Cubic Regression Calculator
Enter x-y pairs to fit a third-degree polynomial model and evaluate predictions and R² goodness of fit.
Cubic model
y = 1.6000 + (0.5476)x + (-0.2857)x² + (0.1667)x³
R² = 0.9997
| x | Actual y | Predicted y | Residual |
|---|---|---|---|
| 1 | 2.0000 | 2.0286 | -0.0286 |
| 2 | 3.0000 | 2.8857 | 0.1143 |
| 3 | 5.0000 | 5.1714 | -0.1714 |
| 4 | 10.0000 | 9.8857 | 0.1143 |
| 5 | 18.0000 | 18.0286 | -0.0286 |
How to Use This Calculator
- Enter at least four data pairs (x, y) representing observed values.
- Review the fitted cubic equation and R² goodness-of-fit metric.
- Use the optional input to predict y for a new x value.
- Inspect the residuals table to assess model performance.
Formula
Model: y = a₀ + a₁x + a₂x² + a₃x³
Coefficients solve (XᵀX)a = Xᵀy where rows of X are [1, x, x², x³]
R² = 1 − (SSres / SStot)
Full Description
Cubic regression fits a third-degree polynomial to capture nonlinear trends with up to two inflection points. Least squares minimizes residual sum of squares, yielding coefficients that best approximate observed data in a linear algebraic framework.
Compare R² and residual patterns to judge fit quality. Consider simpler or more complex models depending on bias-variance trade-offs.
Frequently Asked Questions
Why do I need at least four points?
A cubic has four coefficients; fewer points lead to an underdetermined system. More points improve stability.
Can coefficients be unstable?
Yes. Large x magnitudes or collinearity can cause numerical issues. Standardizing x values can help.
Is R² sufficient to judge fit?
Use R² alongside residual analysis and domain knowledge to avoid overfitting or misinterpretation.
Does the model extrapolate reliably?
Polynomial extrapolation can diverge rapidly. Interpret predictions outside the data range cautiously.