Regression Calculator

Mastering Linear Regression: Complete Guide

Linear regression models the relationship between two variables by fitting a linear equation to observed data. One variable is considered explanatory (x), and the other is the response (y). This Professional Regression Calculator computes the ordinary least squares (OLS) regression line, Pearson correlation coefficient, R‑squared, residuals, and allows you to predict y for any x. A scatter plot with the fitted line is generated for immediate visual feedback.

The Regression Equation

The least squares regression line is y = mx + b, where the slope m and intercept b are chosen to minimize the sum of squared residuals (differences between observed and predicted y). The slope indicates the average change in y per one‑unit increase in x. The intercept is the predicted y when x = 0.

Correlation and R‑Squared

Pearson's correlation coefficient r measures the strength and direction of the linear relationship (−1 ≤ r ≤ 1). Values near ±1 indicate strong linear association; near 0 indicate little to no linear relationship. R‑squared (r²) represents the proportion of variance in y explained by x. For example, R² = 0.85 means 85% of the variability in y is accounted for by the linear model.

Making Predictions

Once the regression line is established, you can predict y for any x value. Predictions within the range of the original x data (interpolation) are generally reliable. Extrapolation (predicting outside the observed x range) should be done with caution, as the linear relationship may not hold beyond the sampled domain.

Residual Analysis

Residuals (observed − predicted) reveal how well the model fits individual data points. A pattern in the residuals (e.g., a curve) suggests that a linear model may not be appropriate. The calculator displays a table of residuals so you can inspect them directly.