Pearson's r

Pearson's r

After looking at the scatter plot and seeing that a linear relationship between two variables seems to exist, what should you do? One option is to use the correlation coefficient to derive a number that represents the direction and exact strength of the relationship between x and y.

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.

The correlation coefficient is calculated

where n = the number of data points.

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 absolutely no linear relationship between x and y (no linear correlation). If r = 1, there is perfect positive correlation. If r = –1, there is perfect negative correlation. When we have a perfect positive or negative correlation, all of the data points lie on a straight line. In the real world, this will generally not 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.

Note

It is important to remember that a correlation does not indicate causation. What this means is that even if there are two variables that are strongly correlated, we still cannot suggest that x causes y or y causes x. We say “correlation does not imply causation.” To apply causation, one has to conduct an experiment.

Click here to go back to main chapter page