Cubic Regression Calculator
Free cubic regression calculator. Fit y = ax³+bx²+cx+d with step-by-step solutions, R², and graphs. Browser-based privacy.
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
Statistics
| Statistic | Value |
|---|---|
| Standard Error | |
| Sample Size (n) | |
| Degrees of Freedom |
Chart
Step-by-Step Solution
इसका उपयोग कैसे करें Cubic Regression Calculator
वृद्धि और क्षय
Fit cubic curves y = ax³ + bx² + cx + d to capture inflection-point patterns.
इनपुट डेटा
Enter paired X and Y values. Minimum 4 points required.
सांख्यिकीय आउटपुट
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
See Also
Polynomial & Nonlinear Regression Calculator
Free polynomial regression calculator. Fit curves of degree 1-5 with step-by-step solutions, R², and graphs. Browser-based privacy.
द्विघात प्रतिगमन कैलकुलेटर
द्विघात प्रतिगमन समीकरण (y = ax² + bx + c) की तुरंत गणना करें। निःशुल्क, चरण-दर-चरण गणितीय विश्लेषण और विश्लेषण के लिए अपने डेटा बिंदु दर्ज करें।
प्रतिगमन वक्र कैलकुलेटर
रैखिक, द्विघात, घातीय और लघुगणक प्रतिगमन मॉडल की एक साथ तुलना करें। हमारे मुफ़्त ऑनलाइन टूल से अपने डेटा के लिए सबसे उपयुक्त वक्र ढूंढें।