KARL PEARSON’S COEFFICIENT OF CORRELATION

                                                      OR

PRODUCT- MOMENT COEFFICIENT OF CORRELATION

When deviations are taken from actual mean

Of the several mathematical methods of measuring correlation, the Karl Pearson’s method, popularly known as Pearsonian coefficient of correlation, is most widely used in practice.The Pearsonian coefficient of correlation is denoted by the symbol r. It is one of the very few symbols that are used universally for describing the degree of correlation between two series.
The formula for computing Pearsonian r is:

x = ( X - X ) ; y = (Y - Y )

σ_{x =} Standard deviation of series X

σ_{y =} Standard deviation of series Y

N = Number of pairs of observations

r  =  the (product moment) Correlation coefficient.

This method is to be applied only where the derivations of items are taken from actual means and not from assumed means.

The value of the coefficient of correlation as obtained by the above formula shall always lie between + 1. When r = +1, it means there is perfect positive correlation between the variables. When r = -1, it means there is perfect negative correlation between the variables. When r = 0, it means there is  no relationship between the two variables. However, in practice such values of r as  +1, -1,  and 0 are rare. We normally get values, which lie between +1 and -1 such as -0.8,  - 0.4, etc. The coefficient of correlation describes not only the magnitude of correlation but also its direction. Thus, + 0.8 would mean that correlation is positive because the sign of r is – and the magnitude of correlation is 0.8. Similarly -0.4 means correlation is negative.

The above formula for computing Pearsonian coefficient of correlation can be transformed to the following form which is easier to apply.

where   x = ( X - X ) ; y = (Y - Y )

It is obvious that while applying this formula we have not to calculate separately the standard deviation of X and Y series as is required by formula (i). This simplifies greatly the task of calculating correlation coefficient.

Steps. (i) Take the deviations of X series from the mean of X and denote these deviations by x.

            (ii)  Square these deviations and obtain the total, i.e., ∑x² .

            (iii) Take the deviations of Y series from the mean of Y and denote these deviations by y.

            (iv)  Square these deviations and obtain the total, i.e., ∑ y².

            (v)  Multiply the deviations of X and Y series and obtain the total, i.e., ∑xy.

            (vi) Substitute the values of ∑xy, ∑ x² and ∑ y² in the above formula.

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