# How to Calculate Covariance in 7 Steps (With Example)

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Covariance measures the linear relationship and joint variability between two variables. Businesses can use the covariance calculation of a set of variables to discover how they relate and influence each other. Knowing how to perform this calculation can help businesses and financial investors make investment and strategic decisions that can yield positive returns. In this article, we explain why knowing how to calculate the covariance of two variables is essential, distinguish between covariance and variance, review the benefits of covariance, and outline how to calculate covariance with an example.

## Why is knowing how to calculate covariance important?

Learning how to calculate covariance can help you measure the relationship between two random variables. Covariance is a statistical tool that can help you evaluate how much and to what extent the variables of two items change together, which can aid strategic business decision-making. Covariance can also help financial advisors determine risks and asset volatility.

The covariance value can be positive, negative, or zero. While a positive covariance indicates that two variables tend to move in the same direction, a negative covariance shows that they are moving in opposite directions. Getting a value of zero generally means the variables don't marry together. You can measure covariance in units and compute the units by multiplying the units of two variables.

## Benefits of covariance

Businesses can use covariance as it has the following benefits:

Provides a historical outlook: Investors can use the covariance tool to analyze past return investment data and pricing for two stock portfolios. With this, they can predict stock market events and movements.

Facilitates diversification: It's possible to diversify your investment portfolio with covariance. This can provide better risk management compared to having investments that are alike and have similar risks.

Aids volatility comparison: You can reduce the volatility of an investment with covariance. It's possible to do this by combining portfolios or other investment option grouping.

Encourages risk management: With covariance, you can measure the directional relationship between two assets prices to facilitate risk management. They can adopt risk management strategies as a positive covariance generally indicates assets move in the same direction, while a negative covariance means the assets are moving in a negative direction.

Helps investors make informed choices: Investors can include assets with negative covariance in their portfolios. This is a strategy they may use to optimize returns for an investment or project.

## Covariance vs. variance

Variance is the measurement of the width of a distribution. It measures the spread between a data set from its mean value. Covariance and variance are mathematical terms businesses frequently use in probability and statistical calculations, but they serve different purposes in businesses and finance. Financial experts use variance to measure an asset's volatility, while covariance refers to two different investment returns over time.

Large variance values typically indicate the numbers in the data set have a considerable distance between them, while a smaller variance means the numbers in the set are closer. With covariance, you can either have a positive or negative value that depicts the returns can be positive or negative. Investors use variance to measure stock volatility, while they use covariance to measure the directional relationship between two random variables. Stock experts also use these tools.

Related: Example of Positive Correlation (And How to Calculate It)

## How to calculate the covariance of two variables

You can establish a relation between two variables by calculating the covariance. Follow these steps to perform this calculation:

### 1. Gather your data

The data is typically from the two variables you're trying to establish a relationship between. It can include information such as prices or stock value. The data can range from one year to another. For example, you may want to calculate the covariance of the stock value for two companies, GoldRose and BlackJet, between 2015 and 2022. Gathering data on the stock value for the specified years for both companies is essential for calculating the covariance of the stock values.

Related: How to Calculate the Median of a Data Set in Statistics

### 2. Calculate the average for both variables

The formula for calculating the covariance of a set of variables involves inputting their average values. You can find the different mean values for both variables once you have the data for them. Continuing with the example from the previous step, to find the average stock value for GoldRose, calculate the stock value from 2015 to 2016 and divide by the number of years in the range, which is seven. This gives you the average value. Repeat this step for BlackJet.

Related: How to Find Critical Value in Two Steps (With Examples)

### 3. Subtract the mean from each value in the two variables

After getting the average for the two variables, subtract it from the values in the variables. For example, if the average stock value for GoldRose is 1,342, subtract this value from each stock value in the different years. Repeat this step for BlackJet.

### 4. Multiply the values for the two variables

After subtracting the average value from each value in the two variables, multiply these values. For GoldRose and BlackJet, you can multiply the values for 2015 after the subtraction. Do this for each year in the date range.

Related: How to Calculate Growth Percentage (With Examples)

### 5. Add the values

After calculating the product of the two variables, you can add the product values from GoldRose and BlackJet to get the summation of all values. You now have a single value representing the sum of other formula parts. You can divide this sum by the value of "n" to get your covariance value.

Related: Guide to Standard Deviation in Finance and How to Calculate

### 6. Write out the formula

The formula for calculating covariance is:

Cov(X, Y) = Σ [(Xi-µ) (Yj-v)] / n

Here are what the symbols in this formula represent:

Cov(X,Y) is the covariance of variables X and Y.

Σ represents the summation of the other parts of the formula.

Xi stands for all values of the X-variable.

µ is the average value of the X variables.

Yj represents all the values of variable Y.

V is the average value of the Y variable.

n is the total amount of data for both variables X and Y.

You can input all these values into the formula to calculate the covariance.

### 7. Substitute your values into the formula

You can substitute the value from the previous steps into the formula to calculate the covariance. In the example above, n equals seven. This is the number of years between 2015 and 2021. You can get the value of the covariance after the substitution.

## Example calculation of covariance

Businesses can measure the covariance of two variables to determine whether they have a positive relationship or not. Here's an example of a business calculating covariance:

BloomEdge is a franchise that wants to offer a new product to its customers. After doing market research, it decides to calculate the covariance of customers from the stores and the weather. Doing this can help the franchise management understand how temperature can influence sales. The temperature readings obtained are 23, 12, 14, 27, and 17. Similarly, the numbers of customers identified are 64, 35,40, 75, and 50.

First, the franchise strategist makes the temperature variable X and the number of customers who visit the stores in those temperatures variable Y. The average of variable X is 18.6, which the franchise strategist gets by adding the readings 23, 12, 14, 27, and 17. Similarly, for variable Y, the average value is 53.

The franchise strategist subtracts each temperature reading from its average. This gives values of 4.4, –6.6, –4.6, 8.4, and –1.6 for variable X. Doing the same for variable Y returns values 12, –18, –13, 22, and –3. Next, they multiply the respective values of the two variables. So, the calculations (4.4 × 12), (–6.6 × –18), (–4.6 × –13), (8.4 × 22), and (–1.6 × –3) give 52.8, 118.8, 59.8, 184.8, and 4.8. The franchise strategist does the sum for the values 52.8, 118.8, 59.8, 184.8, and 4.8 to get a single value. The addition gives a result of 421.

Finally, the franchise strategist substitutes the value of 421 into the formula and divides it by "n," which equals 5 because it's the number of occurrences for the data. The franchise strategist does the division operation to get 84.2. This value of 84.2 is the covariance for the number of customers who visit the store and the temperature.

From the covariance values, BloomEdge finds that more customers visit the store when the temperature is warmer. This deduction is from the positive value of the covariance.

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