3.6 Facts about the correlation coefficient for linear regression

• A positive $r$ means that when $x$ increases, $y$ increases and when $x$ decreases, $y$ decreases (positive correlation) .
• A negative $r$ means that when $x$ increases, $y$ decreases and when $x$ decreases, $y$ increases (negative correlation) .
• An $r$ of zero means there is absolutely no linear relationship between $x$ and $y$ (no correlation) .
• High correlation does not suggest that $x$ causes $y$ or $y$ causes $x$ . We say "correlation does not imply causation." For example, every person who learned math in the 17th century is dead. However, learning math does not necessarily causedeath!

Positive correlation

Negative correlation

Zero correlation

If $r=-1$ or $r=\mathrm{+1}$ , then all the data points lie exactly on a straight line.

If the linear correlation is strong , then the line can be used to predict a $y$ value.