Covariance Calculator
Covariance describes how two numeric data series vary together around their respective means. This tool reports both population and sample covariance.
How to use this calculator
- Enter paired values in the same order, separated by commas. Both series must contain the same number of observations, with at least two pairs.
- Check the units and definitions before comparing results with another source.
- Change one input at a time when comparing scenarios, and keep the reporting period and assumptions consistent.
Formula used by the covariance calculator
Population covariance = Σ((xᵢ − x̄)(yᵢ − ȳ))/n; sample covariance divides by n − 1
- Population covariance = Σ((xᵢ − x̄)(yᵢ − ȳ))/n; sample covariance divides by n − 1
- Covariance is affected by the units and scale of each series. It is not bounded between −1 and 1; use correlation when a normalized association measure is needed.
- Inputs are checked for required values and valid ranges; displayed results are rounded for readability.
Worked examples
Worked example
Series X = 2, 4, 6, 8 and Series Y = 1, 3, 5, 7.
Each series rises together; population covariance is 5 and sample covariance is 6.667.
Each series rises together; population covariance is 5 and sample covariance is 6.667.
Understanding the result
Covariance describes how two numeric data series vary together around their respective means. This tool reports both population and sample covariance.
Covariance is affected by the units and scale of each series. It is not bounded between −1 and 1; use correlation when a normalized association measure is needed.
Important notes
Covariance is affected by the units and scale of each series. It is not bounded between −1 and 1; use correlation when a normalized association measure is needed.
The result applies to the numbers and assumptions entered here; verify specifications for safety-critical or regulated uses.
Frequently asked questions
What does positive covariance mean?
Positive covariance indicates that values tend to be above or below their means together in the observed sample.
Is covariance the same as correlation?
No. Correlation standardizes covariance by the standard deviations and is bounded between −1 and 1.
How do I use this calculator?
Enter paired values in the same order, separated by commas. Both series must contain the same number of observations, with at least two pairs.
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Related explanations
Learn the method behind the number or compare the terminology before using the calculator.
- How to Calculate Standard Deviation — Learn what standard deviation measures, the population and sample formulas, and how to interpret the result.
- How to Calculate a Z-Score — Learn how a z-score shows how far a value is from the mean in standard-deviation units.
- Standard Deviation — Understand standard deviation as a measure of spread and the difference between population and sample standard deviation.
This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.