Quartic Regression Calculator
Enter paired X and Y data and fit a least-squares quartic polynomial y = ex⁴ + dx³ + cx² + bx + a. Inspect R², all five coefficients, the equation, and predict at any new X.
Quartic equation: y = 1x⁴ + 0x³ + 0x² + 0x + 0
How does this calculator work?
Quartic regression fits y = ex⁴ + dx³ + cx² + bx + a by solving the 5×5 normal-equation system built from power sums Σxᵏ (k = 0..8). Requires ≥ 5 data points; more is better. R² = 1 − SSresiduals/SStotal. Higher degree risks overfitting — compare with cubic and quadratic models before committing.
Formula
How this is calculated
Quartic regression finds the five coefficients a, b, c, d, e of the polynomial y = a + bx + cx² + dx³ + ex⁴ that minimise the sum of squared residuals Σ(yᵢ − ŷᵢ)². The optimal coefficients satisfy a 5 × 5 linear system called the normal equations, built from power sums Σxᵏ (k = 0 … 8) and cross-products Σxᵏy (k = 0 … 4). This calculator assembles those sums and solves the system using Gaussian elimination with partial pivoting for numerical stability.
At least five data points are required because the quartic polynomial has five free parameters. With exactly five points the curve passes through all of them (R² = 1 by construction); useful statistical fit quality requires clearly more observations than parameters — at least seven or eight points are recommended for a meaningful R². R² = 1 − SSresidual / SStotal: values close to 1 indicate a tight quartic fit; lower values suggest the data does not follow a quartic pattern or there is substantial scatter around it.
Limitations: the normal-equation method is susceptible to ill-conditioning when X values span a very wide range — consider standardising (centring and scaling) X before fitting. Quartic polynomials can oscillate sharply between data points (Runge's phenomenon) and extrapolate very poorly beyond the observed range. Always compare R² against simpler models (quadratic, cubic) before accepting the added complexity; if a cubic fits equally well, it is the safer choice.
Frequently asked questions
A quartic polynomial has 5 unknown coefficients (a, b, c, d, e). Five data points are the minimum to form a uniquely solvable 5×5 normal-equation system. With more points, least squares finds the curve that best fits all of them simultaneously by minimising total squared error — and R² becomes meaningful only when n > 5.
Use quartic when the scatter plot shows two bends or inflection points — an M- or W-shaped pattern that a lower-degree polynomial cannot capture. Always compare R² (and ideally adjusted R²) against simpler models. If quadratic or cubic fits equally well, prefer the simpler model to avoid overfitting.
R² measures the fraction of variance in Y explained by the fitted quartic. R² near 1 means a tight fit; near 0 means the quartic explains little variance. Caution: adding polynomial terms always raises R² — with exactly 5 points and a degree-4 fit, R² = 1 regardless of the data pattern. Use more data than parameters for meaningful R² values.
Also known as
TG we-Calculate Editorial Team. (2026). Quartic Regression Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/quartic-regression-calculator
TG we-Calculate Editorial Team. "Quartic Regression Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/quartic-regression-calculator.
TG we-Calculate Editorial Team, "Quartic Regression Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/quartic-regression-calculator
@misc{wecalculate_quartic_regression_calculator, title = {Quartic Regression Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/quartic-regression-calculator}}, year = {2026}, note = {TG we-Calculate} }
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