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Kalkulator Regresi Kuadratik
Temukan parabola dengan kecocokan terbaik langkah demi langkah
Kalkulator akan menemukan kurva kuadrat dengan kecocokan terbaik untuk himpunan data berpasangan yang diberikan menggunakan metode kuadrat terkecil, dengan langkah-langkah yang ditunjukkan.
Kalkulator terkait: Kalkulator Regresi Linier, Kalkulator Regresi Kubik
Masukan Anda
Temukan parabola dengan kecocokan terbaik untuk $$$\left\{\left(1, 0\right), \left(4, 5\right), \left(6, 2\right), \left(7, 1\right), \left(3, -3\right)\right\}$$$.
Solusi
Jumlah pengamatan adalah $$$n = 5$$$.
Buat tabel berikut:
| $$$x$$$ | $$$y$$$ | $$$x y$$$ | $$$x^{2}$$$ | $$$x^{2} y$$$ | $$$x^{3}$$$ | $$$x^{4}$$$ | $$$y^{2}$$$ | |
| $$$1$$$ | $$$0$$$ | $$$0$$$ | $$$1$$$ | $$$0$$$ | $$$1$$$ | $$$1$$$ | $$$0$$$ | |
| $$$4$$$ | $$$5$$$ | $$$20$$$ | $$$16$$$ | $$$80$$$ | $$$64$$$ | $$$256$$$ | $$$25$$$ | |
| $$$6$$$ | $$$2$$$ | $$$12$$$ | $$$36$$$ | $$$72$$$ | $$$216$$$ | $$$1296$$$ | $$$4$$$ | |
| $$$7$$$ | $$$1$$$ | $$$7$$$ | $$$49$$$ | $$$49$$$ | $$$343$$$ | $$$2401$$$ | $$$1$$$ | |
| $$$3$$$ | $$$-3$$$ | $$$-9$$$ | $$$9$$$ | $$$-27$$$ | $$$27$$$ | $$$81$$$ | $$$9$$$ | |
| $$$\sum$$$ | $$$21$$$ | $$$5$$$ | $$$30$$$ | $$$111$$$ | $$$174$$$ | $$$651$$$ | $$$4035$$$ | $$$39$$$ |
$$$a = \frac{(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^2)-(\sum x)^2)-(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = - \frac{3}{22}$$$
$$$b = \frac{(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^4)-(\sum x^2)^2)-(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 4035 - 111^{2}\right) - \left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = \frac{3}{2}$$$
$$$c = \frac{(\sum y)-b(\sum x)-a(\sum x^2)}{n} = \frac{5 - \left(\frac{3}{2}\right)\cdot \left(21\right) - \left(- \frac{3}{22}\right)\cdot \left(111\right)}{5} = - \frac{25}{11}$$$
Parabola yang paling cocok adalah $$$y = a x^{2} + b x + c$$$.
Jadi, parabola yang paling sesuai adalah $$$y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}$$$.
Jawaban
Parabola yang paling cocok adalah $$$y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}\approx - 0.136363636363636 x^{2} + 1.5 x - 2.272727272727273.$$$A