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.
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.
Enter your data points
| # | X | Y |
|---|
Results
Equation
R²
Degree
Predicted Y
Coefficients
| Term | Value |
|---|
Statistics
| Statistic | Value |
|---|---|
| Standard Error | |
| Sample Size (n) | |
| Degrees of Freedom |
Chart
Step-by-Step Solution
How to Use This Polynomial & Nonlinear Regression Calculator
Flexible Polynomial Fitting
Fit polynomial curves of degree 1 (linear) through 5 (quintic) to match your data's complexity.
Enter X & Y Data
Input paired values — minimum points required = degree + 1.
Dynamic Coefficients
Get all coefficients, R², standard error, and a complete step-by-step solution.
Best For
Complex curves with multiple turning points, oscillating data, and high-order trends.
Higher-degree polynomials can overfit with small datasets. Prefer the lowest degree that captures the pattern well.
What Is Polynomial Regression?
📐 Polynomial regression generalizes linear and quadratic regression to fit a polynomial of any chosen degree n: y = aₙxⁿ + ... + a₂x² + a₁x + a₀. By selecting the degree (1 through 5), you control how flexibly the curve can bend to match your data.
📊 Key applications include: (1) Physics — projectile trajectories and drag forces, (2) Finance — yield curves and option pricing models, (3) Biology — dose-response curves with multiple thresholds, and (4) Engineering — thermal expansion and vibration modes.
⚠️ Higher degrees increase flexibility but also risk overfitting — the curve may wiggle to fit noise rather than signal. Choose the lowest degree that captures the underlying pattern.
How Polynomial Regression Works
- 1 Choose the degree: Select 1 (linear) through 5 (quintic). The degree determines how many bends the curve can have — degree n allows up to n−1 turning points.
- 2 Build the design matrix: Create matrix X where each row is [1, xᵢ, xᵢ², ..., xᵢⁿ] for each data point, and vector y of observed values.
- 3 Solve normal equations: Compute (XᵀX)·β = Xᵀy using Gaussian elimination with partial pivoting. The solution vector β contains coefficients [a₀, a₁, ..., aₙ].
- 4 Evaluate predictions and fit: Calculate predicted ŷ for every xᵢ, then compute residuals, SSE, SST, and R² = 1 − SSE/SST.
- 5 Assess model complexity: Compare R² across degrees, but favor simpler models. Standard error = sqrt(SSE/(n − degree − 1)) penalizes overfitting.
When to Use Polynomial Regression
- Your data follows a curved but not strictly exponential or logistic pattern
- You need to model multiple turning points (peaks, troughs, inflections)
- You want to compare different complexity levels quickly (degree 1 vs 3 vs 5)
- Domain knowledge suggests a polynomial relationship (physics, engineering)
When to Avoid Polynomial Regression
- When you have very few data points relative to the chosen degree
- When the relationship is naturally exponential, logistic, or periodic
- When higher-degree wiggles are not physically interpretable
- When a simpler model (linear, quadratic, cubic) achieves nearly identical R²
Frequently Asked Questions
What is polynomial regression?
📐 Polynomial regression fits a polynomial of chosen degree n to your data: y = aₙxⁿ + ... + a₁x + a₀. It generalizes linear (degree 1) and quadratic (degree 2) regression.
How do I choose the right degree?
📊 Start with degree 2 or 3. Increase only if R² improves substantially and the curve remains smooth. Use the lowest degree that captures the pattern — overfitting produces meaningless wiggles.
What is the minimum number of data points?
📐 You need at least degree + 1 points. For a cubic (degree 3) you need 4 points; for degree 5 you need 6 points.
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