Linear Regression Calculator

A linear regression model describes the relationship between a predictor variable (X) and a response variable (Y) using a straight line: Ŷ = b0 + b1·X. The slope (b1) indicates how much Y changes for each one-unit increase in X, and the intercept (b0) is the predicted Y when X equals zero. It is one of the most widely used statistical techniques because it both explains relationships and enables prediction.

The slope (b1) quantifies the direction and magnitude of the relationship — a positive slope means Y increases as X increases, while a negative slope means the opposite. The intercept (b0) is the baseline value of Y when X is zero. Together they define the unique line that best fits your data.

R² (the coefficient of determination) measures the proportion of variance in Y that is explained by X, ranging from 0 to 1. An R² of 0.85, for example, means 85% of the variation in Y is accounted for by the linear relationship with X. Higher R² values indicate a better-fitting model, though context matters — some fields accept 0.5 as strong while others require 0.95+.

Simple linear regression assumes: (1) a linear relationship between X and Y, (2) independence of observations, (3) homoscedasticity — residuals have constant variance across all X values, (4) normally distributed residuals, and (5) no significant outliers. Violating these assumptions can lead to unreliable estimates and predictions.

The line is fit using the Ordinary Least Squares (OLS) method, which minimises the sum of squared residuals (the vertical distances between each data point and the line). The slope is calculated as b1 = Σ[(Xi − X̄)(Yi − Ȳ)] / Σ[(Xi − X̄)²], and the intercept as b0 = Ȳ − b1·X̄, where X̄ and Ȳ are the sample means.

The standard error of the estimate (SEE) measures the average distance that observed Y values fall from the regression line. A smaller SEE indicates that the model predictions are more precise. It is calculated as the square root of the mean squared residuals: √[Σ(Yi − Ŷi)² / (n − 2)].

Correlation (r) measures the strength and direction of the linear association between two variables but does not imply causation or provide a predictive equation. Regression goes further by fitting a line that can be used to predict Y from X. R² in regression is simply the square of the Pearson correlation coefficient r in simple linear regression.

You need a minimum of 3 data points for the calculation to work, but reliable estimates generally require at least 10–20 observations. More data points reduce sampling error, improve the stability of the slope and intercept estimates, and increase the statistical power to detect a true linear relationship.