Polynomial Regression Calculator

Polynomial regression is a powerful extension of linear regression that models the relationship between an independent variable x and a dependent variable y as a polynomial of degree n. Unlike simple linear regression, which fits a straight line, polynomial regression can capture curves, bends, and more complex patterns in data — making it useful when real-world relationships are clearly non-linear. The mathematical model takes the form: y = β₀ + β₁x + β₂x² + … + βₙxⁿ, where the coefficients β₀ through βₙ are estimated from the data using the method of least squares. Despite fitting a curve rather than a line, polynomial regression is still classified as a linear model because it is linear in its coefficients. The least squares method minimizes the sum of squared residuals — the differences between observed y values and the values predicted by the polynomial. This is accomplished by solving the normal equations: (XᵀX)β = Xᵀy, where X is the Vandermonde matrix constructed from the x values. This calculator solves those equations using Gaussian elimination, a robust numerical method suitable for polynomials up to degree 10. R-squared (R²), the coefficient of determination, measures how well the fitted polynomial explains the total variability in y. An R² of 1.0 means the curve passes through all data points exactly; 0.0 means the model explains none of the variance. While R² always increases with polynomial degree, a very high R² with a high-degree polynomial may indicate overfitting — the model memorizes the training data rather than capturing the true underlying trend. Choosing the right degree is critical. Degree 1 gives a straight line (equivalent to simple linear regression). Degree 2 (quadratic) captures U-shaped or inverted U-shaped patterns. Degree 3 (cubic) can model S-shaped trends or more complex growth curves. For most practical data sets, degrees 2 or 3 are sufficient, and going beyond degree 5 or 6 often introduces numerical instability and overfitting. Applications of polynomial regression span many fields. Engineers use quadratic models for stress-strain relationships and projectile motion. Economists fit cubic curves to cost functions and production models. Biologists apply polynomial regression to growth curves and dose-response studies. Data scientists use it as a preprocessing step in machine learning pipelines. When using this calculator, be aware of extrapolation risk — polynomial curves can behave wildly outside the range of the observed data. Always verify predictions against domain knowledge, and consider simpler models first before increasing polynomial degree.