1. 15.5 – Scatterplots & Linear Correlation
In statistics, it is often of interest to know if there is a relationship between 2 specific variables:
Does the size of a vehicle affect its gas mileage?
Does the height of a person have any impact on their IQ?
Does the amount of money spent on a school system improve student performance?
A scatterplot (scatter diagram) can be used to visually see if there is any relationship between the 2 variables.

2. Example 1: Scatter Plots and Correlation
The scatter plot shows a downward trend among the data points, with some exceptions.
People with increased education tend to have a lower score on the test measuring prejudice.
From Blitzer, Thinking Mathematically

3. Types of Correlations between 2 variables
No correlation: There doesn’t appear to be any relationship between the 2 variables
Positive correlation: As one variable increases, so does the other. Also, as one variable decreases, so does the other.
Negative correlation: As one variable increases, the other decreases, and vice versa.
Strength of a correlation: The closer to a straight line the data is, the stronger the correlation, regardless of whether it is a positive or negative correlation
In a perfect correlation, all data points lie on a straight line.
Present students with various scatter plots of data and have them discuss the type and strength of correlation as well as possible explanations for the correlation. This may be an excellent opportunity to touch on the difference between correlation and causation.

4. Sci-Fi/Fantasy Movie Data
Production Cost - x (millions of dollars)
Gross Receipts - y (millions of dollars)
Adventures of Pluto Nash
100 5
Armageddon
140
200
Battlefield Earth 80 21
Final Fantasy
137 32
Harry Potter (Chamber)
300
Harry Potter (Sorcerer)
130
320
Lord of the Rings (1)
109
315
Lord of the Rings (2) 94
350
Red Planet 17
Spider-Man
404
Star Wars (Clones)
115
302
Terminator 3
170
Matrix Reloaded
127
X-Men 2
125
215
Waterworld
175 88
From Bennett & Briggs, Using and Understanding Mathematics

5. Sci-Fi/Fantasy Movie Data

6. Old Faithful Eruptions
Waiting time between eruptions and the duration of the eruption for the Old Faithful geyser in Yellowstone National Park, Wyoming, USA. (from wikipedia.com)

7. Regression Lines and Correlation Coefficients
Regression line is a line that best fits the data points in a scatter plot.
Correlation coefficient, designated by r, is a measure that is used to describe the strength and direction of a relationship between variables whose data points lie on or near a line.
Slide 7

8. From Pirnot, Mathematics All Around
Line of Best Fit
From Pirnot, Mathematics All Around

9. Scatter Plots and Correlation Coefficients
Slide 9

10. How good does r have to be?
1. Compute r for n pairs of data.
2. If the absolute value of r exceeds the number in the column labeled α = .05 on line n, then there is less than a .05 (5%) chance that the variables do not have significant linear correlation.
3. If the absolute value of r exceeds the number in the column labeled α = .01 on line n, then there is less than a .01 (1%) chance that the variables do not have significant linear correlation.
© 2010 Pearson Education, Inc. All rights reserved.

11. Correlation and Causal Connections
Although the scatter plot shows a correlation between
education and prejudice, we cannot conclude that
increased education causes a person’s level of prejudice to
decrease.
The correlation could be simply a coincidence.
Education usually involves classrooms with a variety of different kinds of people. Increased exposure to diversity in the classroom might be an underlying cause.
Education requires people to look at new ideas and see things in different ways. Thus, education causes one to be more tolerant and less prejudiced.
Correlation is not necessarily causation.
Slide 11

12. Does Correlation = Causation?
In the early 1800's (1820 to be scientifically accurate according to our accumulated evidence) the number of active pirates was around 35,000. By the end of the twenty century that number had decreased substantially to around 17, give or take a couple of hundred. Coupled with that knowledge, it is obvious that mean global temperatures had also changed from around degrees Centigrade to around 15.9 degrees Centigrade.
The following graph supplied (albeit unknowingly) by the fine blog Mish Mash by Ryan Shea cannot be denied.
                                                                                                                                                      

13. Global Average Temperature vs. # of Pirates

14. Correlation and Causality
Present students with various scatter plots of data and have them discuss the type and strength of correlation as well as possible explanations for the correlation. This may be an excellent opportunity to touch on the difference between correlation and causation.
A scatter diagram for life expectancy and infant mortality.
Copyright © 2005 Pearson Education, Inc.

15. Correlation and Causality
A scatter diagram for diamond weights and prices.
Copyright © 2005 Pearson Education, Inc.

16. © 2010 Pearson Education, Inc. All rights reserved.
Example
An instructor wants to know whether there is a correlation between the number of times students attended tutoring sessions during the semester and their grades on a 50-point examination. The table shows the data for 10 students. Represent these data points by a scatterplot and interpret the graph.
© 2010 Pearson Education, Inc. All rights reserved.

17. More Examples
What type of correlation, if any, would you expect for the following sets of variables?
Cost of a car vs. miles per gallon
Cost of a car vs. age of the car
Level of stress vs. # of days sick
# days absent vs. grade in course