The *Linear Regression Calculator* can be used to derive the linear regression equation. Example is included to demonstrate how to use the calculator.

Please enter your values in the Calculator*.* Each value should be separated by a comma.

**X Values**(Each Value Separated by Comma):

**Y Values**(Each Value Separated by Comma): Enter Value (X) to Make an Estimation:

## How to use the Linear Regression Calculator

Let's now review a simple example to see how to use the Linear Regression Calculator. Suppose that you have the following dataset:

- The X values are: 2, 7, 12
- The Y values are: 4, 11, 15

Plug the above values in the calculator. Each value should be separated by a comma:

Once you're done entering the numbers, click on the **Get Linear Regression Equation **button, and you'll see the Linear Regression equation, as well as the R-squared and the Adjusted R-squared:

To make an estimation, type your value in the third entry box. For example, type the value of 9 in the entry box:

Now, click on the **Estimate** button and you'll get the estimation of **12.2**:

## How to Manually Derive the Linear Regression Equation

The equation of a Simple Linear Regression is:

```
Y = a + bX
```

Where:

- Y = Dependent variable
- a = Y-Intercept
- b = Slope of the regression line
- X = Independent variable

Let's now review an example to demonstrate how to derive the Linear Regression equation for the following data:

- The X values are: 2, 7, 12
- The Y values are: 4, 11, 15

To start, use the following equation to get the Y-Intercept:

(Σy)*(Σx^{2}) - (Σx)*(Σxy) a = n*(Σx^{2}) - (Σx)^{2}

For our example:

- Σy = 4+11+15 = 30
- Σx
^{2}= (2^{2}) + (7^{2}) + (12^{2}) = 197 - Σx = 2+7+12 = 21
- Σxy = (2*4) + (7*11) + (12*15) = 265
- n = 3

Plug the above results in the equation to get the Y-Intercept:

(Σy)*(Σx^{2}) - (Σx)*(Σxy) (30)*(197) - (21)*(265) a = n*(Σx^{2}) - (Σx)^{2}= 3*(197) - (21)^{2}= 2.3

So the Y-Intercept is **2.3**.

Now let's get the Slope of the regression line using this equation:

n*(Σxy) - (Σx)*(Σy) b = n*(Σx^{2}) - (Σx)^{2}

Plug the values into the equation:

n*(Σxy) - (Σx)*(Σy) 3*(265) - (21)*(30) b = n*(Σx^{2}) - (Σx)^{2}= 3*(197) - (21)^{2}= 1.1

You'll then get the slope of **1.1**.

For the final part, let's construct the Linear Regression equation:

```
Y = a + bX = 2.3 + 1.1X
```

### Make an Estimation

Let's now make an estimation, where X = 9:

```
Y = 2.3 + 1.1X = 2.3 + 1.1*9 = 12.2
```

The estimated value is therefore **12.2**.