The document discusses different types of relationships between variables in data sets: - Dichotomous by dichotomous examines the relationship between two variables that can only take two values each, like gender and artichoke preference. - Dichotomous by scaled looks at a dichotomous variable against a scaled variable, for example age groups and hours of sleep. - Ordinal by another variable involves an ordinal variable where numbers represent relative amounts but not equal intervals, such as pole vaulting placements.

1. The Nature of Your Data

2. The purpose of this presentation is to help you
determine if the two data sets you are working
with in this problem are:

3. The purpose of this presentation is to help you
determine if the two data sets you are working
with in this problem are:
Dichotomous by Dichotomous
Dichotomous by Scaled
Ordinal by Another Variable
Scaled by Scaled with at least
one variable Skewed

4. First, let's define
what each of these mean.
Dichotomous by Dichotomous
Dichotomous by Scaled
Ordinal by Another Variable
Scaled by Scaled with at least
one variable Skewed

5. Beginning with
Dichotomous by Dichotomous

6. What is dichotomous data?

7. The "Di" in Dichotomous means "two"

8. . . . and "tomous" or "tomy" as in
“appendec-tomy” means to divide by.

9. . . . and "tomous" or "tomy" as in
“appendec-tomy” means to divide by.

10. So, dichotomous means to divide by two.

11. In this case a variable is divided by two or
specifically it can only take on two values.

12. For example:

13. Gender is a good example of a dichotomous
data.

14. Gender is a good example of a dichotomous
data. It generally takes on two values

15. Gender is a good example of a dichotomous
data. It generally takes on two values
(1) male
(2) female

16. In some cases individuals are divided by
(1) those who received a treatment and
(2) those who did not.

17. For example:

18. You have been asked to determine if those who
eat asparagus score higher on a well-being
scale (1-10) than those who do not.

19. You have been asked to determine if those who
eat asparagus score higher on a well-being
scale (1-10) than those who do not.

20. You have been asked to determine if those who
eat asparagus score higher on a well-being
scale (1-10) than those who do not.
In this case, we are dealing with those
(1) who eat asparagus and those (2) who do not.

21. With dichotomous by dichotomous data you are
examining the relationship between two
dichotomous variables.

22. Here is an example:

23. It has been purported that females prefer
artichokes more than do males.

24. It has been purported that females prefer
artichokes more than do males.

25. It has been purported that females prefer
artichokes more than do males.
Dichotomous variable 1:
Gender
(1)Male
(2)Female

26. It has been purported that females prefer
artichokes more than do males.
Dichotomous variable 1:
Gender
(1)Male
(2)Female

27. It has been purported that females prefer
artichokes more than do males.
Dichotomous variable 1:
Gender
(1)Male
(2)Female

28. It has been purported that females prefer
artichokes more than do males.
Dichotomous variable 2:
Artichoke Preference
(1)Prefer Artichokes
(2)Do not prefer Artichokes

29. It has been purported that females prefer
artichokes more than do males.
Dichotomous variable 2:
Artichoke Preference
(1)Prefer Artichokes
(2)Do not prefer Artichokes

30. It has been purported that females prefer
artichokes more than do males.
Dichotomous variable 2:
Artichoke Preference
(1)Prefer Artichokes
(2)Do not prefer Artichokes

31. Here is what the data set looks like:

32. It has been purported that females prefer
artichokes more than do males.
Study Participant Gender
1 = Male
2 = Female
Artichoke Preference
1 = Prefer Artichokes
2 = Don’t Prefer Artichokes
A 1 2
B 2 1
C 1 2
D 2 1
E 2 1
F 1 2
G 1 2

33. This is an example of:
Dichotomous
Data
Study Participant Gender
1 = Male
2 = Female
Artichoke Preference
1 = Prefer Artichokes
2 = Don’t Prefer Artichokes
A 1 2
B 2 1
C 1 2
D 2 1
E 2 1
F 1 2
G 1 2

34. This is an example of:
Dichotomous
Data
Study Participant Gender
1 = Male
2 = Female
by
Dichotomous
Data
Artichoke Preference
1 = Prefer Artichokes
2 = Don’t Prefer Artichokes
A 1 2
B 2 1
C 1 2
D 2 1
E 2 1
F 1 2
G 1 2

35. As you will learn, there is a specific statistical
method used to calculate the relationship
between two dichotomous variables. It is called
the Phi-coefficient.

36. Note - a dichotomous variable is also a nominal
variable.

37. Note - a dichotomous variable is also a nominal
variable. However, nominal variables can also
take on more than two values:

38. Note - a dichotomous variable is also a nominal
variable. However, nominal variables can also
take on more than two values:
1 = American
2 = Canadian
3 = Mexican
like so

39. Note - a dichotomous variable is also a nominal
variable. However, nominal variables can also
take on more than two values:
1 = American
2 = Canadian
3 = Mexican
Dichotomous nominal variables can only take on
two values - (e.g., 1 = Male, 2 = Female)

40. The next type of relationship involves
dichotomous by scaled variables.

41. The next type of relationship involves
dichotomous by scaled variables.
Dichotomous by Dichotomous
Dichotomous by Scaled
Ordinal by Another Variable
Scaled by Scaled with at least
one variable Skewed

42. Now you already know what a dichotomous
variable is, but what is a scaled variable?

43. A scaled variable is a variable that theoretically
can take on an infinite amount of values.

44. A scaled variable is a variable that theoretically
can take on an infinite amount of values.

45. For example,

46. Let's say a car can go as slow as 0 miles per hour
and as fast as 130 miles per hour.

47. Within those two points (0 and 130mph) it could
go 30 mph, 60 mph, 23 mph, 120 mph, 33.2
mph, 44.302 mph, or even 88.00000000001
mph.

48. The point is that between these two points (0
and 130mph) there are an infinite number of
values that the speed could take.

49. Scaled data also has what are called equal
intervals.

50. Scaled data also has what are called equal
intervals. This means that the basic unit of
measurement (e.g., inches, miles per hour,
pounds) are the same across the scale:

51. Scaled data also has what are called equal
intervals. This means that the basic unit of
measurement (e.g., inches, miles per hour,
pounds) are the same across the scale:
100o - 101o
70o - 71o
40o - 41o
Each set of readings are the same
distance apart: 1o
Slide 51

52. Here is an example of a word problem with
scaled by dichotomous variables:

53. You have been asked to determine the
relationship between age and hours of sleep.
Age is divided into two groups: Middle Age (45-
64) and Old Age (65-94).

54. You have been asked to determine the
relationship between age and hours of sleep.
Age is divided into two groups: Middle Age (45-
64) and Old Age (65-94).
The Scaled Variable is hours of
sleep which can take on values
from 0 to 8+ hours.

55. You have been asked to determine the
relationship between age and hours of sleep.
Age is divided into two groups: Middle Age (45-
64) and Old Age (65-94).
The Dichotomous Variable is age
which in this case can take on two
values (1) middle and (2) old age.

56. Here is what the data set might look like:

57. Here is what the data set might look like:
Study Participant Age
1 = 45-64 years
2 = 65-94 years
Hours of Sleep
A 1 6.2
B 2 9.1
C 1 5.8
D 2 8.2
E 2 7.4
F 1 4.9
G 1 6.8

58. Here is what the data set might look like:
Dichotomous
Data
Study Participant Age
1 = 45-64 years
2 = 65-94 years
Hours of Sleep
A 1 6.2
B 2 9.1
C 1 5.8
D 2 8.2
E 2 7.4
F 1 4.9
G 1 6.8

59. Here is what the data set might look like:
Dichotomous
Data
Study Participant Age
1 = 45-64 years
2 = 65-94 years
Hours of Sleep
A 1 6.2
B 2 9.1
C 1 5.8
D 2 8.2
E 2 7.4
F 1 4.9
G 1 6.8

60. Here is what the data set might look like:
Dichotomous
Data
Study Participant Age
1 = 45-64 years
2 = 65-94 years
Hours of Sleep
by
A 1 6.2
B 2 9.1
C 1 5.8
D 2 8.2
E 2 7.4
F 1 4.9
G 1 6.8

61. Here is what the data set might look like:
Dichotomous
Data
Study Participant Age
1 = 45-64 years
2 = 65-94 years
Scaled
Data
Hours of Sleep
by
A 1 6.2
B 2 9.1
C 1 5.8
D 2 8.2
E 2 7.4
F 1 4.9
G 1 6.8

62. Note, in the strictest sense scaled data should
be like the car example (values are infinite
between 0 and 130 mph).

63. However, in the social sciences many times data
that is technically not scaled (e.g., on a scale of
1-10 how would you rate the ballerina's
performance), are still treated as scaled data.

64. However, in the social sciences many times data
that is technically not scaled (e.g., on a scale of
1-10 how would you rate the ballerina's
performance), are still treated as scaled data.
Yes, it is true there are only 10 values that the
variable can take on, but many researchers will
treat it as scaled data. For the purposes of this
class we will treat variables such as these as
scaled data as well.

65. However, in the social sciences many times data
that is technically not scaled (e.g., on a scale of
1-10 how would you rate the ballerina's
performance), are still treated as scaled data.
Yes, it is true there are only 10 values that the
variable can take on, but many researchers will
treat it as scaled data. For the purposes of this
class we will treat variables such as these as
scaled data as well.

66. However, if we were rating on a scale of 1-2, 1-3
or 1-4 we most likely would not treat such
variables as scaled.

67. As you will learn there is a specific statistical
method used to calculate the relationship
between scaled by dichotomous variables. it is
called the Point Biserial Correlation.

68. Next, let's consider the relationship involving
ordinal data by another variable.

69. Next, let's consider the relationship involving
ordinal data by another variable.
Dichotomous by Dichotomous
Dichotomous by Scaled
Ordinal by Another Variable
Scaled by Scaled with at least
one variable Skewed

70. An ordinal variable is a variable where the
numbers represent relative amounts of a an
attribute. However, they do not have equal
intervals.

71. For example,

72. In this pole vaulting example you will notice that
1st and 2nd place are closer to each other:

73. In this pole vaulting example you will notice that
1st and 2nd place are closer to each other:
3rd
Place
15’ 2”
2nd
Place
18’ 1”
1st
Place
18’ 3”

74. In this pole vaulting example you will notice that
1st and 2nd place are closer to each other:
3rd
Place
15’ 2”
2nd
Place
18’ 1”
1st
Place
18’ 3”

75. In this pole vaulting example you will notice that
1st and 2nd place are closer to each other:
3rd
Place
15’ 2”
2nd
Place
18’ 1”
1st
Place
18’ 3”
2 inches
apart

76. . . . than 2nd and 3rd place, which are much
further apart

77. . . . than 2nd and 3rd place, which are much
further apart
3rd
Place
15’ 2”
2nd
Place
18’ 1”
1st
Place
18’ 3”

78. . . . than 2nd and 3rd place, which are much
further apart
3rd
Place
15’ 2”
2nd
Place
18’ 1”
1st
Place
18’ 3”
3 feet
1” apart

79. Rank ordered or ordinal data such as these do
not have equal intervals.
3rd
Place
15’ 2”
2nd
Place
18’ 1”
1st
Place
18’ 3”

80. Rank ordered or ordinal data such as these do
not have equal intervals.
3rd
Place
15’ 2”
2nd
Place
18’ 1”
1st
Place
18’ 3”

81. Here is what an ordinal by ordinal problem looks
like:

82. In a study, researchers rank order different
breeds of dog based on how high they can jump.
They then rank order them based on the length
of their hind legs. They wish to determine if a
relationship exists between jumping height and
hind leg length.

83. In a study, researchers rank order different
breeds of dog based on how high they can jump.
They then rank order them based on the length
of their hind legs. They wish to determine if a
relationship exists between jumping height and
hind leg length.

84. In a study, researchers rank order different
breeds of dog based on how high they can jump.
They then rank order them based on the length
of their hind legs. They wish to determine if a
relationship exists between jumping height and
hind leg length.

85. Here’s the data set:

86. Here’s the data set:
Breed Participant Jumping Rank Hind-Leg Length
Rank
A 1st 2nd
B 3rd 6th
C 6th 4th
D 4th 3rd
E 7th 7th
F 2nd 1st
G 5th 5th

87. Here’s the data set:
Ordinal or
Ranked Data
Breed Participant Jumping Rank Hind-Leg Length
Rank
A 1st 2nd
B 3rd 6th
C 6th 4th
D 4th 3rd
E 7th 7th
F 2nd 1st
G 5th 5th

88. Here’s the data set:
by
Ordinal or
Ranked Data
Ordinal or
Ranked Data
Breed Participant Jumping Rank Hind-Leg Length
Rank
A 1st 2nd
B 3rd 6th
C 6th 4th
D 4th 3rd
E 7th 7th
F 2nd 1st
G 5th 5th

89. Rank ordered data can also take the form of
percentiles.

90. Percentiles communicate the percentage of
observations or values below a certain point.

91. If my score on the ACT is at the 35th percentile
that means the 35% of ACT takers are below me.

92. If my score on the ACT is at the 35th percentile
that means the 35% of ACT takers are below me.

93. A data set taken from the dog jumping question
might look like this:

94. A data set taken from the dog jumping question
might look like this:
Breed Participant Jumping
Percentile Rank
Hind-Leg
Percentile Rank
A 99% 85%
B 78% 33%
C 54% 64%
D 69% 73%
E 34% 28%
F 84% 97%
G 61% 54%

95. A data set taken from the dog jumping question
might look like this:
Ordinal or
Percentile
Ranked Data
Breed Participant Jumping
Percentile Rank
Hind-Leg
Percentile Rank
A 99% 85%
B 78% 33%
C 54% 64%
D 69% 73%
E 34% 28%
F 84% 97%
G 61% 54%

96. A data set taken from the dog jumping question
might look like this:
Ordinal or
Percentile
Ranked Data
Breed Participant Jumping
Percentile Rank
Ordinal or
Percentile
Ranked Data
Hind-Leg
Percentile Rank
by
A 99% 85%
B 78% 33%
C 54% 64%
D 69% 73%
E 34% 28%
F 84% 97%
G 61% 54%

97. The next example is that of a relationship
between ordinal variable and a scaled variable.

98. You have been asked to determine if there is a
relationship between the height of marathon
runners and their final ranking in a race.

99. You have been asked to determine if there is a
relationship between the height of marathon
runners and their final ranking in a race.

100. Here’s the data set:
Marathon Runners Height in inches Order of Finish
A 73 6th
B 67 4th
C 69 5th
D 64 2nd
E 71 7th
F 62 1st
G 66 3rd

101. Here’s the data set:
Scaled
Data
Marathon Runners Height in inches Order of Finish
A 73 6th
B 67 4th
C 69 5th
D 64 2nd
E 71 7th
F 62 1st
G 66 3rd

102. Here’s the data set:
Scaled
Data
by Ordinal/
Ranked Data
Marathon Runners Height in inches Order of Finish
A 73 6th
B 67 4th
C 69 5th
D 64 2nd
E 71 7th
F 62 1st
G 66 3rd

103. The final example is that of a relationship
between ordinal variable and a nominal
variable.

104. You have been asked to determine if there is a
relationship between gender and spelling bee
competition rankings.

105. You have been asked to determine if there is a
relationship between gender and spelling bee
competition rankings.

106. Here’s the data set:

107. Marathon Runners Gender Spelling Bee Rank
A 1 6th
B 2 4th
C 2 5th
D 2 2nd
E 1 7th
F 1 1st
G 2 3rd

108. Dichotomous/
Nominal Data
Marathon Runners Gender Spelling Bee Rank
A 1 6th
B 2 4th
C 2 5th
D 2 2nd
E 1 7th
F 1 1st
G 2 3rd

109. by Ordinal/
Ranked Data
Dichotomous/
Nominal Data
Marathon Runners Gender Spelling Bee Rank
A 1 6th
B 2 4th
C 2 5th
D 2 2nd
E 1 7th
F 1 1st
G 2 3rd

110. In summary,

111. In summary, when at least one variable in the
relationship is ordinal or rank ordered, then you
choose the final option:

112. In summary, when at least one variable in the
relationship is ordinal or rank ordered, then you
choose the final option:
Dichotomous by Dichotomous
Dichotomous by Scaled
Ordinal by Another Variable
Scaled by Scaled with at least
one variable Skewed

113. As you will learn there are specific statistical
methods used to calculate the relationship
between ordinal by ordinal or ordinal by other
variables.

114. As you will learn there are specific statistical
methods used to calculate the relationship
between ordinal by ordinal or ordinal by other
variables. They are the Spearman Rho and
Kendall Tau.

115. As you will learn there are specific statistical
methods used to calculate the relationship
between ordinal by ordinal or ordinal by other
variables. They are the Spearman Rho and
Kendall Tau. We'll explain their difference in
another presentation.

116. Lastly,

117. Lastly, let's consider the relationship involving
scaled data with at least one variable having a
skewed distribution.

118. For example,

119. You wish to determine the relationship between
daily temperature and ice cream sales.

120. You wish to determine the relationship between
daily temperature and ice cream sales.

121. You wish to determine the relationship between
daily temperature and ice cream sales .
Month
Average Daily
Temperature
Average Daily
Ice Cream Sales
Jan 100 $100
Feb 200 $200
Mar 300 $300
Apr 400 $400
May 500 $500
Jun 600 $300
Jul 700 $200
Aug 600 $100
Sep 500 $300
Oct 400 $200
Nov 300 $400
Dec 800 $1000

122. You wish to determine the relationship between
daily temperature and ice cream sales .
Month
Average Daily
Temperature
Average Daily
Ice Cream Sales
Jan 100 $100
Feb 200 $200
Mar 300 $300
Apr 400 $400
May 500 $500
Jun 600 $300
Jul 700 $200
Aug 600 $100
Sep 500 $300
Oct 400 $200
Nov 300 $400
Dec 800 $1000
The skew is
“0.00”
therefore
temperature is
normally
distributed

123. You wish to determine the relationship between
daily temperature and ice cream sales .
Month
Average Daily
Temperature
Average Daily
Ice Cream Sales
Jan 100 $100
Feb 200 $200
Mar 300 $300
Apr 400 $400
May 500 $500
Jun 600 $300
Jul 700 $200
Aug 600 $100
Sep 500 $300
Oct 400 $200
Nov 300 $400
Dec 800 $1000
The skew is
“0.00”
therefore
temperature is
normally
distributed

124. You wish to determine the relationship between
daily temperature and ice cream sales .
Month
Average Daily
Temperature
Average Daily
Ice Cream Sales
Jan 100 $100
Feb 200 $200
Mar 300 $300
Apr 400 $400
May 500 $500
Jun 600 $300
Jul 700 $200
Aug 600 $100
Sep 500 $300
Oct 400 $200
Nov 300 $400
Dec 800 $1000
The skew is
“+3.23”
therefore ice
cream sales is
Positively
Skewed

125. You wish to determine the relationship between
daily temperature and ice cream sales .
Month
Average Daily
Temperature
Average Daily
Ice Cream Sales
Jan 100 $100
Feb 200 $200
Mar 300 $300
Apr 400 $400
May 500 $500
Jun 600 $300
Jul 700 $200
Aug 600 $100
Sep 500 $300
Oct 400 $200
Nov 300 $400
Dec 800 $1000
The skew is
“+3.23”
therefore ice
cream sales is
Positively
Skewed

126. You wish to determine the relationship between
daily temperature and ice cream sales .
Month
Average Daily
Temperature
Average Daily
Ice Cream Sales
Jan 100 $100
Feb 200 $200
Mar 300 $300
Apr 400 $400
May 500 $500
Jun 600 $300
Jul 700 $200
Aug 600 $100
Sep 500 $300
Oct 400 $200
Nov 300 $400
Dec 800 $1000

127. You wish to determine the relationship between
daily temperature and ice cream sales .
Month
Average Daily
Temperature
Average Daily
Ice Cream Sales
Jan 100 $100
Feb 200 $200
Mar 300 $300
Apr 400 $400
May 500 $500
Jun 600 $300
Jul 700 $200
Aug 600 $100
Sep 500 $300
Oct 400 $200
Nov 300 $400
Dec 800 $1000
This is an example where
one variable is skewed and
the other normal

128. If your problem had one scaled variable that was
skewed and the other normal or if both were
skewed you would select:

129. If your problem had one scaled variable that was
skewed and the other normal or if both were
skewed you would select:
Dichotomous by Dichotomous
Dichotomous by Scaled
Ordinal by Another Variable
Scaled by Scaled with at least
one variable Skewed

130. A final note:

131. Dichotomous data like this:
1 = Catholic
2 = Mormon

132. Dichotomous data like this:
1 = Catholic
2 = Mormon
Study
Participants
Religious
Affiliation
1 = Catholic
2 = Mormon
A 1
B 1
C 1
D 2
E 1
F 2

133. Dichotomous data like this:
1 = Catholic
2 = Mormon
Study
Participants
Religious
Affiliation
1 = Catholic
2 = Mormon
A 1
B 1
C 1
D 2
E 1
F 2

134. Dichotomous data like this:
1 = Catholic
2 = Mormon
. . . can become scaled if we are talking about
the number of Catholics or Mormons.

135. Dichotomous data like this:
1 = Catholic
2 = Mormon
Event Number of
Catholics in
attendance
Number of
Mormons in
attendance
A 120 22
B 322 34
C 401 78
D 73 55
E 80 3
F 392 102
. . . can become scaled if we are talking about
the number of Catholics or Mormons.

136. Dichotomous data like this:
1 = Catholic
2 = Mormon
Event Number of
Catholics in
attendance
Number of
Mormons in
attendance
A 120 22
B 322 34
C 401 78
D 73 55
E 80 3
F 392 102
. . . can become scaled if we are talking about
the number of Catholics or Mormons.

137. Which option is most appropriate for the
problem you are working with:

138. Which option is most appropriate for the
problem you are working with:
Dichotomous by Dichotomous
Dichotomous by Scaled
Ordinal by Another Variable
Scaled by Scaled with at least
one variable Skewed