Quadratic Regression Calculator
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 + cx²where 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 + 0.05x²This 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 + cx²Where:
- 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 + 0.05x²In 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.