1. Summarizing Bivariate Data
Describing relationships

2. Response variable (y) measures an outcome (dependent) of a study
Variables:
Response variable (y) measures an outcome (dependent) of a study
Explanatory variable (x) helps explain or influence changes in a response variable (independent)

3. Identify the Explanatory and Response variables:
Amount of rain
Weed growth
Resting pulse rate
Amount of daily exercise
Amount of sleep
Amount of time watching TV
Winning percentage of basketball team
Attendance at games

4. How to Make a Scatterplot
Displaying Relationships: Scatterplots
The most useful graph for displaying the relationship between two quantitative variables is a scatterplot.
Definition:
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.

5. Scatterplots and Correlation
Displaying Relationships: Scatterplots
Make a scatterplot of the relationship between body weight and pack weight.
Since Body weight is our eXplanatory variable, be sure to place it on the X-axis!
Scatterplots and Correlation
Body weight (lb)
120
187
109
103
131
165
158
116
Backpack weight (lb) 26 30 24 29 35 31 28

6. Suppose we found the age and weight of a sample of 10 adults.
Create a scatterplot of the data below.
Is there any relationship between the age and weight of these adults?
Age 24 30 41 28 50 46 49 35 20 39 Wt
256
124
320
185
158
129
103
196
110
130

7. Does there seem to be a relationship between age and weight of these adults?

8. Create a scatterplot of the data below.
Suppose we found the height and weight of a sample of 10 adults.
Create a scatterplot of the data below.
Is there any relationship between the height and weight of these adults?
Is it positive or negative? Weak or strong? Ht 74 65 77 72 68 60 62 73 61 64 Wt
256
124
320
185
158
129
103
196
110
130

9. Does there seem to be a relationship between height and weight of these adults?

10. And ALWAYS in context of the problem!
When describing relationships between two variables, you should address:
Direction: the overall pattern moving left to right (positive, negative, or neither)
Strength of the relationship: how closely the points follow a clear form (how much scattering?)
Form:overall pattern (linear or some other pattern)
Unusual features: individual values that fall outside overall pattern (outliers or influential points)
And ALWAYS in context of the problem!

11. Identify as having a positive association, a negative association, or no association.+
Heights of mothers & heights of their adult daughters -
Age of a car in years and its current value +
Weight of a person and calories consumed NO
Height of a person and the person’s birth month -
Number of hours spent in safety training and the number of accidents that occur

12. The farther away from a straight line – the weaker the relationship
The closer the points in a scatterplot are to a straight line - the stronger the relationship.
The farther away from a straight line – the weaker the relationship

13. Interpreting Scatterplots
Outlier
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 students are carrying lighter backpacks.

14. Interpreting Scatterplots
Definition:
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
Direction
Form
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.

15. What explains the clusters? What do the two circled points mean?
Interpreting Scatterplots
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.
W. V.
Maine
What explains the clusters?
What do the two circled points mean?

16. Association does not imply causation

17. Correlation
Measures the direction and the strength of a linear relationship between 2 quantitative variables.
Correlation coefficient (r):
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.

18. Properties of r (correlation coefficient)
legitimate values of r is [-1,1]
Strong correlation No
Correlation
Moderate Correlation
Weak correlation

19. Some Correlation Pictures

20. Some Correlation Pictures

21. Some Correlation Pictures

22. Some Correlation Pictures

23. Some Correlation Pictures

24. Some Correlation Pictures

25. GUESS THAT CORRELATION!
Correlation Practice
For each graph, estimate the correlation r and interpret it in context.
GUESS
THAT CORRELATION!

26. Correlation Coefficient (r)
A quantitative assessment of the strength & direction of the linear relationship between bivariate, quantitative data
Pearson’s sample correlation is used most
parameter - r (rho)
statistic - r

27. Heights and weights of 10 students:
Which variable is the explanatory variable? Response variable? Why?
Make a scatterplot. Describe the relationship.
Height (in)
Weight (lb) 74
256 65
124 77
320 72
185 68
158 60
129 62
103 73
196 61
110 64
130 =

28. Find the mean and standard deviation of the heights and weights of the 10 students:
Height (in)
Weight (lb) 74
256 65
124 77
320 72
185 68
158 60
129 62
103 73
196 61
110 64
130 =

29. Find the mean and standard deviation of the heights and weights of the 10 students:
Height (in)
Weight (lb) 74
256
1.06
1.21
1.28 65
124
-.43
-.67
.29 77
320
1.55
2.12
3.29 72
185
.73
.20
.14 68
158
.07
-.19
-.01 60
129
-1.25
-.60
.75 62
103
-.92
-.97
.90 73
196
.89
.35
.32 61
110
-1.09
-.87
.95 64
130
-.59 =
8.24/9=
.9157

30. Heights and weights of 10 students:
Horizontal line through mean of weights
Vertical line through mean of heights
Height (in)
Weight (lb) 74
256 65
124 77
320 72
185 68
158 60
129 62
103 73
196 61
110 64
130 =

31. Calculate r. Interpret r in context.
Speed Limit (mph) 55 50 45 40 30 20
Avg. # of accidents (weekly) 28 25 21 17 11 6
Calculate r. Interpret r in context.
r = .9964
There is a strong, positive, linear relationship between speed limit and average number of accidents per week.

32. Find r. Interpret r in context. x (in mm) 12 15 21 32 26 19 24y
Find r.
Interpret r in context.
.9181

33. value of r is not changed by any transformations
x (in mm) y
Find r.
Change to cm & find r.
Do the following transformations & calculate r
1) 5(x + 14)
2) (y + 30) ÷ 4
.9181
.9181
The correlations are the same.
STILL = .9181

34. value of r does not depend on which of the two variables is labeled x
Switch x & y & find r.
Type: LinReg L2, L1
The correlations are the same.

35. value of r is non-resistantx y
Find r.
Outliers affect the correlation coefficient

36. r = 0, but has a definite relationship!
value of r is a measure of the extent to which x & y are linearly related
Find the correlation for these points: x Y
What does this correlation mean?
Sketch the scatterplot
r = 0, but has a definite relationship!

37. 4) Correlation requires both variables to be quantitative.
Correlation makes no distinction between explanatory and response variable.
2) Correlation coefficient has not unit of measurement. It is just a number.
3) Correlation does not change when we change the units of measurement of x, y, or both.
4) Correlation requires both variables to be quantitative.
5) Correlation does not describe curved relationship between variables, no matter how strong. Only the
linear relationship between variables.
6) Like the mean and standard deviation, the correlation is not resistant: r is strongly affected by a few
outlying observations.

38. Correlation does not imply causation