# Covariance Calculator

## 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.

For finding middle values with different functionalities, try Mean Calculator & Midpoint Calculator.

## What is Covariance Formula?

In the world of statistics and probability, covariance formula calculates 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 covariance formula. To learn about remaining values, use Remainder Calculator.

## Can Covariance be Negative?

Covariance can be either positive, negative or it can be zero as well. If 2 variables vary in the same direction, covariance will be a positive. If they travel in opposite direction, it will be a positive covariance.

If the values do not vary together, than the covariance will be a 0. Variance is not negative. To learn about variance & calculations, try Variance Calculator.

## What is Correlation?

It measures the strength of a linear relationship between 2 variables. The quantitative variables are height & weight.

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 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
• σX = The standard deviation (SD) of the X-variable
• σY = The standard deviation (SD) of the Y-variable

For learning of standard diviation & its calculations on run time, use Standard Deviation Calculator for that purpose.

## 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.

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

## Covariance Equations

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

## Example of Covariance equation & calculation

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.) Thus the covariance is 2.267.

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

## How to calculate Covariance with 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 provides help to high school students to solve covariance problems. If a students don't know how to find covariance, He/She must give a try to our covariance calculator to determine the linear relationship between two variables.

We also present other math calculators like Factor Calculator & Factorial Calculator. Use those if you want, our calculators are absolutely free to use.

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