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Pearson's r (Calculation) Calculator

Detailed calculation of the Pearson correlation coefficient.

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Pearson's r

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Overview

Pearson's r, or the product-moment correlation coefficient, quantifies the strength and direction of the linear relationship between two continuous variables. It is calculated by dividing the sum of products of deviations from the mean by the square root of the product of the squared deviations for each variable.

Symbols

Variables

r = Pearson's r, SP = Sum of Products, SS_x = Sum Squares X, SS_y = Sum Squares Y

Pearson's r
Sum of Products
Sum Squares X
Sum Squares Y

Apply it well

When To Use

When to use: Use this metric when analyzing two interval- or ratio-level variables that appear to share a linear trend. It assumes bivariate normality and is sensitive to outliers, so data should be screened for extreme values before calculation.

Why it matters: In psychological research, it identifies how variables like personality traits and behavioral outcomes co-vary. Understanding these associations allows for the development of predictive models and the validation of measurement scales.

One free problem

Practice Problem

A clinical psychologist is studying the link between hours of meditation (X) and anxiety scores (Y). After calculating the sums, they find the sum of products (sp) is -45, the sum of squares for X (ssx) is 50, and the sum of squares for Y (ssy) is 72. Calculate Pearson's r.

Sum of Products-45
Sum Squares X50
Sum Squares Y72

Solve for:

Hint: Divide the sum of products by the square root of the product of the two sums of squares.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Gravetter, F. J., Wallnau, L. B., Forzano, L. B., & Witnauer, J. E. (2021). Essentials of statistics for the behavioral sciences (10th ed.).
  2. Field, A. (2018). Discovering statistics using IBM SPSS Statistics (5th ed.). SAGE Publications.
  3. Wikipedia: Pearson correlation coefficient
  4. Discovering Statistics Using IBM SPSS Statistics (Field, 2018)
  5. Statistics for the Behavioral Sciences (Gravetter & Wallnau, 2017)
  6. Cohen Statistical Power Analysis for the Behavioral Sciences
  7. Field Discovering Statistics Using IBM SPSS Statistics
  8. Howell Statistical Methods for Psychology