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Quadratic Regression Calculator

Quadratic Regression Calculator

Enter X Numbers: (separated by commas)

Enter Y Numbers: (separated by commas)

An online quadratic regression calculator helps you to determine the quadratic regression equation representing the parabola that best suits the data points. We have arranged a proper guide in this content so that you may not face any hurdle while doing such analysis.

Stay focused!

What Is Quadratic Regression?

In statistical analysis:

“A particular operation that is performed on a set of data points to find the equation of the parabola is known as regression analysis”

You can consider it as an advancement of linear regression.

Quadratic Regression Formula:

You can work for the quadratic regression equations in the following form:

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

Our free quadratic regression calculator determines the equation in the same form but in a fraction of seconds to save your precious time.

So be ready as we are going to start our roller coaster of various formulae that you need to memorize during this analysis. Keep scrolling!

Mean:

As we have x and y values in the points defined, so we have to determine the mean for both x and y values as follows:

$$ \bar{x} = \frac{1}{n}\sum_{i=1}^nx_{i} $$

$$ \bar{x^{2}} = \frac{1}{n}\sum_{i=1}^nx_{i}^{2} $$

$$  \bar{y} = \frac{1}{n}\sum_{i=1}^ny_{i} $$

Summations:

After doing so, we need to calculate a series of sums with the help of the following formulae:

$$ S_{xx} = \sum_{i=1}^n \left(x_{i} – \bar{x}\right)^2 $$

$$ S_{xy} = \sum_{i=1}^n \left(x_{i} – \bar{x}\right) \left(y_{i} – \bar{y}\right) $$

$$ S_{xx^{2}} = \sum_{i=1}^n \left(x_{i} – \bar{x}\right) \left(x_{i}^2 – \bar{x^{2}}\right) $$

$$ S_{x^{2}x^{2}} = \sum_{i=1}^n \left(x_{i}^2 – \bar{x^{2}}\right)^2 $$

$$ S{x^{2}y} = \sum_{i=1}^n \left(x_{i}^2 – \bar{x^{2}}\right) \left(y_{i} – \bar{y}\right) $$

Coefficients:

Next, we need to determine the coefficients of the equation as follows:

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

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

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

How To Find Quadratic Regression Equations?

Let us solve an example so that you understand the proper use of all formulae to determine the equation. Just stay connected!

Example:

Determine quadratic regression equation for the following data set of points:

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

Solution:

From the data set given, we can separate the values of X and Y as follows:

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

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

First of all, we have to determine the mean of both X and Y values:

$$ Mean X = \bar{x} = \frac{1}{n}\sum_{i=1}^nx_{i} $$

$$ mean X = \bar{x} =\frac{1}{n} \left(12 + 11 + 14 + 9 + 2 + 13\right) $$

$$ Mean X = \bar{x} = \frac{\left(12 + 11 + 14 + 9 + 2 + 13\right)}{6} $$

$$ Mean X = \bar{x} = \frac{61}{6} $$

$$ Mean X = \bar{x} = 10.166 $$

Now we have:

$$ Mean Y = \bar{y} = \frac{1}{n}\sum_{i=1}^ny_{i} $$

$$ Mean Y = \bar{y} = \frac{1}{n} \left(13 + 17 + 11 + 12 + 11 + 10\right) $$

$$ Mean Y = \bar{y} = \frac{\left(13 + 17 + 11 + 12 + 11 + 10\right)}{6} $$

$$ Mean Y = \bar{y} = \frac{74}{6} $$

$$ Mean Y = \bar{y} = 12.33 $$

Also we have:

$$ \bar{x^{2}} = \frac{1}{n}\sum_{i=1}^nx_{i}^{2} $$

$$ \bar{x^{2}} = \frac{1}{n} \left(12 + 11 + 14 + 9 + 2 + 13\right)^2 $$

$$ \bar{x^{2}} = \frac{\left(61\right)^2}{6} $$

$$ \bar{x^{2}} = \frac{3721}{6} $$

$$ \bar{x^{2}} = 620.16 $$

Now we need to calculate the following values and arrange them in the table just like below:

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

Calculating important summations as follows:

$$ S_{xx} = \sum_{i=1}^n \left(x_{i} – \bar{x}\right)^2 $$

$$ S_{xx} = 3.36 + 0.694 + 14.692 + 1.3612 + 66.7 + 8.026 $$

$$ S_{xx} = 94.83 $$

$$ S_{xy} = \sum_{i=1}^n \left(x_{i} – \bar{x}\right) \left(y_{i} – \bar{y}\right) $$

$$ S_{xy} = 1.223 + 3.888 + (-5.109) + 0.389 + 10.887 + (-6.609) $$

$$ S_{xy} = 1.223 + 3.888 – 5.109 + 0.389 + 10.887 – 6.609 $$

$$ S_{xy} = 4.67 $$

$$ S_{xx^{2}} = \sum_{i=1}^n \left(x_{i} – \bar{x}\right) \left(x_{i}^2 – \bar{x^{2}}\right) $$

$$ S_{xx^{2}} = 45.519 + 1.527 + 294.501 + 44.541 + 940.569 + 141.177 $$

$$ S_{xx^{2}} = 1467.83 $$

$$ S_{x^{2}x^{2}} = \sum_{i=1}^n \left(x_{i}^2 – \bar{x^{2}}\right)^2 $$

$$ S_{x^{2}x^{2}} = 616.678 + 3.36 + 5903.31 + 1456.72 + 13263.438 + 2483.328 $$

$$ S_{x^{2}x^{2}} = 23726.83 $$

$$ S{x^{2}y} = \sum_{i=1}^n \left(x_{i}^2 – \bar{x^{2}}\right) \left(y_{i} – \bar{y}\right) $$

$$ S{x^{2}y} = 16.564 + 8.555 + (-102.418) + 12.71 + 153.518 + (-116.26) $$

$$ S{x^{2}y} = 16.564 + 8.555 – 102.418 + 12.71 + 153.518 – 116.26 $$

$$ S{x^{2}y} = -27.33 $$

Determining the coefficients of the equation:

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

$$ b = \frac{\left(4.67\right) \left(23726.83\right) + \left(27.33\right) \left(1467.83\right)}{\left(94.83\right) \left(23726.83\right) – \left(1467.83\right)^2} $$

$$ b = \frac{110804.2961 + 40115.7939}{2250015.2889 – 2154524.9089} $$

$$ b = \frac{150920.09}{95490.38} $$

$$ b = 1.580 $$

Now we have:

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

$$ c = \frac{\left(-27.33\right) \left(94.83\right) – \left(4.67\right) \left(1467.83\right)}{\left(94.83\right) \left(23726.83\right) – \left(1467.83\right)^2} $$

$$ c = \frac{-2591.7039 – 6854.7661}{2250015.2889 – 2154524.9089} $$

$$ c = \frac{-9446.47}{95490.38} $$

$$ c = -0.098 $$

Now we have:

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

$$ a = 12.33 – \left(1.580\right) \left(10.167\right) – \left(-0.098\right) \left(103.367889\right) $$

$$ a = 12.33 – 16.06386 + 10.130053122 $$

$$ a = 8.05845 $$

At last, we have to find correlation coefficient as follows:

$$ \text{Correlation Coefficient} = r = \frac{n \left(\sum xy\right) – \left(\sum x\right) \left(\sum y\right)}{\sqrt([n\sum x^{2} – \left(\sum x\right)^2][n\sum y^{2} – \left(\sum y\right)^2])} $$

$$ \text{Correlation Coefficient} = r = 0.3213 $$ (for calculations, click Correlation Coefficient Calculator)

Now the quadratic regression equation is as follows:

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

$$ y = 8.05845x^{2} + 1.57855x – 0.09881 $$

Which is our required answer.

Apart from these lengthy calculations, our free online quadratic regression calculator determines the same results with each step properly performed within seconds.

How Quadratic Regression Calculator Works?

Whenever you are doing complex statistical analysis, you look for better and fast results. No doubt it is a little bit difficult to perform complex calculations by hand. That is why we have programmed this free quadratic regression equation calculator so that you may use it to get accurate outputs and absolutely for free. Want to know proper use of it?

Keep scrolling!

Input:

  • Enter all values for x separated by commas.
  • Enter the values of y in the same way
  • Hit the ‘calculate’ button.

Output:

The free quadratic table calculator performs the following analysis:

  • Mean value of x and y
  • Results against all summations that we need to perform
  • Values corresponding to coefficients a, b, and c
  • Value of correlation coefficient r
  • And the final and most important of all which is the quadratic regression equation.
  • The whole analysis is displayed in the form of well-organized table

FAQ’s:

What is the best value considered for correlation coefficient?

According to the principle defined for correlation coefficients, a value that is more close to the 1 is considered as the best.

What do you mean by the term ‘critical value”?

A specific value that distributes your graph into two section i.e; one is rejected and other one is that which is accepted is known as  critical value.

What is the generic form of the cubic regression equation?

The following equation shows the general form of the cubic regression below:

$$ y = a + bx + cx^{2} + dx^{3} $$

Where:

a, b, c, and d are the coefficients of the regression equation respectively.

What is the purpose of the regression analysis?

The principle purpose of the regression analysis is to analyze a considerable effect of the independent variable on the dependent variable.

Conclusion:

Regression analysis is the best way to depict the trends in any data set. Professionals and scholars use our free online quadratic regression calculator to do various predictions in data schemes. There is a vast use of quadratic regression in the field of finance management and forecasting.

References:

From the source of wikipedia: Quadratic equation, Discriminant, Alternative methods of root calculation

From the source of khan academy: Parabolas, quadratic graph

From the source of lumen learning: Multiple Regression, Null Hypothesis, Importance of Slope, Evaluating Model Utility, Interaction Models, Qualitative Variable Models, ANOVA Models, Polynomial Regression