This document discusses tools for analyzing relationships between variables, including scatterplots and two-by-two tables. It uses an example of a belief that a mother's protectiveness may cause her son to be gay. A two-by-two table is constructed using hypothetical data on 1,000 men, examining mother's protectiveness and son's sexual orientation. The data does not support the belief, as most homosexual men did not have overprotective mothers, and most men with overprotective mothers were heterosexual. The data is not consistent with the mother's protectiveness causing her son's sexual orientation.

1. Thinking about Correlation
Chapter 2 – Third Slideshow

2. Let’s start by revisiting the story of Jen and her son
Tyler.

3. Marie, Jen, and Sara sit and chat
at the local park every Saturday.
Jen keeps a very close eye on her
son, Tyler. She warns him about the
gravel under the swings. Then she tells
Tyler to be careful at the top of the slide.
When all of the kids climb the jungle
gym, she jumps up and runs over, telling
him to get off because it is not safe.

4. Scolded, and feeling sad, Tyler walks
over to his mother for a hug. After
comforting him, Jen tells him to go
play with the children by the swings.
Watching this, Sara and Marie exchange a
disgusted look. They worry that the way Jen treats
Tyler may make him grow up to be gay.

5. How can we research this question?
Can you change a child’s sexuality,
or their gender identity,
by how you parent them?

6. Many beliefs include a relationship between two
variables.
Marie and Sara’s belief is a good example. Their
concern that Jen’s parenting may cause Tyler to
grow up to be gay involves two variables.

7. Although Marie and Sara are reacting to the specific
things that Jen did with Tyler at the park, their belief
is more general. To see the variables, you have to
think more abstractly than just this one case.

8. The two variables are:
1. The degree to which a mother is protective of her son.
2. The probability of a boy growing up to have a
homosexual orientation.

9. Each variable can vary across a continuum of values.
1. The degree to which a mother is protective of her son.
Not Protective Very Protective
2. The probability of a boy growing up to have a
homosexual orientation.
Definitely not Homosexual Definitely Homosexual

10. Marie and Sara are worried that Jen falls high on the first
variable, and that Tyler may be high on the second
variable because of that.
1. The degree to which a mother is protective of her son.
Jen
Not Protective Very Protective
2. The probability of a boy growing up to have a
homosexual orientation.
Tyler
Definitely not Homosexual Definitely Homosexual

11. Marie and Sara believe that these two variables are
related. They believe that the more protective a
mother is, the higher the probability that her son will
grow up to identify as a gay man.

12. Even more, they believe that the first variable causes
the second variable.
They believe that a mother’s protectiveness causes
her son to be gay.

13. As you learned earlier, looking for correlation is the
first step in the search for cause and effect
relationships.
Conditions to establish Cause and Effect.
1. The variables are correlated.
2. The cause comes before the effect.
3. There are no other variables to explain the effect.

14. Correlations Have Two Ends, or Sides
As you learned earlier, if you think two variables are
correlated, be sure to think about both sides of the
relationship, the high end and the low end.

15. Tools to Think About Correlations

16. Scatterplots
There are many tools to study correlation, and I
encourage you to take a course in statistics to learn
more about them. One of these tools is a
scatterplot.

17. Example
Let’s revisit another example as we introduce this new
tool.

18. Remember, we hypothesized that teacher’s niceness
causes his/her students to learn more.
Also remember, the first step when looking for a cause
is to see if there is a correlation.

19. For this example, let’s pretend that you formed this
belief by watching a particular teacher, Mr. Carter,
who was really nice and pleasant. You also noticed
how well students did in Mr. Carter’s class. Perhaps
it was his niceness that caused this.

20. Note, this is a belief formed through personal
experience
(Review: this was one of our ways of knowing!)

21. Review - Thinking about Variables
This experience represents a co-occurrence of two
things: a nice teacher and a class performing well.
Each of those is one possible value on a variable.
Seeing a “nice teacher” can be thought of as seeing a
person who is high in niceness compared to other
teachers (a variable)
Seeing a “class that is doing well” can be thought of as
a class that is scoring high on a test of learning
compared to other classes (a variable).

22. Interpreting a Scatterplot
On the next slide, each dot represents a particular
teacher’s niceness and his/her class’s learning. This
graph shows 18 teacher’s and classes’ scores.

23. Scatterplot Example
Mr. Carter’s
Class
Ms. Stark’s
Class
Not at all Nice Very Nice
Teacher Niceness

24. Interpreting a Scatterplot
One variable is represented as horizontal distances.
Dots to the right represent nicer teachers; dots to the
left represent less nice teachers.

25. Scatterplot Example
Mr. Carter’s
Class
Ms. Stark’s
Class
Not at all Nice Very Nice
Teacher Niceness

26. Interpreting a Scatterplot
The other variable is represented as vertical distances.
Dots that are higher represent classes that are
performing well; lower dots represent classes that
are doing poorly.

27. Scatterplot Example
Mr. Carter’s
Class
Ms. Stark’s
Class
Not at all Nice Very Nice
Teacher Niceness

28. Tools to See Correlations - Scatterplots
You can see correlations by viewing scatterplots. If the
two variables are positively related, you see an oval-
shaped cluster of dots that slopes upward, starting
low on the left and getting higher to the right, which
is what you see below.

29. Tools to See Correlations - Scatterplots
Variables can also be negatively related. If the two
variables are negatively related, you see an oval-
shaped cluster of dots that slopes downward,
starting high on the left and getting lower to the right,
which is what you see below.

30. Tools to See Correlation
Scatterplots are extremely useful tools. However, not
everyone finds graphs to be useful. Here is another
way to try to think about correlations.

31. Tools to See Correlation
– Two-by-Two Tables
We can use the same observations to categorize Mr.
Carter and his excellent class.
This is our second tool for thinking about correlations:
a two-by-two table.

32. Two-by-Two Table Example
Below, you see a two-by-two table, which has two columns,
and two rows (hence the name).

33. Two-by-Two Table Example
We build the table by making columns represent low and
high levels of one variable – in this case, teacher niceness.
Teachers Who are Not Teachers Who are
Nice Nice

34. Two-by-Two Table Example
We make the rows represent low and high levels of the
other variable – in this case, the degree of Student
Learning in the teacher’s class.
Teachers Who are Not Teachers Who are
Nice Nice
Class Mastering
Content
Very Little Learning

35. Two-by-Two Table Example
Our belief was based on a personal experience with Mr.
Carter. That represents one case. His class would
contribute one count to the cell shown below (highlighted
yellow).
Teachers Who are Not Teachers Who are
Nice Nice
Class Mastering
Content 1
Very Little Learning

36. Two-by-Two Table Example
This does not show a correlation, though. This just shows
co-occurrence. We have one example of a nice teacher
with a class of students who master the content he
teaches.
Teachers Who are Not Teachers Who are
Nice Nice
Class Mastering
Content 1
Very Little Learning

37. Two-by-Two Table Example
Let’s add Ms. Stark’s class. She was not nice, and her
students did not learn very much. Her class contributes
one count to the cell shown below (highlighted in yellow).
Teachers Who are Not Teachers Who are
Nice Nice
Class Mastering
Content 1
Very Little Learning 1

38. Two-by-Two Table Example
Next, we go and collect data from an additional 16 classes,
and add them to the counts in our two-by-two table. Below,
you see an example of what this might look like (these are
not real data!).
Teachers Who are Not Teachers Who are
Nice Nice
Class Mastering
Content 2 8
Very Little Learning 7 1

39. For there to be a correlation, you need to see most
cases showing up in two cells diagonal to each
other, and very few counts in the other two cells.

40. Two-by-Two Table Example
Notice that in this example, most of the cases are in the
upper right and lower left cells (yellow). Few cases are
counted in the other two cells. This pattern indicates a
positive correlation between teacher niceness and student
learning.
Teachers Who are Not Teachers Who are
Nice Nice
Class Mastering
Content 2 8
Very Little Learning 7 1

41. Two-by-Two Table Example
Notice this is the same pattern that we had with the
scatterplot for a positive correlation. This isn’t a
coincidence. Math is awesome!
Low on A High on A
High on B Few Lots
Low on B Lots Few

42. Two-by-Two Table, Negative Correlation
You see a correlation by two diagonal cells having large
counts, and the other two having few cases in them. It can
also happen in the pattern below. This is a negative
correlation.
Low on A High on A
High on B Lots Few
Low on B Few Lots

43. Two-by-Two Table, Negative Correlation
Again, notice the downward slope in the scatterplot is the
same pattern of cells in the two-by-two table.
High on
Low on A A
High on B Lots Few
Low on B Few Lots

44. Two-by-Two Table, Negative Correlation
Here is an example of a negative correlation. These are
fictional counts based on 100 students. Real data would
not be this dramatic.
Students Earning Students Earning
Poor Grades in Good Grades in
Classes Classes
Students who Work
Full-time Outside of 19 15
Classes
Students who Work
less than Full-time or 17 49
Do Not Work

45. Two-by-Two Table, Negative Correlation
Note that the biggest counts are on a diagonal, highlighted
in yellow. Also note that the other two cells are lower.
Students Earning Students Earning
Poor Grades in Good Grades in
Classes Classes
Students who Work
Full-time Outside of 19 15
Classes
Students who Work
less than Full-time or 17 49
Do Not Work

46. Two-by-Two Table, Negative Correlation
This negative correlation means that students who work full
time tend to do more poorly in their classes.
Students Earning Students Earning
Poor Grades in Good Grades in
Classes Classes
Students who Work
Full-time Outside of 19 15
Classes
Students who Work
less than Full-time or 17 49
Do Not Work

47. Two-by-Two Table, Benefits
Two-by-two tables are an incredibly useful tool for thinking
about relationships between variables.
Everyday experiences rarely help us observe cases that fit
all four cells. This is one of the first benefits of
systematically collecting evidence with the scientific
method – we can investigate all four cells to find evidence
of a correlation.

48. Two-by-Two Table, Categories
Let’s expand our tool.
Another type of variable is one where something is either
present or absent, or a member of a category.

49. Two-by-Two Table, Categories
For example, you can either wear glasses, or not wear
glasses. This varies across people.
For an example of categories, you could either be left-
handed or right-handed. This is also a variable.

50. Two-by-Two Table, Categories
Perhaps you hypothesize that right-handed people are
more likely to have glasses.
Let’s pretend we purposefully find 100 left-handed and 100
right-handed people and count how many have glasses.
Left-Handed Right-Handed
Has Glasses ? ?
Does Not Have Glasses ? ?

51. Two-by-Two Table, Categories
Pretend you make careful records and get the counts
shown below. These data indicate no relationship. These
two variables are uncorrelated. You can see this because
all of the cells have about the same number of people.
Left-Handed Right-Handed
Has Glasses 50 49
Does Not Have Glasses 50 51

52. Although I’m suggesting that you think about
relationships by using tools such as the scatterplot
and two-by-two tables, psychologists use more tools
than these.

53. Specifically, we need a way of determining when to
conclude that the variables are “correlated” and
when to conclude that there is no relationship. We
accomplish this by using statistics.

54. Without statistics, we can not fully use these tools to
help us make decisions about correlations.
However, we can make a big step toward thinking
more like a psychologist about variables and their
relationships.

55. Let’s look at the example that we started with at the
beginning of the slideshow. Marie and Sara believe
that a mother’s protectiveness causes her son to be
gay.

56. The first variable is a mother’s protectiveness. Let’s
think about this variable for a two-by-two table.
Some mothers are extremely protective, and others
are normal (meaning they have a typical, or
average, level of protectiveness).
(Keep in mind that we are using the term “normal” to mean “typical.”
Psychologists use this word differently. We don’t mean any
judgment of goodness or badness.)

57. The second variable is the son’s adult sexual
orientation. We can think about this as a category,
either Homosexual or Heterosexual.

58. Two-by-Two Table for Sexuality Example
The data below are for 1,000 imaginary men. These are
fictitious data, but they reflect what real studies have found.
Heterosexual Son Homosexual Son
Extremely Protective
Mother 145 6
Normal Mother 825 24

59. Two-by-Two Table for Sexuality Example
Unfortunately, the picture is not immediately clear. Let’s
look at what we can conclude.
Heterosexual Son Homosexual Son
Extremely Protective
Mother 145 6
Normal Mother 825 24

60. Two-by-Two Table for Sexuality Example
1. Most men are heterosexual. Notice that there are far
more men in that column (highlighted yellow),
regardless of row.
Heterosexual Son Homosexual Son
Extremely Protective
Mother 145 6
Normal Mother 825 24

61. Two-by-Two Table for Sexuality Example
2. Most mothers are Normal. Most cases are in the
bottom row, reflecting that “normal mothers” are
common.
So far, these do not tell us anything about a correlation.
Heterosexual Son Homosexual Son
Extremely Protective
Mother 145 6
Normal Mother 825 24

62. Two-by-Two Table for Sexuality Example
3. Most homosexual men did not have overprotective
mothers. You can see this in the highlighted column.
Heterosexual Son Homosexual Son
Extremely Protective
Mother 145 6
Normal Mother 825 24

63. Two-by-Two Table for Sexuality Example
4. Most men with overprotective mothers are
heterosexual. You can see this in the top row below.
This means that knowing Jen is overprotective, we would still predict
that Tyler will be heterosexual, because most men with
overprotective mothers are heterosexual.
Heterosexual Son Homosexual Son
Extremely Protective
Mother 145 6
Normal Mother 825 24

64. Conclusion for Marie and Sara
Data such as these are not consistent with Marie
and Sara’s belief. If a mother’s protectiveness
causes her son’s homosexuality, we would see
the pattern for a correlation.

65. Conclusion for Marie and Sara
Furthermore, if a mother’s protectiveness was the
cause of homosexuality in men, as some people
believe, then we should see a perfect correlation.
The next slide shows what a perfect correlation
would look like for this example.

66. Example of Perfect Correlation
As before, there are far more heterosexual men than
homosexual men. If there is a perfect correlation, all
heterosexual men would have normal mothers and all
homosexual men would have overprotective mothers.
Heterosexual Son Homosexual Son
Extremely Protective
Mother 0 30
Normal Mother 970 0

67. Example of Perfect Correlation
Notice the pattern for the strong correlation:
The diagonal has all cases (or most), and the other two
cells have none (or few).
Heterosexual Son Homosexual Son
Extremely Protective
Mother 0 30
Normal Mother 970 0

68. Summary – Seeing Correlations
 To show a correlation, psychologists use statistics.
 You can use a scatterplot to see a correlation.

69. Summary – Seeing Correlations with
Two-by-Two Tables
 You can also use a two-by-two table to think about
correlations.
 Personal experiences typically only offer us one cell of the
two-by-two table (co-occurrence).
 For a strong correlation to exist, two diagonal cells have
to have most cases, and the other two cells need to have
few cases.

70. What does this mean for Marie and Sara’s belief?

71. Across decades of research, psychologists have been
unable to find any one type of parenting or any
activity that seems to cause homosexuality.

72. Although the belief that parents affect their children’s
sexuality has tenacity(many people continue to
believe it), it is not supported by evidence.

73. Some religious authorities believe that parents have a
moral obligation to behave in certain ways. Scientific
evidence has nothing to say about this. We can not
study ultimate concerns such as one’s salvation or
moral standing with a deity. These are supernatural
questions, outside of what science can study.

74. However, if an authority suggests that a type of
parenting will lead to a child becoming homosexual
(a claim that we can study with the scientific
method), then that claim is inconsistent with
empirical evidence.

75. Practice identifying variables.
Practice trying to identify both sides of a correlation
(e.g., high side: nice teachers have high performing
classes, and low side: mean teachers have low
performing classes).
Practice trying to think about two variables in a
scatterplot, or all four cells in a two-by-two table for a
correlation.