Statistics

Correlation Coefficient Calculator

What this does

Compute Pearson’s r for paired data and get a plain-language strength and direction interpretation.

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Calculator inputs

How to use it

Using the correlation coefficient calculator

  1. 01

    Paste your pairs

    One observation per line as “x,y”.

  2. 02

    Read r and its sign

    Direction first: positive or negative association.

  3. 03

    Then judge strength

    The interpretation row translates |r| into plain language, from negligible to very strong.

Good to know

Calibration for interpreting r

Context sets the bar: |r| = 0.3 is respectable in social science, weak in physics. Common anchors call |r| ≥ 0.9 very strong, 0.7 strong, 0.5 moderate, 0.3 weak; but always compare against typical effect sizes in your field.

Correlation’s classic blind spots

  • Non-linearity: perfect quadratic dependence can yield r ≈ 0.
  • Outliers: one extreme point can manufacture or destroy correlation.
  • Restriction of range: sub-sampling can deflate r dramatically.
  • Lurking variables: ice cream sales correlate with drownings via summer heat.

How it's calculated

The math behind this calculator

r = Σ(xᵢ−x̄)(yᵢ−ȳ) / √(Σ(xᵢ−x̄)² · Σ(yᵢ−ȳ)²)

Pearson’s r standardizes the covariance of x and y by both standard deviations, producing a pure number between −1 and 1. Positive r means the variables rise together; negative means one falls as the other rises; magnitude encodes tightness of the linear relationship.

We also report R²; the square of r; which reads directly as the fraction of variance in one variable explained by the other under a linear model. Correlation is undefined when either variable is constant, since division by zero variance results.

Assumptions & limitations

  • One “x,y” pair per line; at least two pairs.
  • Both variables must vary (no constant columns).
  • r captures linear association only; curved relationships can hide behind a low r.

Worked example

Points (1,2), (2,3.9), (3,6.1), (4,8) track a near-perfect upward line, giving r ≈ 0.9996; a very strong positive correlation.

FAQ

Frequently asked questions

Does r = 0 mean no relationship?
It means no linear relationship. Strong curved, clustered or categorical patterns can coexist with r near zero; plot your data.
Why is correlation capped at ±1?
It is standardized covariance: ±1 corresponds exactly to all points lying on a straight line, the strongest linear association possible.
Is r affected by changing units?
No. Measuring in kilograms versus pounds leaves r unchanged; that unit-free property is its main advantage over covariance.
How many pairs do I need?
Mathematically two suffice (r = ±1 trivially); practically aim for at least 10–20 before trusting the estimate.

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