Home›Glossary›Correlation Coefficient

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 points to measure the strength and direction of their relationship.
Linear Regression Calculator →Calculate the simple linear regression line for paired X, Y data in seconds. Get the slope, intercept, 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 points to measure how the two variables move together instantly.

Related Terms

R-SquaredCoefficient of DeterminationCovarianceStatistical Covariance

Frequently Asked Questions

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.
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.
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.
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.
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.