11.2: Coefficient of Correlation
The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y.
If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.
What the VALUE of r tells us:
The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
The size of the correlation r indicates the strength of the linear relationship between x and y. Values of r close to –1 or to +1 indicate a stronger linear relationship between x and y.
If r = 0, there is likely no linear correlation. It is important to view the scatterplot because data that exhibit a curved or horizontal pattern may have a correlation of 0.
If r = 1, there is a perfect positive correlation. If r = –1, there is a perfect negative correlation. In both these cases, all of the original data points lie in a straight line. Of course, in the real world, this will not generally happen.
What the SIGN of r tells us
A positive value of r means that when x increases, y tends to increase, and when x decreases, y tends to decrease (positive correlation).
A negative value of r means that when x increases, y tends to decrease, and when x decreases, y tends to increase (negative correlation).
The sign of r is the same as the sign of the slope, b, of the best-fit line.
This text is adapted from Openstax, Introductory Statistics, Section 12.3, The Regression Equation