Measuring Correlation
In the previous section, you used words such as strong, weak, positive, and negative to describe correlation.
However, there is also a numerical value that describes the strength and sign of the correlation.
However, there is also a numerical value that describes the strength and sign of the correlation.
It is called the Pearson's product-moment and is usually denoted with the letter .
In the Desmos window above, the value of $\text{corr}\left(x_1,y_1\right)$ is the value of $r$.
When there is perfect positive correlation between two variables, $r=$
Drag the data points so that they form a descending line.
When there is perfect negative correlation between two variables, $r=$
Drag the data points so that they are scattered with no apparent trend.
When there appears to be no correlation between two variables, the value of $r$ is close to
Here are some other values of $r$ and their scatter diagrams.
$r=0.9$
$r=0.4$
$r=-0.3$
$r=-0.65$
When $r=0.9$, there is a
linear correlation between the two variables.
When $r=0.4$, there is a
linear correlation between the two variables.
When $r=-0.3$, there is a
linear correlation between the two variables.
When $r=-0.65$, there is a
linear correlation between the two variables.
This table summarizes how the value of $r$ describes the correlation.
| Pearson's correlation coefficient ($r$) |
Correlation |
|---|---|
| $0 \le r \le 0.25\qquad$ | no / very weak |
| $0.25 \lt r \le 0.5\phantom{0}\qquad$ | weak |
| $0.5 \lt r \le 0.75\qquad$ | moderate |
| $0.75 \lt r \lt 1\phantom{.00}\qquad$ | strong |
| $r = 1\phantom{.00}\qquad$ |