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Pearson Correlation Coefficient Calculator

Measures the linear relationship strength and direction between two continuous variables.

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Pearson's r

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Overview

The Pearson product-moment correlation coefficient (r) quantifies the degree to which two continuous variables are linearly related. Its value ranges from -1 to +1, where +1 indicates a perfect positive linear relationship, -1 a perfect negative linear relationship, and 0 no linear relationship. It's a fundamental statistic for exploring bivariate associations in social science research.

Symbols

Variables

Cov(X,Y) = Covariance of X and Y, = Standard Deviation of X, = Standard Deviation of Y, r = Pearson's r

Cov(X,Y)
Covariance of X and Y
unit²
Standard Deviation of X
unit
Standard Deviation of Y
unit
Pearson's r
dimensionless

Apply it well

When To Use

When to use: Applied when examining the linear association between two continuous, interval, or ratio-level variables. Common in studies exploring relationships between social attitudes, economic indicators, educational attainment, or health outcomes.

Why it matters: Essential for understanding how social phenomena co-vary. It helps sociologists identify potential predictors, explore theoretical relationships, and inform the development of more complex models like regression. It's a key descriptive statistic and a precursor to many inferential analyses.

Avoid these traps

Common Mistakes

  • Inferring causation from correlation.
  • Applying to non-linear relationships.
  • Ignoring outliers or non-normal distributions.

One free problem

Practice Problem

A study on social capital and community engagement found the covariance between the two variables to be 24. The standard deviation of social capital scores was 4, and the standard deviation of community engagement scores was 6. Calculate the Pearson correlation coefficient (r).

Covariance of X and Y24 unit²
Standard Deviation of X4 unit
Standard Deviation of Y6 unit

Solve for:

Hint: Divide the covariance by the product of the two standard deviations.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Pearson correlation coefficient
  2. Agresti, A. (2018). Statistical Methods for the Social Sciences (5th ed.). Pearson.
  3. Andy Field, Discovering Statistics Using IBM SPSS Statistics
  4. Paul F. Velleman, David S. Moore, Richard D. De Veaux, Bock, Stats: Data and Models
  5. Agresti, Statistical Methods for the Social Sciences
  6. Frankfort-Nachmias and Leon-Guerrero, Social Statistics for a Diverse Society
  7. Cohen, Statistical Power Analysis for the Behavioral Sciences
  8. Pearson, K. (1895). Notes on regression and inheritance in the case of two parents. Proceedings of the Royal Society of London, 58, 240-242.