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Correlation Coefficient

General

Pearson Correlation Coefficient (r)

A value between -1 and 1 (Pearson's r) that measures the strength and direction of a linear relationship between two variables.

Definition

The correlation coefficient, most commonly Pearson's r, is a value between -1 and 1 that measures the strength and direction of a linear relationship between two variables. A value near +1 indicates a strong positive relationship (as one variable increases, so does the other), a value near -1 indicates a strong negative relationship (as one increases, the other decreases), and a value near 0 indicates little to no linear relationship.

Correlation coefficients are widely used across finance, science, and social research to quantify how two variables move together, for example, the relationship between advertising spend and sales, or between temperature and ice cream sales. The Correlation Coefficient Calculator computes r directly from paired data points, and the closely related Covariance Calculator computes the unstandardized version of the same underlying relationship.

Correlation is also the foundation of simple linear regression: the Linear Regression Calculator fits a line through the data, and the square of the correlation coefficient gives the R-Squared value describing how much variance that line explains.

Formula

r = Σ[(x − x̄)(y − ȳ)] ÷ √[Σ(x − x̄)² × Σ(y − ȳ)²]

Equivalently, r can be expressed using covariance and standard deviations:

r = Cov(X, Y) ÷ (σₓ × σᵧ)

Worked Example

Consider five paired data points of hours studied (x) and test scores (y): (1, 50), (2, 60), (3, 65), (4, 80), (5, 85).

Calculating the means (x̄ = 3, ȳ = 68), the deviations, and applying the formula yields:

r ≈ 0.99

This indicates an extremely strong positive linear relationship, more hours studied is very strongly associated with higher test scores in this sample.

Key Things to Know

  • Only measures linear relationships: a strong curved (non-linear) pattern can produce a correlation coefficient near 0 even though the variables are clearly related.
  • Squares to give R-Squared: in simple linear regression, r² tells you the proportion of variance explained by the model.
  • Doesn't imply causation: high correlation only shows association, never proof that one variable causes changes in the other.
  • Sensitive to outliers: a single extreme data point can substantially inflate or deflate the correlation coefficient, especially in small samples.
  • Related to covariance: correlation is covariance standardized by the two variables' standard deviations, putting it on a fixed -1 to 1 scale for easy comparison.

Related Calculators

Correlation Coefficient Calculator →Calculate the Pearson correlation coefficient (r) between two variables in seconds. Enter paired X, Y data to measure relationship strength.
Linear Regression Calculator →Calculate the simple linear regression line for paired X, Y data in seconds. Get the slope and R² so you can predict Y from any new X value instantly.
Covariance Calculator →Calculate sample and population covariance between two variables in seconds. Enter paired X, Y data to measure how the two variables move together.

Related Terms

R-SquaredCoefficient of DeterminationCovarianceStatistical Covariance

Frequently Asked Questions

What is considered a strong correlation coefficient?
As a general guideline, |r| values above 0.7 are considered strong, 0.3 to 0.7 moderate, and below 0.3 weak, though these thresholds vary by field, a correlation of 0.3 might be meaningful in social science but weak in physics. The sign (positive or negative) indicates direction, while the magnitude indicates strength, independent of each other.
Does a correlation coefficient of 0 mean there is no relationship at all?
A correlation coefficient of 0 means there is no linear relationship, but it does not rule out a strong non-linear relationship, such as a U-shaped curve. Pearson's r specifically measures linear association, so scatter plots should always be reviewed alongside the coefficient.
What is the difference between correlation and causation?
A high correlation coefficient shows that two variables move together but does not prove that one causes the other, a third variable, reverse causation, or pure coincidence can all produce strong correlations. This is why correlation coefficients are described as measuring association, not proving cause and effect.
How is the correlation coefficient related to R-squared?
R-squared is simply the correlation coefficient squared (r²) in a simple linear regression with one predictor variable, converting the -1 to 1 scale of r into the 0 to 1 scale representing the proportion of variance explained. A correlation of 0.8, for example, corresponds to an R-squared of 0.64.
Can correlation coefficients be negative?
Yes, a negative correlation coefficient, down to -1, indicates an inverse relationship where one variable tends to decrease as the other increases. A coefficient of -1 represents a perfect negative linear relationship, just as +1 represents a perfect positive one.