Covariance Calculator

Our covariance calculator is a statistics tool that estimates the covariance between two random variables X and Y in probability & statistics experiments. Moreover, you need this covariance statistics calculator if you want to:

Calculate Covariance From Dataset
Calculate Covariance From Correlation Coefficient
Compute Covariance Matrix

In this article, you will learn about the covariance formula, how to calculate covariance, and other essential concepts you need to know. Before exploring the covariance calculator, let's start with some basics.

What is Covariance?

In statistics and mathematics, covariance measures the relationship between two random variables, X and Y. Simply put, covariance tells us how much two variables change together. While the concept is similar to variance, the difference is:

Variance: Measures how a single variable varies.
Covariance: Measures how two variables vary together.

Covariance can be either positive or negative:

Positive covariance: The two variables tend to move in the same direction.
Negative covariance: As one variable increases, the other tends to decrease.

Calculating covariance is easy with an online covariance calculator. You can also compute the sum of squares for any dataset using this sum of squares calculator.

Covariance Formula:

Our covariance calculator helps you measure the relationship between two variables using both sample and population covariance formulas.

covariance formula

Sample Covariance Formula:

Sample Cov (X, Y) = Σ (xᵢ - x̄)(yⱼ - ȳ) N - 1

Population Covariance Formula:

Population Cov (X, Y) = Σ (xᵢ - x̄)(yⱼ - ȳ) N

In the above covariance equations;

Σ : Summation notation
xᵢ : Observations of variable X
yⱼ : Observations of variable Y
x̄ and ȳ : Sample means of X and Y
N : Total number of observations

Mean of X and Y:

Mean of X:

\(x̄ = \frac{1}{n}\sum_{i=1}^n x_i\)

Mean of Y:

\(ȳ = \frac{1}{n}\sum_{i=1}^n y_i\)

The covariance calculator helps to find out the statistical relationship between the two sets of population data (X and Y). Also, this sample covariance calculator allows you to calculate covariance matrix and the covariance between two variables X and Y for a given correlation coefficient (Pearson’s) and standard deviations. Don’t fret; covariance calculation is quite easy with this advanced covariance statistics tool.

How to Use This Covariance Calculator?

Our covariance calculator is user-friendly and provides step-by-step solutions. Follow these simple steps:

Input:

Select the sample or population covariance option from the drop-down menu
Enter the values of the data set X separated by commas
Enter the values of the data set Y separated by commas

Once you have added the above values, hit the calculate button, the covariance calculator shows the step-by-step solution in a couple of seconds:

Output:

Automatic Mean Calculation

Auto-calculates and displays the mean of both X and Y datasets.

Covariance Type Selection

Toggle easily between sample and population covariance.

Formula Display

Shows the mathematical formula used for computing covariance.

Mean Calculation

Displays the computed mean values for X and Y datasets.

Data Table

Provides a detailed table showing (Xᵢ - X̄), (Yᵢ - Ȳ), and their product for each observation.

Step-by-Step Solution

Explains how the final covariance is calculated, highlighting the final covariance value clearly.

How to Calculate Covariance (Example)?

Let’s take a look at covariance example:

Calculate the covariance for the given data sets.

X = 4, 7, 10, 13, 16
Y = 1, 3, 5, 7, 9

Step 1: First of all, find the sample mean of data sets X & Y.

➦ For X

x̄ = (4 + 7 + 10 + 13 + 16)/5 = 50/5 = 10

➦ For Y

ȳ = (1 + 3 + 5 + 7 + 9)/5 = 25/5 = 5

Step 2: Now find the deviation (difference of data values from the mean) of sets X & Y and calculate the square of deviations.

xᵢ	xᵢ - x̄	yⱼ	yⱼ - ȳ	(xᵢ - x̄)(yⱼ - ȳ)
4	-6	1	-4	24
7	-3	3	-2	6
10	0	5	0	0
13	3	7	2	6
16	6	9	4	24

Step 3: Calculate the summation of (xᵢ - x̄)(yⱼ - ȳ) terms.

Σ(xᵢ - x̄)(yⱼ - ȳ) = 24 + 6 + 0 + 6 + 24

Σ(xᵢ - x̄)(yⱼ - ȳ) = 60

Step 4: Now divide the above expression by N - 1 to get the result of sample covariance.

Cov(X,Y) = 60 / (5 - 1) = 60 / 4 = 15

✅ Final Answer: Sample Covariance = 15

However, from this example you got a positive covariance, it means that the variables are positively related.

how to calculate covariance

Note:

If you see the given denominator of the above covariance formula, you have the degrees of confidence. However, in the above covariance example, we had more than 2 terms, thus we used the formula n – 1. When you are going to find the covariance of two random variables, then you ought to divide the formula by n only.

Interpretation of Covariance:

From the above example of covariance you will come to know, if you had a positive covariance, which means there is a positive relationship between the variables or that said they are positively related. However, you can use our covariance calculator to calculate covariance from correlation. As a rule of thumb, a large covariance indicates that there may be a strong relationship between variables. Nevertheless, remember that you can’t compare variances over data sets that have several scales. You just have to think about comparing two datasets of variables where one is expressed in inches and the other one in pounds.

interpretation of covariance

This is the problem with the interpretation of covariance outcomes, so as a far better approach is to account the correlation coefficient. So, you have to use the following formula instead:

Corr(X,Y) = Cov(X,Y) / (σX σY)

However, you can confirm your outcomes in our calculate covariance from correlation.

Covariance vs Correlation:

Covariance:

Let’s start with covariance:

Covariance is something that indicates the measurement between two random variables X and Y
Covariance is a measurement of correlation
Values of covariance exist between –x and +x
Change in scale will affects the value of the covariance
Remember that covariance always has units

Correlation:

Now, ahead to Correlation:

Correlation is something that indicates the measurement that how strongly two variables are related
Correlation is said to be a scaled version of covariance
Values of correlation exist between -1 and +1
Change in scale does not affect the value of the correlation
Remember that correlation does not have units

Advantages of the Correlation Coefficient over Covariance:

The Correlation Coefficient has a different number of advantages over covariance for computing strengths of relationships, these are:

Covariance can account on practically any number while a correlation is limited: -1 to +1
Correlation is more useful for finding how strong the relationship is between the two variables, just because of its numerical limitations
Correlation is not at all affected by changes in the center that is said to be the mean or scale of the variable

Frequently Asked Questions:

Can covariance be negative?

Unlike variance, which is non-negative, Covariance is something that can be negative or positive (or zero, of course). A positive covariance indicates that two random variables tend to vary in the same direction; a negative variance indicates that they vary in opposite directions, and zero means they don’t vary together.

What is the symbol for covariance?

The standard symbol is cov(X, Y).

What is the maximum value of covariance?

When it comes to covariance, there is no minimum or maximum value, that’s why the values are more difficult to interpret. For instance, a covariance of 50 may indicate a strong or weak relationship as this actually depends on the units in which covariance is measured.

What is the range of covariance?

Covariance ranges from -∞ to +∞.

Covariance values are not standardized, according to statistical terms, the covariance can range from negative infinity to positive infinity. Thus, the value for a perfect linear relationship all depends on the data.

Should I use correlation or covariance?

When it comes to compare data samples from different populations, the covariance (COV) is considered to find how much two random variables vary together. And, correlation is something that accounts to find when a change in one variable can result in a change in another. Remember that both covariance and correlation determine linear relationships between variables.

Why is correlation preferred over covariance?

Well, when it comes to comparison, which is a better measure of the relationship between two variables, correlation is preferred over covariance as it is the measure that remains unaffected by the change in location and scale – and, also can be accounted to make a comparison between two pairs of variables.

How to create a covariance matrix in Excel?

Just stick to these given steps to create a covariance matrix in Excel or covariance table in Excel:

Step 1: You have to make a click to the data analysis on the top right corner of the data tab
Step 2: You have to choose covariance and click ok
Step 3: In this step, you should have to click in the input range box and choose the range A1:C10, then select the “Labels in first row” tick box and output range and finally click ok

What is variance?

Variance is the mathematical term used in statistics and probability theory, it is referred to the spread of a dataset around its mean value.

Is covariance linear?

Sometimes the covariance is said to be a measure of ‘linear dependence’ between the two random variables. That does not mean the same thing that is in the context of linear algebra.