LaTeX
b_1=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2}Variables
- b1: slope estimate
- x_i,y_i: observations
- x̄,ȳ: sample means
How to use this formula
Computes the least-squares slope in simple linear regression.
Important notes
- The denominator must be nonzero.
- A slope describes association, not necessarily causation.
Quick example
After finding b1, use b0=ȳ−b1x̄.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The denominator must be nonzero.
- For the Simple Regression Slope, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Simple Regression Slope, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- For the Simple Regression Slope, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions.
Worked example
After finding b1, use b0=ȳ−b1x̄.
Common mistakes
- When copying Simple Regression Slope, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Simple Regression Slope, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Simple Regression Slope 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
b_1=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^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 Simple Regression Slope used for?
Computes the least-squares slope in simple linear regression.
Can I copy this formula as LaTeX?
Yes. Copy b_1=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^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.