Correlation Confidence Interval Calculator
Enter two aligned percentage-point return series to calculate Pearson correlation and a Fisher normal-approximation confidence interval. The page exposes centered sums, the Fisher transform and every paired cross-product while withholding dependence, causality, predictive-edge, hedge-ratio, forecast and trading verdicts.
Enter two aligned return series
Each X row must match the Y row from the same observation period and preprocessing convention.
Two-sided confidence from 50% through 99.9%.
Enter 5 to 500 values without percent signs; one complete aligned observation per row.
The Y row count and order must match X exactly.
Pearson and Fisher arithmetic
Entered Return Relationship Diagnostics 1.0.0.
On smaller screens, scroll horizontally to inspect the complete audit table.
| Row | X | Y | X deviation | Y deviation | Cross-product |
|---|
How the correlation confidence interval is calculated
z = atanh(r), SEz = 1 ÷ √(n − 3)
Interval = tanh[z ± Φ−1(1 − α/2) × SEz]
The calculator mean-centers X and Y separately, multiplies each row’s two deviations and adds those cross-products. Dividing that centered cross-sum by the square root of the two centered sums of squares produces the sample Pearson coefficient.
For a coefficient strictly between minus one and one, the Fisher inverse-hyperbolic-tangent transform moves the bounded correlation scale onto an approximately normal scale. Its standard error is one divided by the square root of n minus three.
The selected two-sided confidence level supplies a standard-normal critical value. The model adds and subtracts its z-scale margin, then applies the hyperbolic tangent to return both limits to the correlation scale. Exact minus-one or plus-one sample correlation remains an explicit boundary case.
Worked example from the audited fixture
The audited fixture contains 10 aligned X and Y percentage-point returns. Centered sums are SXX = 9.705, SYY = 4.304 and SXY = 6.34.
- Pearson correlation is 0.98096917. Its Fisher transform is 2.32264048, the Fisher standard error is 0.37796447 and the 95% normal critical value is 1.95996398.
- Transforming the two z-scale limits back gives a 95% interval from 0.91888934 to 0.99564272. This interval is conditional on the sample and assumptions; it is not a forecast that the next returns will move together.
Reproduce it: select “Load audited example” above. The governed engine retains full precision and rounds only the visible interface.
How to interpret the result
- The sample coefficient describes the direction and strength of linear association in the entered paired rows only.
- The two bounds express repeated-sampling uncertainty for population correlation under the Fisher approximation; they are not probabilities assigned to these realized endpoints.
- Higher confidence widens the interval when the observations remain unchanged.
- More genuinely comparable independent pairs usually narrow the approximation, while duplicated or overlapping rows do not create independent evidence.
- A positive interval does not prove a stable, causal, tradeable or hedge-worthy relationship.
Assumptions and limits
- Enter 5 to 500 aligned equal-frequency percentage-point observations under consistent source, timezone, return and cost conventions.
- Pearson correlation detects linear association and can miss nonlinear dependence.
- The Fisher approximation is sensitive to non-normality, serial dependence, regime changes, heavy tails and outliers.
- The calculator cannot verify alignment, independence, stationarity, source completeness or whether the sample was selected after viewing results.
- Correlation can change materially across windows and does not imply causation.
- No dependence verdict, predictive edge, hedge ratio, forecast, grade, signal, position instruction or recommendation is generated.
Which relationship diagnostic answers which question?
Pearson correlation, partial correlation, directional regression and a historical multi-pair matrix describe different evidence. The comparison keeps direction, control variables, uncertainty and source boundaries visible instead of presenting one coefficient as a universal dependence or strategy-validation score.
| Tool | Evidence entered | Primary output | Main boundary |
|---|---|---|---|
| Correlation Confidence Interval | Two aligned series | Symmetric Pearson coefficient plus Fisher limits | No causal, stable-dependence or forecast verdict. |
| Partial Correlation | X, Y and one control Z | Residual X–Y linear association after controlling Z | One linear control does not remove all confounding. |
| Linear Regression | Predictor X and response Y | Directional slope, intercept, residual scale and coefficient intervals | No alpha/beta label, extrapolation or hedge advice. |
| Historical Correlation Matrix | Governed multi-pair market history | Many pairwise historical coefficients | Source window and currency coverage differ from manual inference. |
Frequently asked questions
- It estimates a Fisher normal-approximation interval for population Pearson correlation from one manually entered aligned return sample.
- The model divides the centered X-Y cross-product sum by the square root of the two centered sums of squares.
- The inverse-hyperbolic-tangent transform moves a correlation strictly between minus one and one onto an approximately normal scale for interval construction.
- The Fisher transform is infinite at those boundaries, so version 1.0.0 retains the exact coefficient as a disclosed point interval instead of emitting infinity.
- Yes. Each X row must correspond meaningfully to the Y row from the same observation period and preprocessing convention.
- No. Pearson correlation measures entered linear association and cannot establish causal direction, economic mechanism or a stable future relationship.
- No. The result does not account for position size, value sensitivity, costs, regime stability or execution and generates no hedge ratio.
- No. It creates no dependence label, predictive-edge conclusion, forecast, grade, signal, position instruction or recommendation.
Sources and methodology
- NIST — Correlation — Published centered-sum Pearson product-moment formula.
- NIST — Correlation Confidence Limits — Published Fisher normal-approximation method and worked output table.
Version 1.0.0 was locked only after the formulas and assumptions were checked against the cited NIST material. Canonical fixtures were independently recomputed with SciPy 1.13.1 before comparison with the browser engine. The calculator performs arithmetic locally and does not upload the entered observations.
Continue the return-association review
Verify the paired return evidence before interpreting relationships
Reconcile the exact statement window, sampling frequency, timestamps, missing rows, realized P&L, spread, commission, financing, currency conversion and preprocessing across every series. Correct relationship arithmetic cannot repair mismatched, selected or cost-inconsistent evidence.
Risk and affiliate disclosure: Leveraged forex and CFD trading can result in substantial losses. These are affiliate links, so ForexMT4Indicators.com may receive compensation if you register or trade through them, at no additional cost to you. Availability and terms vary by jurisdiction and broker entity.