Understanding Interpolation & Extrapolation
Educational Tool
This tool demonstrates basic curve fitting concepts for learning purposes. It is not designed for professional forecasting, financial analysis, or safety-critical applications.
Interpolation vs Extrapolation
Interpolation
Estimating values within the range of your known data points. Generally more reliable because we're working in a region where we have information.
Extrapolation
Estimating values beyond the range of your data. Inherently riskier — the model may behave unpredictably outside known regions.
Think of it this way: interpolation is like guessing what happened between data points you've observed, while extrapolation is predicting what might happen in uncharted territory.
Fitting Methods
Linear Fit (Straight Line)
Model: ŷ = c₀ + c₁x (intercept + slope)
- • Best for data with roughly linear trends
- • Simple and robust — less prone to overfitting
- • Cannot capture curvature in the data
- • Extrapolates predictably (straight line continues)
Polynomial Fit
Model: ŷ = c₀ + c₁x + c₂x² + ... + cₐxᵈ
- • Higher degree = more flexibility to fit curves
- • Quadratic (d=2): Parabolic patterns
- • Cubic (d=3): S-shaped curves
- • ⚠ High degrees can oscillate wildly (Runge's phenomenon)
- • ⚠ Risk of overfitting, especially with few data points
Understanding R² (Coefficient of Determination)
R² measures how much of the variance in your data is explained by the fitted model:
- • R² = 1: Perfect fit — model explains all variance
- • R² = 0: Model is no better than a horizontal line at the mean
- • 0 < R² < 1: Model explains some but not all variance
Warning: High R² doesn't guarantee good predictions! A high-degree polynomial can achieve R² ≈ 1 by passing through every point, but may perform terribly for interpolation at new x-values or extrapolation.
Common Pitfalls
- Overfitting: Using too high a polynomial degree for the amount of data. The curve fits your points perfectly but generalizes poorly.
- Trusting extrapolation: Polynomials can behave wildly outside the data range. A quadratic might predict negative values where none make sense.
- Ignoring residuals: If residuals show a pattern (e.g., all positive for low x, all negative for high x), your model may be wrong.
- Too few data points: With only 3 points, a quadratic will fit exactly, but you have no idea if the true relationship is quadratic.
- Confusing correlation with prediction: A good fit to past data doesn't guarantee future predictions will be accurate.
Choosing the Right Fit
| Scenario | Recommended | Why |
|---|---|---|
| Roughly linear trend | Linear fit | Simple, robust, interpretable |
| Clear curvature (one bend) | Quadratic (d=2) | Captures parabolic shape |
| S-shape or inflection | Cubic (d=3) | Can model inflection points |
| Few data points (< 6) | Low degree | Avoid overfitting |
| Need to extrapolate | Linear or low degree | High degrees are unstable outside data |
Tool Limitations
- • Maximum polynomial degree is 6 for numerical stability
- • Maximum 20 data points (educational tool)
- • Does not compute confidence intervals or prediction bands
- • Least-squares fitting assumes errors are normally distributed
- • For serious work, use proper statistical software (R, Python/SciPy, MATLAB)
- • Do NOT use for financial, medical, or safety-critical predictions
Frequently Asked Questions
Related Math & Statistics Tools
Regression Calculator
Fit linear and polynomial regression models with R² and statistics
Smoothing & Moving Average
Apply SMA, EMA, and WMA to time series data for trend analysis
Numerical Root Finder
Find function roots using Newton-Raphson and Bisection methods
Descriptive Statistics
Calculate mean, median, standard deviation, and more
Linear Algebra Helper
Compute determinant, rank, trace, and eigenvalues
Calculus Calculator
Compute derivatives and integrals of functions
Probability Toolkit
Compute probabilities for various distributions