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Correlation
Prerequisites
Values
of the Pearson Correlation, Sampling
Distribution of Pearson's r, Confidence
Intervals
Learning Objectives
- State the standard error of z'
- Compute a confidence interval on ρ
The computation of a confidence interval on the
population value of Pearson's correlation (ρ) is complicated
by the fact that the sampling distribution of r is not normally
distributed. The solution lies with Fisher's z' transformation
described in the section on the sampling
distribution of Pearson's r. The steps in computing a confidence
interval for ρ are:
- Convert r to z'
- Compute a confidence interval in terms of z'
- Convert the confidence interval back to r.
Let's take the data from the case study Animal
Research as an example. In this study, students were asked
to rate the degree to which they thought animal research is wrong
and the degree to which they thought is was necessary. As you
might have expected, there was a negative relationship between
these two variables: the more a student thinks animal research
is wrong the less they think it is necessary. The correlation
based on 34 observations is -0.654. The problem is to compute
a 95% confidence interval on ρ based on this r of -.654.
The conversion of r to z' can be done using a table
or calculator.
The table contains only positive value of r, but that is not a
problem. The value of z' associated with an r of 0.654 is 0.78.
Therefore, the z' associated with an r of -0.654 is -0.78.
The sampling distribution of z' is approximately
normally distributed and has a standard error of
For this example, N = 34 and therefore the standard
error is 0.180. The Z for a 95% confidence interval (Z.95) is
1.96 as can be found using the normal
distribution calculator (setting the shaded area to .95 and
clicking on the "Between" button). The confidence interval
is therefore computed as:
Lower limit = -0.78 - (1.96)(0.18)= -1.13
Upper limit = -0.78 + (1.96)(0.18)= -0.43
The final step is to convert the endpoints of the
interval back to r using a table
or calculator.
The r associated with a z' of -1.13 is -0.81 and the r associated
with a z' -0.43 is -0.40. Therefore, the population correlation
(ρ) is likely to be between -0.81 and -0.40. The 95% confidence
interval is:
-0.81≤ ρ ≤ -0.40
To calculate the 99% confidence interval, you
use the Z for a 99% confidence interval of 2.58 as follows:
Lower limit = -0.775 - (2.58)(0.18)= -1.24
Upper limit = -0.775 + (2.58)(0.18)= -0.32
Converting back to r, the confidence interval
is:
-0.84≤ ρ ≤ -0.31
Naturally, the 99% confidence interval is wider
than the 95% confidence interval.
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