Enter Paired Data
Example Data Table
This sample uses five paired observations. Sample covariance equals 10. Population covariance equals 8.
| Observation | X | Y |
|---|---|---|
| 1 | 2 | 1 |
| 2 | 4 | 3 |
| 3 | 6 | 5 |
| 4 | 8 | 7 |
| 5 | 10 | 9 |
Formula Used
Cov(X, Y) = Σ[(xi - x̄)(yi - ȳ)] / n
Cov(X, Y) = Σ[(xi - x̄)(yi - ȳ)] / (n - 1)
x̄ = mean of X, ȳ = mean of Y, and n = number of pairs. The graph also shows a fitted trend line using slope = Σ(dx·dy) / Σ(dx²).
Positive covariance means both variables usually rise together. Negative covariance means one tends to rise while the other falls. Values near zero suggest little linear co-movement.
How to Use This Calculator
- Enter all X values in the first box.
- Enter matching Y values in the second box.
- Keep both lists the same length.
- Choose sample or population covariance.
- Set your preferred decimal precision.
- Optionally rename both chart axes.
- Click the calculate button.
- Review covariance, correlation, trend line, and paired table.
- Use CSV or PDF buttons for reporting.
Frequently Asked Questions
1. What does covariance measure?
Covariance measures how two variables move together. Positive values suggest they rise together. Negative values suggest opposite movement. Values near zero suggest weak linear co-movement.
2. What is the difference between sample and population covariance?
Sample covariance divides by n - 1. Population covariance divides by n. Use sample covariance when data represents a subset. Use population covariance when all observations are included.
3. Can covariance be negative?
Yes. Negative covariance means one variable tends to increase while the other decreases. It signals opposite directional movement across the paired observations.
4. Does zero covariance mean no relationship?
Not always. Zero covariance suggests no linear relationship, but nonlinear relationships may still exist. A curved pattern can produce near-zero covariance.
5. Why is correlation also shown?
Correlation standardizes the relationship on a scale from -1 to 1. It helps compare strength across datasets that use different measurement units.
6. Must X and Y contain the same number of values?
Yes. Covariance uses paired observations. Every X value must match one Y value in the same position, or the result becomes invalid.
7. Can I enter decimals, negatives, or scientific notation?
Yes. The calculator accepts common numeric formats, including decimals, negative numbers, and scientific notation, as long as each entry is numeric.
8. Why can covariance look large or small?
Covariance depends on the units and scales of both variables. Large units can create large covariance values. Correlation is better for scale-free comparison.