A correlation coefficient without a confidence interval is a point estimate pretending to be a conclusion. Use this page to get the interval, read it correctly, and copy an APA-ready report.
The interval shows the range where the true population correlation likely lives. Wider intervals mean less precision.
Two studies can share the same r value but have very different certainty depending on sample size. CI exposes that difference immediately.
We are 95% confident that the true population correlation lies between 0.250 and 0.766.
Width of CI: 0.516. The estimate is usable, but a larger sample would narrow the interval.
Correlation strength: moderate.
r(28) = .56, 95% CI [.25, .77], p = .001Same point estimate. Three very different confidence intervals. That is the hidden information a point estimate cannot show.
CI includes 0, so no-correlation remains plausible.
CI stays positive, so the result is much more reliable.
CI is tight, so the population estimate is precise.
r is bounded between -1 and +1, so its sampling distribution becomes skewed near the edges. Fisher's z transformation stretches the scale near ±1 and makes the distribution behave much more like a normal curve.
| r | z |
|---|---|
| 0.00 | 0.000 |
| 0.30 | 0.310 |
| 0.50 | 0.549 |
| 0.70 | 0.867 |
| 0.90 | 1.472 |
| 0.95 | 1.832 |
| 0.99 | 2.647 |
r = 0.10 produces an approximately symmetric sampling distribution.
r = 0.90 produces a skewed distribution that needs Fisher z.
Fisher z stretches the bounded correlation scale so we can apply a normal confidence interval safely. That is why the interval is computed in z space first and only then converted back to r.
Hold r at 0.50 and move the sample size. The interval width shrinks fast at first, then more slowly as n becomes large.
A confidence interval for a correlation coefficient is a range that estimates where the true population correlation likely sits. The point estimate r gives one number. The interval tells you how uncertain that number is.
| Interval pattern | What it means |
|---|---|
| CI entirely positive | The population correlation is likely positive and statistically significant at the matching alpha level. |
| CI includes 0 | The data do not rule out no correlation. The estimate may still be useful, but it is not definitive. |
| CI is very wide | The sample does not estimate the population correlation precisely enough for a strong conclusion. |
The calculation happens in Fisher z space because z is approximately normally distributed. After the interval is built, the bounds are transformed back to the correlation scale.
| Confidence | z critical |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
The final interval can be asymmetric around r. That asymmetry is correct, especially when r is far from 0.
Suppose a study reports , , and a 95% confidence level.
The entire interval is positive. You can be confident that the population correlation is positive, and the result is statistically significant at alpha = .05 for a 95% CI.
The interval includes no correlation. You cannot rule out a zero or even opposite-direction population effect, so the result is not statistically significant at the matching alpha level.
The estimate is precise. Your sample is large enough to locate the population correlation within a small practical range.
A p-value tells you whether to reject . A confidence interval tells you where the true effect likely lives. Journals increasingly expect both because the interval carries more information.
| Aspect | p-value | Confidence interval |
|---|---|---|
| Question answered | Is this correlation statistically significant? | Where might the true population correlation be? |
| Output | A single probability, such as p = .023 | A range, such as [.25, .77] |
| Information carried | Mostly a reject/do-not-reject decision | Direction, strength, precision, and significance |
| Equivalent rule | p < .05 | 95% CI does not include 0 |
Report the degrees of freedom, r value, confidence interval, and p-value. For a correlation, .
A Pearson correlation was conducted to assess the
relationship between [Variable X] and [Variable Y]
(n = 30). There was a strong positive correlation
between the two variables, r(28) = .56, 95% CI
[.25, .77], p = .001.Wrong: 95% of possible r values fall between .25 and .77.
Right: We are 95% confident that the population correlation lies between .25 and .77.
Wrong: CI = r +/- margin of error.
Right: Build the interval in Fisher z space, then transform back to r. The final interval can be asymmetric.
Wrong: The CI excludes 0, so the estimate is good enough.
Right: A CI like [.01, .89] technically excludes 0, but it is too wide for a precise conclusion.
Wrong: The same Fisher z formula is fine for any n.
Right: Use at least n >= 5, and prefer n >= 10. For very small samples, bootstrap methods are safer.
A confidence interval for a correlation coefficient is a range of values that likely contains the true population correlation, based on your sample r and sample size n. A 95% CI means that repeated studies would produce intervals containing the true correlation about 95% of the time.
The sampling distribution of r is skewed, especially when r is close to -1 or +1. Fisher's z transformation converts r into a statistic that is approximately normal, so standard normal confidence interval formulas work better.
Larger samples produce narrower intervals. With r = .50, a small sample can leave the interval crossing 0, while a large sample can narrow the range enough to support a precise conclusion.
No. It means the result is not statistically significant at alpha = .05, often because the sample is too small. The point estimate still provides directional information, but the study is not conclusive.
Yes, Fisher's z transformation is often used as an approximation for Spearman's rho, especially with larger samples. For small samples or many tied ranks, bootstrap confidence intervals can be more reliable.
A confidence interval is one part of the correlation workflow. These related tools help you calculate, test, interpret, and plan the full study.