Correlation Confidence Interval Calculator
Uses Fisher’s z transformation to form an approximate interval for a Pearson correlation. The example keeps the method and inputs visible so the result can be checked independently.
Set the comparison values
Correlation confidence interval
Which values belong in the fields during independent review
Check that counts are whole observations, scales refer to the same measurement, and standard errors or deviations come from the population or sample named on the page. A percentage and a proportion differ by a factor of 100. The labeled fields make the assumption auditable.
If a critical value is entered, it must match the intended tail convention and reference degrees of freedom. Changing confidence level without changing that value creates a mislabeled result. This check belongs before rounding.
Before accepting correlation confidence interval, compare the result with the scale of the raw measurement or event rate. A numerically small difference can matter on a tightly controlled scale, while a larger difference may be uninformative when ordinary variation is much wider. The substantive benchmark belongs beside the statistical calculation.
A boundary-case check
Recalculate one intermediate quantity from tanh(atanh(r) ± z*/√(n−3)) and then work backward from the displayed endpoint or statistic. This catches swapped groups, reversed quantiles, and copied denominators. The report should state this boundary plainly.
Vary one credible input while holding the rest fixed. The direction and size of the change should agree with the formula before the result is carried into a report. That distinction remains visible in the worked case.
For correlation confidence interval, a useful audit begins with the numerator and denominator rather than the final display. Write the observed quantity, its reference value, and the uncertainty term on separate lines. That layout makes a misplaced square root, reversed group order, or percentage-scale error visible before rounding.
Coverage, evidence, and context under the stated design
A confidence interval describes a procedure’s long-run coverage under its assumptions; it is not the probability that this fixed interval contains the parameter. This point matters before the result enters another model.
Practical importance requires the effect size, measurement scale, uncertainty, and consequences of a decision. A threshold crossing by itself does not supply that context. The answer should retain that convention.
A method choice made from design in the worked condition
Sparse cells, strong skew, influential observations, clustering, pairing, estimated nuisance parameters, or unequal variances can change the reference distribution. The method assumes independent paired observations and is not robust to influential outliers or nonlinear association. That is a design choice, not a display setting.
Do not choose among methods by selecting the answer that looks most favorable. Choose from the data-generating design, then preserve the method name and convention. A reviewer should not have to infer that choice.
A complete statistical record before the result is reused
Keep the raw counts or summaries, units, group order, exclusions, formula version, and unrounded output. For correlation confidence interval, another analyst should be able to reconstruct the same numerical result.
Round only after downstream calculations are finished. Extra display digits cannot restore precision absent from the measurements or correct selection and measurement bias. The worked values provide a baseline for the comparison.
Two coherent versions of the analysis during independent review
Create a second scenario that changes one uncertain input rather than mixing optimistic values from unrelated cases. Compare both the center and the uncertainty or test statistic. This is where a group-order error is easiest to catch.
If the interpretation reverses under a small defensible change, report that sensitivity. It is more informative than presenting one apparently exact interval result. A changed sample requires the same check again.
The target parameter
Uses Fisher’s z transformation to form an approximate interval for a Pearson correlation. The displayed result follows tanh(atanh(r) ± z*/√(n−3)), with every symbol tied to a labeled input. This condition can be checked without relying on the final display.
For r=0.42 and n=80, the approximate 95% interval is 0.22 to 0.59. This worked condition is a reproducible arithmetic check, not evidence that the model fits every dataset. This prevents a plausible number from carrying the wrong meaning.
The same source summary can also support standard deviation confidence interval, regression slope confidence interval, and variance confidence interval.
Assumptions carried by the formula under the stated design
The method assumes independent paired observations and is not robust to influential outliers or nonlinear association. That step separates arithmetic from interpretation.
The unit of analysis, sampling frame, dependence structure, and treatment of missing values remain outside the final number. Record those choices before interpreting this interval. That choice determines which comparison is defensible.
Checks for this analysis
When the result is reused, what does the reported p-value mean?
It describes how unusual this statistic or a more extreme one would be under the stated null model; it is not the probability that the null is true. For this page, the reported quantity is correlation confidence interval.
When the sample changes, why can another program give a different answer?
Tail conventions, critical values, continuity corrections, treatment of ties, and numerical approximations can differ. Preserve the stated method with the result. For this page, the reported quantity is correlation confidence interval.
When should correlation confidence interval be repeated?
Repeat it when an input, exclusion, group definition, confidence level, tail choice, or model assumption changes. For this page, the reported quantity is correlation confidence interval.
With the stated model, how many digits should be reported?
Retain guard digits during checking, then round to a level justified by the source measurement and the decision that follows. For this page, the reported quantity is correlation confidence interval.
In a reproducible analysis, can a missing value be entered as zero?
Only when zero was observed. Missingness and a measured zero have different statistical meanings. For this page, the reported quantity is correlation confidence interval.
When software results differ, does a narrow interval prove the estimate is unbiased?
No. Precision under a model does not repair selection, measurement, nonresponse, or specification bias. For this page, the reported quantity is correlation confidence interval.