Quadratic Regression
Fit a quadratic regression model y = a + bx + cx² to paired X-Y data online. Free calculator with R², adjusted R², predicted values, residuals, scatter plot, and residual analysis.
About This Calculator
The Quadratic Regression Calculator fits a second-degree polynomial (parabola) to your paired X-Y data using the least-squares method. This tool is ideal for students, researchers, data analysts, and anyone needing to model curved relationships where a straight line is insufficient. Enter your data as comma-separated X and Y values, and the calculator finds the coefficients a, b, c of the quadratic equation y = a + bx + cx² that best fits your points.
The least-squares method minimizes the sum of squared vertical distances from each data point to the fitted parabola. The calculator solves a 3×3 system of normal equations derived from six sums: Σx, Σx², Σx³, Σx⁴, Σxy, and Σx²y. Results include the full regression equation, R² (coefficient of determination), adjusted R², sum of squared errors (SSE), and a detailed breakdown table with predicted values and residuals for every data point. A scatter plot with the fitted parabola and a separate residual plot help you visually assess model quality.
Regional Notes
Global: Quadratic regression is a universal statistical method used across all regions for data analysis in fields such as economics, physics, biology, engineering, and social sciences. The mathematical formulation — solving normal equations via Cramer's rule or Gaussian elimination — is identical worldwide. This tool uses standard notation where x is the explanatory variable and y is the response variable, with no region-specific adaptations required.
Frequently Asked Questions
What is quadratic regression?
Quadratic regression is a statistical method that fits a parabola y = a + bx + cx² to a set of paired data points using the least-squares approach. It models relationships where the rate of change is not constant but follows a curved (quadratic) pattern.
How many data points do I need for quadratic regression?
You need at least three distinct paired data points (X, Y) to perform quadratic regression. With exactly three points, the parabola passes through every point perfectly (R² = 1.0). More points provide a statistically meaningful fit with a proper assessment of goodness-of-fit via R².
What is R² in quadratic regression?
R² (coefficient of determination) measures how well the quadratic model fits your data, ranging from 0 to 1. A value close to 1 indicates the parabola explains most of the variation in Y. For example, R² = 0.95 means the model explains 95% of the variance in the response variable.
What is the difference between quadratic regression and linear regression?
Linear regression fits a straight line y = a + bx, while quadratic regression fits a parabola y = a + bx + cx². Quadratic regression captures curved relationships with one bend, such as acceleration, diminishing returns, or parabolic trajectories. If the cx² term is near zero, the model reduces to linear regression.
What does adjusted R² mean in quadratic regression?
Adjusted R² modifies R² to account for the number of predictors in the model. Since quadratic regression uses two predictors (x and x²), adjusted R² penalizes the extra complexity, making it a more honest measure of fit especially for small sample sizes. It is always less than or equal to R².
What are residuals in quadratic regression?
Residuals are the differences between the actual Y values and the values predicted by the quadratic model. A good model has residuals randomly scattered around zero with no obvious pattern. The residual plot in this calculator helps you visually check whether the quadratic model is appropriate for your data.
Can I use quadratic regression for time series data?
Yes, quadratic regression can model time series with a parabolic trend, such as accelerating growth or a peak-and-decline pattern. However, extrapolating beyond the observed data range is risky because parabolas eventually turn around — always consider domain knowledge before forecasting.
How is the quadratic regression equation calculated?
The coefficients a, b, c are found by solving a 3×3 system of normal equations built from the sums of x, x², x³, x⁴, xy, and x²y. This is solved using Cramer's rule for a unique least-squares solution that minimizes the sum of squared vertical distances from data points to the parabola.