1. 3.1

2. Explanatory and Response variables
Explanatory variables
Also known as “predictor variable” or “x-variable” or “independent variable”
Response variables
Also known as “outcome variable” or “y-variable” or “dependent variable”
An explanatory variable may help to explain/predict/influence changes in the response variable
Note: many studies aim to show that changes in the explanatory variable cause changes in the response variable. We will delve into this distinction more in chapter 4

3. Examples
We believe that the GPA of AP Stats students can help us to predict their grade in the course
Explanatory variable:
Response variable:
We believe that taller trees are more likely to be struck by lightning
Predictor variable:
Outcome variable:

4. Examples
We believe that the GPA of AP Stats students can help us to predict their grade in the course
Explanatory variable: GPA
Response variable: Grade in AP Stats
We believe that taller trees are more likely to be struck by lightning
Predictor variable: Tree height
Outcome variable: Probability of being struck by lightning

5. Scatterplots Plots values for two quantitative variables together
One on each axis
Note: must be values from the same individual
Cannot plot the scores of 30 AP Stats students on the x-axis with the height of 30 bears on the y-axis
Explanatory variable typically goes on x-axis
Response variable typically goes on the y-axis

6. Example

7. Interpretation

8. Interpretation Strength (subjective) Direction (usually obvious) Form
Strong
Moderately Strong
Weak
Direction (usually obvious)
Positive
Negative
No association
Form
Linear
Non-linear

9. Direction (slope)

10. Strength

11. Interpretation

12. Interpretation
There is a moderately strong, negative, curved relationship between the percent of students in a state who take the SAT and the average SAT math score in the state

13. Provide a brief description of the relationship between murder rate and number of ice creams sold

14. Provide a brief description of the relationship between murder rate and number of ice creams sold
There is a moderately strong positive linear relationship between the murder rate and the number of ice creams sold

15. Correlation
We can measure the strength and direction of this relationship
Usually better than just estimating by looking at it
We use the letter r to represent correlation (also known as the correlation coefficient)
r is always between -1 and 1
Positive values represent a positive association
Negative values represent a negative association
Closer to 0 indicate a weaker correlation; closer to 1 or -1 indicate a stronger relationship
r= 1 or r= -1 indicate a perfect linear relationship

17. Calculating r

18. Toy Example—calculate r
GPA
AP Stats Grade
Student #1
3.317
94.37
Student #2
2.500
76.06
Student #3
4.024
100.73
Student #4
4.238
97.11

19. Toy Example—calculate r
GPA
AP Stats Grade
Student #1
3.317
94.37
Student #2
2.500
76.06
Student #3
4.024
100.73
Student #4
4.238
97.11
Mean: 3.52 st. dev: mean: st. dev:
𝑟= 1 3 [ − − − ]
𝑟= 1 3 [− ]
𝑟= 1 3 [2.762]
𝑟=.92

20. Correlation Cautions
It is more important to understand what the correlation means than to be able to calculate it by hand
We will learn how to do it on our calculator next class
Correlation makes no distinction between explanatory and response variables
So it is the correlation for the relationship between the two, not for the effect that one has on the other
r does not change when you change the units of measurement
If the correlation between weight in pounds and height in feet is .74, then the correlation between weight in grams and height in inches will also be .74
r has no units—it is just a number

21. More Correlation Cautions
Requires that both variables be quantitative
So you cannot calculate the correlation coefficient between blue eyes and brown hair
Correlation only measures the strength of a linear relationship, not a curved relationship
If the data looks like it is curved, correlation not super meaningful
Correlation is strongly affected by outliers
Correlation is not a complete summary
So if a question asks you to describe the relationship between two variables, saying “r=.45” is not sufficient

22. HW Time