Page 168 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes 2
P.E. = 0.6745 1–.8 = .06
r 16

The limits of the correlation in the population would be ± P.E.r , i.e., .8 ± .06 or .74—.86.
Instances are quite common wherein a correlation coefficient of 0.5 or even 0.4 has been considered
to be a fairly high degree of correlation by a writer or research worker. Yet a correlation coefficient of
0.5 means that only 25 per cent of the variation is explained. A correlation coefficient of 0.4 means
that only 16 per cent of the variations is explained.
Conditions for the Use of Probable Error

The measure of probable error can be properly used only when the following three conditions exist:
1. The data must approximate a normal frequency curve (bell-shaped curve).
2. The statistical measure for which the P.E. is computed must have been calculated from a sample.
3. The sample must have been selected in an unbiased manner and the individual items must be
independent.
However, these conditions are generally not satisfied and as such the reliability of the correlation
coefficient is determined largely on the basis of exterior tests of reasonableness which are often
of a statistical character.
Example 9: If r = 0.6 and N = 64, find out the probable error of the coefficient of correlation and
determine the limits for population r.

1– r 2
Solution: P.E. = 0.6745
r N
r = 0.6 and N = 64
()
1– .6 2 0.6745 × 0.64
P.E. = 0.6745 = = 0.054
r 64 8
Limits of population correlation
= 0.6 ± 0.054 = 0.546—0.654.

Coefficient of Determination

One very convenient and useful way of interpreting the value of coefficient of correlation between
two variables is to use the square of coefficient of correlation, which is called coefficient of
determination. The coefficient of determination thus equals r . The coefficient r expresses the
2
2
proportion of the variance in y determined by x; that is, the ratio of the explained variance to total
variance. Therefore, the coefficient of determination expresses the proportion of the total variation
that has been ‘explained’, or the relative reduction in variance when measured about the regression
equation rather than about the mean of the dependent variable. If the value of r = 0.9, r will be 0.81
2
and this would mean that 81 per cent of the variation in the dependent variable has been explained
by the independent variable. The maximum value of r is unity because it is possible to explain all of
2
the variation in Y, but it is not possible to explain more than all of it.
2
It is much easier to understand the meaning of r than r and, therefore, the coefficient of determination
is to be preferred in presenting the results of correlation analysis. Tuttle has beautifully pointed out
that “the coefficient of correlation has been grossly overrated and is used entirely too much. Its
square, the coefficient of determination, is a much more useful measure of the linear covariation of
two variables. The reader should develop the habit of squaring every correlation coefficient he finds
cited or stated before coming to any conclusion about the extent of the linear relationship between
the two correlated variables.”

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