Fisher Z-Transformation: Definition & Example

The Fisher Z transformation is a formula we can use to transform Pearson’s correlation coefficient (r) into a value (z_r) that can be used to calculate a confidence interval for Pearson’s correlation coefficient.

The formula is as follows:

z_r = 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 z_r to be:

z_r = ln((1+r) / (1-r)) / 2
z_r = ln((1+.55) / (1-.55)) / 2
z_r = 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 z_r = ln((1+r) / (1-r)) / 2 = ln((1+.56) / (1-.56)) / 2 = 0.6328

Step 2: Find log upper and lower bounds.

Let L = z_r – (z_1-α/2 /√n-3) = .6328 – (1.96 /√60-3) = .373

Let U = z_r + (z_1-α/2 /√n-3) = .6328 + (1.96 /√60-3) = .892

Step 3: Find confidence interval.

Confidence interval = [(e^2L-1)/(e^2L+1), (e^2U-1)/(e^2U+1)]

Confidence interval = [(e^2(.373)-1)/(e^2(.373)+1), (e^2(.892)-1)/(e^2(.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.