Multiple Linear Regression Calculator

Perform ordinary least squares (OLS) multiple regression with up to 10 predictors. Obtain coefficient estimates, R², adjusted R², F-statistic, p-values, and diagnostic residual plots. Upload CSV or enter data directly.

First column is Y (dependent variable). Columns X1, X2, … are predictors. Minimum 3 rows required.
Upload CSV or drag & drop
First row as header (Y, X1, X2, …).
? Advertising: Sales ~ TV + Radio
? Housing: Price ~ SqFt + Bedrooms + Age
? Student: GPA ~ StudyHrs + SleepHrs
? Auto: MPG ~ Horsepower + Weight + Year
Privacy first: All computations run locally in your browser. No data is sent to any server.

What Is Multiple Linear Regression?

Multiple linear regression (MLR) is a statistical technique that models the linear relationship between a dependent (response) variable Y and two or more independent (predictor) variables X1, X2, …, Xk. The model takes the form:

Y = β₀ + β₁·X₁ + β₂·X₂ + … + βk·Xk + ε

where β₀ is the intercept, βⱼ are the slope coefficients, and ε is the error term.

The coefficients are estimated using the ordinary least squares (OLS) method, which minimizes the sum of squared residuals. This calculator computes the OLS solution via matrix algebra: β = (X'X)−1 X'Y, providing unbiased estimates under the Gauss–Markov assumptions.

Why Use This Interactive Regression Tool?

  • Educational Clarity: See how each predictor contributes to the outcome. Ideal for students learning regression diagnostics.
  • Data Exploration: Quickly test hypotheses with your own data or built-in examples.
  • Diagnostic Visualization: The residual vs fitted plot helps detect heteroscedasticity, outliers, and non-linearity.
  • Research & Business: Use for forecasting, pricing models, customer analytics, and more.

How the Calculator Works

The tool constructs the design matrix X with a column of ones for the intercept, followed by the predictor columns. The response vector Y is the first column of your data. The normal equations X'X β = X'Y are solved using Gaussian elimination with partial pivoting. From the coefficient vector, we compute:

  • Fitted values: Ŷ = Xβ
  • Residuals: e = Y − Ŷ
  • R² = 1 − SSE/SST (coefficient of determination)
  • Adjusted R² = 1 − (1−R²)·(n−1)/(n−k−1)
  • F-statistic = (SSR/k) / (SSE/(n−k−1)) with p-value from F-distribution
  • Standard errors: SE(βⱼ) = √(σ² · diag((X'X)−1)) where σ² = SSE/(n−k−1)
  • t-statistics: tⱼ = βⱼ / SE(βⱼ) with p-values from t-distribution

The residual plot displays standardized residuals against fitted values, with a horizontal zero line. Patterns in this plot can indicate model misspecification.

Assumptions & Diagnostics

For valid inference, multiple linear regression relies on several key assumptions:

  • Linearity: The relationship between predictors and the response is linear.
  • Independence: Observations are independent of each other.
  • Homoscedasticity: Constant variance of residuals across all levels of fitted values.
  • Normality: Residuals are approximately normally distributed (especially important for small samples).
  • No perfect multicollinearity: Predictors are not perfectly correlated.

Use the residual plot to check for heteroscedasticity (fanning or funnel shape) and non-linearity (curvature). The calculator also flags potential multicollinearity if the matrix (X'X) is near-singular.

Real-World Applications

Case Study: Advertising Spend vs Sales

A marketing team wants to understand how TV and radio advertising budgets influence product sales. Using the Advertising example dataset, the regression model estimates:

Sales = 2.94 + 0.045·TV + 0.188·Radio

The model explains 89.7% of the variance in sales (R² = 0.897). Both TV and radio coefficients are statistically significant (p < 0.001), with radio having a larger marginal effect. This insight helps allocate budget efficiently: radio advertising yields a higher return per dollar than TV in this dataset.

Case Study: Housing Price Prediction

A real estate analyst builds a model to predict home prices using square footage, number of bedrooms, and age. The model reveals that square footage is the strongest predictor, while age has a negative coefficient (older homes sell for less). The adjusted R² indicates the model generalizes well to new data. This tool enables rapid what-if analysis: “What is the predicted price for a 2,000 sq ft, 3-bedroom, 10-year-old house?”

Common Misconceptions

  • “R² close to 1 means a good model.” Not necessarily — high R² can result from overfitting, especially with many predictors. Adjusted R² penalizes model complexity.
  • “Significant predictors imply causation.” Regression shows association, not causation. Confounding variables may exist.
  • “More predictors always improve the model.” Adding variables can increase R² but may reduce predictive power (overfitting) and inflate standard errors.
  • “The intercept has a meaningful interpretation.” Not always — if the data range does not include zero for predictors, the intercept is a mathematical artifact.

Advanced Topics

  • Multicollinearity: When predictors are highly correlated, coefficient estimates become unstable. The calculator detects this via the condition number of X'X.
  • Heteroscedasticity: If residual variance increases with fitted values, consider robust standard errors or transformations (log, Box-Cox).
  • Interaction terms: Include products of predictors (e.g., X₁·X₂) to model synergistic effects. This tool supports any number of predictors, so you can add interaction columns manually.
  • Polynomial regression: Add squared or cubic terms to capture curvature. This is a special case of multiple regression.

Frequently Asked Questions

A general rule is at least n ≥ k + 10 observations, where k is the number of predictors. The calculator requires at least 3 rows, but more is recommended for reliable inference.

The F-statistic follows an F-distribution with (k, n−k−1) degrees of freedom. The p-value is the probability of observing an F-statistic as extreme as the one computed, under the null hypothesis that all coefficients (except intercept) are zero.

It plots residuals vs fitted values. Ideally, points are randomly scattered around zero with no clear pattern. A funnel shape suggests heteroscedasticity; a curve suggests a non-linear relationship.

Yes, by encoding them as dummy variables (0/1). You can manually add dummy columns in the data table. The calculator treats all columns as numeric.

The calculator uses Lentz's continued fraction algorithm for the incomplete beta function, providing high accuracy (relative error < 1e-12) for all parameter values. Results are typically accurate to 6 decimal places.

Excellent resources include Wikipedia: Linear Regression, Khan Academy, and textbooks like “Introduction to Statistical Learning” by James et al.

Built on sound statistical foundations – This tool implements OLS estimation using matrix methods verified against R and Python (statsmodels). The numerical algorithms use double-precision floating point with pivoting for stability. Reviewed by the GetZenQuery tech team, last updated June 2026.

References: Wikipedia: Linear Regression; James, G. et al. “An Introduction to Statistical Learning” (2021); StatsModels Documentation.