What this quadratic regression calculator does
This tool fits a second-degree polynomial model to your data in the form y = ax² + bx + c. It uses least squares estimation, which finds coefficient values for a, b, and c that minimize total squared error between observed and predicted values.
In practical terms, this helps when your scatter plot is clearly curved rather than linear. Instead of forcing a straight line, quadratic regression can capture acceleration, deceleration, and turning points in real-world processes like growth curves, dosage effects, pricing response, and simple motion trajectories.
How to use the calculator
1) Enter paired data
Put all x-values in the first box and all y-values in the second box. Keep the order consistent: the first x belongs to the first y, second x to second y, and so on.
2) Click “Calculate Regression”
The calculator returns:
- The fitted equation coefficients a, b, c
- The full model equation
- R² (coefficient of determination)
- RMSE (root mean squared error)
- Vertex location and concavity (if applicable)
3) Optional prediction
If you provide a single x in the prediction field, the tool also computes the corresponding predicted y.
Interpreting the output
- a (quadratic term): controls curvature. Positive means opens upward; negative means opens downward.
- b (linear term): tilts the curve left/right and contributes to slope.
- c (intercept): expected y when x = 0.
- R²: proportion of variance explained by the model (closer to 1 is better fit).
- RMSE: average magnitude of prediction error in y-units (smaller is better).
When quadratic regression is a good choice
Use a quadratic model when:
- Your data shows one clear bend (single turning point).
- A linear fit leaves systematic curved residual patterns.
- You need a simple nonlinear model with interpretable parameters.
You may need a different model when:
- There are multiple bends or periodic patterns (consider cubic, spline, or sinusoidal models).
- Response grows exponentially (consider log-linear or exponential models).
- Variance changes strongly across x (consider weighted regression).
Example dataset
Click Load Example Data to see a sample where y rises faster as x increases. The fitted equation should produce a positive a, indicating an upward-opening parabola.
Tips for accurate results
- Use at least 5-6 data points for more stable estimates.
- Check for obvious outliers before fitting.
- Avoid making predictions far outside your observed x-range.
- Plot your data and fitted curve whenever possible.
Frequently asked questions
Can I use negative x-values?
Yes. The calculator accepts any real numeric x and y values.
What if all x-values are the same?
Then the system is singular and coefficients cannot be estimated reliably. You need variation in x for regression.
Does this support weighted quadratic regression?
No, this page performs standard unweighted least squares. Weighted versions require additional inputs and equations.