The Fisher Z transformation is a formula we can use to transform Pearson’s correlation coefficient (r) into a value (zr) that can be used to calculate a confidence interval for Pearson’s correlation coefficient.
The formula is as follows:
zr = ln((1+r) / (1-r)) / 2
For example, if the Pearson correlation coefficient between two variables is found to be r = 0.55, then we would calculate zr to be:
- zr = ln((1+r) / (1-r)) / 2
- zr = ln((1+.55) / (1-.55)) / 2
- zr = 0.618
It turns out that the sampling distribution of this transformed variable follows a normal distribution.
This is important because it allows us to calculate a confidence interval for a Pearson correlation coefficient.
Without performing this Fisher Z transformation, we would be unable to calculate a reliable confidence interval for the Pearson correlation coefficient.
The following example shows how to calculate a confidence interval for a Pearson correlation coefficient in practice.
Example: Calculating a Confidence Interval for Correlation Coefficient
Suppose we want to estimate the correlation coefficient between height and weight of residents in a certain county. We select a random sample of 60 residents and find the following information:
- Sample size n = 60
- Correlation coefficient between height and weight r = 0.56
Here is how to find a 95% confidence interval for the population correlation coefficient:
Step 1: Perform Fisher transformation.
Let zr = ln((1+r) / (1-r)) / 2 = ln((1+.56) / (1-.56)) / 2 = 0.6328
Step 2: Find log upper and lower bounds.
Let L = zr – (z1-α/2 /√n-3) = .6328 – (1.96 /√60-3) = .373
Let U = zr + (z1-α/2 /√n-3) = .6328 + (1.96 /√60-3) = .892
Step 3: Find confidence interval.
Confidence interval = [(e2L-1)/(e2L+1), (e2U-1)/(e2U+1)]
Confidence interval = [(e2(.373)-1)/(e2(.373)+1), (e2(.892)-1)/(e2(.892)+1)] = [.3568, .7126]
Note: You can also find this confidence interval by using the Confidence Interval for a Correlation Coefficient Calculator.
This interval gives us a range of values that is likely to contain the true population Pearson correlation coefficient between weight and height with a high level of confidence.
Note the importance of the Fisher Z transformation: It was the first step we had to perform before we could actually calculate the confidence interval.
Introduction to the Pearson Correlation Coefficient
The Five Assumptions for Pearson Correlation
How to Calculate a Pearson Correlation Coefficient by Hand