 # Covariance Calculator

Covariance calculator is an online statistical tool that is used to calculate the relationship between the two sets of variables X and Y that are commonly described. The variation between these two variables X and Y is measured using this tool. It enables us to understand the relationship between two data sets.

It also calculates the mean value for X and Y with calculating covariance. Here, we will explain what is covariance, How to use covariance calculator, how to find covariance, formula for covariance, and much more.

Covariance Calculator

## How to use covariance calculator?

Covariance calculations involve a complex and lengthy process but our sample covariance calculator eliminates the need to engage in that process. However, if you want to explore the concept of covariance in detail, be our guest. We will explain, how do you calculate covariance with examples in the later section.

This calculator only takes two input from user and produce desired results in no time.

Follow the below steps to calculate covariance using this calculator:

• Enter the data set for the variable X in the given input box.
• Enter the data set for the variable Y in the next input box.
• Note that, all values should be separated using comma.
• Press the Calculate button to get covariance of the given data sets.
• You can hit the Reset button to reset all values in the calculator.

It instantly provides you the covariance(X, Y), X mean, Y mean, and total number of inputs. The results are accurate and tested repeatedly to deliver one hundred percent precision.

Variance, covariance, and coefficient of variance are vitally associated to each other. You can use our variance and coefficient of variance calculator if you are looking at statistics problem related to them.

Now let’s move on to the covariance definition.

## What is covariance?

Covariance measures how many random (X, Y) variables in each population are distinct. When the population is greater or random, the matrix represents the relation between different dimensions. The covariance matrix can be easier to understand by describing this relationship as the relationship between the two random variables in the entire dimension.

### Properties of covariance

Covariance properties can be stated as:

• The smaller X and greater Y values give the positive covariance.

X < Y è +ve covariance

• The greater X and the smaller Y values shows covariance is negative.

X > Y è -ve covariance

• The covariance would be negative or non-linear if all random variables are not statistically dependent.

Covariance may be used to quantify variables not having the same measuring units. We can decide whether units increase or decrease by using covariance. It cannot be measured to what point the variables change together due to various measuring units for covariance.

### Covariance formula

Sample and population covariance have different formulas. The x and y samples have both random X and Y values.

x1, x2,..., xn represents the elements of first sample. X mean refers to the mean value of these sample elements.

y1, y2, ..., yn represents the elements of the second sample. Y mean refers to the mean value of second sample elements.

The following covariance equation is the sample covariance formula Cov(x,y) if two sample sizes are available.

Covsam (x, y) = sum (xi – x mean) (yi – y mean) / n

Where:

• xi– x mean represents the difference between sample elements for X and the mean value of the sample.
• yi– y mean represents the difference between sample elements for Y and the mean value of the sample.
• n represents the size of the sample for both X and

## How to calculate covariance?

As mentioned above, covariance is a complex topic. That’s why, will directly jump into a real life example to understand the covariance calculation.

### Example:

Betty is a stock market expert and invests in stock market and forex. She has purchased the stocks of Retro plastics recently. She wants to purchase one more company’s stocks to broaden her investment but in different industry. She has got two great options after some research, i.e, sun autos and DM textiles. But she doesn’t know which option she should opt out.

She can decide by calculating the covariance for return on both houses.

Betty gets the five closing prices for both of the companies, sun autos and DM textiles.

Prices of sun autos represents xi, and prices of DM textiles represents yi.

 i xi yi 1 8.47 10.63 2 11.22 9.21 3 11.99 10.71 4 11.45 8.01 5 10.92 5.03 Mean value 10.81 8.718

Follow these steps carefully to get the covariance:

Step 1: Calculate the mean value for xi. Add all values and divide the sum by size of the sample (5).

X mean = 10.81

Step 2: Calculate the mean value for yi too. Add all values and divide the sum by size of the sample (5).

Y mean = 8.718

Step 3: Calculate the x diff. Difference of X can be calculated by subtracting each element of x from x mean.

x diff = xi – x mean

To find differences for all x values, use the above equation and place them in a column.

Step 4: Repeat the previous step for y. Calculate ydiff by subtracting all values of y from the y mean.

Y diff = yi – y mean

Step 5: Multiply all values of x diff and y diff and place them in the next column.

Step 6: Add the values of last column, which are the product of the two differences. After adding, divide the sum by the sample size. After dividing the sum by sample size, the resulting value will be the covariance.

Sum of differences = -1.65911

Cov sam (x, y) = -1.65911 / 5

Cov sam (x, y) = -0.3318

 i xi yi x diff y diff x diff × y diff 1 8.47 10.63 -2.34 1.912 -4.47 2 11.22 9.21 -0.144 0.958 -0.1380 3 11.99 10.71 0.626 2.458 1.5387 4 11.45 8.01 0.086 -0.242 -0.02081 5 10.92 5.03 -0.444 -3.222 1.431 Mean value 10.81 8.718 -1.65911

The covariance value -0.3318 gives us the idea about variation in the prices of both companies. By using this value, Betty can decide to buy stocks of one of them.

## What is population covariance?

Usually, the whole population data is not available to us. Only small amounts of samples are available most of the times. Such samples can provide an estimate of population covariance for random variables X and Y.

The below formula is used to calculate the population covariance:

Cov pop (X, Y) = sum (xi – x mean) (yi – y mean) / (n-1)

The value n-1 is slightly better than the sample covariance. As small samples are not actually the entire variance between entire populations, it is natural to infer that denominator N-1 is the best correction factors.

Relationship between sample and population covariance

The relationship between sample and population variance can be expressed as:

Cov pop (X, Y) = (n / n-1) × Cov sam (x, y)

Nevertheless, the difference between n and n-1 becomes less when the sample size increases. Therefore, the population covariance and sample covariance formula give comparable results for large samples.

## Covariance vs Correlation

The relationship between covariance and correlation can be stated as:

• Another way to express the difference between two random variables X and Y is called correlation, while,
• Covariance computes the difference between X and Y, two random variables.

The relationship between both can be express in equation form as:

Corr (X, Y) = Cov (X, Y) / (σX × σY)

Where:

σX represents the standard deviation of X, and

σY represents the standard deviation of Y.

The correlation is considered as a stable form of covariance. The correlation should be between 1 and -1 according to the above equation. That is why correlation is used more often than covariance, even if they perform the same function.

### Can covariance be negative?

Yes. In a case, where two variables are inversely related, covariance can be negative.

### What does a negative covariance mean?

Negative covariance depicts that the both variables X and Y are inversely proportional to each other. If X goes down, Y goes up and if Y goes down, X increases.

If the covariance is negative, the variables are considered related in reverse.

For example:

If temperature goes up, use of heating instruments also goes up. It demonstrates that the relationship between both variables is inversely proportional.

### What does a covariance of 0 mean?

The covariance is zero if the two random variables are not mutually dependent. However, a covariance at zero does not imply the variables are independent. A non-linear relationship can still occur, resulting in a zero covariance value.

## What does covariance tell us?

Covariance shows us the extent to which two variables differ. It shows us how different a variable is from a related one. It calculates, how two variables relate monotonically to each other.

## What is a high covariance?

A high covariance indicates a strong relation between the two variables. In the case of low covariance it is the other way around. The weaker relationship between two variables are represented by low covariance.

## What does a covariance of 1 mean?

Covariance of 1 means that the both variables are directly proportional to each other. If one variable reduces other will increase, and in case, one variable gets a boost, other goes down.

## What does analysis of covariance mean?

Covariance analysis (ANCOVA) is used to analyze differences in the mean values of the variable that are dependent on each other. These variables are related to the impact of the independent controlled variables thus taking into account the effects of the independent uncontrolled variables.