Should I use sample or population covariance?

Use sample covariance when your observations represent a subset of a larger population. Use population covariance when your data contains every observation in the population.

Can covariance be greater than 1?

Yes. Covariance is not restricted to the range from minus one to plus one because its magnitude depends on the units and scale of the datasets.

Can covariance prove causation?

No. Covariance describes how variables vary together, but it does not prove that one variable causes changes in another.

Covariance Calculator

Calculate sample or population covariance between two numerical datasets. Review averages, deviations, products, correlation, and complete calculation steps.

Instant Results Detailed Calculations Private and Browser-Based

Calculate Covariance

Enter an equal number of values in both datasets.

Calculator Ready

Enter Your Data

Dataset X Example: 2, 4, 6, 8

Dataset Y Example: 5, 7, 9, 11

Covariance Type

Sample Covariance Use when your values represent a sample from a larger population. Divides by n − 1. Population Covariance Use when your values contain the complete population. Divides by n.

Statistical Results

Your Results Will Appear Here

Add two equal-length datasets, select sample or population covariance, and click the calculate button.

Sample Covariance

No relationship detected.

Number of Pairs 0

Mean of X 0

Mean of Y 0

Correlation 0

Applied Formula

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

Calculation Details

Values are rounded for display.

#	X	Y	X − Mean X	Y − Mean Y	Deviation Product
Sum of deviation products					0

What Is Covariance?

Covariance is a statistical measurement that indicates how two variables change in relation to each other. It helps identify whether larger values in one dataset tend to occur with larger or smaller values in another dataset.

A positive covariance generally means the two variables move in the same direction. A negative covariance means they tend to move in opposite directions. A result near zero suggests little or no linear relationship.

Sample Covariance Formula

s_xy = Σ[(x_i − x̄)(y_i − ȳ)] ÷ (n − 1)

Sample covariance uses n − 1 in the denominator. It is appropriate when the entered values represent only a sample from a larger population.

Population Covariance Formula

σ_xy = Σ[(x_i − μ_x)(y_i − μ_y)] ÷ n

Population covariance uses n in the denominator. Select it when your datasets contain every observation in the population you want to study.

What the Symbols Mean

xᵢ and yᵢ: Individual paired observations.
x̄ and ȳ: Sample means of X and Y.
μx and μy: Population means of X and Y.
n: Total number of paired observations.
Σ: The sum of all calculated values.

Interpreting the Result

The sign of covariance describes the general direction of the relationship. Its magnitude depends on the units and scale of the original variables.

Positive Covariance Higher X values tend to occur with higher Y values.

Negative Covariance Higher X values tend to occur with lower Y values.

Covariance Near Zero There may be little or no linear relationship.

Covariance vs Correlation

Covariance identifies direction, but its value changes with measurement units. Correlation standardizes the relationship between −1 and +1, which makes relationships easier to compare across different datasets.

How to Use the Covariance Calculator

Calculate covariance and review every step in three simple stages.

Enter Dataset X and Y

Add numerical values separated by commas, spaces, semicolons, or new lines. Both datasets must contain the same number of values.

Select the Covariance Type

Choose sample covariance for sampled observations or population covariance when your data represents the complete population.

Review the Results

View covariance, means, correlation, dataset size, formula, individual deviations, products, and the final calculation.

Frequently Asked Questions

Common questions about covariance calculations and interpretation.

Positive covariance means the variables generally move in the same direction. When values in one dataset increase, values in the other dataset also tend to increase.

Negative covariance means the variables generally move in opposite directions. As one variable increases, the other tends to decrease.

Use sample covariance when your observations represent a subset of a larger group. Use population covariance when your data includes every observation in the population being studied.

Yes. Covariance is not limited to a range between −1 and +1. Its value depends on the units and scale of both datasets. Correlation, however, always remains between −1 and +1.

No. Covariance only describes how two variables vary together. It does not prove that one variable causes changes in the other.

Covariance is calculated from paired observations. Each X value must correspond to one Y value, so both datasets must contain the same number of valid numerical entries.

No. This calculator performs the calculation inside your browser using JavaScript. The entered datasets are not uploaded or stored by this PHP file.