Cubic Regression Calculator

Free cubic regression calculator. Fit y = ax³+bx²+cx+d with step-by-step solutions, R², and graphs. Browser-based privacy.

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Cubic Regression Calculator

Free cubic regression calculator. Fit y = ax³+bx²+cx+d with step-by-step solutions, R², and graphs. Browser-based privacy.

Enter your data points

#	X	Y

Results

Equation

a (x³)

b (x²)

c (x)

d (intercept)

R²

Predicted Y

For X = —

Statistics

Statistic	Value
Standard Error
Sample Size (n)
Degrees of Freedom

Chart

Step-by-Step Solution

Как использовать Cubic Regression Calculator

Рост и распад

Fit cubic curves y = ax³ + bx² + cx + d to capture inflection-point patterns.

Входные данные

Enter paired X and Y values. Minimum 4 points required.

Статистический вывод

Get all four coefficients, R², standard error, and predictions.

Cubic regression captures one inflection point. Use quartic for two inflection points.

What Is Cubic Regression?

📐 Cubic regression fits a third-degree polynomial y = ax³ + bx² + cx + d to your data. Unlike quadratic curves (parabolas), cubic curves can have one inflection point where the curvature changes from concave to convex or vice versa.

📊 Common applications include: (1) Economics — cost functions that initially decrease then increase due to scale effects, (2) Engineering — material stress-strain curves with yield points, (3) Ecology — population growth that accelerates, then plateaus, then crashes, and (4) Psychology — learning curves that improve rapidly, then slow, then improve again.

📐 Cubic regression captures richer patterns than quadratic while remaining more interpretable than higher-degree polynomials. Use it when your data clearly changes direction once.

How Cubic Regression Works

1
Enter data: Provide paired X and Y values. At least 4 points are required to fit 4 unknown coefficients.
2
Build normal equations: Construct the 4×4 symmetric matrix XᵀX and the vector Xᵀy using sums of powers of X from x⁰ up to x⁶.
3
Solve the linear system: Use Gaussian elimination with partial pivoting to solve (XᵀX)·β = Xᵀy for the coefficient vector β = [d, c, b, a]ᵀ.
4
Compute predictions and residuals: Calculate predicted ŷ = ax³ + bx² + cx + d and residuals e = y − ŷ for each data point.
5
Evaluate goodness-of-fit: Compute R² = 1 − SSE/SST, standard error = sqrt(SSE/(n−4)), and plot the fitted cubic curve against the scatter.

When to Use Cubic Regression

Your data changes direction exactly once (one inflection point)
A quadratic parabola is too simple but a 5th-degree polynomial would overfit
You suspect an S-shaped or N-shaped pattern in the data
You need to model behavior with both acceleration and deceleration phases

When to Avoid Cubic Regression

When you have fewer than 4 data points (insufficient to fit 4 coefficients)
When a straight line or simple curve already explains the data well
When higher-order wiggles are not physically meaningful
When a simpler model (linear, quadratic, or exponential) achieves similar R²

Cubic Regression Calculator