Page 173 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes 2. The two variables may act upon each other: Cause and effect relation exists in this case
also but it may be very difficult to find out which of the two variables is independent. For
example, if we have data on price of wheat and its cost of production, the correlation
between them may be very high because higher price of wheat may attract farmers to
produce more wheat and more production of wheat may mean higher cost of production,
assuming that it is an increasing cost industry. Further, the higher cost of production may
in turn raise the price of wheat. For the purpose of determining a relationship between the
two variables in such situations, we can take any one of them as independent variable.
3. The two variables may be acted upon by the outside influences: In this case we might get a
high value of correlation between the two variables, however, apparently no cause and
effect type relation seems to exist between them. For example, the demands of the two
commodities, say X and Y, may be positively correlated because the incomes of the
consumers are rising. Coefficient of correlation obtained in such a situation is called a
spurious or nonsense correlation.
4. A high value of the correlation coefficient may be obtained due to sheer coincidence (or
pure chance): This is another situation of spurious correlation. Given the data on any two
variables, one may obtain a high value of correlation coefficient when in fact they do not
have any relationship. For example, a high value of correlation coefficient may be obtained
between the size of shoe and the income of persons of a locality.

8.1.3 Properties of Coefficient of Correlation

1. The coefficient of correlation is independent of the change of origin and scale of
measurements.
This property is very useful in the simplification of computations of correlation. On the
basis of this property, we can write a short-cut formula for the computation of r :
XY
n u v u v
i i i i
r XY
2 2 2 2
n u u n v v
i i i i
2. If X and Y are independent they are uncorrelated, but the converse is not true: This property
points our attention to the fact that r is only a measure of the degree of linear association
XY
between X and Y. If the association is non-linear, the computed value of r is no longer a
XY
measure of the degree of association between the two variables.
8.1.4 Scatter Diagram

Let the bivariate data be denoted by (X , Y ), where i = 1, 2 ...... n. In order to have some idea about
i i
the extent of association between variables X and Y, each pair (X , Y ), i = 1, 2......n, is plotted on
i i
a graph. The diagram, thus obtained, is called a Scatter Diagram.
Each pair of values (X , Y ) is denoted by a point on the graph. The set of such points (also known
i i
as dots of the diagram) may cluster around a straight line or a curve or may not show any
tendency of association. Various possible situations are shown with the help of following
diagrams:

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