Enter the values of X and Y variavles to calcualte the quadratic regression equation through this calculator.
In statistical evaluation: “a selected operation that is achieved on a fixed of information factors to discover the equation of the parabola is called regression analysis”
you could work for the quadratic regression equations in the following form: $$ y = ax^{2} + bx + c $$
Mean:
As we've x and y values inside 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 assist 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:
subsequent, we need to decide the coefficients of the equation as follows:
$$ a = \bar{y}-b\bar{x}-c\bar{x^2} $$
$$ b = \dfrac{S_{xy}S_{x^2x^2}-S_{x^2y}S_{xx^2}}{S_{xx}S_{x^2x^2}-(S_{xx^2})^2} $$
$$ c = \dfrac{S_{x^2y}S_{xx}-S_{xy}S_{xx^2}}{S_{xx}S_{x^2x^2}-(S_{xx^2})^2} $$
Instance:
decide quadratic regression for the following facts set of factors:
$$ (12, 13), (11, 17), (14, 11), (9, 12), (2, 11), (13, 10) $$
Solution:
From the records set given, we will separate the values of X and Y as follows:
$$ X = 12, 11, 14, 9, 2, 13 $$
$$ y = 13, 17, 11, 12, 11, 10 $$
to start with, we have to determine the imply 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 want to calculate the subsequent values and set up 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 crucial 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've got:
$$ c = \dfrac{S_{x^2y}S_{xx}-S_{xy}S_{xx^2}}{S_{xx}S_{x^2x^2}-(S_{xx^2})^2} $$
$$ c = \frac{\left(-27.33\right) \left(94.83\right) - \left(4.67\right) \left(1467.83\right)}{\left(94.83\right) \left(23726.83\right) - \left(1467.83\right)^2} $$
$$ c = \frac{-2591.7039 - 6854.7661}{2250015.2889 - 2154524.9089} $$
$$ c = \frac{-9446.47}{95490.38} $$
$$ c = -0.098 $$
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 closing, we must 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 is as follows:
$$ y = ax^{2} + bx + c $$
$$ y = 8.05845x^{2} + 1.57855x - 0.09881 $$
Quadratic regression pertains to a polynomial regression method utilized for depicting associations between a variable in dependence and one in independence, assuming they display non-linear connectivity. It sets up a basic equation with squared values plus a non-squared value plus a constant for the data.
As opposed to linear regression, which aligns a straight line, quadratic regression aligns a parabolic curve. It helps to show data that has a curve more accurately, not just a straight line.
Quadratic regression applies when variables exhibit a nonlinear connection, such as an uptick and downtick dependency.
Quadratic regression involves three key parameters.
a (quadratic coefficient) – Determines the curvature of the parabola. b (linear coefficient) – Controls the slope of the function. k (steady term) - Denotes the y-intercept, the spot where the graph touches the y-axis. How is Quadratic Regression Calculated. Quadratic regression uses the least squares method to find the best-fitting parabola. The algorithm aims to reduce the squared amounts between what we see and what our quad equation says should be there.
Quadratic regression is widely used in various fields, including.
Physics: Modeling projectile motion, acceleration, and forces.
Economics: Analyzing diminishing returns and cost functions. Biology: Studying population growth and decay rates. Engineering: Predicting material stress and structural deformations. How Can You Interpret the Quadratic Regression Equation. If a > 0, the parabola opens upward, indicating a minimum value. If a < 0, the parabola opens downward, indicating a maximum value. The vertex of the parabola represents the turning point of the function.
The R² metric quantifies the quadratic model's accuracy in relating to the dataset. A correlation closer to 1 means a strong relationship, and a value near 0 shows a weak relationship.
Affirmative, quadratic extrapolation can be utilized for predicting impending figures, although it proves most efficacious when the dataset inherently conforms to an arched progression. If the data is very unpredictable or does not look like a smooth line, the forecasts may not be trustworthy.
It assumes a quadratic relationship, which may not always be accurate. Overfitting or underfitting may occur if an insufficient number of data points are present. It fails to deliver well if the connection exceeds a second-degree polynomial level. How Does Quadratic Regression Compare to Polynomial Regression. Quadratic analysis stands as an exclusive model within the polynomial method where the greatest degree is twofold. In place of, quadratic approximation can incorporate bigger-power figures (like cube, fourth) for more complicated links.
Suppose a business is analyzing the effect of advertising spend on sales. If spending initially elevates sales numbers, however, they taper off as markets reach equilibrium, quadratic fitting offers a way to determine the inflection point where further expenses fail to escalate income.
Ignoring residual analysis: Not checking if errors are randomly distributed. Overfitting: Using a quadratic model when a simpler linear model would suffice. Extending predictions beyond what's seen can lead to wrong results.
Use a larger dataset for better generalization. Ensure the data is preprocessed and cleaned to remove noise. Consider other polynomial degrees if the quadratic model does not fit well. What Are the Steps to Perform Quadratic Regression. Collect and plot the data to identify a potential quadratic pattern. Apply prediction methods to formulate the equation y = alpha times x squared plus beta times x plus gamma. Calculate the R² value to evaluate model accuracy. Interpret coefficients to understand the data trend. Use the equation for predictions while staying within the data range.
