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Unformatted text preview: egression lines are based on two equations known as regression
equations. These equations show best estimate of one variable for
the known value of the other. The equations are linear.
Linear regression equation of Y on X is
Y = a + bX …. (1)
And X on Y is
X = a + bY…. (2)
a, b are constants.
219 From (1) We can estimate Y for known value of X.
(2) We can estimate X for known value of Y.
9.3.1 Regression Lines:
For regression analysis of two variables there are two
regression lines, namely Y on X and X on Y. The two regression
lines show the average relationship between the two variables.
For perfect correlation, positive or negative i.e., r = + 1,
the two lines coincide i.e., we will find only one straight line. If r =
0, i.e., both the variables are independent then the two lines will cut
each other at right angle. In this case the two lines will be parallel
to X and Y-axes.
r =+1 O XO X O
Lastly the two lines intersect at the point of means of X and
Y. From this point of intersection, if a straight line is drawn on Xaxis, it will touch at the mean value of x. Similarly, a perpendicular
drawn from the point of intersection of two regression lines on Yaxis will touch the mean value of Y.
( x, y ) O X O 220 X 9.3.2 Principle of ‘ Least Squares’ :
Regression shows an average relationship between two
variables, which is expressed by a line of regression drawn by the
method of “least squares”. This line of regression can be derived
graphically or algebraically. Before we discuss the various methods
let us understand the meaning of least squares.
A line fitted by the method of least squares is known as the
line of best fit. The line adapts to the following rules:
The algebraic sum of deviation in the individual
observations with reference to the regression line may be
equal to zero. i.e.,
∑(X – Xc) = 0 or ∑ (Y- Yc ) = 0
Where Xc and Yc are the values obtained by regression analysis.
The sum of the squares of these deviations is less than
the sum of squares of deviations from any other line...
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- Winter '08