By p Nitin

Feb 16, 2013

Linear Regression Definition states that it can be measured by using lines of regression. Regression measures the amount of average relationship or mathematical relationship between two variables in terms of original units of data. Whereas, correlation measures the nature of relationship between two variables. i.e.., positive or negative or uncorrelated.

Regression is used for estimating the value of one variable if you know the value of other variable. i.e.., One of the variable is independent variable and other variable is dependent variable.

Let ( Xi , Yi ) ; i = 1, 2, 3, ...................n the n pairs of observations are given now plot all these points in XY-plane which reserves a scatter diagram.

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Let ( Xi , Yi ) ; i = 1,2,3,................ n be n-pairs of observations are given and there are representing a linear regression.

We know that, coefficient of correlation

r = cov ( X , Y ) / (sigmaX sigmaY ) where cov ( X , Y ) = ( 1/n) sum XiYi - barX barY

and X2 = (1/n) sum ( Xi - barX

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( 5 )

cov ( X, Y ) = ( 1/n)sum Xi Yi - barX barY

sigma X2 = ( 1/n )sum X2i - barX 2

Multiplying equation ( 4 ) with X and sub from ( 5 )

cov ( X , Y ) = b sigma X2

b = cov ( X , Y ) / sigma X2

Substitute the value of b in ( 4 )

rArr barY = a + cov ( X , Y ) barX / sigma X2

a = barY - cov ( X , Y ) barX / sigma X2

Substitute the value of a in ( 1 )

Y = barY - cov ( X , Y ) barX / sigma X2 + cov ( X , Y ) X / sigma X2

Y - barY = cov ( X , Y ) ( X - barX ) / sigma X2

Y - barY = rsigma X sigma Y ( X -barX ) / sigma X2

Y - barY = r sigma Y ( X - barX ) / sigma X

Y - barY = bYX ( X - barX )

Similarly, we can prove that regression equation of X-on-Y is

X - barX = bXY ( Y - barY )

Types of Linear Regression

There are two different types of linear regression. They are,

Simple Linear Regression

Multiple Linear Regression

Examples

Given below are some examples on how to calculate linear regression.

Example:

10 observations on price X and supply "Y" the following data was obtained sum X = 130, sum Y = 220, sum X2 = 2288, sum2 = 5506, sum XY = 3467 Find the line of regression of Y on X. Y

Solution:

The line of regression of Y on