Linear Regression & Line of Best Fit Calculator

Linear Regression Guide

Least-Squares Regression

The least-squares regression line minimises the sum of squared vertical distances from each data point to the line (residuals). Gradient m = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)². Y-intercept c = ȳ − m×x̄. The line always passes through the point (x̄, ȳ) — the mean of both variables. This is the LSRL or line of best fit. It minimises Σ(yᵢ − ŷᵢ)² where ŷᵢ = predicted values. There is only one least-squares line for a given dataset (unlike other fitting methods).

R-squared Interpretation

R² = square of the correlation coefficient r. It represents the proportion of variance in Y explained by X. R² = 0.85 means 85% of the variation in Y is explained by the linear relationship with X. The remaining 15% is due to other factors (lurking variables, random variation). R² = 1: perfect fit. R² = 0: no linear relationship. Important: a high R² does not mean causation, and does not mean the linear model is appropriate — always plot the data. R² can be high even when the relationship is cle

Interpolation vs Extrapolation

Interpolation: using the regression line to predict Y for X values within the range of the data — generally reliable. Extrapolation: predicting outside the data range — unreliable and potentially very wrong. The linear relationship that holds between x=10 and x=100 may not hold at x=500. Example: a linear model relating height and weight in children aged 5-10 is reasonable for predicting weight at age 7. Predicting weight at age 40 from the same model would be nonsense. Always state when predict

Assumptions of Linear Regression

Linearity: the relationship between X and Y is actually linear — check with a scatterplot. Independence: each observation is independent (no repeated measurements). Homoscedasticity: variance of residuals is constant across all X values — if residuals are larger at high X, this assumption is violated. Normality of residuals: for hypothesis testing, residuals should be approximately normally distributed (less critical with large samples). Residual plot: plot residuals against fitted values. Rando