StatReason

Correlation Confidence Interval Calculator

A single correlation coefficient from a sample is just a point estimate. This calculator gives it a confidence interval using the Fisher z transform, the standard method that accounts for the fact that r's own sampling distribution is not symmetric.

Already have raw paired data instead of a known r? Use the Pearson Correlation Calculator, which reports both r and this same interval.

What this answers

This calculator answers "given my sample's correlation, what range of true population correlations is plausible?" A correlation of exactly .50 in one sample and .50 in a much larger sample carry very different amounts of certainty, and this interval makes that difference explicit.

How it is calculated

Pearson's r is not symmetrically distributed, especially as it approaches negative 1 or 1, so a plain normal interval on r itself would perform poorly. The Fisher z transform maps r onto a scale where the sampling distribution is approximately normal with a known standard error of 1 divided by the square root of n minus 3. The interval is built in that transformed z space using a standard normal critical value, then converted back to the r scale with the inverse hyperbolic tangent (tanh) function, which is why the resulting interval is not symmetric around r itself even though it is symmetric in z.

Worked example

For r equal to .5 with n equal to 30, the Fisher z value is about 0.549, with a standard error of 1 divided by the square root of 27, about 0.192. At 95% confidence, the z interval runs from about 0.172 to 0.926, which converts back through tanh to an r interval of roughly .17 to .73. Notice this interval is noticeably wider on the low side than a naive symmetric guess would suggest, exactly the asymmetry the Fisher transform is built to handle correctly.

Assumption audit

Calculated from your data: whether n is large enough (greater than 3) for the Fisher transform to be defined, and whether r sits strictly between negative 1 and 1, since the transform is undefined exactly at those bounds.
Evidence to review: whether the underlying relationship between your two variables is actually linear and free of extreme leverage points, since this interval inherits every assumption of the Pearson correlation it is built around.
You must verify: that your pairs were sampled independently of one another. Repeated measurements on the same subjects, or pairs drawn from a clustered design, will make this interval's stated coverage unreliable.

What this interval does not mean

A correlation confidence interval does not tell you whether the relationship is linear, causal, or driven by a third variable; it only quantifies sampling uncertainty around the specific linear-association number you supplied. A narrow interval around a small r still describes a weak relationship precisely, not a strong one.

Limitations

This calculator assumes the Pearson correlation model is appropriate for your data; if the relationship is strongly nonlinear or driven by a handful of outliers, consider the Spearman Rank Correlation Calculator instead, whose interval this page does not provide.