Quadratic Regression Calculator

Enter the values of X and Y variables to calculate the quadratic regression equation using this calculator.

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Quadratic regression calculator helps you determine the quadratic regression equation that best fits a given set of data points. This guide provides step-by-step instructions to make the analysis easier.

What Is Quadratic Regression?

In statistics: “Regression analysis is used to find the equation of a parabola that best fits the data points.”

Quadratic Regression Formula:

The quadratic regression equation can be expressed as:

$$ y = ax^{2} + bx + c $$

Mean:

For a set of X and Y values, calculate the means:

$$ \bar{x} = \frac{1}{n}\sum_{i=1}^n x_i $$
$$ \bar{x^2} = \frac{1}{n}\sum_{i=1}^n x_i^2 $$
$$ \bar{y} = \frac{1}{n}\sum_{i=1}^n y_i $$

Summations:

Compute these summations for regression coefficients:

$$ S_{xx} = \sum (x_i - \bar{x})^2 $$

$$ S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) $$

$$ S_{xx^2} = \sum (x_i - \bar{x})(x_i^2 - \bar{x^2}) $$

$$ S_{x^2 x^2} = \sum (x_i^2 - \bar{x^2})^2 $$

$$ S_{x^2 y} = \sum (x_i^2 - \bar{x^2})(y_i - \bar{y}) $$

Coefficients:

Determine the coefficients of the quadratic equation:

$$ b = \frac{S_{xy} S_{x^2 x^2} - S_{x^2 y} S_{xx^2}}{S_{xx} S_{x^2 x^2} - (S_{xx^2})^2} $$

$$ c = \frac{S_{x^2 y} S_{xx} - S_{xy} S_{xx^2}}{S_{xx} S_{x^2 x^2} - (S_{xx^2})^2} $$

$$ a = \bar{y} - b\bar{x} - c\bar{x^2} $$

How to Find Quadratic Regression?

Example:

Determine the quadratic regression for the dataset:

$$ (12, 13), (11, 17), (14, 11), (9, 12), (2, 11), (13, 10) $$

Step 1: Calculate the mean of X and Y

$$ X = 12, 11, 14, 9, 2, 13 $$

$$ Y = 13, 17, 11, 12, 11, 10 $$

$$ \bar{x} = \frac{12+11+14+9+2+13}{6} = 10.166 $$

$$ \bar{y} = \frac{13+17+11+12+11+10}{6} = 12.33 $$

Step 2: Calculate mean of X squared

$$ \bar{x^2} = \frac{12^2 + 11^2 + 14^2 + 9^2 + 2^2 + 13^2}{6} = 119.166 $$

Arrange intermediate calculations in the table:

$(x_i - \bar{x})^2$	$(x_i - \bar{x})(y_i - \bar{y})$	$(x_i - \bar{x})(x_i^2 - \bar{x^2})$	$(x_i^2 - \bar{x^2})^2$	$(x_i^2 - \bar{x^2})(y_i - \bar{y})$
3.36	1.223	45.519	616.678	16.564
0.694	3.888	1.527	3.36	8.555
14.692	-5.109	294.501	5903.31	-102.418
1.362	0.389	44.541	1456.72	12.71
66.7	10.887	940.569	13263.438	153.518
8.026	-6.609	141.177	2483.328	-116.26