Free Quadratic Regression Calculator

What This Tool Offers

This quadratic regression calculator — also known as a parabola of best fit calculator, a second degree polynomial regression tool, or a quadratic least squares regression engine — finds the parabola that best fits your dataset. Whether you are working on a homework problem, a research project, or a data analysis task, this free online quadratic curve fitting calculator delivers the coefficients of the quadratic model along with a clear measure of how well the curve represents the points.

Understanding the Quadratic Regression Model

Quadratic regression (sometimes called parabola regression or second‑degree polynomial regression) is used when the relationship between an explanatory variable $x$ and a response variable $y$ follows a curved path that a straight line cannot describe. The objective is to determine an equation of the form

$$y = a + b\,x + c\,x^{2}$$

that minimizes the sum of squared residuals (the least‑squares principle). If the coefficient $c$ equals zero, the model reduces to simple linear regression.

The goodness of fit is assessed by the coefficient of determination $R^{2}$. It varies between 0 and 1, with values closer to 1 indicating that the parabola explains a high proportion of the total variation in $y$.

Obtaining the Coefficients

The least‑squares estimates of $a$, $b$, and $c$ can be derived by solving a set of three normal equations:

\begin{aligned} n\,a + b\sum x_i + c\sum x_i^{2} &= \sum y_i \$$4pt] a\sum x_i + b\sum x_i^{2} + c\sum x_i^{3} &= \sum x_i y_i \$$4pt] a\sum x_i^{2} + b\sum x_i^{3} + c\sum x_i^{4} &= \sum x_i^{2} y_i \end{aligned}

An equivalent approach uses explicit formulas based on deviations from the means. Define

$$\begin{aligned} S_{xx} &= \sum (x_i - \bar{x})^{2}, \\ S_{xy} &= \sum (x_i - \bar{x})(y_i - \bar{y}), \\ S_{xx^{2}} &= \sum (x_i - \bar{x})(x_i^{2} - \overline{x^{2}}), \\ S_{x^{2}x^{2}} &= \sum (x_i^{2} - \overline{x^{2}})^{2}, \\ S_{x^{2}y} &= \sum (x_i^{2} - \overline{x^{2}})(y_i - \bar{y}), \end{aligned}$$

where $\bar{x}$ and $\bar{y}$ are the sample means and $\overline{x^{2}} = \frac{1}{n}\sum x_i^{2}$. The coefficients then become

$$c = \frac{S_{x^{2}y}S_{xx} - S_{xy}S_{xx^{2}}}{S_{xx}S_{x^{2}x^{2}} - (S_{xx^{2}})^{2}}, \qquad b = \frac{S_{xy} - c\,S_{xx^{2}}}{S_{xx}}, \qquad a = \bar{y} - b\,\bar{x} - c\,\overline{x^{2}}.$$

These formulas constitute the core of quadratic least‑squares regression and produce identical results to solving the matrix form of the normal equations.

How to Use This Parabola of Best Fit Calculator

Enter your data points (up to 30 pairs) in the input fields; new rows will appear automatically as you type. A minimum of three distinct points is needed to determine a unique parabola (with exactly three points the fit will be perfect). The calculator instantly generates a scatter plot with the fitted curve overlaid, and displays the quadratic regression equation together with the $R^{2}$ value. You can adjust the number of significant figures in the output to increase or decrease the precision.

Step‑by‑Step Example: Fitting a Parabola by Hand

Suppose we have the points $(4,5),\; (3,6),\; (2,4),\; (1,8)$.

1. Basic statistics

$$n = 4,\quad \bar{x} = 2.5,\quad \bar{y} = 5.75,\quad \overline{x^{2}} = 7.5$$

2. Deviation‑based sums

Term	$S_{xx}$	$S_{xy}$	$S_{xx^{2}}$	$S_{x^{2}x^{2}}$	$S_{x^{2}y}$
(1,8)	2.25	-1.125	12.75	72.25	-6.375
(2,4)	0.25	0.125	0.75	2.25	0.375
(3,6)	0.25	0.875	1.75	12.25	6.125
(4,5)	2.25	-3.375	9.75	42.25	-14.625
Total	5	-3.5	25	129	-14.5

3. Coefficient calculations

$$c = \frac{(-14.5)(5) - (-3.5)(25)}{(5)(129) - (25)^{2}} = \frac{-72.5 + 87.5}{645 - 625} = \frac{15}{20} = 0.75$$ $$b = \frac{-3.5 - 0.75 \cdot 25}{5} = \frac{-3.5 - 18.75}{5} = \frac{-22.25}{5} = -4.45$$ $$a = 5.75 - (-4.45)(2.5) - 0.75(7.5) = 5.75 + 11.125 - 5.625 = 11.25$$

4. Regression equation

$$y = 11.25 - 4.45\,x + 0.75\,x^{2}$$

This quadratic function produces the best (least‑squares) fit for the given data. By substituting the computed coefficients back into the normal equations, you can verify that they satisfy the system and yield a high $R^{2}$.