Is there a relationship (difference) or isn't there a relationship (difference)?

Once you have learned the correlation coefficient
(__r__) for your sample, you need to determine what the likelihood is that the __r__
value you found occurred by chance. In other words, does the relationship you
found in your sample really exist in the population or
were your results a fluke?** -- OR -- **In the case of a *t*-test,
did the difference between the two means in your sample occurred by chance and
not really exist in your population.

Prior to collecting data, researchers predetermine
an alpha level, which is how willing they are to be wrong when they state that there is a
relationship (in the case of correlation research) or difference (in the case of a
*t*-test)
between the two variables they measured. A common alpha level for educational research is
.05.

Let’s assume that you recollected your data
with 100 different samples from the same population and calculated __r__ (or __t__) each
time (you would not normally do this). You would assume that the value for __r__ (or __t__) would not be the same for each of the 100
samples that you collected. When you set your alpha level to .05, you are saying that you
are willing to be wrong (say there was a relationship in your sample when there
was not one in your population 5 times out of 100). In other words, a maximum of 5 of those 100 samples
might show a relationship (__r__ <> 0) when there really was no
relationship in the population (__r__ = 0). Another way of looking at
it is at least 95 times out of a 100 the relationship (difference in the case of
a *t*-test) you found with your
sample probably also exists in the populations from which you drew your sample. On the other hand, the
relationship (difference in the case of a *t*-test) you found in the sample
might occur by chance (__r__ = 0 in the population, but you found __r__
<> 0 in the sample) 5 times out of a 100 times. For example, if 100 times you repeatedly drew samples of 27 pairs of scores from a population where the correlation was exactly 0, by chance five of those times you would get a correlation of .381 or higher.

**Use the following for Correlation...**

For this example, we have set the alpha level
(likelihood of being incorrect when we say the relationship we found in our
sample reflects a relationship in the population) at .05. In order to
determine if the __r__ value we found with our sample meets that requirement, we will
use a critical value table for Pearson’s Correlation
Coefficient. To use the table, you need two pieces of information, how many subjects you
had and the correlation coefficient __r__ for your study.

First you must determine something called degrees
of freedom (__df__). For a correlation study, the degrees of freedom is equal to 2 less
than the number of subjects you had. If you collected data from 27 pairs, the degrees of
freedom would be 25. Use the critical value table to find the
intersection of alpha .05 (see the columns) and 25 degrees of freedom (see rows). The
value found at the intersection (.381) is the minimum correlation coefficient __r__
that you would need to confidently state 95 times out of a hundred that the relationship
you found with your 27 subjects exists in the population from which they were drawn.

If the absolute value of your correlation
coefficient is above .381, you reject your
null hypothesis (there is no relationship) and accept the alternative hypothesis: There is
a statistically significant relationship between arm span and height, __r__ (25) = .87,
__p__ < .05.

If the absolute value of your correlation
coefficient were less than .381, you would fail to reject your null hypotheses: There is
not a statistically significant relationship between arm span and height, __r__ (25) =
.12, __p__ > .05.

Note: The number in parentheses following the r is
the degrees of freedom and the number following the equal sign is your correlation
coefficient __r__. __p__<.05 means your correlation coefficient exceeded the
critical value found on the table and you are 95% confident that a relationship exists. __p__
> .05 means that your correlation coefficient was less than the critical value on the
table and you cannot be 95% confident that a relationship exists.

When using the critical value table, use the
absolute value of your __r__ (in other words...ignore the negative sign of your __r__
if you have a negative relationship). The sign tells the DIRECTION of the
relationship...not the STRENGTH. Since we are wondering if there is a strong enough
relationship to be statistically significant, we

are only concerned about strength when using the table. An __r__ = -.85 has the same
strength as __r__ = .85. You

do need to report the direction in your answer and must place the negative sign in front
of the __r__ value.

If your degree of freedom is not on the correlation table, go to the next lowest degree of freedom (df) that is.

Del Siegle,
Ph.D.

Neag School of Education - University of Connecticut

del.siegle@uconn.edu

www.delsiegle.com

Last revised 10/14/2009