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Pearson's Correlation (r) Calculator

Strength and direction of the linear relationship between two variables.

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Pearson's r

Formula first

Overview

Pearson's correlation coefficient, often denoted as r, measures the strength and direction of the linear relationship between two continuous variables. It is calculated by dividing the covariance of the two variables by the product of their individual standard deviations, effectively scaling the result to a range between -1 and +1.

Symbols

Variables

r = Pearson's r, Cov = Covariance, SD_x = SD of X, SD_y = SD of Y

Pearson's r
Covariance
SD of X
SD of Y

Apply it well

When To Use

When to use: Apply this metric when examining the linear association between two interval or ratio-level variables that follow a normal distribution. It is essential for datasets where you assume a constant rate of change and want to ignore the specific units of measurement. It should not be used for non-linear relationships or data with extreme outliers.

Why it matters: In psychological research, r allows for the quantification of relationships between abstract constructs like intelligence and academic performance. This coefficient is vital for assessing the reliability of diagnostic tools and predicting behavioral outcomes based on known variables. It serves as the foundation for more complex multivariate analyses like regression.

Avoid these traps

Common Mistakes

  • Using it on non-linear data.
  • Assuming correlation implies causation.

One free problem

Practice Problem

A psychologist finds that the covariance between study hours (X) and test scores (Y) is 12.0. If the standard deviation for study hours is 4.0 and for test scores is 5.0, what is the correlation coefficient?

Covariance12
SD of X4
SD of Y5

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. Statistics for Psychology (8th ed.) by Arthur Aron, Elaine N. Aron, and Elliot Coups
  3. Pearson product-moment correlation coefficient (Wikipedia article)
  4. Discovering Statistics Using IBM SPSS Statistics by Andy Field
  5. Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). SAGE Publications.
  6. Howell, D. C. (2013). Statistical Methods for Psychology (8th ed.). Wadsworth Cengage Learning.
  7. AQA Psychology for A Level Year 2 by Cara Flanagan, Dave Berry, Jo Hayward, and Rob Liddle