Quartic Regression Formula
A quartic function expresses a relationship between variables using a degree-4 polynomial. It contains five terms, each governed by a coefficient that the regression algorithm solves for based on your data points.
y = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀
y— The dependent variable (output)x— The independent variable (input)a₄— Coefficient of the x⁴ terma₃— Coefficient of the x³ terma₂— Coefficient of the x² terma₁— Coefficient of the x term (linear slope)a₀— Constant term (y-intercept)
Understanding Quartic Regression
Quartic regression extends polynomial fitting into the 4th order, allowing you to model complex, non-linear relationships that simpler functions cannot capture. Unlike linear regression, which assumes a straight-line relationship, or quadratic regression, which models a single curve, quartic models accommodate up to three inflection or turning points within your data range.
The five coefficients (a₄ through a₀) are derived using the method of least squares, minimising the sum of squared residuals between observed and predicted values. The more coefficients a polynomial has, the more flexibility it gains to fit wiggly data—but this flexibility can also lead to overfitting if your dataset is small or noisy.
Quartic regression is particularly valuable in:
- Engineering: modelling stress-strain curves and material deformation
- Physics: fitting projectile trajectories and oscillatory systems
- Economics: capturing non-monotonic relationships in supply-demand data
- Biology: describing growth rates with acceleration and deceleration phases
How to Use This Calculator
Enter paired x and y coordinates for each observation. You need a minimum of 5 data points; the tool accepts up to 30 pairs. Type values into the input fields labelled x₁, y₁, x₂, y₂, and so on.
Once you've entered your data, the calculator automatically computes the five coefficients. You can adjust the precision setting to display results as whole numbers, decimals, or scientific notation. The results section will show:
- The five coefficients (a₄, a₃, a₂, a₁, a₀)
- Goodness-of-fit statistics (R², correlation coefficient)
- The complete equation in standard form
If you have gaps in your data or suspect a point is an outlier, use the No fit checkbox to exclude specific rows from the regression calculation.
Common Pitfalls and Considerations
Quartic models are powerful but require careful interpretation.
- Overfitting with sparse data — A quartic polynomial has five parameters to estimate. With fewer than 10–15 data points, the model may fit noise rather than signal, producing excellent R² values that fail on new data. Always validate with a hold-out test set.
- Extrapolation beyond your range — Quartic functions grow rapidly at extreme x values. Predictions far outside your observed data range become unreliable. The model describes your data, not universal truth.
- Coefficient interpretation is non-trivial — Unlike linear regression, individual quartic coefficients don't have straightforward real-world meanings. Focus on the overall fit and predicted values, not on interpreting a₃ or a₄ in isolation.
- Check residuals for systematic patterns — A high R² does not guarantee the model is correct. Plot residuals (actual minus predicted) against x. If patterns remain, try a different functional form, such as exponential or logarithmic regression.
Quartic vs. Other Polynomial Regressions
Regression fits range from linear (degree 1) through cubic (degree 3) to quartic (degree 4) and beyond. Each step increases flexibility:
- Linear: One slope, no curvature. Best for simple trends.
- Quadratic: One turning point. Good for parabolic data (e.g., projectile motion at a glance).
- Cubic: Two turning points. Captures S-shaped or oscillatory behaviour in one phase.
- Quartic: Three turning points. Handles complex, multi-phase behaviour.
Start with the simplest model that fits well. If residuals are randomly scattered and R² exceeds 0.90, you likely have adequate fit. If residuals show a pattern, increase the polynomial degree.