Page 346 - DMTH404_STATISTICS

Statistics

Notes III. Relation of r with b and d

s Y s X 2
r
b d = × r × × = r
´
s X s Y
or r = b d
´

23.7 Keywords

Coefficient of correlation: If the coefficient of correlation calculated for bivariate data (X , Y ),
i i
i = 1,2, ...... n, is reasonably high and a cause and effect type of relation is also believed to be
existing between them, the next logical step is to obtain a functional relation between these
variables.
Term regression: The term regression was first introduced by Sir Francis Galton in 1877.
Independent variable: For a bivariate data (X , Y ), i = 1,2, ...... n, we can have either X or Y as
i i
independent variable.

23.8 Self Assessment

1. Fill in the blanks :
(i) The two regression coefficients are of ........ sign.

(ii) If a regression coefficient is negative then the correlation between the variables
would also be ........
(iii) The coefficient of determination is a real number lying between ........ and ........ .

(iv) Regression analysis is used to study ........ between the variables.
(v) If correlation between two variables is zero, the two regression lines are ........ to
each other and if it is equal to ± 1, the two lines are the ........ .

(vi) The smaller is the angle between the two lines of regression, the ........ is correlation
between the variables.
(vii) If r  ± 1, the two regression lines are ........ .

23.9 Review Questions

1. Distinguish between correlation and regression. Discuss least square method of fitting
regression.
2. What do you understand by linear regression ? Why there are two lines of regression?
Under what condition(s) can there be only one line ?
3. Define the regression of Y on X and of X on Y for a bivariate data (X , Y ), i = 1, 2, ...... n. What
i i
would be the values of the coefficient of correlation if the two regression lines (a) intersect
at right angle and (b) coincide?
4. (a) Show that the proportion of variations explained by a regression equation is r 2

(b) What is the relation between Total Sum of Squares (TSS), Explained Sum of Squares
(ESS) and Residual Sum of squares (RSS). Use this relationship to prove that the
coefficient of correlation has a value between –1 and +1.
Hint: See § 23.3

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