Chapter 6: Correlational Research
• Examine whether variables are related to one
another (whether they vary together).
Correlation coefficient: statistic indicating how well
two variables are related to one another (how well
they vary together) in a linear fashion.
• Must obtain a score on each variable for each
participant.
• Pearson correlation coefficient (r): most
common. Values range from -1.00 to +1.00
• The direction of the relationship is indicated by
the sign of the correlation coefficient.
• Positive correlation: indicates a direct, linear,
positive relationship (as one variable increases
the other variable also increases).
• Negative correlation: indicates a direct, linear,
negative relationship (as one variable increases
the other variable decreases)
• Magnitude of the correlation: the numerical value
(ignoring the sign) which expresses the strength
of the relation
• Correlation of .33, indicates that the variables are
not a strongly related as variables with a
correlation of .65
• The stronger the correlation the more tightly the
data cluster around the mean
• Two variables may be related in a curvilinear
fashion.
• The correlation will be 0 but the variables may still
be related in a non-linear way.
Coefficient of determination: represents the
proportion of the variance in one variable (x) that
is accounted for by the other variable (y).
• r2 (square the correlation coefficient).
• If the correlation between two variables (x and y)
is 0.3. Then 0.3 squared = 0.09, or 9% is the
variance in x is accounted for y
• Proportion of variance in x that is systemic
variance shared with y.
Practice correlation calculation:
In this study, 12 participants were given as much time
as they needed to memorize a poem. When they
thought they had memorized the poem, the
participants recited it, and the number of errors they
made were counted. Calculate the correlation
between the amount of time participants worked on
memorizing the poem and the number of errors they
made.
Practice correlation calculation:
• x and y represent the variables of interest.
• xy means you multiply each participant’s x and
y score and then sum all the products across
participants
• (x)(y) means that you sum all the participant’s
x scores, sum all the y scores, and then multiply
these two sums together.
Time
6
11
10
5
3
12
9
4
4
1
7
9
Errors
3
0
5
2
1
6
0
9
5
7
4
1
Statistical significance of r
• exists when the correlation coefficient has a very
low chance of being 0 in the population.
• Statistically significant: means the chance that our
correlation is truly 0 in the population is very low
(usually less than .05). Meaning there is a 5%
probability that our result is not really significant
but happened by chance.
Statistical significance can be influenced by:
• sample size: the larger the sample size the more
likely you are to conclude that a correlation is
statistically significant.
• The magnitude of the correlation: the larger the
more confident you are in concluding that the
correlation is statistically significant
• P value: the level of significance you set before
you calculate the correlation.
• Most common is .05
• Some researchers are more conservative and use
.01 meaning there is only a 1% probability the
correlation could be found significant even if it
really is not significant (or due to chance).
• With a P value of .01 you must have a larger
correlation than with a P value of .05 for it to be
significant.
Factors that distort correlation coefficients
1) Restricted range: the size of the correlation may
be reduced by a restriction of the range in the
variables being correlated.
• A restricted range occurs when most participants
have similar scores (less variability).
• This can occur when you are correlating scores
that are either either high or low on one variable.
• E.g. If you correlate SAT scores of people who get
into college with their college GPA, you may be
dealing with a restricted range because usually
those with higher SAT scores get in to college.
• Must ensure you have a broad range of scores.
2) Outliers:
• Outliers that are far off the correlation line (high on
x but lower on y) tend to deflate the value of r.
• Outliers that are on the correlation line but to the
extreme on both x and y tend to inflate the value
of r.
3) Reliability of measures: the less reliable the
measures the lower the correlation coefficients.
Correlation and Casualty: you can not infer that
one variable causes the other in a correlation.
• The variables may be related: a correlation between
obesity and depression (more obese people are
more depressed) does NOT mean that obesity
causes depression, or that depression causes
people to become obese.
Experimental studies must be conducted to infer
causality in which there must be:
• Covariation: changes in the value of one variable
are associated with changes in the value of
another variable
• Directionality: the presumed cause must precede
the effect in time. Very difficult to do in
correlational research.
• Elimination of extraneous variables: eliminate all
other factors that may influence the relationship
between the two variables.
• Two variables may be correlated only because they
are actually correlated with a third variable.
• E.g. There is a correlation between eating ice
cream and drowning. But these variable are only
correlated because they are both correlated with a
third variable called summer (heat). People eat
more ice cream in the summer (when it is hotter)
and people drown more in the summer (swim more
when it is hotter).
Partial Correlation: The correlation between two
variables after the influence of the third variable is
statistically removed.
• E.g. Correlation between viewing violent TV and
childhood aggression (children who watch more
violent TV are more aggressive in their play)
– But, parent discipline style may also be related to
childhood aggression. More harsh and mean
parents may have more aggressive children.
• So with a partial correlation we can determine the
correlation between violent TV viewing (x) and
childhood aggression (y) once we statistically
remove the influence of parent’s discipline style (z).
Aggression
(y)
Violent TV
(x)
Parental
Discipline
(z)
If the correlation between x and y is still significant
after removing z:
• we can conclude that x and y are correlated even
after we account for parent discipline style (z)
• and the relationship between x and y is unlikely
due to parent discipline style (z).
Aggression
(y)
Violent
TV (x)
Parental
Discipline
(z)
If the correlation between x and y is no longer
significant after you remove z:
• then we conclude that the previous observed
correlation between x and y was likely due to
another variable: parent discipline style (z).
Sometimes after removing another variable (z)
the correlation between x and y is smaller but
still significant, which means that z did have
an influence, but x and y are still related.
Aggression
(y)
Violent TV
(x)
Parental
Discipline
(z)
Other indices of correlation:
• Spearman rank-order correlation: correlation
between two variables when one or both of the
variables is on an ordinal scale (the numbers
reflect rank ordering).
• E.g. Correlation between teachers ranking of the
best to worst students (ordinal scale) and the
students IQ scores (interval scale).
• Point biserial correlation: used when one
variable is dichotomous
• Gender is dichotomous (male or female). To
correlate gender with spatial memory you would
assign all males a 1 and all females a 2.
• If you get a significant positive correlation that
would mean that females tend to score higher on
spatial memory than males. A significant negative
correlation would mean that males score higher.
• Phi coefficient: used when both variables
being correlated are dichotomous (e.g.,
gender, handedness, yes/no answer)
Group Task: Single People Attract Crime
Statistics show that people who are not married are
three to four times more likely to be victims of violent
crime as people who are currently married. The
number of violent crimes per 1,000 people age 12
years or older are shown in the following list. Clearly,
marital status correlates with victimization.
Marital Status
Violent Crimes per 1,000 people
Married
13
Widowed
8
Divorced or separated
42
Never married
51
1. Speculate regarding possible explanations of
this relationship. Suggest at least three reasons
that marital status and victimization may be
linked.
2. Consider how you would conduct a
correlational study to test each of your
explanations. You will probably want to design
studies that allow you to partial out variables
that may mediate the relationship between
marital status and victimization.
Class Discussion
1. Imagine you predicted a moderate correlation
between people’s scores on a measure of anxiety and
the degree to which they report having insomnia. You
administered measures of anxiety and insomnia to a
sample of 30 participants, and obtained a correlation
of .28. Because this correlation is not statistically
significant (the critical value is .30), you must treat it
as if it were zero. Yet you still think that anxiety and
insomnia are correlated. If you were going to conduct
the study again, what could you do to provide a more
powerful test of your hypothesis?
2. Imagine you obtained a point biserial correlation of
.35 between gender and punctuality, showing that
men arrived later to class than women. You think that
this correlation might be due to the fact that more
women wear watches, so you calculate the partial
correlation between gender and punctuality while
removing the influence of watch-wearing. The
resulting correlation was .35
Interpret the partial correlation.
What if the correlation was .10 and no longer
significant?
What if the correlation was .25 and still significant?
3. Following the rash of school shootings that occurred
in the late 1990’s, some individuals suggested that
violent video games were making children and
adolescents more aggressive. Imagine that you
obtained a sample of 150 15-years-old males and
correlated their level of aggressiveness with the
amount of time per week that they played violent
video games. The correlation coefficient was .56
(highly significant). Does this finding support the idea
that playing violent video games increases
aggression?