Why the Correlation Coefficient r ranges between -1 and +1.?

Image by Author – Correlation Matrix

Correlation Coefficient is a statistical measure of the strength of the relationship between the relative movements of two variables.

We know that the equation of Sum of Squared Errors is as below.

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If there is a perfect Linear Regression exists between 2 variables x & y, then the line of regression runs through all the points and so there will be no error.

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This is the maximum value of r² which is 1.

The maximum SSE occurs only when there is no correlation. i.e minimum correlation.

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This means that r² is 0 which is the minimum value of r².

If the gradient (i.e. Slope) of line of regression is -ve, then Sxy <0. So r² < 0.

If it is +ve, then Sxy > 0. So r > 0.

Hence -1≤ r ≤+1

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