1. Aim – How can we analyze bivariate data using scatterplots?
H.W. – read pages 141 – 152 Do page 159 #1 – 6, 11

2. CHAPTER 3 Describing Relationships
3.1
Scatterplots and Correlation

3. Scatterplots and Correlation
IDENTIFY explanatory and response variables in situations where one variable helps to explain or influences the other.
MAKE a scatterplot to display the relationship between two quantitative variables.
DESCRIBE the direction, form, and strength of a relationship displayed in a scatterplot and identify outliers in a scatterplot.
INTERPRET the correlation.
UNDERSTAND the basic properties of correlation, including how the correlation is influenced by outliers
USE technology to calculate correlation.
EXPLAIN why association does not imply causation.

4. Explanatory and Response Variables
Most statistical studies examine data on more than one variable. In many of these settings, the two variables play different roles.
A response variable measures an outcome of a study.
An explanatory variable may help explain or influence changes in a response variable.
Note: In many studies, the goal is to show that changes in one or more explanatory variables actually cause changes in a response variable. However, other explanatory-response relationships don’t involve direct causation.

5. Explanatory vs. Response
Julie asks, “Can I predict a state’s mean SAT math score if I know its mean SAT critical reading score?”
Identify the explanatory variable and the response variable..
Solution: :
Julie is treating the mean SAT math score as the explanatory variable and the mean SAT critical reading score as the response variable.
Jim wants to know how the mean SAT Math and critical reading scores this year in the 50 states are related to each other.
b. Identify the explanatory variable and the response variable, if possible.
solution: Jim is interested in exploring the relationship between the two variables, therefore, there is no clear explanatory or response variable.

6. Explanatory vs. Response
1. Tim wants to know if there is a relationship between height and weight. Kelly wants to know if she can predict a student’s weight from his or her height. Information about height is easier to obtain than information about weight!
Problem: For each student, identify the explanatory and response variables, if possible.
Solution: Tim is just interested in the relationship between the two variables, so there is no clear explanatory or response variable. Kelly is treating a student’s height as the explanatory variable and the student’s weight as the response variable.

7. Displaying Relationships: Scatterplots
A scatterplot shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as a point on the graph.
How to Make a Scatterplot
Decide which variable should go on each axis.
• Remember, the eXplanatory variable goes on the X-axis!
Label and scale your axes.
Plot individual data values.

8. Making a scatterplot:
At the end of the 2011 college football season, the University of Alabama defeated Louisiana State University for the national championship. Both teams were from the Southeastern conference (SEC). The table below shows the points per game scored and the number of wins for each of the twelve teams in the SEC section.
Elink:

9. Practice making scatterplots (if necessary)
Track and field day!
Each member of a small statistics class ran a 40-yard sprint and then did a long jump (with a running start). The table below shows the sprint time (in seconds) and the long-jump distance (in inches)
Problem: Make a scatterplot of the relationship between sprint time and long-jump distance.
Sprint time (s)
5.41
5.05
7.01
7.17
6.73
5.68
5.78
6.31
6.44
6.50
6.80
7.25
Long-jump distance (in)
171
184 90 65 78
130
173
143 92
139
120
110

10. Re-do practice on a graphing calculator
Track and field day!
Explanatory variable (Spring time) = L1
Response variable (Long jump) = L2
2nd y= to access statplot, turn it on, highlight the first one for type.
Xlist:L1 Ylist: L2, then zoom 9
Problem: Make a scatterplot of the relationship between sprint time and long-jump distance.
Sprint time (s)
5.41
5.05
7.01
7.17
6.73
5.68
5.78
6.31
6.44
6.50
6.80
7.25
Long-jump distance (in)
171
184 90 65 78
130
173
143 92
139
120
110

11. Using scatterplot on a graphing calculator
Notice that there are no scales on the axes and that the axes are not labeled. If you copy a scatterplot from your calculator to paper, make sure you used scaled axes with labels.

12. Describing Scatterplots
To describe a scatterplot, follow the basic strategy of data analysis from Chapters 1 and 2. Look for patterns and important departures from those patterns.
How to Examine a Scatterplot
As in any graph of data, look for the overall pattern and for striking departures from that pattern.
• You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship.
• An important kind of departure is an outlier, an individual value that falls outside the overall pattern of the relationship.

13. Example: Describing a scatterplot
Direction: In general, it appears that teams that score more points per game have more wins and teams that score fewer points per game have fewer wins. We say that there is a positive association between points per game and wins.
Form: There seems to be a linear pattern in the graph (that is, the overall pattern follows a straight line).
Strength: Because the points do not vary much from the linear pattern, the relationship is fairly strong. There do not appear to be any values that depart from the linear pattern, so there are no outliers.
Try Exercise 7

14. Describing Scatterplots
Two variables have a positive association when above-average values of one tend to accompany above-average values of the other and when below-average values also tend to occur together.
Two variables have a negative association when above-average values of one tend to accompany below-average values of the other.
Strength
Describe the scatterplot.
Direction
There is a moderately strong, negative, curved relationship between the percent of students in a state who take the SAT and the mean SAT math score.
Further, there are two distinct clusters of states and two possible outliers that fall outside the overall pattern.
Form

15. Describing Scatterplots:

16. For the A.P. Exam
When it comes to scatterplots, you are expected to discuss the direction, form, and strength of the association as well as identify any possible outliers. To help remember this, here comes another acronym:
DOFS:
Direction
Outliers
Form
Strength

17. Exit Ticket – make a scatterplot between sodium and fat
Exit Ticket – make a scatterplot between sodium and fat. Don’t forget DOFS
McDonald’s restaurants offer a variety of salads. The table below lists 10 different salads, along with the amount of sodium (in mg) and the amount of fat (in grams). source:
Salad
Sodium
Fat
Southwest Salad with Grilled Chicken
650 8
Southwest Salad with Crispy Chicken
820 21
Southwest Salad without chicken
150
4.5
Bacon Ranch Salad with Grilled Chicken
700 9
Bacon Ranch Salad with Crispy Chicken
870 22
Bacon Ranch Salad without chicken
300 7
Caesar Salad with Grilled Chicken
580 5
Caesar Salad with Crispy Chicken
740 18
Caesar Salad without chicken
180 4
Side Salad 10

18. Aim – How do we calculate and analyze the correlation between bivariate data?
H.W. read section 3.1 entirely Do page. 161 – 163 #15 – 25 (odd only), *31

19. Measuring Linear Association: Correlation
A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. Linear relationships are important because a straight line is a simple pattern that is quite common. Unfortunately, our eyes are not good judges of how strong a linear relationship is.
The correlation r measures the direction and strength of the linear relationship between two quantitative variables.
r is always a number between -1 and 1
r > 0 indicates a positive association.
r < 0 indicates a negative association.
Values of r near 0 indicate a very weak linear relationship.
The strength of the linear relationship increases as r moves away from 0 towards -1 or 1.
The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship.

20. Measuring Linear Association: Correlation

21. Calculating Correlation
The formula for r is a bit complex. It helps us to see what correlation is, but in practice, you should use your calculator or software to find r.
How to Calculate the Correlation r
Suppose that we have data on variables x and y for n individuals. The values for the first individual are x1 and y1, the values for the second individual are x2 and y2, and so on. The means and standard deviations of the two variables are x-bar and sx for the x-values and y-bar and sy for the y-values.
The correlation r between x and y is:

22. Facts About Correlation
How correlation behaves is more important than the details of the formula. Here are some important facts about r.
Correlation makes no distinction between explanatory and response variables.
r does not change when we change the units of measurement of x, y, or both.
The correlation r itself has no unit of measurement.
Cautions:
Correlation requires that both variables be quantitative.
Correlation does not describe curved relationships between variables, no matter how strong the relationship is.
Correlation is not resistant. r is strongly affected by a few outlying observations.
Correlation is not a complete summary of two-variable data.

23. Example of calculating correlation.
Step by step: Recall the data from previous example: Problem: calculate the correlation coefficient: Note – . L1 = sprint time, L2 = long jump distance
Sprint time (s)
5.41
5.05
7.01
7.17
6.73
5.68
5.78
6.31
6.44
6.50
6.80
7.25
Long-jump distance (in)
171
184 90 65 78
130
173
143 92
139
120
110

24. Example continued
2 – vars stat for L1 and L2 reveal the following, respectively: Average = 6.34, and sample deviation = 0.72 (L1, x) Average = and sample deviation = 39.1 (L2, y) N = 12, and now we do the calculations
Sprint time (s)
5.41
5.05
7.01
7.17
6.73
5.68
5.78
6.31
6.44
6.50
6.80
7.25
Long-jump distance (in)
171
184 90 65 78
130
173
143 92
139
120
110

25. Check on the graphing calculator:
Make sure statdiagnostic is turned on. Ti-84: mode, turn on stat diagnostic Now click on stat Scroll over to calc Click 4 Make sure x list: L1 y list: L2 Keep pressing enter

26. Correlation Practice
For each graph, estimate the correlation r and interpret it in context.

27. Scatterplots and Correlation
IDENTIFY explanatory and response variables in situations where one variable helps to explain or influences the other.
MAKE a scatterplot to display the relationship between two quantitative variables.
DESCRIBE the direction, form, and strength of a relationship displayed in a scatterplot and identify outliers in a scatterplot.
INTERPRET the correlation.
UNDERSTAND the basic properties of correlation, including how the correlation is influenced by outliers
USE technology to calculate correlation.
EXPLAIN why association does not imply causation.