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 Regressionis a statistical technique used to model the relationship between adependent variable(the variable being predicted or explained) and one or moreindependent variables(the variables used to predict or explain the dependent variable). It is a type ofmultiple regressionanalysis, where the relationship between the variables is described by asecond-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 = 1is they-intercept, or the value ofywhenx = 0b = 0.5is thelinear coefficient, which represents the change inyfor a one-unit increase inxc = 0.05is thequadratic coefficient, which represents the curvature of the relationship betweenxandy

## 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.**