Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Use the correlation coefficient as another indicator (besides the
scatterplot) of the strength of the relationship between *x* and *y*.

The **correlation coe****f****f****i****cient**, **r,** developed by Karl Pearson in the early 1900s, is a numerical measure of the strength of association between the independent variable x and the dependent variable y.

The correlation coefficient is calculated as

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. In both these cases, all of the original data points lie on 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.

The formula for *r* looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate *r*. The correlation coefficient r is the
bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).

- 1708 reads