## What is Covariance?

As covariance definition elaborates, in statistics and mathematics, the measurement of the relationship between two random variables (X, Y) is called covariance.

These variables are either positive or negative numbers and denoted by $$\text{Cov(X, Y)}$$

The positive value indicates the positive relationship whereas the negative value indicates the negative relationship.

Positive covariance reveals that each of the two variables tend to move in the same direction while negative covariance value indicates that each of the two variables tend to move in the opposite direction.

**Check an example.**

In this example, you will see how variables vary together as shown in the above given graph. In the middle graph (near zero covariance), these dots have no relation and that is practically zero covariance.

If you have a very strong negative covariance, the dots are going to travel together in the same negative direction as shown in the left graph.

If you have a large positive covariance, the dots are going to travel together in the same positive direction as shown in the right graph.

## What is Covariance Calculator?

The covariance calculator determines the statistical relationship, a measurement between the two population data sets (x, y) and finds their sample mean as well. The variance of one variable is equivalent to the variance of the other variable because these are changeable values.

The covariance calculator is featured to generate steps for corresponding input to complete the whole given task which may provide help to high school students to solve covariance problems and worksheets. It is also used to determine the linear relationship between two variables.

## Covariance vs Correlation

Points | Covariance | Correlation |
---|---|---|

Meanings of Covariance and Correlation | It indicates the measurement between two random variables X and Y | It indicates the measurement that how strongly two variables are related |

What is it? | It is a measurement of correlation | It is a scaled version of covariance |

Values of Covariance and Correlation | It exists between -∞ and + ∞ | It exists between -1 and +1 |

Change in scale | Affects the value of the covariance | Does not affect the value of the correlation |

Unit | No | Yes |

The relation between both concepts can be known by a given formula:

$$_ρ(X,Y) = \frac{cov(X,Y)}{_ρX_ρY}$$

- ρ X, Y = The correlation between variables X and Y
- Cov (X,Y) = The covariance between variables X and Y
_{σ}X = The standard deviation (SD) of the X-variable_{σ}Y = The standard deviation (SD) of the Y-variable

In covariance, correlation is obtained when the data is standardized. The correlation remains the same when the change occurs in scale or location whereas covariance would be changed.

## Covariance Formula

In the world of statistics and probability, there is a covariance formula to calculate the covariance between two random changeable variables X and Y. By using this formula, after calculation, you can verify the result of such calculations by using our covariance calculator.

Formula to determine the covariance between two variables

$$Cov (X,Y) =\sum_{i=1}^n (X - \overline X)(Y - \overline Y)$$

cov (X,Y) = Covariance between X and Y

x and y = components of X and Y

$$\overline x \; and \; \overline y =\;mean\; of \; X \; and \;Y $$

n = number of members

Covariance calculator works at this above given formula.

## How to Calculate Covariance?

To understand the working of covariance calculator, here we will work out step by step calculation. It’ll make easy to understand the calculation of covariance for beginners, learners and our school going students.

## Calculation Summary

Covariance for two random variables X = 2, 4, 6, 8 and Y = 1, 3, 5, 7

Estimate the strength of linear interdependence between them.

Calculation Summary | |
---|---|

Dataset X | 2, 4, 6, 8 |

Dataset Y | 1, 3, 5, 7 |

cov (X,Y) | 5 |

## Workout:

Input parameters and values

$$X = 2, 4, 6, 8$$

$$Y = 1, 3, 5, 7$$

$$\text{Number of inputs} = 4$$

$$Cov(X,Y)=\frac{1}{n}\sum_{i=1}^n (X - \overline X)(Y - \overline Y)$$

By putting values into covariance formula

$$ cov(X, Y) = \frac{1}{4} \; { (2 - 5) x (1 - 4) + (4 - 5) x (3 - 4) + (6 - 5) x (5 - 4) + (8 - 5) x (7 - 4)}$$

$$= \frac{1}{4} \; {(-3) x (-3) + (-1) x (-1) + (1) x (1) + (3) x (3)}$$

$$= \frac{1}{4} \; {(9) + (1) + (1) + (9)}$$

$$= \frac{20}{4}=5$$

Thus 5 is covariance of X = 2, 4, 6, 8 and Y = 1, 3, 5, 7

## Example:

In this example we will know about that how to calculate covariance. Let’s move on to an example to find the covariance for this set of four data points.

X = 2.1, 2.5, 3.6, 4.0

Y = 8, 10, 12, 14

$$Cov(X,Y)=\frac{\sum(X - \overline X)(Y - \overline Y)}{n-1}$$

Here ∑ is sum of X values subtract the mean of x (`x ) multiplied by Y subtract the mean of Y (`Y ). This all equation divided by n – 1

The first thing we need to find that is means mean of X and mean of Y. Well If I add these together and divide by 4. Then I get:

X = 2.1, 2.5, 3.6, 4.0 (`X ) = 3.1

Y = 8, 10, 12, 14 (`Y ) = 11

Now I got all the values to put into the covariance formula.

First, we will just solve this portion (X -`X) (Y - `Y)of our equation.

(Here in first part we take X values and subtract the mean of X and multiply them by corresponding Y values and subtract the mean of Y and so on. Just have a look.)

$$=\frac{(2.1 – 3.1)(8 - 11) + (2.5 – 3.1) (10 - 11) + (3.6 – 3.1) (12 - 11) + (4 – 3.1) (14 - 11)}{n-1}$$

$$=\frac{(-1) (-3) + (-0.6) (-1) + (0.5) (1) + (0.9) (3)}{n-1}$$

$$=\frac{3 + 0.6 + 0.5 + 2.7}{n-1}$$

$$=\frac{6.8}{n-1}$$

**As**

$$n = 4$$

$$=\frac{6.8}{4-1}$$

$$=\frac{6.8}{3}$$

$$=2.267$$

Thus the covariance is 2.267.

This answer is positive and tells us that these values tend in a positive direction together.

Hope so our covariance calculator will also be helpful and easy to use for you like our other tools. Thanks for staying with us.