What is the difference between population and sample covariance?

Population covariance divides by N (all data points) and is used when you have the entire population. Sample covariance divides by N−1 (Bessel's correction) and is used when your data is a sample drawn from a larger population, giving an unbiased estimate.

What does a positive covariance mean?

A positive covariance means the two variables tend to increase together — when X is above its mean, Y tends to be above its mean as well.

What does a negative covariance mean?

A negative covariance means the variables move in opposite directions — when X is above its mean, Y tends to be below its mean.

Can covariance be zero?

Yes. A covariance of zero indicates no linear relationship between the two variables. Note that this does not rule out non-linear relationships.

How is covariance different from correlation?

Correlation is covariance divided by the product of the two standard deviations. It normalizes covariance to a −1 to +1 scale, making it easier to compare across datasets with different units.

Both datasets must have the same number of values?

Yes. Covariance is computed pair-by-pair (x₁y₁, x₂y₂, …), so both lists must have equal length. This calculator will show an error if the lengths differ.

Covariance Calculator

Enter two datasets and instantly compute their covariance — population or sample — with a full step-by-step breakdown.

Dataset Inputs

Dataset X

Comma-separated numbers

Dataset Y

Must have the same count as X

Covariance Type

Sample (N−1) Population (N)

Enter two datasets and click Calculate

Results and step-by-step working will appear here

Sample Covariance

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Mean X

—

Mean Y

—

N (pairs)

—

Formula Used

Step-by-Step Breakdown

i	xᵢ	yᵢ	(xᵢ − x̄)	(yᵢ − ȳ)	(xᵢ−x̄)(yᵢ−ȳ)

Summary

Covariance measures the directional relationship between two variables — whether they tend to increase together (positive covariance) or move in opposite directions (negative covariance). This calculator accepts two comma-separated datasets, computes population covariance (divides by N) or sample covariance (divides by N-1), and shows a full step-by-step breakdown so you can verify every calculation. Results also include mean values and the number of data points used.

How it works

Enter your first dataset (X values) as comma-separated numbers.
Enter your second dataset (Y values) — it must have the same count as X.
Choose Population covariance (divide by N) or Sample covariance (divide by N−1).
Click Calculate to see the result and a detailed step-by-step breakdown.
Optionally load an example dataset to see how the tool works.

Use cases

Determine whether two financial assets move together for portfolio analysis.
Verify hand-calculated covariance for statistics homework or exams.
Explore linear relationships between two variables before computing correlation.
Understand the direction and magnitude of co-movement in experimental data.
Compare population vs. sample covariance for the same dataset.
Teach or learn the covariance formula with a live worked example.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu