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# Linear Regression Calculator

To find linear regression equation, input **x **& **y **values and press the calculate button

## Linear Regression Calculator

Use the line regression calculator to find the regression equation. This linear regression calculator uses **X **and **Y **values to determine the regression equation. This tool also computes the following components required in the regression equation:

- Y-intercept
- Mean of the dependent variables (
**y**) - Mean of the independent variables (
**x**) - Slope

Users can add more readings of **x **and **y **by clicking on “Add more” and on the way, these rows can be deleted as well.

## What is linear regression?

**Regression **is a technique or ability to establish a mathematical relationship between two variables. One of these is dependent and the other is independent.

**For example**, the estimation of the dependence of food consumption on the monthly income of families. Linear regression involves only two **variables**. There is another type of regression, known as** multiple regression**. It includes more than two variables.

There is another objective of linear regression in statistics and that is the forecasting of the new observation.

**For example**, from the previous data of a family’s increase in consumption with the increase in income, the prediction of the consumption increases if the income rises more this year.

## Regression Equation:

Before looking at the regression equation, it is important to know about the variables that lay down the foundation of it.

**Variable X** is usually taken as an independent variable and this is the one that explains the dependent variable. The dependent **variable is Y** in a bi-dimensional plane. It is one that we forecast.

The general equation is:

**Y _{i} = ꞵ_{0} + ꞵ_{1}X_{i} + Ɛ_{i} **

Where

**Y**= Dependent variable_{i}**ꞵ**= Population y-intercept_{0}**ꞵ**= Population Slope_{1 }**X**= Independent variable_{i}**Ɛ**= Error term_{i}

In statistics, most of the techniques are designed to apply to the sample data. So the fitted equation will be;

`ŷ = a + bx`

Here:

**ŷ**is the estimated value of y**a**is the sample y-intercept**b**is the sample slope/regression coefficient**x**is the independent variable

Now let’s move on to the computation of this equation.

## How to calculate the regression equation?

There are two main values that you have to calculate to make the regression equation; y-intercept(a) and slope(b).

The slope has three formulas. The basic and easiest one is the one written below.

In this formula, the numerator is the covariance of x and y and the denominator is the variance of x.

To find the y-intercept, use the given formula.

`a = ͞y - b ͞x`

The ͞**y **and ͞**x **represent the mean of **y** and **x **respectively.

After finding both values, all you have to do is put them in the sample equation. To clear your concept, read the solved example below.

**Example:**

Given these then pairs of **(X, Y) **values

X | 1 | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 6 | 7 |

Y | 2.1 | 2.5 | 3.1 | 3.0 | 3.8 | 3.2 | 4.3 | 3.9 | 4.4 | 4.8 |

- Find the regression equation.
- Predict the value of
**Y**when**X=10**.

**Solution:**

**(a) Regression equation**

**Step 1:** Find the slope.

It will be easy to make a table and find the necessary values through it.

n= ?

Summation X =?

Summation Y =?

XY =?

Summation XY =?

X^{2} = ?

Summation X^{2} = ?

n | X | Y | (X)(Y) | X^{2} |

1 | 1 | 2.1 | 2.1 | 1 |

2 | 1 | 2.5 | 2.5 | 1 |

3 | 2 | 3.1 | 6.2 | 4 |

4 | 3 | 3.0 | 9 | 9 |

5 | 4 | 3.8 | 15.2 | 16 |

6 | 4 | 3.2 | 12.8 | 16 |

7 | 5 | 4.3 | 21.5 | 25 |

8 | 6 | 3.9 | 23.4 | 36 |

9 | 6 | 4.4 | 26.4 | 36 |

10 | 7 | 4.8 | 33.6 | 49 |

`n = 10`

∑x = 1 + 1 + 2 + 3 + 4 + 4 + 5 + 6 + 6 + 7

` = 39`

∑y = 2.1 + 2.5 + 3.1 + 3.0 + 3.8 + 3.2 + 4.3 + 3.9 + 4.4 + 4.8

`= 35.1`

∑xy = 2.1 + 2.5 + 6.2 + 9 + 15.2 + 12.8 + 21.5 + 23.4 + 26.4 + 33.6

`= 152.7`

∑x^{2} = 1 + 1 + 4 + 9 + 16 + 16 + 25 + 36 + 36 + 49

`= 193`

Putting these values in the equation, we have;

b = [(10)(152.7) - (39)(35.1)] / [10 (193) - (39)2]

= 158.1 / 409

`= `

**0.387**

**Step 2:** Calculate the y-intercept.

`a = ͞y - b ͞x`

Mean of Y = 35.1 / 10

= 3.51

Mean of X = 39 / 10

= 3.9

Now, use this data in the intercept equation.

a = 3.51 - (0.387)(3.9)

`= `

**2.00**

**Step 3:** Put both values in the regression equation.

ŷ = a + bx

**ŷ = 2.001 + 0.387x**

**(b) Predict Y **

**Step 1: **Put the value of **X **in the computed regression equation.

ŷ = 2.001 + 0.387x

ŷ = 2.001 + 0.387 (10)

ŷ = 2.001 + 3.87

**ŷ = 5.871**