Confidence Interval for a Correlation Coefficient

Methodology

This calculator uses the standard Fisher z transformation for a Pearson product-moment correlation coefficient. PASS documents a dedicated procedure for confidence intervals for Pearson's correlation, while nQuery includes a dedicated one-correlation confidence-interval procedure. The implementation here uses the conventional two-sided large-sample Fisher transformation interval.

1. Fisher z transformation

The observed correlation r is transformed to the approximately normally distributed Fisher scale:

z = atanh(r) = ½ ln[(1 + r) / (1 − r)]

For a sample of size n, the standard error of the transformed correlation is:

SE(z) = 1 / √(n − 3)

2. Construct the interval on the z scale

For a two-sided confidence level of 1 − α, the interval is formed using the corresponding standard-normal critical value:

z_lower = z − z_1−α/2 × SE(z)
z_upper = z + z_1−α/2 × SE(z)

3. Transform back to the correlation scale

The Fisher-scale limits are converted back to correlation coefficients using the hyperbolic tangent:

r_lower = tanh(z_lower)
r_upper = tanh(z_upper)

The resulting limits are generally asymmetric around the observed correlation because the Fisher transformation is nonlinear. The procedure is appropriate for the usual Pearson-correlation setting in which the large-sample Fisher approximation is reasonable.

Validation example

The PASS documentation gives a two-sided 95% confidence-interval example with n = 161 and observed Pearson correlation r = −0.9. Its reported confidence limits are approximately −0.926 and −0.866.

r = −0.900 · n = 161 · confidence level = 95%
Expected CI = [−0.926, −0.866]

The JavaScript implementation reproduces those limits to three decimal places.

References

Fisher, R. A. (1915). Frequency Distribution of the Values of the Correlation Coefficient in Samples from an Indefinitely Large Population. Biometrika, 10(4), 507–521.

NCSS, LLC. PASS 2026: Confidence Intervals for Pearson's Correlation, Procedure 801. The PASS documentation describes the Pearson product-moment correlation confidence-interval procedure and provides validation/output examples.

Statistical Solutions / Statsols. nQuery, table AOC12, "Confidence Interval for One Correlation." The current nQuery procedure listing identifies this as a dedicated correlation confidence-interval procedure.

Confidence Interval for a Correlation Coefficient

Correlation Information