# Correlation Calculator

> Pearson product-moment linear correlation of paired observations. No significance test or causation claim.

Page: https://freecalculator.tech/tools/pearson-correlation-calculator
Last reviewed: 2026-10-23

## Formula

r = Σ((xᵢ − x̄)(yᵢ − ȳ)) ÷ √(Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²).

- Methodology type: scientific equation.
- Methodology: NIST Pearson product-moment linear correlation, not Spearman rank correlation.
- No p-value or normal-distribution inference is calculated. Nonlinear association may exist even when r is near zero.
- r = Σ((xᵢ − x̄)(yᵢ − ȳ)) ÷ √(Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²).
- Inputs used: Observations, Paired values y.

## Inputs

- Observations
- Paired values y

## Worked example: Pearson r

Sxy = 4, Sxx = 2, Syy = 8; r = 4 ÷ √16 = 1.
1. Sxy = 4, Sxx = 2, Syy = 8; r = 4 ÷ √16 = 1.

## Common questions

### Is this Spearman correlation?

No. Pearson r uses numeric values rather than ranks.

### Why reject a constant list?

Its variance is zero, so the correlation denominator is zero. Check the formula, example, and limitations on this page before using the result for a real statistics decision.

### Does r = 0 mean no relationship?

No. It indicates no linear correlation; a nonlinear relationship may remain.

### How do I use the Correlation Calculator?

Enter the required values for Observations, Paired values y. The calculator applies the formula on this page and shows the main result with any supporting breakdown so you can check the arithmetic.

### What formula does the Correlation Calculator use?

r = Σ((xᵢ − x̄)(yᵢ − ȳ)) ÷ √(Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²). The visible formula section above lists the calculation path and the edge cases the page handles, so the result can be checked without relying on the form alone.

### Can the Correlation Calculator be used for exact decisions?

Use it as a calculation aid, not as a substitute for checking the underlying rule, contract, policy, or professional advice that applies to your situation. When a result depends on local rules, personal details, prices, or dates, enter those values yourself and confirm them before acting.

## Limitations and assumptions

- Paired finite values; sensitive to outliers.
- No significance test or causation claim.
- For education and operational analysis only. No diagnosis, treatment recommendations, prescribing, dosing instructions, or reimbursement entitlement. Scientific calculations require appropriate laboratory verification; operational estimates do not replace local clinical, safety or financial policies.
- The result depends on the values you enter for this correlation calculator; it does not supply missing rates, rules, prices, dates, or assumptions for you.

## Sources

- NIST — https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/corr.htm

## Cite this page

- APA: Free Calculator. (2026, October 23). Correlation Calculator. https://freecalculator.tech/tools/pearson-correlation-calculator
- MLA: "Correlation Calculator." Free Calculator, 23 Oct. 2026, https://freecalculator.tech/tools/pearson-correlation-calculator.
- Link: Correlation Calculator — Free Calculator, https://freecalculator.tech/tools/pearson-correlation-calculator