The
Correlation
The
correlation is one of the easiest descriptive
statistics to understand and possibly one of the
most widely used.
The term correlation literally means
corelate and refers to the measurement of a
relationship between two or more variables.
A correlational coefficient is used to
represent this relationship and is often abbreviated
with the letter r.
A correlational coefficient typically ranges
between 1.0 and +1.0 and provides two important
pieces of information regarding the relationship:
Intensity and Direction.
Intensity
refers to the strength of the relationship and is
expressed as a number between zero (meaning no
correlation) and one (meaning a perfect
correlation). These
two extremes are rare as most correlations fall
somewhere in between.
In the social sciences, a correlation of 0.30
may be considered significant and any correlation
above 0.70 is almost always significant.
The absolute value of r represents the
intensity of any correlation.
Direction
refers to how one variable moves in relation to the
other. A
positive correlation (or direct relationship) means
that two variables move in the same direction,
either both moving up or both moving down.
For instance, high school grades and college
grades are often positively correlated in that
students who earn high grades in high school tend to
also earn high grades in college. A negative correlation (or inverse relationship) means that
the two variables move in opposite directions; as
one goes up, the other tends to go down.
For instance, depression and selfesteem tend
to be inversely related because the more depressed
an individual is the lower his or her selfesteem. As depression increases, then, selfesteem tends to decrease.
The sign in front of the r represents
the direction of a correlation.
Figure
8.6: Scatter plots for sample correlations
Scatter
Plot.
Correlations are graphed on a special type of
graph called a scatter plot (or scatter gram).
On a scatter plot, one variable (typically
called the X variable) is placed on the horizontal
axis (abscissa) and the Y variable is placed on the
vertical axis (ordinate).
For example, if we were measuring years of
work experience and yearly income, we would likely
find a positive correlation.
Imagine we looked at ten subjects and found
the hypothetical results listed in Table 8.2.
Table
8.2: Sample Correlation Data
Subject
Number

Experience
in Years

Income
in Thousands

Subject
Number

Experience
in Years

Income
in Thousands

1

0

20

6

15

50

2

5

30

7

20

60

3

5

40

8

25

50

4

10

30

9

30

70

5

10

50

10

35

60

Notice
how each subject has two pieces of information (years
of experience and income).
These are the two variables that we are looking
at to determine if a relationship exists. To place this information in a scatter plot we will consider
experience the X variable and income the Y variable
(the results will be the same even if the variables
are reversed) and then each dot will represent one
subject. The
scatter plot in Figure 8.7 represents this data.
Notice how the line drawn through the data
points has an upward slope.
This slope represents the direction of the
relationship and tells us that as experience increases
so does income.
Figure
8.7: Scatter Plot for Sample Data
Correlation
and Causality.
One common mistake made by people
interpreting a correlational coefficient refers to
causality. When
we see that depression and low selfesteem are
negatively correlated, we often surmise that
depression must therefore cause the decrease in
selfesteem. When
contemplating this, consider the following
correlations that have been found in research:

Positive
correlation between ice cream consumption and
drownings

Positive
correlation between ice cream consumption and
murder

Positive
correlation between ice cream consumption and
boating accidents

Positive
correlation between ice cream consumption and
shark attacks
If
we were to assume that every correlation represents
a causal relationship then ice cream would most
certainly be banned due to the devastating effects
it has on society.
Does icecream consumption cause people to
drown? Does
ice cream lead to murder?
The truth is that often two variables are
related only because of a third variable that is not
accounted for within the statistic.
In this case, the weather is this third
variable because as the weather gets warmer, people
tend to consume more ice cream.
Warmer weather also results in an increase in
swimming and boating and therefore increased
drownings, boating accidents, and shark attacks.
So
looking back at the positive correlation between
depression and selfesteem, it could be that
depression causes selfesteem to go down, or that
low selfesteem results in depression, or that a
third variable causes the change in both. When looking at a correlational coefficient, be sure to
recognize that the variables may be related but that
it in no way implies that the change in one causes
the change in the other.
Specific
Correlations.
Up to this point we have been discussing a
specific correlation known as the Pearson Product
Moment Correlation (or Pearsons r) which is
abbreviated with the letter r.
Pearson is the most commonly cited
correlation but can only be used when there are only
two variables that both move in a continuous linear
(straight line) direction.
When there are more than two variables, when
the variables are dichotomous (true/false or yes/no)
or rank ordered, or when the variables have a
nonlinear or curved direction, different types of
correlations would be used.
The
Biserial and Point Biserial Correlations are used
when one variable is dichotomous and the other is
continuous such as gender and income.
The phi or tetrachoric correlations are used
when both variables are dichotomous such as gender
and race. And
finally, Spearmans rho correlation is used with
two rank ordered variables and eta is used when the
variables are nonlinear.
