Quadratic Regression Calculator

The calculator will find the quadratic of best fit for the given set of paired data using the least squares method, with steps shown.

Related calculator: Linear Regression Calculator

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Find the parabola of best fit for $$$\left\{\left(1, 0\right), \left(4, 5\right), \left(6, 2\right), \left(7, 1\right), \left(3, -3\right)\right\}$$$.

Solution

The number of observations is $$$n = 5$$$.

Generate the following table:

	$$$x$$$	$$$y$$$	$$$x y$$$	$$$x^{2}$$$	$$$x^{2} y$$$	$$$x^{3}$$$	$$$x^{4}$$$	$$$y^{2}$$$
	$$$1$$$	$$$0$$$	$$$0$$$	$$$1$$$	$$$0$$$	$$$1$$$	$$$1$$$	$$$0$$$
	$$$4$$$	$$$5$$$	$$$20$$$	$$$16$$$	$$$80$$$	$$$64$$$	$$$256$$$	$$$25$$$
	$$$6$$$	$$$2$$$	$$$12$$$	$$$36$$$	$$$72$$$	$$$216$$$	$$$1296$$$	$$$4$$$
	$$$7$$$	$$$1$$$	$$$7$$$	$$$49$$$	$$$49$$$	$$$343$$$	$$$2401$$$	$$$1$$$
	$$$3$$$	$$$-3$$$	$$$-9$$$	$$$9$$$	$$$-27$$$	$$$27$$$	$$$81$$$	$$$9$$$
$$$\sum$$$	$$$21$$$	$$$5$$$	$$$30$$$	$$$111$$$	$$$174$$$	$$$651$$$	$$$4035$$$	$$$39$$$

The parabola of best fit is $$$y = a x^{2} + b x + c$$$.

$$$a = \frac{(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^2)-(\sum x)^2)-(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = - \frac{3}{22}$$$

$$$b = \frac{(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^4)-(\sum x^2)^2)-(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 4035 - 111^{2}\right) - \left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = \frac{3}{2}$$$

$$$c = \frac{(\sum y)-b(\sum x)-a(\sum x^2)}{n} = \frac{5 - \left(\frac{3}{2}\right)\cdot \left(21\right) - \left(- \frac{3}{22}\right)\cdot \left(111\right)}{5} = - \frac{25}{11}$$$

Thus, the parabola of best fit is $$$y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}$$$.

Answer

The parabola of best fit is $$$y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}\approx - 0.136363636363636 x^{2} + 1.5 x - 2.272727272727273.$$$A

Quadratic Regression Calculator

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