Cubic Regression Calculator

What Is Cubic Regression?

Cubic regression fits a third-degree polynomial to paired data so you can model curved relationships that a straight line cannot capture. The model is:

$$y = a + bx + cx^{2} + dx^{3}$$

The coefficients $a$, $b$, $c$, and $d$ are chosen by least squares to minimize the sum of squared residuals between observed $y$ values and predictions $\hat{y}$.

How the Calculator Works

Enter matching lists of $X$ (independent) and $Y$ (dependent) values. You need at least four points. The tool builds the model matrix with columns $[1, x, x^{2}, x^{3}]$, solves the normal equations $\beta = (X^{\top}X)^{-1}X^{\top}y$, then reports the equation, coefficients, $R^{2}$, fitted values, and residuals.

When to Use Cubic Regression

Use a cubic model when a scatter plot shows an S-shaped bend or when theory suggests a cubic trend. Prefer a simpler model when it fits nearly as well. Compare with the Linear Regression Calculator before adding higher-order terms.

Example

For $X = \{0,1,2,3,4,5\}$ and $Y = \{2,1,3,8,14,20\}$, the fitted cubic curve tracks the upward bend in $Y$. Inspect $R^{2}$ and residuals to judge fit quality.

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