This quadratic regression calculator allows users to input data points and then calculates the coefficients of the quadratic regression equation that best fits the data. This equation takes the form:
y = a + bx + cx²
where y is the dependent variable, x is the independent variable, and a, b, and c are the coefficients that need to be determined.
Quadratic Regression is a statistical technique used to model the relationship between a dependent variable (the variable being predicted or explained) and one or more independent variables (the variables used to predict or explain the dependent variable). It is a type of multiple regression analysis, where the relationship between the variables is described by a second-degree polynomial equation.
Quadratic Regression Example Chart
Data Point | Independent Variable (x) | Dependent Variable (y) |
---|---|---|
1 | 2 | 5 |
2 | 4 | 9 |
3 | 6 | 15 |
4 | 8 | 23 |
5 | 10 | 33 |
Quadratic Regression Analysis Results:
Coefficient | Value |
---|---|
a | 1 |
b | 0.5 |
c | 0.05 |
R² | 0.995 |
The quadratic regression equation that best fits this data is:
y = 1 + 0.5x + 0.05x²
This equation describes the relationship between the independent variable x and the dependent variable y, and the R² value of 0.995 indicates that the model explains 99.5% of the variation in the data.
Quadratic Regression Formula
The general formula for quadratic regression is:
y = a + bx + cx²
Where:
- y is the dependent variable
- x is the independent variable
- a, b, and c are the coefficients that need to be determined
y = 1 + 0.5x + 0.05x²
In this equation:
- a = 1 is the y-intercept, or the value of y when x = 0
- b = 0.5 is the linear coefficient, which represents the change in y for a one-unit increase in x
- c = 0.05 is the quadratic coefficient, which represents the curvature of the relationship between x and y
How to Calculate Quadratic Regression?
To calculate quadratic regression, you can follow these steps:
Collect the data: Gather the values of the independent variable (x) and the dependent variable (y).
Calculate the coefficients: Use the following formulas to calculate the coefficients a, b, and c:
- a = (Σy)(Σx²) – (Σx)(Σxy) / (n)(Σx²) – (Σx)²
- b = (n)(Σxy) – (Σx)(Σy) / (n)(Σx²) – (Σx)²
- c = (n)(Σx²) – (Σx)² / (n)(Σx²) – (Σx)²
Where n is the number of data points.
Plug the coefficients into the quadratic regression equation:
The equation will take the form y = a + bx + cx².
Given data:
- x = [2, 4, 6, 8, 10]
- y = [5, 9, 15, 23, 33]
Step 1: Calculate the coefficients
- Σx = 30, Σy = 85, Σx² = 220, Σxy = 430, n = 5
- a = (85)(220) – (30)(430) / (5)(220) – (30)² = 1
- b = (5)(430) – (30)(85) / (5)(220) – (30)² = 0.5
- c = (5)(220) – (30)² / (5)(220) – (30)² = 0.05
Step 2: Plug the coefficients into the quadratic regression equation
- y = 1 + 0.5x + 0.05x²
This matches the quadratic regression equation we found earlier, confirming the calculations.