# 9.2 Measures of Association

Besides looking at the scatter plot and seeing that a linear relationship seems reasonable, and identifying a positive or negative trend, how can you tell more about this relationship? While it is always good practice to first examine things visually, you may find that deciphering a scatterplot, especially the strength of a relationship can be tricky. The next step is then to then calculate numerical measures of this association.

# The Correlation Coefficient, *r*

The correlation coefficient, *r*, developed by Karl Pearson in the early 1900s, is a numerical measure that provides a measure of strength and direction of the linear association between the independent variable *x* and the dependent variable *y*.

The correlation coefficient can be calculated using the formula:

where *n* = the number of data points.

The formula for *r* is formidable, so I would not recommend doing this by hand, however technology can make quick work of the calculation.

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, however, because data that exhibit a curved or horizontal pattern may have a correlation of 0. - 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.

*x*causes

*y*or

*y*causes

*x*. We say “correlation does not imply causation.”

Example

A random sample of 11 statistics students produced the following data, where *x* is the third exam score out of 80, and *y* is the final exam score out of 200.

x (third exam score) | y (final exam score) |
---|---|

65 | 175 |

67 | 133 |

71 | 185 |

71 | 163 |

66 | 126 |

75 | 198 |

67 | 153 |

70 | 163 |

71 | 159 |

69 | 151 |

69 | 159 |

Find the correlation coefficient:

Your turn!

Match the following scatter plots with their description of correlation coefficient

- –1 <
*r*< 0 *r*= 0- 0 <
*r*< 1

# The Coefficient of Determination, *r*^{2 }

The coefficient of determination, *r*^{2 }, is (obviously) the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. It has an interpretation in the context of the data:

- , when expressed as a percent, represents the percent of variation in the dependent (predicted) variable
*y*that can be explained by variation in the independent (explanatory) variable*x*using the regression (best-fit) line. - 1 – , when expressed as a percentage, represents the percent of variation in
*y*that is NOT explained by variation in*x*using the regression line. This can be seen as the scattering of the observed data points about the regression line.

Example

Recall our previous example using a student’s third exam scores to predict their final exam scores:

We found the correlation coefficient is *r* = 0.6631.

Find the coefficient of determination:

Interpret of *r*^{2} in the context of this example:

### Image References

Figure 9.8: Kindred Grey via Virginia Tech (2020). “Figure 9.8” CC BY-SA 4.0. Retrieved from https://commons.wikimedia.org/wiki/File:Figure_9.8.png . Adaptation of Figure 12.9 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/12-3-the-regression-equation

Figure 9.9: Kindred Grey via Virginia Tech (2020). “Figure 9.9” CC BY-SA 4.0. Retrieved from https://commons.wikimedia.org/wiki/File:Figure_9.9.png . Adaptation of Figure 12.13 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/12-3-the-regression-equation

A numerical measure that provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y