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

Name: Covariance Calculator
Availability: InStock
Author: Nham Vu

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

—

Mean X

—

Mean Y

—

N (pairs)

—

Formula Used

Step-by-Step Breakdown

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

Summary

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

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: 2026-06-10 · Reviewed by Nham Vu