LaTeX
r=\frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2\sum(y_i-\bar y)^2}}Variables
- r: correlation coefficient
- x̄,ȳ: sample means
How to use this formula
Measures linear association between two variables.
Important notes
- Correlation does not imply causation.
Quick example
r=1 indicates perfect positive linear association.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Correlation does not imply causation.
- For the Pearson Correlation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.
- For the Pearson Correlation, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
Worked example
r=1 indicates perfect positive linear association.
Common mistakes
- When copying Pearson Correlation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Pearson Correlation, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Pearson Correlation in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
r=\frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2\sum(y_i-\bar y)^2}}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Pearson Correlation used for?
Measures linear association between two variables.
Can I copy this formula as LaTeX?
Yes. Copy r=\frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2\sum(y_i-\bar y)^2}} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.