Table of Contents
What Is Quadratic Regression?
Quadratic regression fits a second-degree polynomial curve (parabola) to a set of data points. Unlike linear regression which fits a straight line, quadratic regression can model curved relationships where the rate of change itself changes. The model has the form y = ax² + bx + c.
This technique is widely used in physics (projectile motion), economics (diminishing returns), biology (growth curves), and engineering. If your data shows curvature that a straight line cannot capture, quadratic regression may provide a significantly better fit.
The Least Squares Method
The coefficients a, b, and c are found by solving a system of three normal equations derived from minimizing the sum of squared residuals:
The R² (coefficient of determination) measures how well the quadratic model fits the data, ranging from 0 (no fit) to 1 (perfect fit).
When to Use Quadratic vs Linear
| Feature | Linear Regression | Quadratic Regression |
|---|---|---|
| Model | y = mx + b | y = ax² + bx + c |
| Shape | Straight line | Parabola |
| Min data points | 2 | 3 |
| Best for | Constant rate of change | Accelerating/decelerating trends |
Frequently Asked Questions
How many data points do I need?
You need a minimum of 3 data points to fit a quadratic model (since there are 3 coefficients). However, for reliable results with meaningful R² values, at least 5-10 data points are recommended. More data generally yields a more reliable model.
What does R² tell me?
R² (coefficient of determination) indicates how well the quadratic model explains the variation in your data. An R² of 0.95 means 95% of the variation is explained by the model. Values above 0.9 generally indicate an excellent fit. If R² is low, consider whether the relationship is truly quadratic or requires a different model.