Linear Regression

The regression line is the line that best fits or represents the data on the scatter plot.

Line of Regression of Y on X

The regression line of y on x is used to estimate the values of y from x.

The slope of the line is the quotient between the covariance and variance of the variable X.

Line of Regression of Y on X


Line of Regression of X on Y

The regression line of x on y is used to estimate the values of x from the values of y.

The slope of the line is the quotient between the covariance and variance of the variable y.

Line of Regression of X on Y


If r = 0 the regression lines are perpendicular to each other, and their equations are:

y = media de y

x = media de x


Example

The scores of 12 students in their mathematics and physics classes are:

Mathematics 2 3 4 4 5 6 6 7 7 8 10 10
Physics 1 3 2 4 4 4 6 4 6 7 9 10

Find the regression lines and represent them.

xi yi xi ·yi xi2 yi2
2 1 2 4 1
3 3 9 9 9
4 2 8 16 4
4 4 16 16 16
5 4 20 25 16
6 4 24 36 16
6 6 36 36 36
7 4 28 49 16
7 6 42 49 36
8 7 56 64 49
10 9 90 100 81
10 10 100 100 100
72 60 431 504 380

1. Find the arithmetic means.

Arithmetic Means Calculations

2. Calculate the covariance.

Covariance Operations

3. Calculate the variances.

Variance Calculations

4.Linear regression of y on x.

Linear Regression of Y on X

5.Linear regression of x on y.

Linear Regression of X on Y

Linear Regression Graph




  • Linear Correlation Coefficient
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