EDUC 5113 Study Guide Materials
Week 5
CORRELATIONAL ANALYSIS: PEARSON’S r
READING
Dancey and Reidy (4th edition, 2007), Chapter 6
(or Chapter 5 in the 3rd Edition)
CHAPTER OVERVIEW
In the earlier chapters we have introduced to you the statistical bases that you will
need to understand the statistical analyses presented in the remainder of this course.
We also introduced to you the idea of looking at relationships between variables. For
example the relationship between hours spent studying and performance in an exam.
In its simplest form, correlational analysis is a measure of the relationship between
two variables. It is a most important statistic that all researchers use a great deal. In
the present chapter we will discuss the following:



Pearson’s correlation coefficient (also known as product-moment correlation)
Partial correlation
Effect size
KEY TERMS
1- Correlation does not mean causality.
2- Linear relationship.
2- Directionality of a relationship.
3- Magnitude of a relationship.
4- Non-linear relationship.
5- Variance explained in a correlation coefficient.
6- Scatterplot. (visual display of two factors)
7- Partial correlation
KEY POINTS
1- Correlation is a bivariate measure of association or strength of the relationship
between two variables. For example, the relationship between a student
mathematical self-efficacy and achievement in maths.
2- Correlation varies from 0 (i.e., random relationship) to 1 (i.e., perfect positive
linear relationship) or -1 (i.e., perfect negative linear relationship).
3- Correlation is usually reported in terms of its square (r2), interpreted as percent
of variance explained. For example, if r2 is .25, then variable A is said to
explain 25% of the variance in the variable B and vice versa (i.e., correlation
is bidirectional).
4- Coefficient of determination, r2,, is the square of the Pearson’s correlation
coefficient. It is a measure of effect size. That is the effect of one variable on
another variable.
5- Correlation is the covariance of standardized variables.
6- Standardization is the process of making variables comparable in magnitude
and dispersion: one subtracts the mean from each variable and divides by its
standard deviation, giving all variables a mean of 0 and a standard deviation of
1.
7- Standardization is built into correlation and need not be requested explicitly in
SPSS or other programs.
8- Pearson’s r is a measure of correlation, sometimes called Pearson’s productmoment correlation. Pearson’s r is a measure of association which varies from
-1 to +1, with 0 indicating no relationship (i.e., random pairing of values) and
1 indicating perfect relationship, taking the form, "The more the x, the more
the y, and vice versa." A value of -1 is a perfect negative relationship, taking
the form "The more the x, the less the y, and vice versa."
Further readings
Correlation coefficient
http://mathbits.com/MathBits/TISection/Statistics2/correlation.htm
Interpretation of correlation analysis output from SPSS
http://wwwstage.valpo.edu/other/dabook/ch8/c8-3.htm
9- The relationship between two variables is represented visually by a scatterplot.
You can show the regression line (i.e., goodness of fit line or linear
relationship) which represents a perfect relationship. The relationship between
the two variables is represented by dots scattered around this regression line.
The closer these dots are to the line, the stronger the relationship between the
two variables.
An interactive tools for correlation coefficients and scatterplots is found at
http://academic.udayton.edu/GregElvers/psy216/activex/CorReg.htm
This gives 25 points on two dimensions, first randomly. But with this one you can
play with it by dragging points on the small Java screen to see what happens when
you change their positions. A clever little simulation: Just play with it, and it helps
you see how data relates to coefficients.
10- When two variables correlate with each other, this does not mean that one
variable causes the other. There could be other external variables that
determine the relationship between the two variables.
Further readings
Misinterpretation of the correlation coefficient Correlation and causation
http://www.upa.pdx.edu/IOA/newsom/pa551/lectur16.htm
11- The relationship between two variables may be non-linear. An example of
such relationship is between anxiety and exam performance. Certain level of
anxiety can improve performance till a certain level of anxiety when exam
performance starts going down. This relationship can also be represented by a
scatterplot which will take a curve shape.
12- Partial correlation is the correlation of two variables while controlling for a
third or more other variables. Specifically, we control for the effect of a
variables on both variables that we want to test whether there is a relationship
between them.
SPSS video-type activities
1- Scatterplot (Good, but takes a few minutes)
http://distdell4.ad.stat.tamu.edu/spss_1/Scatter.html
2- Scatterplot (interactive mode in SPSS graphs)
http://distdell4.ad.stat.tamu.edu/spss_1/scatter_i.html
3- Matrix scatterplot
http://distdell4.ad.stat.tamu.edu/spss_1/MatrixScatter.html
4- Correlation using SPSS (some fast clicking on this one)
http://wwwstage.valpo.edu/other/dabook/ch8/c8-4.htm
ACTIVE LEARNING AND OPPORTUNITIES
1- Suppose that a researcher found a correlation coefficient of 1.6 between variable
X and variable Y in a sample of 80 students. Please choose from the following:
A- The relationship between the X and Y is very strong.
B- The sample size is not enough.
C- The researcher has made a computational mistake.
2- The following table represents a hypothetical relationship between students’
mastery goal (i.e., development of competency and task mastery) and academic
achievement. Please answer the following questions:
Correlations
MASTERY
ACHIEVE
MASTERY
Pears on Correlation
1
Sig. (2-tailed)
.
N
93
Pears on Correlation
.350
Sig. (2-tailed)
.010
N
93
ACHIEVE
.350
.010
93
1
.
93
A- What is the relationship between mastery goal and academic
achievement?
B- What does the correlation coefficient value mean with regard to the
directionality of the relationship?
C- Is the relationship between the two variables significant? Why? And if
so, what does the significance of the relationship mean?
D- What is the percentage of variance that mastery goal can explain in
academic achievement?