Quadratic Regression

Quadratic Regression Calculator

The Quadratic regression calculator helps you evaluate the quadratic regression equation using data sets X and Y. It also shows a graph with a step-by-step solution. This quadratic equation models the relationship between two variables in the form of a parabola which is U shaped curve. This equation is used to make predictions or analyze trends.

What is Quadratic Regression?

Quadratic regression is a statistical method used to model the relationship between a dependent variable (y) and an independent variable (x) by fitting a quadratic (parabolic) equation. This technique is useful when the data follows a curved rather than a straight line.

Formula of Quadratic Regression:

A quadratic equation is a mathematical expression that describes a parabolic relationship between two variables. The quadratic equation takes the form:

Y = a + b(x) + c(x²)  

Where:

• Y is the dependent variable.
• X is the independent variable.
• a, b, and c are the coefficients determined by the regression analysis.

The coefficients a, b, and $c$ can be calculated using the following formulas:

b = (S_xy S_x²_x² – S_x²_y S_xx² ) / S_xy S_x²_x² – (S_xx²)²

c = (S_x²_y S_xx – S_xy S_xx² ) / S_xy S_x²_x² – (S_xx²)²

a = y - bx - cx²

• S_xy​ is the sum of the product of x and y values.
• S_x²​ is the sum of the squared $x$ values.
• S_x²_x² is the sum of the fourth power of x values.
• S_x²_y​ is the sum of the product of squared x and y values.
• S_xx²​ is the sum of the product of $x$ and squared x values.
• x̄ is the mean of the x values.
• ȳ is the mean of the y values.
• x̄² is the mean of the squared x values.

Method to calculate the quadratic regression:

In the following example, the procedure to calculate the quadratic regression is explained briefly.  

Example 1:

Determine a quadratic regression equation for the following data set of points.

(4,5),(3,6),(2,4),(1,8)

Solution:

Step 1: Separate the values

X= 4, 3, 2, 1

Y= 5, 6, 4, 8

Step 2: Calculate the mean of the datasets

Mean X = (4 + 3 + 2 + 1) / 4 = 5 / 2 = 2.5

Mean Y = (5 + 6 + 4 + 8) / 4 = 23 / 4 = 5.75    

Step 3: Draw a table.

Step 4: Calculate a, b, and c.  

For “b”:

b = (S_xy S_x²_x² – S_x²_y S_xx² ) / S_xy S_x²_x² – (S_xx²)²

b = {(-3.5) (129) – (-14.5) (25)} / {(5) (129) – (25)²}  

b= -4.45

For “c”:

c = (S_x²_y S_xx – S_xy S_xx² ) / S_xy S_x²_x² – (S_xx²)²

c = {(-14.5) (5) – (-3.5) (25)} / {(5) (129) – (25)²} 

c = 0.75

For “a”:

a = y - bx - cx²

a = 5.75 – (-4) (2.5) – (0.75) (7.5)

a = 11.25

Step 5: Put the values in the formula.

y = ax² + b(x) + c, where a ≠ 0

y = (11.25) x² + (-4.45) x + (0.75)

References:

Quadratic regression | Voxco

A brief introduction to Quadratic regression | Varsitytutors