Correlation Confidence Interval Calculator (Fisher z Transformation)
A confidence interval for the true correlation behind a sample r, by Fisher's z transformation, with an approximate p-value for r = 0 and the shared variance r².
Results
What this tool does
A correlation measured on a sample is only an estimate, and its error is lopsided: near ±1 the possible values bunch up against the limit. Fisher's transformation z′ = atanh r makes the error roughly normal with a standard error of 1 ÷ √(n − 3), so the interval is built there and mapped back with tanh. With r = 0.62 from 30 pairs the true value could plausibly lie anywhere from about 0.33 to 0.80. The p-value uses the same normal approximation and is a little different from the exact t-test.
Formula
z′ = atanh r = ½ ln((1 + r) ÷ (1 − r)) · SE = 1 ÷ √(n − 3) · interval = tanh(z′ ± z × SE)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
r | Sample correlation r | — |
n | Number of pairs (n) | — |
cl | Confidence level | — |
LO | Interval for the true correlation, lower end | — |
HI | Interval for the true correlation, upper end | — |
ZF | Fisher z (atanh r) | — |
SE | Standard error of z | — |
PV | p-value for r = 0 (two-sided, approximate) | — |
R2 | Shared variance (r²) | % |
Worked example
- Sample correlation r0.62
- Number of pairs (n)30
- Confidence level95
- Interval for the true correlation, lower end0.334
- Interval for the true correlation, upper end0.801
- Fisher z (atanh r)0.7250
- Standard error of z0.1925
- p-value for r = 0 (two-sided, approximate)0.0002
- Shared variance (r²)38.4 %
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Standard deviation and variance are calculated for a sample (dividing by n − 1). For a full population, divide by n instead.