Statistics calculator

Correlation Coefficient Calculator

Calculate Pearson correlation from matched x-y pairs, inspect the complete working, and interpret positive, negative, weak, and undefined linear relationships.

Pearson correlation

Check paired observations before interpreting Pearson r

Pearson's correlation coefficient measures the direction and strength of a linear relationship. It ranges from -1 to 1. The sign gives the direction; the absolute value gives the strength of the linear association in the entered pairs.

The calculation uses matched x-y observations. A row cannot combine an x value from one subject, date, or item with a y value from another. Constant x or y values make the denominator zero, so the coefficient is undefined rather than zero.

A high absolute correlation does not establish causation. Plot the pairs and investigate outliers, restricted ranges, curved relationships, repeated measurements, and confounding variables before drawing a conclusion.

ReferenceValue or meaning
r = 1Perfect positive linear relationship
r = 0No linear relationship; a nonlinear pattern may still exist
r = -1Perfect negative linear relationship
r squaredShare of linear variance described in a simple two-variable model

Checks before using the result

  • Keep each x value paired with the y value from the same observation.
  • Inspect a scatterplot for curvature and influential outliers.
  • Report the number of pairs together with r.
  • Use a statistical package for p-values, confidence intervals, missing-data rules, or large datasets.

Sources and related tools

Statistics calculatorReview center and spread for each variable.Z-score calculatorStandardize an observation against a distribution.Confidence interval calculatorEstimate uncertainty around a sample mean.Sample size calculatorPlan data collection before analysis.
Formular = sum((x - mean x)(y - mean y)) / sqrt(sum(x - mean x)^2 sum(y - mean y)^2)

Formula

Pearson correlation

Correlation measures linear association, not causation.

r = sum((x - mean x)(y - mean y)) / sqrt(sum(x - mean x)^2 sum(y - mean y)^2)

Five x-y pairs can show positive, negative, or weak linear association.

AssumptionsReview the assumptions and exclusions that define what this calculator does and does not include.

Assumptions and exclusions

Calculation notes

Review the assumptions and exclusions that define what this calculator does and does not include.

Effective year
Current rules
Last verified
2026-09-18

Correlation measures linear association, not causation.

Assumptions

    Not included

      FAQ8 common questions for this calculator.

      FAQs

      What does r near 1 mean?+

      It means a strong positive linear relationship.

      What does r near 0 mean?+

      It means little linear relationship in the entered data, but a curved or other nonlinear relationship can still exist.

      Why is correlation undefined for constant values?+

      If every x value or every y value is the same, that variable has zero variation and the Pearson denominator is zero.

      Does correlation prove causation?+

      No. A strong correlation can reflect confounding, selection, time trends, reverse direction, or chance rather than a causal effect.

      How does the Correlation Coefficient Calculator calculate the result?+

      For the Correlation Coefficient Calculator, it uses the Pearson correlation: r = sum((x - mean x)(y - mean y)) / sqrt(sum(x - mean x)^2 sum(y - mean y)^2). Five x-y pairs can show positive, negative, or weak linear association.

      What information do I need to use the Correlation Coefficient Calculator?+

      For the Correlation Coefficient Calculator, enter matched x-y pairs, inspect the complete working, and interpret positive, negative, weak, and undefined linear relationships. Keep the units consistent with the calculator fields and compare your setup with the worked example on the page.

      How accurate is the Correlation Coefficient Calculator?+

      Correlation Coefficient Calculator calculates the selected statistic from the sample values provided. It uses the Pearson correlation, but missing data, outliers, sample bias, and distribution assumptions can affect what the result means.

      What should I check before using the Correlation Coefficient Calculator result?+

      For the Correlation Coefficient Calculator, check matched x-y pairs, inspect the complete working, and interpret positive, negative, weak, and undefined linear relationships, sample definition, missing values, outliers, and rounding precision. A statistic can be numerically correct but still describe a weak sample.

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      Statistics guide

      How to use the Correlation Coefficient Calculator

      Calculate Pearson correlation from matched x-y pairs, inspect the complete working, and interpret positive, negative, weak, and undefined linear relationships. The page also explains the pearson correlation and shows a practical example: Five x-y pairs can show positive, negative, or weak linear association.

      1. 1

        Enter your details

        Enter matched x-y pairs, inspect the complete working, and interpret positive, negative, weak, and undefined linear relationships, then complete any other fields shown in the calculator.

      2. 2

        Check the calculation

        Review the result alongside the pearson correlation: r = sum((x - mean x)(y - mean y)) / sqrt(sum(x - mean x)^2 sum(y - mean y)^2).

      3. 3

        Compare scenarios

        Change one or more inputs to see how they affect the correlation Coefficient Calculator result before you use the estimate.