Regression is a statistical tool used to predict unknown values of a dependent variable from known values of one or more independent variables. It estimates the average change in the dependent variable given a change in the independent variable(s). There are two regression lines - one with Y as the dependent variable (Y on X) and one with X as the dependent variable (X on Y). The regression equation expresses these lines algebraically. The constants a and b are estimated using the method of least squares, which finds the line that minimizes the vertical differences between actual and estimated Y values. Multiple regression uses more than one independent variable to increase prediction accuracy.

1. UNIT III
REGRESSION

2. Meaning
 The dictionary meaning of regression is “the
act of returning or going back”;
 First used in 1877 by Francis Galton;
 Regression is the statistical tool with the help
of which we are in a position to estimate
(predict) the unknown values of one variable
from the known values of another variable;
 It helps to find out average probable change
in one variable given a certain amount of
change in another;

3. Importance

4. Regression lines
 For two variables X and Y, we will have two
regression lines:
1. Regression line X on Y gives values of Y for
given values of X;
2. Regression line Y on X gives values of X for
given values of Y;

5. Regression Equation
 Regression equations are algebraic expressions
of regression lines;
Y on X
Regression equation expressed as
Y=a+bX
Y is dependent variable
X is independent variable
„a‟ & „b‟ are constants/parameters of line
„a‟ determines the level of fitted line (i.e. distance of
line above or below origin)
„b‟ determines the slope of line (i.e change in Y for
unit change in X)

6.  Regression equations are algebraic expressions
of regression lines;
X on Y
Regression equation expressed as
X=a+bY
X is dependent variable
Y is independent variable
„a‟ & „b‟ are constants/parameters of line
„a‟ determines the level of fitted line (i.e. distance of
line above or below origin)
„b‟ determines the slope of line (i.e change in Y for
unit change in X)

7. Method of Least Square
 Constant “a” & “b” can be calculated by method of
least square;
 The line should be drawn through the plotted
points in such a manner that the sum of square of
the vertical deviations of actual Y values from
estimated Y values is the least i.e. ∑(Y-Ye)2
should be minimum;
 Such a line is known as line of best fit;
 with algebra & calculus:
For Y on X For X on Y
∑Y=Na+b ∑X ∑X=Na+b ∑Y
∑XY=a ∑X + b ∑X2 ∑XY=a ∑Y + b ∑Y2

8. Multiple Regression
 When we use more than one independent
variable to estimate the dependent variable in
order to increase the accuracy of the estimate;
the process is called multiple regression
analysis.
 It is based on the same assumptions &
procedure that are encountered using simple
regression.
 The principal advantage of multiple regression
is that it allows us to use more of the
information available to us to estimate the

9. Estimating equation describing
relationship among three variables
Y= a+b1X1+b2X2
 where, Y = estimated value corresponding to
the dependent variable
 a= Y intercept
 b1 and b2 = slopes associated with X1 and X2,
respectively
 X1 and X2 = values of the two independent
variables

10. Normal Equations:
 we use three equations (which statistician call
the “normal equation”) to determine the values
of the constants a, b1 and b2
 ∑Y=Na+b1∑X1 + b2∑X2
 ∑X1Y=a ∑X1 + b1 ∑X1
2 + b2∑X1 X2
 ∑X2Y=a ∑X2 + b2 ∑X2
2 + b1∑X1 X2

11. Difference between regression &
correlation
 Correlation coefficient (r)
between x & y is a
measure of direction &
degree of linear
relationship between x &
y;
 It does not imply cause &
effect relationship
between the variables.
 It indicates the degree of
association
 bxy & byx are
mathematical measures
expressing the average
relationship between the
two variables
 It indicates the cause &
effect relationship between
variables.
 It is used to forecast the
nature of dependent
variable when the value of
independent variable is
Correlation Regression