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What is Quadratic Regression?
Quadratic regression is a statistical technique used to find the equation of the parabola that best fits a set of data. This type of regression is an extension of simple linear regression that is used to find the equation of the straight line that best fits a set of data.
When illustrated on a scatter plot, a quadratic equation will form a “U” shape that is either concave down or concave up. Linear regression can be performed even with just two points, while quadratic regression requires many more data points. This is due to the fact that quadratic regression requires more data points to ensure that the data falls into the “U” shape.
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R-Squared in Quadratic Regression
In quadratic regression, R-squared is the coefficient of the determination and it illustrates the degree to which the variation in y can be explained by x-variables. The r-squared value, therefore, allows us to evaluate how the differences in one variable can be explained by a difference in the second variable. R-squared can take values between 0 to 1, where 0 reflects 0% variation and 1 reflects a 100% variation.
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Quadratic Regression Equation
The equation used in quadratic regression is as follows:
y = ax2 + bx + c
The following formulas can be used to calculate the values of a, b, and c:
a = { [ Σ x2 y * Σ xx ] – [Σ xy * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] – [Σ xx2 ]2 }
b = { [ Σ xy * Σ x2x2 ] – [Σ x2y * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] – [Σ xx2 ]2 }
c = [ Σ y / n ] – { b * [ Σ x / n ] } – { a * [ Σ x 2 / n ] }
Where,
- x and y: The Variables
- a, b, and c: The Coefficients of the Quadratic Equation
- n = Number of Values/Elements
- Σ x= Sum of First Scores
- Σ y = Sum of Second Scores
- Σ x2 = Sum of Square of First Scores
- Σ x 3 = Sum of Cube of First Scores
- Σ x 4 = Sum of Power Four of First Scores
- Σ xy= Sum of the Product of First and Second Scores
- Σ x2y = Sum of Square of First Scores and Second Scores
- Σ x x = [ Σ x 2 ] – [ ( Σ x )2 / n ]
- Σ x y = [ Σ x y ] – [ ( Σ x * Σ y ) / n ]
- Σ x x2 = [ Σ x 3 ] – [ ( Σ x 2 * Σ x ) / n ]
- Σ x2 y = [ Σ x 2 y] – [ ( Σ x 2 * Σ y ) / n ]
- Σ x2 x2 = [ Σ x 4 ] – [ ( Σ x 2 )2 / n ]
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FAQs on Quadratic Regression
Quadratic regression is a kind of statistical technique used to find the equation of the parabola that best fits a set of data.
Simple linear regression is used to find the equation of the straight line that best fits a set of data while quadratic regression is used to find the equation of the parabola that best fits a set of data.
The quadratic regression equation is;
y = ax2 + bx + c
Where,
a ≠ 0
Quadratic regression is used to model the relationship between two sets of variables so that the model can be used to explain or predict certain outcomes.
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