Quadratic Regression Calculator

Fit quadratic curves to data and calculate regression coefficients with R²

About the Quadratic Regression Calculator

Quadratic regression is a statistical technique used to find the equation of a parabola that best represents a set of data points. Unlike linear regression, which assumes a constant rate of change, quadratic regression is ideal for modeling relationships where the data reaches a maximum or minimum value and then reverses direction. This creates a U-shaped or inverted U-shaped curve on a scatter plot. Scientists, engineers, and economists use this tool to analyze trends that exhibit curvature, such as the trajectory of a projectile, the growth rate of a population facing resource limits, or the relationship between speed and fuel efficiency in vehicles.

This calculator processes your (x, y) coordinate pairs to determine the specific values for the coefficients a, b, and c in the standard quadratic form. Beyond just providing the equation, the tool calculates the R-squared value, which serves as a statistical measure of how close the data is to the fitted regression line. This is essential for determining if a curved model is truly appropriate for your dataset or if the relationship might be better explained by a different mathematical function.

Formula

y = ax² + bx + c

In this equation, 'y' is the dependent variable and 'x' is the independent variable. The coefficients are calculated using the method of least squares to minimize the sum of the squares of the vertical deviations between each data point and the curve. 'a' represents the quadratic coefficient (determining the curvature), 'b' is the linear coefficient, and 'c' is the y-intercept (the constant term).

Worked examples

Example 1: A researcher measures the height of a small projectile at three intervals: (1, 3), (2, 4), and (3, 3).

1. Set up the system of normal equations based on the sums of x, x², x³, x⁴, y, xy, and x²y.\n2. Calculate the sums: Σx=6, Σx²=14, Σx³=36, Σx⁴=98, Σy=10, Σxy=20, Σx²y=50.\n3. Solve the resulting matrix for coefficients a, b, and c.\n4. Resulting 'a' is -0.5, 'b' is 2.5, and 'c' is 1.0.

Result: y = -0.5x² + 2.5x + 1.0 with R² = 1.0. This is a perfect fit representing an inverted parabola.

Common use cases

Predicting the yield of a crop based on the amount of fertilizer applied, where too much fertilizer eventually decreases yield.
Estimating the height of a ball over time after it has been thrown into the air.
Analyzing the optimal temperature for a chemical reaction where efficiency peaks at a specific point.
Modeling the relationship between a person's age and their average annual income.

Pitfalls and limitations

Extrapolating data far beyond the range of your input points can lead to highly inaccurate predictions because parabolas grow or decay rapidly at their ends.
A high R-squared value does not guarantee that your data is actually quadratic; it only means the curve fits the specific points provided.
Outliers can disproportionately pull the curve away from the rest of the data, significantly skewing the 'a' coefficient.

Frequently asked questions

How do I know if a quadratic model fits my data well?

You can use the coefficient of determination, or R-squared (R²). A value close to 1.0 indicates that the quadratic curve is a very strong fit for your data points, while a value near 0 means the model does not explain the variability of the data.

difference between linear and quadratic regression

Linear regression fits a straight line (y = mx + b), while quadratic regression fits a curved parabola (y = ax² + bx + c). You should choose quadratic regression when your data shows a 'U' shape or a change in direction that a straight line cannot capture.

what does a negative leading coefficient mean in regression?

A negative 'a' coefficient means the parabola opens downward, suggesting the data reaches a peak and then decreases. An 'a' value of zero would mean the relationship is actually linear, not quadratic.

is quadratic regression the same as polynomial regression?

While both are types of polynomial regression, quadratic regression is specifically a polynomial of degree two. Higher-order polynomials (cubic, quartic) can fit more complex wiggles in data but are more prone to overfitting than a simple quadratic curve.

minimum number of points for quadratic regression

Quadratic regression requires at least three distinct data points to solve for the three unknown coefficients (a, b, and c). However, using only three points will always result in a perfect R² of 1.0, so more points are needed for meaningful statistical analysis.