What Is Curve Fitting?
Curve fitting means choosing a function and estimating its parameters so that it follows your data. Different shapes suit different patterns: a polynomial for general bends, an exponential for growth or decay, a power law for scaling and a logarithmic curve for slowing growth.
Comparing models by R² alone favours complex models, so the calculator also reports adjusted R² and AICc, which penalise extra parameters, and tells you which model AICc prefers. For a fit that is not reducible to a straight line, see the nonlinear regression calculator.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as how to choose between models and why extrapolation is risky.
How to Use the Curve Fitting Calculator
- Enter x and y values in matching order (at least 4 points).
- Leave the model on “Automatic” to feature the best by AICc, or choose a specific model.
- Check the comparison table and the scatter plot. A good fit follows the points without wild swings.
- Optionally enter an x to predict y. Be cautious outside the range of your data.
Curve Fitting Formula
Polynomial of degree d
Exponential
Power
Logarithmic
Corrected Akaike criterion
Curve Fitting Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x, y | Your paired data | |
| ŷ | Fitted value from the model | |
| R² | Share of variation explained | 1 − SSE ÷ SST |
| AICc | Small-sample corrected AIC | Lower is better |
| RMSE | Root mean squared error | Same units as y |
Curve Fitting Worked Example
With the example values loaded in the calculator above, the steps are:
- Each model is fitted by least squares: polynomials directly, and exponential, power and logarithmic models by transforming the data with logarithms and fitting a straight line.
- Models are then compared on the original y scale using SSE, R², RMSE and AICc, which penalises extra parameters to avoid overfitting.
- Selected model: Exponential. y = 0.99798 · e^(1.0003x).
- R² = 1 − SSE ÷ SST = 1 − 340.56 ÷ 439,675,520 = 1.
Result: Best model (lowest AICc): Exponential = y = 0.99798 · e^(1.0003x); R² = 1.
How to Read the Model Comparison
The sample data grow by a factor of about 2.7 at each step, and the exponential model recovers that: y = 0.998 · e^(1.0003x), a multiplier of e^1.0003 = 2.72 per unit of x. R² is 1, yet the RMSE is 5.84. There is no contradiction: the largest y value is 22,026, so an error of a few units is tiny in relative terms.
R² alone always favours a model with more parameters. AICc, shown here, adds a penalty for extra parameters and is the sensible way to choose between models of different sizes. Before trusting a curve outside the range of your data, remember that a good fit says nothing about whether the same pattern continues.
Choosing the right type of curve
Draw the scatter plot first. The shape usually tells you which families are plausible, and theory about the process should guide the choice more than R² alone.
| Pattern in the scatter plot | Try | Typical example |
|---|---|---|
| Straight line | Linear | Cost vs quantity |
| One bend, parabola | Quadratic | Projectile height, profit vs price |
| Rapid growth or decay by constant percentage | Exponential | Population, compound interest, radioactive decay |
| Straight on a log-log plot | Power | Area vs length, allometric scaling |
| Fast then levelling off | Logarithmic | Learning curves, diminishing returns |
| S-shaped | Logistic (use non-linear regression) | Adoption, saturation |
Overfitting: why R² alone is not enough
A polynomial of degree n − 1 passes exactly through n points and gives R² = 1, yet it is useless for prediction because it wiggles between the points. Adjusted R² and AICc penalise complexity. Prefer the simplest model that captures the pattern, and when you can, test the model on data it was not fitted to.
How the fits are computed
Polynomials are fitted by least squares on centred data for numerical stability. Exponential and power curves are fitted by taking logarithms and fitting a straight line, which is fast and robust but minimises error on the log scale; for the exact least squares fit on the original scale, use the non-linear regression calculator. All models are then compared using errors measured on the original y values.
Cautions
- Extrapolating polynomial curves beyond the data can give absurd values.
- Exponential and power fits require positive y (and positive x for power and log).
- Outliers can distort a fit; check residuals.
- A good fit does not prove the model reflects the underlying mechanism.
Where Curve Fitting Is Used
Choosing a model
Compare several curves by AICc, R² and RMSE.
Prediction
Estimate y at a new x with the best model.
Growth and decay
Test whether data follow an exponential or power law.
Avoiding overfitting
Use AICc to penalise extra parameters.
Common Curve Fitting Mistakes to Avoid
- Use the lowest degree that fits well.
- Compare models by AICc, not R² alone.
- Transform checks: a straight line on semi-log paper suggests exponential; on log-log suggests a power law.
Curve Fitting Terms and Definitions
- Overfitting
- A model so flexible that it fits noise rather than the underlying pattern.
- AICc
- Akaike information criterion corrected for small samples; lower is better.
- RMSE
- Root mean squared error of the fit, in the units of y.
Curve Fitting FAQs
What is curve fitting?
Finding a function whose graph follows your data points as closely as possible, using a criterion such as least squares.
How do I choose between a polynomial and an exponential fit?
Look at the plot and the physical process. Compare AICc and adjusted R². If growth is by a constant percentage, exponential is usually right.
Why does a high-degree polynomial fit so well but predict badly?
It adapts to random noise in the points and oscillates between and beyond them. This is overfitting.
What does R² mean for a curve fit?
The share of the variation in y explained by the curve. It is 1 for a perfect fit, though that may be overfitting.
Why do some models not appear?
Exponential and power models need positive y, and power and logarithmic models need positive x.
Can I get the exact least squares exponential fit?
This tool fits via logarithms. For an exact non-linear least squares fit of y = Ae^{Bx}, use the non-linear regression calculator.
How does the calculator choose the best model?
It fits several curve types and compares them with AICc, which rewards fit but penalises extra parameters. The lowest AICc wins.
Why not just pick the model with the highest R²?
More flexible models can always raise R² by fitting noise. AICc guards against this overfitting.
Can I fit one specific model instead of auto?
Yes. Choose the model type directly and the tool fits it and reports R², RMSE and SSE.
Can I predict a y value from a new x?
Yes. Enter an x value and the tool evaluates the fitted equation there. Be careful predicting outside the range of your data.