1. Two-Variable Data Analysis
AP Statistics
Two-Variable Data Analysis

2. Key Ideas Scatterplots Lines of Best Fit The Correlation Coefficient
Least Squares Regression Line
Coefficient of Determination
Residuals
Outliers and Influential Points
Transformations to Achieve Linearity

3. Two-Variable Data
When looking at two-variable data, we are primarily interested in whether or not two variables have a linear relationship and how changes in one variable can predict changes in the other variable.
Two-variable data is also called bivariate data.

4. Response, Explanatory
A response variable measures an outcome of a study.
An explanatory variable helps explain or influences changes in a response variable.
Exercises: page 173 problems 3.1 – 3.4

5. Example STUDENT HOURS STUDIED SCORE ON EXAM A 0.5 65 B 2.5 80 C 3.0 77
1.5 60 E
1.25 68 F
0.75 70 G
4.0 83 H
2.25 85 I J
6.0 96 K
3.25 84 L M
0.0 51 N
1.75 63 O
2.0 71

6. Example, cont.
A teacher wanted to know if additional studying resulted in higher grades. In other words, does studying have an effect on test performance?
Draw a scatterplot – putting one variable on the horizontal axis and the other on the vertical axis. If we have an explanatory variable, it should go on the horizontal axis and the response variable on the vertical axis.

7. Interpreting a Scatterplot
Direction, form and strength of the relationship
Striking deviations
Outliers

8. Association?
If the variable on the vertical axis tends to increase as the variable on the horizontal axis increases, we say that the two variables are positively associated.
If one of them decreases as the other increases, we way they are negatively associated.

9. Calculator Tip
To draw a scatterplot on your calculator, enter the data in two lists (L1 and L2).
Go to STAT PLOT and choose the scatterplot icon.
Enter L1 for Xlist and L2 for Ylist.
Do ZOOM: ZoomStat.
See Technology Toolbox on page 183

10. Exercises
Page 179, problems 3.5 – 3.10

11. Correlation
We are primarily interested in determining the extent to which two variables are linearly associated.
The first statistic we have to determine a linear relationship is the Pearson product moment correlation, or more simply, the correlation coefficient, denoted by the letter r.
The correlation coefficient is a measure of the strength of the linear relationship between two variables as well as an indicator of the direction of the linear relationship.

12. Formula
If we have a sample of size n of paired data, say (x,y), and assuming that we have computed summary statistics for x and y (means and standard deviations), the correlation coefficient r is defined as follows:
Find r for the previous example.

13. Properties of r
If r is positive, it indicates that the variables are positively associated. If r is negative, the variables are negatively associated.
If r = 0, it indicates that there is no linear association that would allow us to predict y from x. It doesn’t mean that there is no relationship – just not a linear one.
It doesn’t matter which variable you call x and which one you call y.
r doesn’t depend on units of measurements.
r is not resistant to extreme values because it is based on the mean.

14. Guidelines
There are no hard and fast rules about how strong a relationship is based on the numerical value of r.
VALUE OF r
STRENGTH OF RELATIONSHIP
-1 < r < -0.8
0.8 < r < 1
strong
-0.8 < r < -0.5
0.5 < r < 0.8
moderate
-0.5 < r < 0.5
weak

15. Calculator Tip
You will first need to turn “Diagnostic On”. You can find this in CATALOG. Choose it and press ENTER twice.
Enter values in L1 and L2.
STAT: CALC: LinReg(a + bx) then ENTER
Enter L1, L2 and press ENTER

16. Correlation and Causation
Association does not imply causation!
Just because two things seem to go together does not mean that one caused the other.
Some third variable, called a lurking variable, may be influencing them both.

17. YEAR
Number of Methodist Ministers in New England
Number of barrels of Cuban rum imported to Boston
1860 63
8376
1865 48
6406
1870 53
7005
1875 64
8486
1880 72
9595
1885 80
10,643
1890 85
11,265
1895 76
10.071
1900
10,547
1905 83
11,008
1910
105
13,885
1915
140
18,559
1920
175
23,024
1925
183
24,185
1930
192
25,434
1935
221
29.238

18. Line of Best Fit
Once we have determined that two variables have a strong linear relationship, we can find a line of best fit so that we can predict values.
This is called linear regression. In this situation, it matters which variable we call x and which one we call y.
The line we are looking for is called the least squares regression line.

19. Example STUDENT HOURS STUDIED SCORE ON EXAM A 0.5 65 B 2.5 80 C 3.0 77
1.5 60 E
1.25 68 F
0.75 70 G
4.0 83 H
2.25 85 I J
6.0 96 K
3.25 84 L M
0.0 51 N
1.75 63 O
2.0 71