Covariance Calculator

A Covariance Calculator is a free online statistics tool that helps you measure how two numerical variables change together. It tells you whether two datasets generally move in the same direction or in opposite directions.

Calculator Pool’s free Covariance Calculator online supports both:

Population Covariance

and:

Sample Covariance

You can enter two datasets, X and Y, with the same number of paired observations. The calculator then calculates covariance along with useful supporting statistics such as:

• Mean of X
• Mean of Y
• Standard deviation of X
• Standard deviation of Y
• Variance of X
• Variance of Y
• Sum of cross-deviations
• Number of paired observations

For example, consider:

X = 10, 20, 30, 40, 50

Y = 8, 18, 27, 39, 52

Both variables generally increase together, so their covariance is positive.

Important: Covariance describes the direction of joint variation, but its magnitude depends on the units of measurement. Unlike correlation, covariance is not standardized and therefore should not be interpreted simply by looking at its size alone.

What Is Covariance?

Covariance is a measure that describes how two variables vary together.

When the variables tend to increase and decrease together, covariance is positive.

When one variable tends to increase while the other decreases, covariance is negative.

When there is little systematic joint movement, covariance may be close to zero.

In simple terms:

Positive covariance → variables tend to move together

Negative covariance → variables tend to move in opposite directions

Covariance near zero → little linear co-movement

What Is a Covariance Calculator?

A covariance calculator is an online statistics calculator that calculates the covariance between two numerical datasets.

Calculator Pool allows you to enter:

Dataset X

and:

Dataset Y

and choose:

Population Covariance

or:

Sample Covariance

The calculator also reports:

Mean X

Mean Y

Standard Deviation X

Standard Deviation Y

Variance X

Variance Y

and:

Paired Observation Count

This makes it useful for statistics, probability, data analysis, research and educational calculations.

How to Use the Covariance Calculator

Using Calculator Pool’s online covariance calculator is simple.

Step 1: Enter Dataset X

Enter the first numerical dataset.

For example:

10, 20, 30, 40, 50

Step 2: Enter Dataset Y

Enter the corresponding observations.

For example:

8, 18, 27, 39, 52

The values should be paired by position:

X₁ ↔ Y₁

X₂ ↔ Y₂

X₃ ↔ Y₃

and so on.

Step 3: Select the Covariance Type

Choose:

Population Covariance

or:

Sample Covariance

Step 4: Calculate

Click:

Calculate Covariance

The calculator displays covariance and the related statistics.

Population Covariance

Population covariance measures the joint variation for an entire population.

The formula is:

Cov(X,Y) = Σ[(x − μx)(y − μy)] / N

Where:

• x = X observation
• y = Y observation
• μx = mean of X
• μy = mean of Y
• N = population size

Population covariance is appropriate when the data represents the complete population for the analysis being performed.

Sample Covariance

Sample covariance estimates the covariance of a larger population using sample data.

The standard sample covariance formula is:

Cov(X,Y) = Σ[(x − x̄)(y − ȳ)] / (n − 1)

Where:

• x̄ = sample mean of X
• ȳ = sample mean of Y
• n = number of paired observations

The denominator:

n − 1

is used in the standard sample covariance estimator.

Population Covariance vs Sample Covariance

The main difference between the two calculations is the denominator.

Population Covariance

Divide by N

Sample Covariance

Divide by n − 1

For the same dataset, a non-zero sum of cross-deviations divided by n−1 will generally have a larger magnitude than when divided by n.

Covariance Formula

The covariance formula depends on whether you are working with a population or a sample.

Population

σxy = Σ[(x − μx)(y − μy)] / N

Sample

sxy = Σ[(x − x̄)(y − ȳ)] / (n − 1)

The cross-product:

(x − mean X)(y − mean Y)

is central to the calculation.

How to Calculate Covariance

Let’s calculate covariance manually using:

X = 10, 20, 30, 40, 50

Y = 8, 18, 27, 39, 52

Step 1: Calculate the Means

Mean X:

30

Mean Y:

28.8

Step 2: Calculate Deviations

For each pair, calculate:

x − mean X

and:

y − mean Y

Step 3: Multiply the Deviations

Calculate:

(x − x̄)(y − ȳ)

for every pair.

Step 4: Add the Cross-Deviations

The calculator calls this:

Sum of Cross-Deviations

Step 5: Divide by the Appropriate Denominator

Population:

Divide by N

Sample:

Divide by n − 1

The result is covariance.

Covariance Example

Consider the paired data:

Mean X:

30

Mean Y:

28.8

The variables generally increase together.

The sample covariance for this dataset is:

272.5

while population covariance is:

218.0

This demonstrates how changing the denominator affects the covariance result.

What Does Positive Covariance Mean?

A positive covariance indicates that the two variables tend to move in the same direction.

For example:

Study hours ↑

and:

Exam scores ↑

may produce positive covariance in an appropriate dataset.

Another example could be:

Advertising expenditure ↑

and:

Sales ↑

Again, this is only an example of a possible relationship. Actual covariance must be calculated from data.

What Does Negative Covariance Mean?

A negative covariance indicates that the variables tend to move in opposite directions.

For example:

Price ↑

while:

Quantity demanded ↓

may produce negative covariance in some datasets.

The sign tells you the direction of joint variation.

What Does Covariance Near Zero Mean?

A covariance close to zero suggests little systematic linear co-movement.

However, covariance near zero does not necessarily mean that there is absolutely no relationship.

Non-linear relationships can exist without producing a strong linear covariance pattern.

Covariance vs Correlation

Covariance and correlation are closely related, but they are not the same.

Covariance

Measures how two variables vary together and depends on their units.

Correlation

Standardizes the relationship and produces a unit-free value between:

−1 and +1

The relationship is:

Correlation = Covariance / (SDx × SDy)

Calculator Pool provides a dedicated Correlation Calculator for calculating Pearson correlation.

Covariance vs Correlation Example

Suppose:

Covariance = 272.5

The numerical magnitude by itself is difficult to interpret without knowing the units and scales of X and Y.

Correlation solves this by dividing covariance by the product of the standard deviations.

This produces a standardized coefficient:

−1 ≤ r ≤ +1

Therefore:

Covariance → Unit-dependent

Correlation → Unit-free

Why Is Correlation Often Easier to Interpret?

Imagine one dataset is measured in:

Dollars

and another in:

Units sold

Covariance would have combined units involving both measurements.

Its magnitude would change if the units changed.

Correlation removes this unit dependence through standardization.

For relationship strength comparisons, correlation is therefore often easier to interpret.

Covariance and Standard Deviation

Covariance is related to the standard deviations of both variables.

For Pearson correlation:

r = Cov(X,Y) / (SDx × SDy)

Therefore, standard deviation helps convert covariance into a standardized measure.

Calculator Pool displays:

SD of X

and:

SD of Y

alongside covariance.

Covariance and Variance

Variance measures how a single variable varies around its mean.

Covariance measures how two variables vary together.

In fact, variance is a special case of covariance:

Var(X) = Cov(X,X)

This is an important connection in statistics.

Calculator Pool also provides a Variance Calculator.

Covariance and Mean

Covariance is calculated from deviations from the mean.

For each observation:

x − x̄

and:

y − ȳ

are calculated.

These deviations are then multiplied.

Therefore, the mean of each variable is essential to the covariance calculation.

Calculator Pool displays:

Mean X

and:

Mean Y

to make the result easier to understand.

Covariance and Standard Error

Covariance and standard error describe different statistical properties.

Covariance

Describes joint variation between two variables.

Standard Error

Describes sampling variability of an estimator.

The two concepts can appear together in advanced statistics, but they should not be confused.

Calculator Pool provides a separate Standard Error Calculator.

Covariance and Z-Score

Z-scores standardize individual observations.

Correlation can be understood as a standardized relationship between X and Y, and covariance is one of the quantities used before standardization.

The connection is:

Covariance → Divide by SDx × SDy → Correlation

Calculator Pool also provides a Z-Score Calculator.

Covariance and Confidence Intervals

In advanced statistical analysis, covariance matrices and estimated covariance parameters can appear in calculations involving uncertainty and confidence intervals.

However, covariance itself is not a confidence interval.

Calculator Pool provides a separate Confidence Interval Calculator for normal-approximation intervals around a mean.

Covariance and Probability

Covariance is an important concept in probability theory.

For random variables X and Y:

Cov(X,Y) describes their tendency to vary jointly.

If:

Cov(X,Y) > 0

the variables tend to move together.

If:

Cov(X,Y) < 0

they tend to move in opposite directions.

Covariance is also used in multivariate probability distributions and statistical models.

Calculator Pool provides a Probability Calculator for basic probability calculations.

Covariance and Normal Distribution

In a multivariate normal distribution, covariance is part of the covariance matrix that describes relationships among multiple variables.

For example, two variables can have:

Positive covariance

Negative covariance

or:

Zero covariance

The covariance structure is an important part of multivariate statistical analysis.

Covariance Matrix

When there are multiple variables, covariance values can be organized into a covariance matrix.

For two variables:

[Var(X)Cov(X,Y)​ Cov(X,Y)Var(Y)]

The diagonal contains variances.

The off-diagonal elements contain covariances.

This idea becomes especially important in multivariate statistics, portfolio analysis, principal component analysis, and machine learning.

Covariance Matrix Example

For two variables:

X

and:

Y

the matrix is:

Since:

Cov(X,Y) = Cov(Y,X)

the matrix is symmetric.

Covariance Is Symmetric

One important property of covariance is:

Cov(X,Y) = Cov(Y,X)

Switching the order of the two variables does not change the covariance.

However, the data must still remain correctly paired.

Covariance and Scaling

Covariance changes when variables are rescaled.

Suppose:

X’ = aX

and:

Y’ = bY

Then:

Cov(X’,Y’) = ab × Cov(X,Y)

This demonstrates why covariance magnitude depends on units.

Correlation, by contrast, is unchanged by positive linear rescaling of either variable.

Covariance and Units

Suppose X is measured in:

meters

and Y in:

seconds

Then covariance has units:

meter-seconds

If you change meters to centimeters, the numerical covariance changes.

This is another reason why raw covariance values should not be compared across variables without considering units.

Covariance With Negative Values

Negative observations are completely valid.

For example:

X = −5, −2, 0, 3, 6

Y = −10, −4, 0, 5, 8

The covariance calculation works normally.

The sign of the individual observations does not determine covariance directly; the deviations from the respective means matter.

Covariance With Decimal Values

Covariance also works with decimal datasets.

For example:

X = 1.5, 2.5, 3.5, 4.5

Y = 2.1, 3.2, 5.0, 6.4

The calculator can process decimal observations normally.

Covariance With Repeated Values

Repeated observations are allowed.

For example:

X = 10, 10, 20, 20, 30

and:

Y = 5, 5, 15, 20, 25

The covariance uses every paired observation.

Why Must Datasets Have the Same Number of Values?

Covariance is based on paired observations.

Each X value needs a corresponding Y value.

For example:

X₁ → Y₁

X₂ → Y₂

X₃ → Y₃

If X contains five values and Y contains four values, one observation cannot be paired.

Calculator Pool therefore requires both datasets to contain the same number of observations.

Covariance and Paired Observations

Consider:

Here there are:

5 paired observations

The covariance calculation uses each pair.

Incorrectly rearranging one dataset can change the covariance substantially.

Covariance and Scatter Plots

A scatter plot can help visualize the relationship behind a covariance value.

An upward pattern generally corresponds to positive covariance.

A downward pattern generally corresponds to negative covariance.

A random cloud can produce covariance near zero.

Visualizing data before interpretation can help reveal:

• Trends
• Outliers
• Clusters
• Non-linear patterns

Covariance and Outliers

Covariance can be sensitive to extreme observations because it uses products of deviations.

A single unusual pair can significantly influence:

(x − x̄)(y − ȳ)

and therefore change the final covariance.

Outliers should therefore be investigated before drawing conclusions from covariance.

Covariance and Linear Relationships

Covariance is particularly useful when studying how two variables vary together in a linear statistical framework.

However, like Pearson correlation, a covariance close to zero does not prove the absence of every possible relationship.

Non-linear relationships require additional analysis.

Covariance vs Pearson Correlation

Both measures involve two variables.

Covariance

Unit-dependent and can have any positive or negative magnitude.

Pearson Correlation

Unit-free and always lies between:

−1 and +1

Correlation is therefore better suited for comparing the strength of relationships across datasets with different units.

Covariance vs Regression

Covariance is one component used in some regression calculations.

For simple linear regression, the slope can be expressed using:

Slope = Cov(X,Y) / Var(X)

under compatible sample/population conventions.

Regression goes further by producing an equation for predicting or modeling a response variable.

Covariance alone does not provide a full regression model.

Covariance and Slope

Covariance is connected to the slope of a simple linear regression line.

For a standard sample-based formulation:

b = sxy / sx²

where:

sxy = sample covariance

and:

sx² = sample variance of X

This relationship shows how covariance contributes to the direction and magnitude of a regression slope.

Covariance and Data Analysis

A covariance calculator for data analysis can provide a quick measure of how two numerical variables move together.

It can be useful during exploratory analysis before moving to:

Correlation

Regression

Prediction

or:

Multivariate analysis

Covariance is especially useful when constructing covariance matrices or working with several related variables.

Covariance Calculator for Students

A covariance calculator for students can help explain:

• Covariance formula
• Positive covariance
• Negative covariance
• Population covariance
• Sample covariance
• Paired observations
• Relationship between covariance and correlation
• Variance and standard deviation

It can also help verify manual homework calculations.

Covariance Calculator for Statistics

A statistics covariance calculator can quickly calculate covariance from paired datasets.

It eliminates repetitive manual calculations involving:

Means

Deviations

Cross-products

and:

Appropriate denominators

Covariance Calculator for Research

A covariance calculator for research can be useful for exploratory statistical analysis.

Researchers may examine covariance when studying:

• Experimental variables
• Measurements
• Financial returns
• Scientific observations
• Survey data
• Time-series variables

For formal research, the sampling design and appropriate statistical assumptions should always be considered.

Covariance Calculator for Finance

Covariance is particularly important in finance because portfolio analysis often examines how asset returns move together.

For two assets:

Positive covariance

suggests returns tend to move in the same direction.

Negative covariance

suggests they tend to move in opposite directions.

Covariance contributes to portfolio variance calculations.

Covariance in Portfolio Analysis

Portfolio risk depends not only on the variance of individual assets but also on how their returns co-move.

For two assets, a portfolio variance formula can include a covariance term:

σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂)

where:

• w₁, w₂ = portfolio weights
• σ₁², σ₂² = individual variances
• Cov(R₁,R₂) = covariance between returns

This is one reason covariance is important in quantitative finance.

Covariance in Machine Learning

Covariance appears in several machine-learning techniques.

For example, Principal Component Analysis (PCA) commonly works with a covariance matrix to understand the directions of greatest variation in multivariate data.

Covariance can therefore be an important building block in dimensionality-reduction workflows.

Covariance in Multivariate Statistics

When analyzing several variables simultaneously, a covariance matrix summarizes pairwise relationships.

For example, with:

X₁, X₂, X₃

the covariance matrix contains:

Var(X₁)

Var(X₂)

Var(X₃)

and all pairwise covariances.

This provides a compact description of multivariate variability.

How to Interpret Covariance

Interpreting covariance starts with its sign.

Positive

Variables tend to move together.

Negative

Variables tend to move in opposite directions.

Near Zero

Little linear co-movement.

The magnitude requires context because it depends on units and scale.

For relationship strength, Pearson correlation is generally easier to interpret.

Why Covariance Magnitude Is Difficult to Interpret

Suppose one analysis produces:

Covariance = 100

and another:

Covariance = 1,000

It would be incorrect to automatically say the second relationship is ten times stronger.

Different units or scales could explain the difference.

Correlation solves this comparability problem by standardizing covariance.

Why Use an Online Covariance Calculator?

Manually calculating covariance can be tedious because you need to calculate two means, many deviations, cross-products, and the appropriate denominator.

An online covariance calculator automates these steps.

It can help you:

• Calculate covariance quickly
• Compare population and sample covariance
• Check manual calculations
• Understand paired datasets
• Calculate means and standard deviations
• View variances
• Explore relationships between variables

Benefits of Calculator Pool’s Covariance Calculator

Calculator Pool’s free Covariance Calculator online lets you enter two datasets and choose:

Population Covariance

or:

Sample Covariance

The result includes:

Covariance

Mean X

Mean Y

Standard Deviation X

Standard Deviation Y

Variance X

Variance Y

Paired Observation Count

and:

Sum of Cross-Deviations

This gives you a useful statistical summary instead of only returning one number.

Common Covariance Calculation Mistakes

Using Different Dataset Sizes

X and Y must contain the same number of paired observations.

Mixing Up Pair Order

X₁ must correspond to Y₁.

Using the Wrong Denominator

Population uses:

N

Sample uses:

n − 1

Confusing Covariance With Correlation

Covariance is unit-dependent.

Correlation is standardized.

Interpreting Magnitude Without Considering Units

Covariance values cannot be compared meaningfully without considering measurement scales.

Assuming Covariance Proves Causation

It does not.

Ignoring Outliers

Extreme observations can strongly affect covariance.

Assuming Covariance Near Zero Means No Relationship

A non-linear relationship may still exist.

Covariance Calculation Example

Let’s use:

X = 2, 4, 6, 8

Y = 1, 3, 5, 7

Mean X:

5

Mean Y:

4

The deviations are:

X:

−3, −1, 1, 3

Y:

−3, −1, 1, 3

Cross-products:

9, 1, 1, 9

Sum:

20

Population Covariance

20 ÷ 4 = 5

Sample Covariance

20 ÷ 3 ≈ 6.6667

Therefore:

Population Covariance = 5

Sample Covariance ≈ 6.6667

Another Covariance Example

Suppose:

X = 1, 2, 3, 4, 5

Y = 5, 4, 3, 2, 1

As X increases, Y decreases.

The covariance is negative.

This is a clear example of negative covariance.

Positive Covariance Example

Suppose:

X = 1, 2, 3, 4, 5

Y = 2, 4, 6, 8, 10

Both variables increase together.

The covariance is positive.

The corresponding Pearson correlation is:

+1

because the relationship is perfectly linear and positive.

Negative Covariance Example

For:

X = 1, 2, 3, 4, 5

and:

Y = 10, 8, 6, 4, 2

the variables move in opposite directions.

Therefore:

Covariance < 0

and:

Pearson r = −1

under the standard calculation.

Zero Covariance Example

A covariance of zero can occur when positive and negative cross-deviations cancel.

For example, certain non-linear relationships can have:

Covariance = 0

while still showing a clear relationship in a scatter plot.

This is why zero covariance should not automatically be interpreted as complete independence or absence of association.

Covariance and Independence

If two random variables are independent and the relevant expectations exist, their covariance is zero.

However:

Covariance = 0

does not generally imply independence.

Uncorrelated variables can still have non-linear dependence.

This is an important distinction in probability and statistics.

Covariance and Correlation Relationship

The Pearson correlation coefficient is:

r = Cov(X,Y) / (SDx × SDy)

This means correlation is essentially a standardized covariance.

Because standard deviations are positive, covariance and correlation have the same sign.

Therefore:

Positive covariance → Positive correlation

Negative covariance → Negative correlation

when the correlation is defined.

Frequently Asked Questions

What is a Covariance Calculator?

A Covariance Calculator calculates how two numerical variables vary together using population or sample covariance.

What is covariance?

Covariance measures the direction of joint variation between two variables.

What does positive covariance mean?

It means the variables tend to move in the same direction.

What does negative covariance mean?

It means the variables tend to move in opposite directions.

What does covariance near zero mean?

It indicates little linear co-movement, although non-linear relationships may still exist.

What is the covariance formula?

Population covariance:

Σ[(x − μx)(y − μy)] / N

Sample covariance:

Σ[(x − x̄)(y − ȳ)] / (n − 1)

What is population covariance?

It calculates covariance for a complete population using N in the denominator.

What is sample covariance?

It estimates population covariance from a sample using n − 1 in the denominator.

What is the difference between population and sample covariance?

Population covariance divides by N. Sample covariance divides by n − 1.

Why does sample covariance use n − 1?

The standard sample covariance estimator uses n − 1 to account for estimating the sample means and to provide the usual unbiased estimator under standard assumptions.

Is covariance the same as correlation?

No. Covariance depends on measurement units, while correlation is standardized between −1 and +1.

What is the relationship between covariance and correlation?

r = Cov(X,Y) / (SDx × SDy)

Can covariance be negative?

Yes.

Can covariance be positive?

Yes.

Can covariance be zero?

Yes.

Can covariance be greater than 1?

Yes. Unlike correlation, covariance is not restricted to the range −1 to +1.

Can covariance be less than −1?

Yes. There is no universal −1 to +1 restriction on covariance.

Does covariance have units?

Yes. Its units are the product of the units of X and Y.

Why is covariance not as easy to interpret as correlation?

Its magnitude depends on units and scale.

What is a covariance matrix?

A covariance matrix contains variances on the diagonal and pairwise covariances off the diagonal.

Is a covariance matrix symmetric?

Yes, because:

Cov(X,Y) = Cov(Y,X)

What is the difference between covariance and variance?

Variance measures variability of one variable. Covariance measures joint variability between two variables.

Is variance a covariance?

Yes.

Var(X) = Cov(X,X)

Does covariance measure causation?

No. Covariance measures statistical co-movement and does not establish causation.

Can covariance detect non-linear relationships?

Covariance primarily captures linear co-movement and may be close to zero even when a strong non-linear relationship exists.

Does covariance prove independence?

No. Independence implies zero covariance under suitable conditions, but zero covariance does not generally imply independence.

Does covariance use means?

Yes. Covariance is calculated from deviations from the means of both variables.

Does covariance use standard deviation?

Not directly in its basic formula, although standard deviation is used when converting covariance into correlation.

Does covariance use variance?

Covariance and variance are closely related. Variance is covariance of a variable with itself.

Does covariance appear in regression?

Yes. Sample covariance and sample variance can be used to express the slope of simple linear regression.

Does covariance appear in finance?

Yes. Covariance is used in portfolio variance and other quantitative-finance calculations.

Does covariance appear in machine learning?

Yes. Covariance matrices appear in techniques such as Principal Component Analysis.

Why must two datasets have equal length?

Covariance requires paired observations, so each X value needs a corresponding Y value.

Can I use decimal values?

Yes.

Can I use negative values?

Yes.

Can I use repeated values?

Yes.

Can I enter values separated by commas?

Yes.

Can I enter values separated by spaces?

Yes.

Can I enter values on separate lines?

Yes.

What is paired covariance?

It refers to calculating covariance from corresponding X and Y observations.

What happens if one dataset has zero variance?

If one variable has no variability, its standard deviation is zero and Pearson correlation becomes undefined. Covariance itself can still be calculated.

What is cross-deviation?

A cross-deviation is the product:

(x − mean X)(y − mean Y)

The calculator sums these products before dividing by the selected denominator.

Why are cross-deviations multiplied?

Multiplying paired deviations shows whether the two variables deviate from their means in the same or opposite directions.

What is positive cross-deviation?

When both deviations have the same sign, their product is positive.

What is negative cross-deviation?

When one deviation is positive and the other negative, their product is negative.

Is covariance useful for data analysis?

Yes. It is a fundamental measure of joint variation between numerical variables.

Is covariance useful for research?

Yes. It appears in exploratory analysis, multivariate statistics and many statistical models.

Can covariance be used for exam scores?

Yes. For example, covariance can measure how study hours and exam scores vary together in a dataset.

Can covariance be used for financial returns?

Yes. It is commonly used to understand how asset returns move together.

Is the Covariance Calculator free?

Yes. Calculator Pool’s Covariance Calculator is free to use online.

Related Calculators

You may also find these Calculator Pool tools useful:

Correlation Calculator

Calculate Pearson correlation coefficient, relationship direction, relationship strength, covariance, means, standard deviations and r².

Variance Calculator

Calculate population or sample variance along with mean and standard deviation.

Standard Deviation Calculator

Calculate population or sample standard deviation.

Average Calculator

Calculate the arithmetic mean, sum, count, minimum and maximum.

Z-Score Calculator

Calculate how many standard deviations a value is above or below the mean.

Standard Error Calculator

Calculate standard error from standard deviation and sample size.

Confidence Interval Calculator

Calculate a normal-approximation confidence interval using mean, SD, sample size and confidence level.

Percentile Calculator

Calculate percentiles from 0 to 100.

Quartile Calculator

Calculate Q1, Q2, Q3 and IQR.

Probability Calculator

Calculate basic probability from favorable and total possible outcomes.

Covariance Calculator – Quick Summary

A Covariance Calculator measures how two numerical variables vary together.

It can produce either:

Population Covariance

or:

Sample Covariance

The population covariance formula is:

Cov(X,Y) = Σ[(x − μx)(y − μy)] / N

The sample covariance formula is:

Cov(X,Y) = Σ[(x − x̄)(y − ȳ)] / (n − 1)

A positive covariance indicates that the two variables tend to move in the same direction.

A negative covariance indicates that they tend to move in opposite directions.

A value close to zero indicates little linear co-movement, although a non-linear relationship may still exist.

Unlike Pearson correlation, covariance is not restricted to −1 and +1. Its magnitude depends on the units and scales of the two variables.

The relationship between covariance and Pearson correlation is:

r = Cov(X,Y) / (SDx × SDy)

Calculator Pool’s free Covariance Calculator online also provides:

Mean X

Mean Y

Standard Deviation X

Standard Deviation Y

Variance X

Variance Y

Count

and:

Sum of Cross-Deviations

For example, with:

X = 10, 20, 30, 40, 50

and:

Y = 8, 18, 27, 39, 52

the sample covariance is:

272.5

while the population covariance is:

218.0

Covariance is widely used in statistics, probability, data analysis, finance, regression, multivariate statistics, research and machine learning.

Use Calculator Pool’s online covariance calculator to quickly calculate population or sample covariance from paired datasets and understand how two variables vary together.