1. Basic Practice of Statistics - 3rd Edition
Stat 1510: Statistical Thinking and Concepts Scatterplots and Correlation
Chapter 5 1

2. Agenda Explanatory and Response Variables2
Explanatory and Response Variables
Displaying Relationships: Scatterplots
Interpreting Scatterplots
Adding Categorical Variables to Scatterplots
Measuring Linear Association: Correlation
Facts About Correlation

3. Explanatory and Response Variables3
Example:- A medical study finds that short women are
More likely to have heart attacks than women of average
height, while tall women the fewer heart attacks.
Here this is just a study which looks at the relationship
between two variables. We ignored some other variables
like weight and exercise habits.
We assume that height of a woman has influence in
heart attack chances.

4. Explanatory and Response Variables4
A response variable measures an outcome of a study. It is
also called dependent variable.
An explanatory variable may explain or influence changes
in a response variable. You will often find explanatory
variables called independent variables or predictor variables.
For the previous example;
Response Variable:- A woman's Chance to have heart attack.
Explanatory Variable:- Height of a woman.

5. How to Make a Scatterplot5
The most useful graph for displaying the relationship between two quantitative variables is a scatterplot.
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. If a distinction exists, plot the explanatory variable on the x-axis and the response variable on the y-axis.
Label and scale your axes.
Plot individual data values. 5

6. Scatterplot 6
Example: Make a scatterplot of the relationship between body weight and pack weight for a group of hikers.
Body weight (lb)
120
187
109
103
131
165
158
116
Backpack weight (lb) 26 30 24 29 35 31 28

7. How to Examine a Scatterplot
Interpreting Scatterplots 7
To interpret a scatterplot, follow the basic strategy of data analysis discussed earlier. 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.

8. Interpreting Scatterplots8
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.
There is one possible outlier, the hiker with the body weight of 187 pounds seems to be carrying relatively less weight than are the other group members.
Strength
Direction
Form
There is a moderately strong, positive, linear relationship between body weight and pack weight.
It appears that lighter hikers are carrying lighter backpacks.

9. Southern states highlighted
Adding Categorical Variables 9
Consider the relationship between mean SAT verbal score and percent of high-school grads taking SAT for each state.
Southern states highlighted
To add a categorical variable, use a different plot color or symbol for each category. 9

10. Measuring Linear Association10
A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables.
How can we understand the relation is linear from a scatterplot?
A linear relation is strong if the points lie close to a arbitrary straight line, and its weak if they are widely scattered about a line.
Make sure you are not fooled by scaling difference in the plot.
If there is a linear relationship, you can use correlation to measure the direction and strength of the relation.

11. Correlation 11
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 toward -1 or 1.
The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship.

12. Correlation 12

13. Facts About Correlation13
Correlation makes no distinction between explanatory and response variables.
r has no units and does not change when we change the units of measurement of x, y, or both.
Positive r indicates positive association between the variables, and negative r indicates negative association.
The correlation r is always a number between -1 and 1.
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.

14. Correlation Practice 14
For each graph, estimate the correlation r and interpret it in context.

15. Case Study Per Capita Gross Domestic Product15
Per Capita Gross Domestic Product
and Average Life Expectancy for
Countries in Western Europe

16. Case Study Country Per Capita GDP (x) Life Expectancy (y) Austria 21.417
Country
Per Capita GDP (x)
Life Expectancy (y)
Austria
21.4
77.48
Belgium
23.2
77.53
Finland
20.0
77.32
France
22.7
78.63
Germany
20.8
77.17
Ireland
18.6
76.39
Italy
21.5
78.51
Netherlands
22.0
78.15
Switzerland
23.8
78.99
United Kingdom
21.2
77.37

17. Case Study 17 x y
21.4
77.48
-0.078
-0.345
0.027
23.2
77.53
1.097
-0.282
-0.309
20.0
77.32
-0.992
-0.546
0.542
22.7
78.63
0.770
1.102
0.849
20.8
77.17
-0.470
-0.735
0.345
18.6
76.39
-1.906
-1.716
3.271
21.5
78.51
-0.013
0.951
-0.012
22.0
78.15
0.313
0.498
0.156
23.8
78.99
1.489
1.555
2.315
21.2
77.37
-0.209
-0.483
0.101
= 21.52 =
sum = 7.285
sx =1.532
sy =0.795

18. Case Study 18

19. Correlation simplified formulas