4 (98) – 20 (20)) / (4 (114) – 400)

= -8/56

= -0.14

So, the linear regression equation is, 5.71 – 0.14 x.

Problem 2: Find the linear regression equation for the given data:

x	y
4	6
7	5
3	8
1	3

Solution:

Calculate the intercept and slope value.

x	y	x2	xy
4	6	16	24
7	5	49	35
3	8	9	24
1	3	1	3
∑x = 15	∑y = 22	∑x2 = 75	∑xy = 134

Using the formula we get,

= (15 (75) – 15 (134)) / (4 (75) – 225)

= -885/75

= -11.8

= (4 (134) – 15 (22)) / (4 (75) – 225)

= -206/75

= -0.14

So, the linear regression equation is, -11.8 – 2.74 x.

Problem 3: Find the intercept of linear regression line if ∑x = 25, ∑y = 20, ∑x² = 90, ∑xy = 150 and n = 5.

Solution:

Using the formula we get,

= (20 (90) – 25 (150)) / (5 (90) – 625)

= -1950/-175

= 11.14

Problem 4: Find the intercept of linear regression line if ∑x = 30, ∑y = 27, ∑x² = 110, ∑xy = 190 and n = 4.

Solution:

Using the formula we get,

= (27 (110) – 25 (190)) / (5 (110) – 900)

= -1780/-350

= 5.08

Problem 5: Find the slope of linear regression line if ∑x = 10, ∑y = 16, ∑x² = 60, ∑xy = 120 and n = 4.

Solution:

Using the formula we get,

= (4 (120) – 10 (16)) / (4 (60) – 100)

= 320/140

= 2.28

Problem 6: Find the slope of linear regression line if ∑x = 40, ∑y = 32, ∑x² = 130, ∑xy = 210 and n = 4.

Solution:

Using the formula we get,

= (4 (210) – 40 (32)) / (4 (130) – 1600)

= -440/-1080

= 0.404

Problem 7. Find the slope of linear regression line if ∑x = 50, ∑y = 44, ∑x² = 150, ∑xy = 230 and n = 4.

Solution:

Using the formula we get,

= (44 (150) – 50 (230)) / (4 (150) – 2500)

= -4900/-1900

= 2.57

= (4 (230) – 50 (44)) / (4 (150) – 2500)

= -1280/-1900

= 0.67