# Covariance Calculator

The following Covariance Calculator can be used to derive both the Population Covariance as well as the Sample Covariance. Each value entered in the calculator should be separated by a comma. Example is included below.

X Values (each value separated by comma): Y Values (each value separated by comma):

## How to use the Covariance Calculator

Let's review a simple example to see how to use the Covariance Calculator in practice.

Suppose that your goal is to derive the Covariance, given the following data:

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

To start, enter the above values in the calculator, and then click on the Calculate Covariance button:

You'll then get the Population Covariance of 18.33, and the Sample Covariance of 27.50:

## How to manually derive the Covariance

You may use the following formula to get the Population Covariance:

```                                                  Σ [(xi - x̅) * (yi - ȳ)]
Population Covariance  =                        n
```

Alternatively, you may use the formula below to derive the Sample Covariance:

```                                                  Σ [(xi - x̅) * (yi - ȳ)]
Sample Covariance  =                         n - 1
```

For example, let's say that your goal is to derive the Population and Sample Covariances, given the following data:

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

To start, calculate the averages:

```                          2 + 7 + 12
Average ( x̅ )  =        3          =  7
```
```                          4 + 11 + 15
Average ( ȳ )  =         3          =  10
```

Next, calculate the Population Covariance :

`    [(2 - 7) * (4 - 10)] + [(7 - 7) * (11 - 10)] + [(12 - 7) * (15 - 10)]                                                      3                                                   = 18.33`

You'll then get the Population Covariance of 18.33.

Finally, calculate the Sample Covariance:

`    [(2 - 7) * (4 - 10)] + [(7 - 7) * (11 - 10)] + [(12 - 7) * (15 - 10)]                                                      3 - 1                                               = 27.50`

You'll now get the Sample Covariance of 27.50.