Mathematical Properties of Coefficient of Correlation

The following are the important mathematical properties of the Coefficient of Correlation or r. .
The Coefficient of Correlation lies between —1 and +1. It cannot exceed unity.

Symbolically —1 ≤ r ≤ +1

Proof of the property is given below:

Let x and y denote the deviation of x and y series from their actual arithmetic average and ax and ay be their standard deviations respectively. Then,
The Coefficient Of Correlation.JPG

But (∑x2)/σ_(x2 ) = n because σ_(x2 ) = (∑x2)/n ∴ (∑x2)/σ_(x2 ) = (∑x2)/(∑x2 ) × n = n
Similarly (∑x2)/σ_(x2 ) = n and

(2∑xy)/σ_(x σ_y ) = 2nr because r = (2∑xy)/σ_(x σ_y )

As such ∑ ((∑x)/σ_x +(∑y)/σ_y )2 = n + n + 2nr = 2n + 2nr = 2n (1 + r)

But ∑ (x/σ_x +y/σ_y )2 is the sum of square of real quantities and as such

cannot be negative. At best it can be 0.
Now 2n (1+r) ≥ 0
Therefore r cannot be less than —1 or —1≤ r.similarly by expanding
∑ (x/σ_x +y/σ_y )2 it can be proved that this value would be 2n(1—r) and hence, r cannot be greater than + 1 or r≤ +1.

Hence—1 ≤ r ≤ - 1.

Sometimes it appears that the values of the various variables so obtained are inter-related. It is likely that such relationship may be obtained in two series relating to the heights and weights of a group of persons. It may be observed that weights increase with increase in heights- so that tall people are heavier than short sized people. Similarly, if the data are collected about the prices of a commodity and the quantities sold at different prices, two series would be obtained. One variable would be the various prices of the commodity and the other variable would be the quantities sold at these prices. In two such series we are again likely to find some relationship. With increase in the price of the commodity the quantity sold is bound to decrease. We can thus conclude that there is some relationship between price and demand. Such relationships can be found in many types of series, for example, prices and supply, heights and weights of persons, prices of sugar and sugarcane, ages of husbands and wives, etc.