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Statistics

Notes The other term Y - Y gives the deviation of i th observed value from the regression line and
i Ci
2
å Y - Y ) gives the variations in Y about the line of regression.
thus the magnitude of the term ( i Ci
These variations are also known as unexplained variations in Y.
Adding the two types of variations, we get the magnitude of total variations in Y. Thus, equation
(2) can also be written as
Total variations in Y = Unexplained variations in Y + Explained variations in Y.

2
å Y - Y ) , we have
Dividing both sides of equation (2) by ( i

Y -
å ( i Y Ci ) 2 å ( Ci Y ) 2
Y -
1 = 2 + 2 .... (3)
Y -
Y -
å ( i Y ) å ( i Y )
or 1 = Proportion of unexplained variations + Proportion of variations explained by the regression
equation.
The proportion of variation explained by regression equation is called the coefficient of
determination.

å Y - Y ) 2
Thus, the coefficient of determination ( Ci
= 2
Y -
å ( i Y )
2 2 é ) ù 2
b å ( X - X ) ë å ( X - X Y - Y û 2
)( i
i
i
= 2 = 2 2 = r
Y -
å ( i Y ) å ( X - X å Y - Y )
) ( i
i
This result shows that the coefficient of determination is equal to the square of the coefficient of
correlation, i.e., r gives the proportion of variations explained by each regression equation.
2
Remarks:
(i) It should be obvious from the above that it is desirable to calculate the coefficient of
2
correlation prior to the fitting of a regression line. If r is high enough, the fitted line will
explain a greater proportion of the variations in the dependent variable. A low value of r 2
would, however, indicate that the proposed fitting of regression would not be of much
use.
(ii) The expression for the coefficient of determination for regression of X on Y can be written

å ( X - X ) 2
Ci
2
in a similar way. Here we can write r = 2 .
å ( X - X )
i
23.3.1 The Coefficient of Non-Determination

The proportion of unexplained variations is also termed as the coefficient of non-determination.
2
2
2
It is denoted by k , where k = (1 - r ). The square root of k is termed as the coefficient of
2
2
alienation, i.e., k = ( 1- r ) .

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