# Correlation Coefficient Calculator

Free Pearson correlation coefficient calculator. Calculate r and R² from summary statistics to measure the strength and direction of linear relationships.

## What this calculates

Calculate the Pearson correlation coefficient (r) to measure the strength and direction of the linear relationship between two variables. Also computes R-squared and tests for significance.

## Inputs

- **Number of Data Points (n)** — min 3 — Total number of paired observations.
- **Sum of X (Σx)** — Sum of all x values.
- **Sum of Y (Σy)** — Sum of all y values.
- **Sum of XY (Σxy)** — Sum of the products of each x and y pair.
- **Sum of X² (Σx²)** — Sum of each x value squared.
- **Sum of Y² (Σy²)** — Sum of each y value squared.

## Outputs

- **Pearson r** — The Pearson correlation coefficient.
- **R²** — Coefficient of determination.
- **Strength** — formatted as text — Interpretation of the correlation strength.
- **Direction** — formatted as text — Whether the correlation is positive or negative.
- **T-Statistic** — T-statistic for testing significance of r.
- **Formula** — formatted as text — The correlation formula and calculation.

## Details

The Pearson correlation coefficient (r) ranges from -1 to +1.

Formula:
r = (nΣxy - ΣxΣy) / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]

Interpretation

- r = +1: Perfect positive linear relationship

- r = -1: Perfect negative linear relationship

- r = 0: No linear relationship

- |r| > 0.7: Strong correlation

- |r| 0.3-0.7: Moderate correlation

- |r| < 0.3: Weak correlation

Important: Correlation does not imply causation.

## Frequently Asked Questions

**Q: What is the difference between r and R²?**

A: r (Pearson correlation coefficient) measures the strength and direction of a linear relationship (-1 to +1). R² (coefficient of determination) is r squared and represents the proportion of variance in one variable explained by the other (0 to 1). For example, r = 0.8 means R² = 0.64, so 64% of the variance is explained.

**Q: Does correlation imply causation?**

A: No. A strong correlation between two variables does not mean one causes the other. There may be a confounding variable, the relationship may be coincidental, or the causation may be reversed. Controlled experiments are needed to establish causation.

**Q: When is Pearson correlation not appropriate?**

A: Pearson correlation assumes a linear relationship and continuous data. If the relationship is non-linear, use Spearman's rank correlation instead. Pearson r is also sensitive to outliers, which can inflate or deflate the correlation.

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Source: https://vastcalc.com/calculators/statistics/correlation
Category: Statistics
Last updated: 2026-04-21