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
x^{2}
xy
3
9
9
27
5
3
25
15
8
6
64
48
4
2
16
8
∑x = 20
∑y = 20
∑x^{2} = 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
x^{2}
xy
4
6
16
24
7
5
49
35
3
8
9
24
1
3
1
3
∑x = 15
∑y = 22
∑x^{2} = 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, ∑x^{2} = 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, ∑x^{2} = 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, ∑x^{2} = 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, ∑x^{2} = 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, ∑x^{2} = 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
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