204 PART 5 Looking for Relationships with Correlation and Regression
correlation coefficient r, based on N observations, by the formula
t
r
r
N
/
/
1
2
2
. Because t fluctuates in accordance with the Student t dis-
tribution with N
2 degrees of freedom (df), it is useful for statistical testing (see
Chapter 11 for more about t).
For example, if r
0 500
.
for a sample of 12 participants, then t
0 5. /
1
0 5
12
2
2
.
/
, which works out to t
1 8257
.
, with 10 degrees of freedom. You
can use the online calculator at https://statpages.info/pdfs.html and calcu-
late the p by entering the t and df values. You can also do this in R by using
the code:
2 * pt(q = 1.8257, df = 10, lower.tail = FALSE).
Either way, you get p
0 098
.
, which is greater than 0.05. At α = 0.05, the r value
of 0.500 is not statistically significantly different from zero (see Chapter 12 for
more about α).
How precise is an r value?
You can calculate confidence limits around an observed r value using a somewhat
roundabout process. The quantity z, calculated by the Fisher z transformation
z
r
r
1
2
1
1
log
/
, is approximately normally distributed with a standard
deviation of 1
3
/ N
. Therefore, using the formulas for normal-based confidence
intervals (see Chapter 10), you can calculate the lower and upper 95 percent con-
fidence limits around z z
z
N
:
.
Lower
/
1 96
3 and Z
z
N
Upper
/
1 96
3
.
. You
can turn these into the corresponding confidence limits around r by the reverse of
the z transformation: r
e
e
x
z
2
2
1
1
/
for z
zLower and z
zUpper.
Here are the steps for calculating 95 percent confidence limits around an observed
r value of 0.05 for a sample of 12 participants (N = 12):
1.
Calculate the Fisher z transformation of the observed r value:
z
1
2
1
0 5
1
0 5
0 549
log
.
.
.
/
2.
Calculate the lower and upper 95 percent confidence limits for z:
z
z
Lower
Upper
/
/
0 549
1 96
12
3
0 104
0 549
1 96
12
3
1 203
.
.
.
.
.
.