Covariance Calculator
Enter paired observations for variables X and Y to compute their sample covariance and inspect mean values.
Covariance
10.5000
Mean of X
5.2000
Mean of Y
8.0000
Pairs: 5
Covariance values rely on centered deviations around each mean.
How to Use This Calculator
- Enter paired observations for X and Y (one pair per line).
- Review the sample covariance to see whether the variables co-vary positively or negatively.
- Use the mean values to understand centers of each distribution.
- Combine with correlation to gauge standardized association if needed.
Formula
Cov(X, Y) = Σ[(xi − μx)(yi − μy)] / (n − 1)
μx, μy = sample means of X and Y
n = number of paired observations
Positive covariance indicates X and Y increase together; negative values suggest opposite movement.
Full Description
Covariance quantifies the joint variability of two variables. While its magnitude depends on measurement units, its sign reveals whether the variables move together. Analysts often compute covariance before normalizing to correlation.
The sample covariance divides by n − 1 to provide an unbiased estimate when working with sample data.
Frequently Asked Questions
How many pairs do I need?
At least two pairs are required to compute sample covariance.
Can covariance be zero?
Yes. Zero covariance means no linear co-movement, but other relationships may still exist.
What if one variable has no variance?
If all X or Y values are identical, covariance is zero because deviations from the mean vanish.
How does covariance relate to correlation?
Correlation divides covariance by the product of standard deviations, yielding a dimensionless measure between −1 and +1.