Enter your X data points and Y data points as comma-separated values to fit a quadratic regression curve (Y = ax² + bx + c) to your data. The Quadratic Regression Calculator returns the coefficients a, b, and c, plus the R² value so you can assess how well the parabola fits your dataset.
Quadratic Equation (Y = ax² + bx + c)
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Coefficient a
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Coefficient b
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Coefficient c
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R² (Goodness of Fit)
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Number of Data Points
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Quadratic regression is a statistical method that fits a second-degree polynomial curve (a parabola) to a set of data points. The resulting equation takes the form Y = ax² + bx + c, where a, b, and c are constants determined to minimize the sum of squared differences between the observed and predicted Y values.
The calculator uses the method of least squares to find the coefficients a, b, and c that best fit your data. It sets up a system of normal equations using sums of powers of X and solves a 3×3 linear system. The result is the parabola that minimizes total squared error across all your data points.
Enter your X values in the first field and your corresponding Y values in the second field, both separated by commas. Make sure the number of X and Y values match. The calculator will immediately display the equation coefficients a, b, c, the R² value, and a table of fitted vs. actual values.
Quadratic regression is best when your data follows a curved, parabolic trend rather than a straight line — for example, projectile motion, profit optimization, or any phenomenon that rises and then falls (or vice versa). If a scatter plot of your data looks like a U-shape or an inverted U, quadratic regression is a strong candidate.
R² (the coefficient of determination) measures how well the fitted parabola explains the variability in your Y data. A value of 1.0 means a perfect fit, while 0.0 means the model explains none of the variance. Generally, R² above 0.9 indicates a strong fit for most practical applications.
Linear regression fits a straight line (Y = mx + b) and assumes a constant rate of change. Quadratic regression fits a curved parabola (Y = ax² + bx + c) and captures non-linear relationships where the rate of change itself increases or decreases. If your data curves, quadratic regression will produce a much better fit than linear regression.
Advantages include the ability to model curved, non-linear data that linear regression cannot capture, and relative simplicity compared to higher-degree polynomials. Disadvantages include potential overfitting with small datasets, sensitivity to outliers, and the risk of extrapolating wildly outside the range of your input data.
You need at least 3 data points to uniquely determine the three coefficients a, b, and c. However, for statistically meaningful results, using 6 or more data points is strongly recommended. More data points generally lead to a more reliable and stable regression equation.