Quadratic Regression Calculator
The best-fit parabola through a set of points.
At least three points, and the two lists must be the same length.
What it does
Quadratic regression finds the parabola y = ax² + bx + c that comes closest to a set of points, where 'closest' means the sum of the squared vertical distances is as small as possible. Those distances are the residuals.
Least squares
Squaring the residuals does two jobs: it stops positive and negative errors cancelling, and it penalises large misses more heavily than small ones. Minimising that sum produces three simultaneous equations in a, b and c — the normal equations — which the calculator solves by Gaussian elimination with partial pivoting.
Reading R squared
R² is the proportion of the variation in y that the curve accounts for. 1.0 is a perfect fit through every point; 0 means the curve does no better than a horizontal line at the mean.
A high R² is not proof the model is right. Adding curvature will always fit at least as well as a straight line, so R² can rise simply because the model has more freedom. Look at the residuals: if they show a pattern rather than scatter, the model is missing something.
When a quadratic is the wrong choice
Fit a parabola only when the relationship genuinely turns — projectile height, area against length, a rate with a maximum. Exponential growth fitted with a quadratic will look convincing over a short range and then fail badly outside it. Extrapolating a regression beyond the data is where most of the damage happens.
Three points exactly
Three non-collinear points with distinct x values determine one parabola exactly, so R² will be 1 by construction. That is interpolation, not evidence of a relationship.