Linear Regression Calculator

Linear Regression: A Friendly Guide

Linear regression finds the best-fitting straight line through your data, modeling the relationship between an independent variable X and a dependent variable Y. The model is y = a + b·x, where a is the intercept and b is the slope.

What the numbers mean

• Slope (b): Change in y for each 1-unit increase in x.
• Intercept (a): Value of y when x = 0.
• R²: Proportion of variance in y explained by x (0 to 1).

How it works (light math)

The line minimizes the sum of squared vertical distances from each point to the line (ordinary least squares). Given x and y:

• b = Sxy / Sxx, where Sxy = Σ(x−x̄)(y−ȳ) and Sxx = Σ(x−x̄)²
• a = ȳ − b·x̄
• R² = 1 − SSE/SST, with SSE = Σ(ŷ−y)² and SST = Σ(y−ȳ)²

When to use it

• Predicting outcomes (sales vs. ads, weight vs. height)
• Measuring strength and direction of linear relationships
• Building simple forecasting models

Tips for better fits

• Plot your data first to confirm a roughly linear trend.
• Watch out for outliers—one extreme point can tilt the line.
• Avoid extrapolation far beyond your data range.

FAQ

Deep Dive: Mastering Linear Regression

This deep dive builds on the basics and helps you interpret, validate, and communicate linear regression results confidently. It focuses on assumptions, diagnostics, intervals, and best practices so your conclusions are reliable.

1) Core ideas and terminology

• Model: y = a + b·x + ε, where ε is random error around the line.
• Fitted line: ŷ = â + b̂·x (what the model predicts).
• Residual: e = y − ŷ (what’s left after prediction).
• R²: Fraction of variance in y explained by x (0–1).

2) Assumptions to keep in mind

• Linearity: The relationship between x and y is approximately straight.
• Independence: Observations (and residuals) are independent of each other.
• Homoscedasticity: Residual spread is roughly constant across x (no funnel shape).
• Normality (often for inference): Residuals are roughly normal, especially for smaller samples.

3) Diagnostic checklist

• Scatter plot: Confirm a straight trend and spot outliers.
• Residual vs. fitted: Look for random scatter. Patterns suggest nonlinearity or heteroscedasticity.
• Outliers and leverage: A single extreme point can tilt the slope. Investigate, don’t just delete.
• Nonlinearity: If curved, consider transformations (log, square) or another model.

4) Intervals you’ll care about

• Confidence interval for slope/intercept: Range of plausible parameter values.
• Confidence interval for the mean response: Uncertainty in the average ŷ at a given x.
• Prediction interval: Wider, for an individual future y at a given x (includes randomness).

5) Practical guidance

• Scale and clean data: Fix obvious errors, ensure consistent units, and treat missing values.
• Check range: Avoid extrapolating far beyond observed x; predictions can be misleading.
• Context matters: Combine statistical fit with domain knowledge for decisions.
• Compare models: Try forcing the origin only if theory requires a = 0.

6) Mini worked example (conceptual)

Suppose you have x = [1,2,3,4,5] and y = [2.1, 2.9, 3.8, 4.2, 5.1]. After fitting, you get â ≈ 0.3 and b̂ ≈ 0.95. Interpretation: each +1 in x adds ~0.95 in y. If x = 6, ŷ ≈ 0.3 + 0.95·6 = 6.0. R² tells you how much of y’s variation your line explains.

7) Common pitfalls

• Correlation ≠ causation: A strong fit doesn’t prove x causes y.
• Over-reliance on R²: High R² with a bad model shape is still a bad model.
• Ignoring outliers: Outliers may reflect data quality or real phenomena—investigate!
• Extrapolation: Predictions far outside observed x can be very wrong.

8) Beyond simple linear regression

• Multiple regression: Model y with several predictors (y = a + b₁x₁ + b₂x₂ + …).
• Polynomial terms: Capture curves by adding x², x³ terms when justified.
• Transformations: Log/Box-Cox transforms can stabilize variance or linearize relationships.

9) Quick FAQ