Calculate the line of best fit using Ordinary Least Squares (OLS): slope (b), y-intercept (a), R² value, and interactive Y predictions.
Click "Calculate Regression Line" to fit linear model.
A real estate appraiser predicting home values based on square footage analyzes paired historical sales data (square feet vs. sale price): applying Ordinary Least Squares regression yields the best fit equation ŷ = 150x + 50,000 with R² = 0.88. Using this linear model, an appraiser predicting the market value of a 2,000 sq ft home evaluates ŷ = 150(2000) + 50,000 = $350,000. An economist modeling consumer spending against disposable income uses linear regression to estimate marginal propensity to consume.
Linear regression is a fundamental statistical modeling method used to model the linear relationship between a independent explanatory variable X and a dependent outcome variable Y.
The standard simple linear regression equation takes the slope-intercept form:
ŷ = β₁ x + β₀ (or y = bx + a)
Where β₁ (b) is the slope (the average change in Y per unit increase in X), and β₀ (a) is the y-intercept (the predicted value of Y when X = 0). This calculator computes OLS slope, y-intercept, Pearson's r, R² variance explained, and predicts Y for custom X values. The following guide covers residual minimization, R² interpretation, and real-world predictive modeling applications.
When paired X and Y inputs are submitted, the engine calculates sample means x̄ and ȳ, evaluates cross-product sums, and solves the Ordinary Least Squares normal equations.
1. Slope Formula (β₁ or b):
b = SP_xy / SS_xx = ∑ (xᵢ - x̄)(yᵢ - ȳ) / ∑ (xᵢ - x̄)²
2. Y-Intercept Formula (β₀ or a):
a = ȳ - (b × x̄)
3. Regression Line Equation:
ŷ = b x + a
4. Coefficient of Determination (R²):
R² = r² = (SP_xy / √(SS_xx × SS_yy))²
5. Prediction Evaluation:
For any custom input X_target:
ŷ_predicted = b(X_target) + a
Sales forecasting and financial planning. Business analysts project future quarterly sales revenue based on advertising expenditure trends.
Real estate property appraisal. Appraisers model property sales price as a function of home size, lot size, or neighborhood location metrics.
Machine learning baseline modeling. Data engineers train simple linear regression models as interpretable benchmarks before deploying complex neural networks.
Medical dosing and pharmacology. Clinical researchers model blood drug concentration levels as a function of patient body weight.
Ensure X and Y input fields have equal numbers of observations (paired data points).
Use the Predict Y field to estimate dependent variable values for any hypothetical X inputs.
For standalone correlation strength analysis, pair this tool with our Correlation Coefficient Calculator.
The calculation engine operates client-side in JavaScript using IEEE 754 double-precision floats. Paired datasets with up to 20,000 observations process in under 10 milliseconds.
| Metric | Symbol | Formula / Meaning |
|---|---|---|
| Slope | b (β₁) | SP_xy / SS_xx (Change in Y per unit X) |
| Y-Intercept | a (β₀) | ȳ - b x̄ (Value of Y when X = 0) |
| Best Fit Line | ŷ | ŷ = bx + a |
| R² Value | R² | Percentage of total Y variance explained by X |
| Pearson r | r | Linear correlation strength (-1 to +1) |
Simple linear regression models Y using a single predictor X (ŷ = bx + a). Multiple linear regression models Y using two or more predictors (ŷ = b₁x₁ + b₂x₂ + ... + a).
An R² of 0.85 means that 85% of the variance in the dependent variable Y is explained by the linear relationship with independent variable X.
Yes. A negative slope (b < 0) indicates an inverse relationship: as X increases, Y decreases.
Correlation Coefficient Calculator — Computes Pearson's r and covariance.
Standard Deviation Calculator — Computes sample and population standard deviations.
P-Value Calculator — Evaluates statistical significance p-values.