Cubic Regression Calculator
Enter paired X and Y data and fit a least-squares cubic polynomial y = dx³ + cx² + bx + a. Inspect R², all four coefficients, the equation, and predict at any new X.
Cubic equation: y = 1x³ + 0x² + 0x + 0
How does this calculator work?
Cubic regression fits y = dx³ + cx² + bx + a to your data by solving the 4×4 normal-equation system Σxᵏ·coefficients = Σxᵏy (k = 0..3). Requires ≥ 4 data points. R² = 1 − SSresiduals/SStotal measures fit quality. Extrapolation beyond the data range is unreliable.
Formula
How this is calculated
Cubic regression finds the four coefficients a, b, c, d of the polynomial y = a + bx + cx² + dx³ that minimise the sum of squared residuals Σ(yᵢ − ŷᵢ)². The optimal coefficients satisfy a 4 × 4 linear system called the normal equations, built from the power sums Σxᵏ (k = 0 … 6) and cross-products Σxᵏy (k = 0 … 3). This calculator assembles that system and solves it using Gaussian elimination with partial pivoting for numerical stability.
At least four data points are required because the cubic polynomial has four free parameters. With exactly four points the curve passes through all of them (R² = 1 by construction); useful statistical fit quality requires more data than parameters, so at least five or six points are recommended for a meaningful R² interpretation. R² = 1 − SSresidual / SStotal: values close to 1 indicate a tight cubic fit; lower values suggest the data does not follow a cubic pattern or there is scatter.
Limitations: the normal-equation method can suffer from ill-conditioning when X values span a very large range (e.g., millions vs thousands) — consider centring or scaling your X values first. Cubic regression can overfit small samples and extrapolate very poorly outside the observed X range. For exploration and hypothesis testing the fit is reliable; for engineering models, validate against held-out data.
Frequently asked questions
A cubic polynomial has 4 unknown coefficients (a, b, c, d). Four points are the minimum to solve a unique cubic. With more points, least squares finds the curve that best fits all of them simultaneously by minimising the total squared error.
R² measures the fraction of the variation in Y explained by the fitted cubic. R² = 1 − SSresiduals / SStotal. An R² near 1 means the cubic fits well; near 0 means it fits poorly. Note that adding terms always raises R² — use adjusted R² or cross-validation to compare against simpler models.
Use cubic regression when the relationship has an inflection point — rising then falling (or vice versa) and then rising again. If a linear or quadratic model fits just as well (similar R²), prefer the simpler model to avoid overfitting.
Also known as
TG we-Calculate Editorial Team. (2026). Cubic Regression Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cubic-regression-calculator
TG we-Calculate Editorial Team. "Cubic Regression Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/cubic-regression-calculator.
TG we-Calculate Editorial Team, "Cubic Regression Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cubic-regression-calculator
@misc{wecalculate_cubic_regression_calculator, title = {Cubic Regression Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cubic-regression-calculator}}, year = {2026}, note = {TG we-Calculate} }
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