Covariance Calculator

Our covariance calculator is a statistics tool that estimates the covariance between two random variables X and Y in probability & statistics experiments. It shows whether high values of one variable tend to align with the high or low values of the other variable. You can easily calculate the covariance from sample (n-1) or population (N) data sets along with the mean and sum of squares.

What is Covariance?

Covariance is a statistical measure to check how two variables change together. It measures how two random variables, “X” and “Y,” affect each other. Covariance values can be positive, negative, or zero. Each value has special meaning and is critical in the decision-making process.

Interpretation of Covariance:

There are 3 possibilities for covariance calculations and their interpretation. These are based on the values of the data sets “X” and “Y”.

Positive Covariance (> 0): If the covariance value is positive, both variables are proportional to each other. (e.g., study hours and test scores).
Negative Covariance (< 0): In this case, the variables are inverse proportioinal to each other(e.g., product price and product demand).
Zero Covariance (≈ 0): In this scenario, no relationship exists between the variables, or they have no effect on each other's values. (e.g., number of pages in a book and number of students in a class).

How to Use the Covariance Calculator?

Follow these simple steps to find the statistical covariance of data sets “X” and “Y” with our covariance calculator online:

Choose Option: Select the sample or population option
Enter Dataset X: Input your first set of numerical values
Enter Dataset Y: Input your second set of numerical values
Click Calculate: Press the “CALCULATE” button to instantly view the final covariance
Switch the Covariance Type: If needed, use the Sample/Population switch in the result section to view the results in either mode without re-entering your data

How to Find Covariance?

For a sample dataset of size n, the sample covariance between two variables X and Y is calculated using the following formula:

The Sample Mean Covariance Formula

Sample Covariance: Cov(X, Y) = Σ(x_i − x̄)(y_i − ȳ) N − 1

The Population Mean Covariance Formula

Population Covariance: Cov(X, Y) = Σ(x_i − x̄)(y_i − 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:

Mean of X: x̄ = 1 n ∑_{i = 1}ⁿ x_i

Mean of Y:

Mean of Y: ȳ = 1 n ∑_{i = 1}ⁿ y_i

A Practical Example:

Imagine you own an ice cream shop and want to know how daily temperature affects your sales.

Variable X (Temperature in °C)= 20, 25, 30, 35
Variable Y (Daily Sales in $)= 200, 250, 400, 500

Step 1:

Find the means of temperature and sales

Mean temperature (x̄) = (20 + 25 + 30 + 35) / 4 = 27.5
Mean sales (ȳ) = (200 + 250 + 400 + 500) / 4 = 337.5

Step 2:

Now calculate deviations from the mean for each day for both variables

x_i	x_i - X̄	y_i	y_i - Ȳ	(x_i - X̄)(y_i - Ȳ)
20	-7.5	200	-137.5	1031.25
25	-2.5	250	-87.5	218.75
30	2.5	400	62.5	156.25
35	7.5	500	162.5	1218.75
∑ x_i = 110		∑ y_i = 1350		∑ (x_i - X̄)(y_i - Ȳ) = 2625

Step 3:

Now sum and divide the values to find the covariance

Sum of products: 1031.25 + 218.75 + 156.25 + 1218.75 = 2625
Divide by (n - 1): 2625 / 3 = 875

Result:

The positive covariance indicates that the ice cream sales tend to increase as temperature rises. For faster results, use our Sample Covariance Calculator. Simply enter the two data sets and get your result with just one click.

Covariance vs. Correlation:

Both covariance and correlation measure the relationship between two variables, so it is necessary to know the exact difference between the two concepts.

The table below elaborates on the difference between covariance and corelation and

Feature	Covariance	Correlation
Definition	Measures the directional relationship between two variables.	Measures both the strength and direction of the linear relationship.
Value Range	-∞ to +∞	-1 to +1
Unit Dependency	Affected by the units of the variables.	Not affected by the units as a unitless (standardized).
Comparability	Hard to compare across different scales or units.	Easily comparable across different datasets and variable types.

FAQ's:

Can I Enter Negative Or Decimal Values In This Calculator?

Yes, you can enter both negative numbers and decimal values into this covariance calculator. The online tool is able to deal with diverse statistical datasets, so it can easily calculate the dataset's value, including fractions or negative figures.

What Is The Difference Between Sample And Population Covariance?

Sample covariance estimates the relationship in a subset of data values, which is calculated from the sum of the products of deviations divided by “N - 1”. In contrast, population covariance measures the value of data against the entire population and is therefore divided by “N”.

What Does A Covariance Of Zero Mean?

It indicates that there is no linear relationship between the two variables. So the change in one does not mean a certain increase or decrease in the other variable.

Can Covariance Be Greater Than 1 Or Less Than -1?

Yes, covariance can easily exceed 1 or drop below -1, as its value ranges from -∞ to +∞. On the other hand, the correlation coefficient ranges from -1 to +1.

Can Covariance Be Negative?

Unlike variance, which is non-negative, covariance can be negative, positive, or zero.

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 +∞.

Should I Use Correlation Or Covariance?

When it comes to comparing 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 for how a change in one variable can result in a change in another. Remember that both covariance and correlation determine linear relationships between variables.

How To Create A Covariance Matrix In Excel?

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

Step 1: You have to click Data Analysis in 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 the Output Range, and finally click ok

What Is Variance?

Variance is the mathematical term used in statistics and probability theory; it refers 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. This does not mean the same thing as in the context of linear algebra.