MATH 375 -- Probability Theory 

The Correlation Coefficient and Its Geometric Meaning
November 16, 2011 

The following plot shows points where   

    x = age in years 

    y = weight in grams 

for a collection of smallmouth bass taken in a stream 

in Washington State:  

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Given a scatter plot, the correlation coefficient  r 

is a way to quantify how close the points (x, y) are to 

lying along a single straight line.  The coefficient
takes values in the range  to 1: 

A value of r = -1:  indicates that the points all
                             lie along a line of negative slope

                 r = 0:    indicates no linear relationship at all

                 r = +1:  indicates that the points all
                              lie along a line of positive slope

Values between 0 and 1 or between -1 and 0  indicate 

the ``strength'' of the tendency toward a linear relation
between  x  and  y.  Here are some illustrative examples:     

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(1)

 

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(2)