Calculate the confidence interval for a correlation coefficient using sample correlation, sample size, and confidence level.

## Understanding Confidence Intervals for a Correlation Coefficient

A confidence interval for a correlation coefficient estimates the range within which the true population correlation coefficient is likely to fall, based on sample data and a chosen confidence level.

### Using Fisher's Z-Transformation (When Sampling Distribution is Unknown)

When the sampling distribution of the correlation coefficient \( r \) is unknown, Fisher's Z-transformation helps approximate normality by transforming \( r \) to a Z-score. This transformation allows us to use normal theory to calculate the confidence interval:

#### Steps to Compute the Confidence Interval

**Step 1:**Transform \( r \) to \( z' \) using Fisher's Z-transformation:$$ z' = \frac{1}{2} \ln\left(\frac{1 + r}{1 - r}\right) $$

**Step 2:**Calculate the standard error for \( z' \):$$ \sigma_{z'} = \frac{1}{\sqrt{n - 3}} $$

where \( n \) is the sample size.

**Step 3:**Determine the Z-score critical value for the desired confidence level. For example, for a 95% confidence level, use \( Z = 1.96 \).**Step 4:**Construct the confidence interval in \( z' \)-space:$$ CI_{z'} = z' \pm Z \cdot \sigma_{z'} $$

**Step 5:**Convert back to the correlation scale by applying the inverse Z-transformation:$$ CI_{r} = \left( \frac{e^{2 \cdot (z' - ME)} - 1}{e^{2 \cdot (z' - ME)} + 1}, \frac{e^{2 \cdot (z' + ME)} - 1}{e^{2 \cdot (z' + ME)} + 1} \right) $$

where \( ME \) is the margin of error calculated in \( z' \)-space.

This method allows us to compute a confidence interval for \( r \) even when its exact sampling distribution is unknown.

### Example Calculation Using Fisher's Transformation

Suppose you have a sample correlation coefficient \( r = 0.6 \) from a sample of \( n = 200 \) and you want a 95% confidence interval:

**Step 1:**Apply the Fisher \( z \)-transformation: \[ z' = \frac{1}{2} \ln\left(\frac{1 + 0.6}{1 - 0.6}\right) = 0.6931 \]**Step 2:**Calculate the standard error of \( z' \): \[ \sigma_{z'} = \frac{1}{\sqrt{200 - 3}} = 0.071 \]**Step 3:**Determine the margin of error in \( z' \)-space: \[ ME = 1.96 \times 0.071 = 0.1392 \]**Step 4:**Construct the confidence interval in \( z' \)-space: \[ CI_{z'} = 0.6931 \pm 0.1392 = [0.5539, 0.8323] \]**Step 5:**Back-transform to get the confidence interval for \( r \): \[ CI_{r} = \left( \frac{e^{2 \cdot 0.5539} - 1}{e^{2 \cdot 0.5539} + 1}, \frac{e^{2 \cdot 0.8323} - 1}{e^{2 \cdot 0.8323} + 1} \right) = [0.504, 0.681] \]

This results in a 95% confidence interval for the population correlation coefficient of approximately \([0.504, 0.681]\).

### Calculating the CI with Known Sampling Distribution

If the sampling distribution of \( r \) is known, we can calculate the confidence interval directly using the distribution's properties. Knowing the sampling distribution of \( r \) provides two main advantages:

- We can apply critical values directly based on the exact distribution, rather than relying on approximations.
- We can account for any skewness, kurtosis, or other characteristics specific to the distribution, which improves the accuracy of the interval estimation.

#### Example Calculation with Known Sampling Distribution

Assume we have a large sample (e.g., \( n = 200 \)) and know the sampling distribution of \( r \) is approximately normal, with a mean of \( \rho \) (the population correlation) and standard error \( \sigma_r = \sqrt{\frac{1 - r^2}{n - 1}} \). If \( r = 0.6 \) and we want a 95% confidence interval:

**Step 1:**Calculate the standard error: \[ \sigma_r = \sqrt{\frac{1 - 0.6^2}{200 - 1}} = 0.035 \]**Step 2:**Use the Z-score for a 95% confidence level, which is \( Z = 1.96 \).**Step 3:**Calculate the margin of error: \[ ME = 1.96 \times 0.035 = 0.0686 \]**Step 4:**Construct the confidence interval: \[ CI = r \pm ME = 0.6 \pm 0.0686 = [0.5314, 0.6686] \]

This gives a 95% confidence interval of approximately \([0.5314, 0.6686]\) for the population correlation coefficient, using the known properties of the sampling distribution.

### Comparison of Confidence Intervals

In the two examples, the confidence intervals differ slightly due to the different approaches:

- Using Fisher’s Z-transformation, we obtained a confidence interval of \([0.504, 0.681]\). This method relies on approximating normality through a transformation since the exact sampling distribution is unknown.
- With a known sampling distribution, we calculated a narrower interval of \([0.5314, 0.6686]\). This is because the exact distribution allows for more precise critical values, avoiding the need for approximation through transformation.

Knowing the sampling distribution generally results in a more accurate and often narrower confidence interval, as it reflects the actual characteristics of the data without relying on transformation approximations.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.