Mathematics · Statistics
Covariance Calculator
Calculate population covariance for three paired observations.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Means: x̄=2, ȳ=4.
- Average cross-deviation=1.3333333333333333.
Problem → model → reason → result
What problem does this model solve?
Calculate population covariance for three paired observations.
Why does the model apply?
Covariance indicates whether paired variables tend to move in the same or opposite directions.
What assumptions does it make?
The entered observations belong to the dataset being described. A sample statistic should not be treated automatically as a complete population description.
Formula
cov(X,Y)=Σ(xᵢ−x̄)(yᵢ−ȳ)/N
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
Pairs (1,2),(2,4),(3,6) have positive covariance 4/3.
Common mistake
Covariance magnitude depends on units and is not a standardized strength measure.
When does this model not apply?
A summary statistic describes the supplied values but does not establish causation, representativeness or statistical significance.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Covariance do?
Calculate population covariance for three paired observations.
How does the Covariance work?
The calculator applies cov(X,Y)=Σ(xᵢ−x̄)(yᵢ−ȳ)/N. Covariance indicates whether paired variables tend to move in the same or opposite directions.
What can I learn from the Covariance?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.