CORRELATIONAL
ANALYSES
EDRS 5305
EDUCATIONAL RESEARCH &
STATISTICS
 Focus
will be on the Pearson r, most
commonly used correlation statistic.
 When reading research studies, likely to
encounter studies in which rs are
reported without reference to type of
correlation.
 Most likely a Pearson r or Pearson
product-moment correlation coefficient.
 Even
though taught Pearson r should
be used only on interval/ratio level data,
some social science methodologists
have argued, convincingly, that the
Pearson r may be used even when data
satisfy assumptions of ordinal data.
 Commonly used this way with
questionnaire/survey data.
What is null hypothesis for r?
 Null
hypothesis, in dealing with a single
correlation, will simply be a pinpoint
statement as to a possible value of the
correlation in the population.
 Typically the pinpoint value is no
relationship or a .00 correlation.
Meaning of Correlation
r
indicates degree of relationship
between two variables
 does NOT indicate the strength of
association in the data
 strength of association MORE important
than degree of relationship
Strength of Association
 r2
= the r squared
 r2 = extent to which variables share
common properties or characteristics
 r2 = measure of the proportion of
variability in one variable that can be
determined from (explained by) the
relationship of the other
r = .9, then r2 = .81 or 81%
 81% of the variance in variable A is
explained or determined by B and 81%
of the variance in variable B is
explained or determined by A
 In this case, the variables, 81% worth,
are essentially measuring the same
construct(s).
 If
r = .5, then r2 = .25 or 25%
 25% of the variance in variable A is
explained or determined by B and 25%
of the variance in variable B is
explained or determined by A
 In this case, the variables, 75% worth,
are essentially measuring different
construct(s).
 If
 Even
if a correlation coefficient of .2 is
statistically significant, it only accounts
for, explains, or determines 4% of the
variance---most likely [not always--medical research] a trivial amount.
Reporting Correlation Results
r value
sample size
p value
r2 value
 r (167) = -.63, p < .001, 39.69% of
variance accounted for
Example of Reporting r Results
 A statistically
significant relationship was found
between students’ study skills and their locus
of control, r (153) = -.63, p < .001. Squaring
the correlation revealed that these two
variables had 39.69% of the variance in
common. Thus, students who exhibited good
study skills tended to report more of an internal
locus of control than did students with poor
study skills.
Another Example
 Use
of a Pearson r yielded a
statistically significant relationship, r
(235) = +.75, p < .01, between scores
on the Wechsler IQ and Wechsler
achievement measures. The IQ and
achievement measures had 56.25% of
variance in common, figures which are
supported by previous researchers.
Another Example
 A Pearson
r, calculated between scores
on the Woodcock Basic Reading Test
and the WIAT Basic Reading Test, was
statistically significant, r (96) = +.35, p
< .05. Even so, only 12.25% of the
variance in test scores was shared by
these measures in which the same
construct is supposedly measured.
Reliability and Validity rs
 Can
have statistically significant rs for
reliabilities and validities that are NOT
important NOR meaningful
 Important to examine not only p level
but, more importantly, the magnitude of
the relationship
 The lower the reliability of measuring
instrument, lower the validity must be.
 For
internal consistency reliability or
Cronbach’s coefficient alpha, .9 is desirable.
 .9 means that 90% of the test score is true
score variance and 10% is error.
 For test-retest reliability, .8 is desirable.
 .8 means that 80% of the test score is true
score variance and 20% is error.
 Remember that ERROR is always present.
 For
research purposes
 Nunnally (1978) stated that coefficient
alphas above .75 may be viewed as
evidence that a scale has acceptable
reliability for use in research.