The document discusses questions of independence, which examine whether changes in one variable are related to or independent of changes in another variable. It provides examples of determining if two variables, such as number of cigarette packs smoked per day and cognitive impairment, or anger survey scores and number of racing accidents, are independent. The goal is to analyze data sets to see if higher or lower scores on one variable are unrelated to groups or values on the other variable.

1. Is the problem you are working on a question
of Independence?

2. Is the problem you are working on a question
of Independence?

3. Questions of Independence are actually the flip side of
questions of relationship.

4. If a variable is independent of another variable,

5. If a variable is independent of another variable, then an
increase or decrease in one WILL NOT be accompanied

6. If a variable is independent of another variable, then an
increase or decrease in one WILL NOT be accompanied
by an increase or decrease in the other.

7. A question of independence generally can be
detected by an equation like this one:

8. A question of independence generally can be
detected by an equation like this one:
Variable 1
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Variable 2

9. Questions of Independence are often posed as
questions of bias.

10. Here is an example:

11. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.

12. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.

13. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.
Variable 1
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Variable 2

14. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.
Variable 1
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Variable 2

15. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.
Variable 1
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Variable 2

16. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.
Packs
smoked per
day
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Variable 2

17. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.
Packs
smoked per
day
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Variable 2

18. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.
Packs
smoked per
day
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Variable 2

19. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.
Packs
smoked per
day
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Response
Time

20. In the 1970s tobacco companies spent resources on
research that showed there was no link between the
amount of cigarette smoking (packs per day) and
cognitive impairment. Cognitive impairment was
measured in terms of the amount of time it took
someone to respond to certain questions.
Packs per
day smoked
An Increase
or decrease in
IS NOT
accompanied
by an increase
or decrease in
Response
This is a question of independence Time

21. Here is a data set for this problem:

22. Here is a data set for this problem:
Research
Subjects
A
B
C
D
E
F
G

23. Here is a data set for this problem:
Research
Subjects
Amount of Packs
Smoked each Day
A 3
B 2
C 3
D 1
E 0
F 0
G 4

24. Here is a data set for this problem:
Research
Subjects
Amount of Packs
Smoked each Day
Minutes to respond to
a cognitive question
A 3 10
B 2 4
C 3 1
D 1 2
E 0 3
F 0 14
G 4 3

25. Here is a data set for this problem:
Research
Subjects
Amount of Packs
Smoked each Day
Minutes to respond to
a cognitive question
A 3 10
B 2 4
C 3 1
D 1 2
E 0 3
F 0 14
G 4 3
Researchers are hoping NO LINK between
these two variables will be found.

26. Here is a data set for this problem:
Research
Subjects
Amount of Packs
Smoked each Day
Minutes to respond to
a cognitive question
A 3 10
B 2 4
C 3 1
D 1 2
E 0 3
F 0 14
G 4 3
Researchers are hoping NO LINK between
these two variables will be found.

27. Here is a data set for this problem:
Research
Subjects
Amount of Packs
Smoked each Day
Minutes to respond to
a cognitive question
A 3 10
B 2 4
C 3 1
D 1 2
E 0 3
F 0 14
G 4 3
Researchers are hoping NO LINK between
these two variables will be found.

28. Here is a data set for this problem:
Research
Subjects
Amount of Packs
Smoked each Day
Minutes to respond to
a cognitive question
A 3 10
B 2 4
C 3 NO LINK
1
D 1 2
E 0 3
F 0 14
G 4 3
Researchers are hoping NO LINK between
these two variables will be found.

29. You will be shown in future presentations how to analyze
the data to determine if the variables are independent of
one another or not.

30. As with relationship questions there will be some
situations where you are determining independence
between a variable with unlimited values (e.g., speed,
height, age) and a variable with limited values (e.g.,
gender, year in school).

31. As with relationship questions there will be some
situations where you are determining independence
between a variable with unlimited values (e.g., speed,
height, age) and a variable with limited values (e.g.,
gender, year in school).
Variable 1
Higher and
lower scores
in
tend to be
UNRELATED to
certain groups in
Variable 2

32. For example:

33. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.

34. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.

35. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate.

36. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate.

37. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of 0
to 30 to approximately forty drivers.

38. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of 0
to 30 to approximately forty drivers. You then add up the
number of accidents they are in during the period of a
month.

39. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of 0
to 30 to approximately forty drivers. You then add up the
number of accidents they are in during the period of a
month. You wish to determine if anger scores are
independent of accidents.

40. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of 0
to 30 to approximately forty drivers. You then add up the
number of accidents they are in during the period of a
month. You wish to determine if anger scores are
independent of accidents.
Variable 1
Higher and
lower scores
in
tend to be
UNRELATED to
certain groups in
Variable 2

41. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of
0 to 30 to approximately forty drivers. You then add up
the number of accidents they are in during the period of
a month. You wish to determine if anger scores are
independent of accidents.
Variable 1
Higher and
lower scores
in
tend to be
UNRELATED to
certain groups in
Variable 2

42. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of
0 to 30 to approximately forty drivers. You then add up
the number of accidents they are in during the period of
a month. You wish to determine if anger scores are
independent of accidents.
Anger
Survey
Scores
Higher and
lower scores
in
tend to be
UNRELATED to
certain groups in
Variable 2

43. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of 0
to 30 to approximately forty drivers. You then add up the
number of accidents they are in during the period of a
month. You wish to determine if anger scores are
independent of accidents.
Anger
Survey
Scores
Higher and
lower scores
in
tend to be
UNRELATED to
certain groups in
Variable 2

44. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of 0
to 30 to approximately forty drivers. You then add up the
number of accidents they are in during the period of a
month. You wish to determine if anger scores are
independent of accidents.
Anger
Survey
Scores
Higher and
lower scores
in
tend to be
UNRELATED to
certain groups in
Variable 2

45. Complaints have arisen that a local raceway has more
accidents because of more “hot headed” (angry) drivers.
You have been asked to investigate. You administer a
“easily provoked to anger” survey with a score range of 0
to 30 to approximately forty drivers. You then add up the
number of accidents they are in during the period of a
month. You wish to determine if anger scores are
independent of accidents.
Anger
Survey
Scores
Higher and
lower scores
in
tend to be
UNRELATED to
certain groups in
Number of
Accidents

46. Here is what a data set might look like for this problem:

47. Here is what a data set might look like for this problem:
Drivers
A
B
C
D
E
F
G

48. Here is what a data set might look like for this problem:
Drivers Anger Survey
Scores
A 28
B 5
C 21
D 14
E 25
F 18
G 1

49. Here is what a data set might look like for this problem:
Drivers Anger Survey
Scores
Number of Accidents
in a Month
A 28 4
B 5 5
C 21 12
D 14 7
E 25 4
F 18 7
G 1 6

50. Here is what a data set might look like for this problem:
Drivers Anger Survey
Scores
Number of Accidents
in a Month
A 28 4
B 5 5
C 21 12
D 14 7
E 25 4
F 18 7
G 1 6
Owners of the speedway hope that NO LINK
between these two variables will be found.

51. Here is what a data set might look like for this problem:
Drivers Anger Survey
Scores
Number of Accidents
in a Month
A 28 4
B 5 5
C 21 12
D 14 No link
7
E 25 4
F 18 7
G 1 6
Owners of the speedway hope that NO LINK
between these two variables will be found.

52. You will be shown in future presentations how to analyze
the data to determine if the variables are independent
from one another or not.

53. Finally, there will be some situations where you will
determine independence between two variables with
limited values (e.g., gender, year in school).

54. For example:

55. A claim has been made that admission decisions are
biased toward minority groups.

56. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.

57. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.
You have decided to state your research question as
follows: “To what degree are admission decisions
independent of minority group status?”

58. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.
You have decided to state your research question as
follows: “To what degree are admission decisions
independent of minority group status?”
Is Variable 1 independent of Variable 2

59. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.
You have decided to state your research question as
follows: “To what degree are admission decisions
independent of minority group status?”
Is Variable 1 independent of Variable 2

60. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.
You have decided to state your research question as
follows: “To what degree are admission decisions
independent of minority group status?”
admissions
Is independent of Variable 2
status

61. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.
You have decided to state your research question as
follows: “To what degree are admission decisions
independent of minority group status?”
admissions
Is independent of Variable 2
status

62. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.
You have decided to state your research question as
follows: “To what degree are admission decisions
independent of minority group status?”
admissions
Is independent of Variable 2
status

63. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.
You have decided to state your research question as
follows: “To what degree are admission decisions
independent of minority group status?”
admissions
Is independent of Variable 2
status

64. A claim has been made that admission decisions are
biased toward minority groups. You have been asked by
an outside evaluation agency to look into this allegation.
You have decided to state your research question as
follows: “To what degree are admission decisions
independent of minority group status?”
admissions
Is independent of
status
minority
group
status

65. Here is a data set for this problem:

66. Here is a data set for this problem:
Research
Subjects
A
B
C
D
E
F
G

67. Here is a data set for this problem:
Research
Subjects
Admitted
1 = yes, 2 = no
A 1
B 1
C 2
D 1
E 2
F 1
G 1

68. Here is a data set for this problem:
Research
Subjects
Admitted
1 = yes, 2 = no
Minority Group Status
1 = yes, 2 = no
A 1 1
B 1 1
C 2 2
D 1 2
E 2 2
F 1 1
G 1 1

69. Here is a data set for this problem:
Research
Subjects
Admitted
1 = yes, 2 = no
Minority Group Status
1 = yes, 2 = no
A 1 1
B 1 1
C 2 2
D 1 2
E 2 2
F 1 1
G 1 1
The admissions office hopes that NO LINK
between these two variables will be found.

70. Here is a data set for this problem:
Research
Subjects
Admitted
1 = yes, 2 = no
Minority Group Status
1 = yes, 2 = no
A 1 1
B 1 1
C 2 2
D 1 No link
2
E 2 2
F 1 1
G 1 1
The admissions office hopes that NO LINK
between these two variables will be found.

71. Here are more examples of independence-oriented
questions:

72. To what degree is -

73. To what degree is -
• math proficiency independent of gender?

74. To what degree is -
• math proficiency independent of gender?

75. To what degree is -
• math proficiency independent of gender?
• news reporting void of political bias?

76. To what degree is -
• math proficiency independent of gender?
• news reporting void of political bias?

77. To what degree is -
• math proficiency independent of gender?
• news reporting void of political bias?
• the risk of lung cancer independent of cigarette
smoking?

78. To what degree is -
• math proficiency independent of gender?
• news reporting void of political bias?
• the risk of lung cancer independent of cigarette
smoking?

79. To what degree is -
• math proficiency independent of gender?
• news reporting void of political bias?
• the risk of lung cancer independent of cigarette
smoking?
• incarceration independent of race?

80. To what degree is -
• math proficiency independent of gender?
• news reporting void of political bias?
• the risk of lung cancer independent of cigarette
smoking?
• incarceration independent of race?

81. Note – that any independence question can be worded as
a relationship question.

82. For example,

83. Is math proficiency independent of gender?

84. Is math proficiency independent of gender?
can be worded as –

85. Is math proficiency independent of gender?
can be worded as –
Is there a relationship between math proficiency and
gender?

86. The key takeaway – relationship questions can be worded
as independence questions and independence questions
can be worded as relationship questions.

87. The key takeaway – relationship questions can be worded
as independence questions and independence questions
can be worded as relationship questions.
It all depends on the nature of the research question!

88. Examine the question or problem you are
working on.

89. Is it a question of independence?

90. If so, select INDEPENDENCE