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.

**Additional Resources**

Introduction to the Pearson Correlation Coefficient

The Five Assumptions for Pearson Correlation

How to Calculate a Pearson Correlation Coefficient by Hand