1. Describing Scatterplots
Section 3.1

2. Data
Univariate data:

3. Data Univariate data: data that involve a single variable per case
This is the type of data we have been working with so far.

4. Data
Bivariate data:

5. Data
Bivariate data: data that involve two variables per case

6. Data Bivariate data: data that involve two variables per case
For quantitative variables, often displayed as a scatterplot.

7. Scatterplot
Scatterplot shows relationship between two quantitative variables.

8. Scatterplot
Scatterplot shows relationship between two quantitative variables.
y vs x

9. Describing Scatterplots
Recall, for univariate data we use shape, center, and spread to summarize data.

10. Describing Scatterplots
Recall, for univariate data we use shape, center, and spread to summarize data.
For bivariate data, we use shape, trend, and strength.

11. Describing Scatterplots
Describing scatterplots is a 6-step process.

12. 6-Step Process
Identify cases and variables

13. 6-Step Process
Identify cases and variables
Describe overall shape

14. 6-Step Process Identify cases and variables Describe overall shape
Describe trend

15. 6-Step Process Identify cases and variables Describe overall shape
Describe trend
Describe strength

16. 6-Step Process Identify cases and variables Describe overall shape
Describe trend
Describe strength
Does pattern generalize to other cases?

17. 6-Step Process Identify cases and variables Describe overall shape
Describe trend
Describe strength
Does pattern generalize to other cases?
Plausible explanation for pattern – or is there a lurking variable

18. 1. Identify Cases and Variables

19. 1. Identify Cases and Variables
Need these to put the data in context; otherwise, data are meaningless

20. Cases and Variables
Scatterplot: each point represents one case

21. Cases and Variables Scatterplot: each point represents one case
x-coordinate equal to value of one variable and y-coordinate equal to value of the other variable
Describe scale (units of measurement) and range of each variable

22. 2. Describe Overall Shape

23. 2. Describe Overall Shape
Linearity: is pattern linear, curved, or none at all?

24. 2. Describe Overall Shape
Linearity: is pattern linear, curved, or none at all?
Clusters: is there just one cluster or more than one?

25. 2. Describe Overall Shape
Linearity: is pattern linear, curved, or none at all?
Clusters: is there just one cluster or more than one?
Outliers: any striking exceptions to the pattern?

26. Shape : Linear
Points do not have to form a line to have a linear pattern!

27. Shape : Linear

28. Shape: Curved

29. Shape: Curved

30. Shape: Curved

31. Shape: None

32. Shape: None

33. Outliers : Extreme x-value

34. Outliers : Extreme x-value

35. Outlier: Extreme y-value

36. Outlier: Extreme y-value

37. Outliers: Not Follow General Trend

38. Outliers: Not Follow General Trend

39. Clusters
Is there just one cluster or is there more than one?

40. 3. Describe Trend
If as x gets larger, y tends to get larger, there is a positive trend.

41. Trend

42. Trend
Think of slope!

43. Trend
If as x gets larger, y tends to get smaller, there is a negative trend.

44. Trend

45. Trend
If there is no shape, then there is no trend.

46. Trend

47. 4. Describe Strength
If the points cluster closely around an imaginary line or curve, the strength is strong.

48. Strength

49. 4. Describe Strength
If the points are scattered farther away from an imaginary line or curve, the strength decreases.

50. Strength

51. Strength

52. Strength

53. Variability of Strength
Is strength constant or does the strength vary?

54. Variability: Fairly Uniform

55. Variability: Fairly Uniform
Could be strong. moderate, or weak and still be fairly uniform

56. Variability: Heteroscedasticity

57. Variability: Heteroscedasticity

58. 5. Pattern Generalize?
Does pattern generalize to other cases or is relationship a case of “what you see is all you get?”

59. 5. Pattern Generalize?
Does pattern generalize to other cases or is relationship a case of “what you see is all you get?”

60. 6. Plausible Explanation
Are there plausible explanations for pattern?

61. 6. Plausible Explanation
Are there plausible explanations for pattern?
Is it reasonable to conclude that change in one variable causes a change in the other?

62. 6. Plausible Explanation
Are there plausible explanations for pattern?
Is it reasonable to conclude that change in one variable causes a change in the other?
Is there a third or lurking variable that might be causing both variables to change?

63. 6. Plausible Explanation

64. Generalization vs Explanation
Prices of houses sold in Gainsville, Fl in one month

65. Example
Turn to page 110 and look at problem P1.

66. Page 110, P1

67. Page 110, P1 b. These data are not very interesting to describe.
The x-axis shows ages 2 to 7 years, and the y-axis
shows the median height of children at each age.
The shape is linear, the trend is positive, and the
strength is very strong. That is, the scatterplot
shows a very strong positive linear trend.
Students may mention that a typical child grows about 2.7 inches per year.

68. Page 110, P1 c. The linear trend could reasonably be
expected to hold for another year. However,
median height could not be expected to
increase at this rate to age 50, as people
typically stop growing around age 20.

69. Page 110, P1 d. In the background is something called
“growing up” that happens over time during
the early years of life.
That is, an increase in age is associated
with an increase in height.

70. Page 111, E1

71. Page 111, E1 Plot a shows a positive relationship that is
strong and linear.
There is fairly uniform variation across all
values of x.

72. Page 111, E1 Plot b shows a negative relationship that is
strong and linear, again with fairly uniform
variation across all values of x.

73. Page 111, E1 Plot c shows a positive relationship that is
moderate and linear with fairly uniform
variation across all values of x.
One point lies a short distance from the bulk
of the data.

74. Page 111, E1 Plot d shows a negative relationship that is
moderate and linear with fairly uniform
variation across all values of x.
Again, there is one outlier.

75. Page 111, E1 Plot e shows a positive relationship that
is strong and linear except for the outlier.
The one outlier has dramatic influence on
the strength of this relationship.
There is fairly uniform variation across all
values of x.

76. Page 111, E1 Plot f shows a negative relationship that is
very strong and curved. One point on the far
right lies in the general pattern but far away
from the remainder of the data, which
accentuates the strong relationship.
Another outlier lies below the bulk of the
data on the left .

77. Page 111, E1 Plot g shows a negative relationship that is
strong and curved. The two points at either
end of the array accentuate the curvature.
There is a bit more variability among values
of y for smaller values of x than for larger
values of x.

78. Page 111, E1 Plot h shows a positive relationship that is
strong and curved. Again, the outlier on the
extreme right accentuates the curved
pattern and would have dramatic influence
on where a trend line might be placed.
The variability in y is fairly constant across
all values of x.

79. Page 113, E5

80. Page 113, E5 a. Plots A, B, and C are the most linear. Plot
D is not linear because of the seven universities
in the lower right, which may be different from
the rest. Plot A, of graduation rate versus alumni
giving rate, gives some impression of downward
curvature. However, if you disregard the point in
the upper right, the impression of any curvature
disappears.

81. Page 113, E5 Plots A, B, and C all have just one cluster.
However, plot D, the plot of graduation rate versus
top 10% in high school, has two clusters. Most of
the points follow the upward linear trend, but the
cluster of seven points in the lower right with the
highest percentage of freshmen in the top 10%
shows little relationship with the graduation rate.

82. Page 113, E5 Plots A and C have possible outliers. Plot A,
the plot of graduation rate versus alumni giving
rate has a possible outlier in the upper right. The
point is below the general trend and its x-value
(but not its y-value) is unusually large. In plot C,
the plot of graduation rate versus SAT 75th
percentile, the points toward the upper left and the
middle right should be examined because they
are farther from the general trend than the other
points, although neither their x-values nor their
y-values are unusual. The point in the lower right
of plot D should also be examined, along with the
other six points nearby.

83. Page 113, E5 b. Plots A and C have similar moderate
Positive linear trends. Plot D shows wide
variation in both variables, with little or no
trend.
Plot B is the only plot that shows a negative

84. c. Among these four variables, it appears that
the alumni giving rate is the best predictor of the
graduation rate, and SAT scores (as measured by
the 75th percentile) is second best. However, both
of these relationships are moderate and neither is
a strong predictor of graduation rate. Ranking in
high school class (as measured by the top 10%) is
almost useless as a predictor of college graduation
rate. Plot A, of graduation rate versus alumni
giving rate, owes part of the impression of a strong
relationship to the point in the upper right. This
plot shows some heteroscedasticity, with the
graduation rate varying more with smaller alumni
giving rates.

85. Questions?