Calculator Form
Enter paired values using commas, spaces, or new lines. The form uses three columns on large screens, two on medium screens, and one on mobile.
Plotly Graph
The scatter plot and trend line appear after calculation. They help you inspect direction, spread, and the linear pattern between both variables.
Example Data Table
This sample dataset is preloaded in the form, so you can test the calculator immediately.
| Index | Feature X | Feature Y |
|---|---|---|
| 1 | 2 | 3 |
| 2 | 4 | 5 |
| 3 | 6 | 7 |
| 4 | 8 | 9 |
| 5 | 10 | 10 |
| 6 | 12 | 14 |
Formula Used
Sample Covariance
Cov(X,Y) = Σ[(xi - x̄)(yi - ȳ)] / (n - 1)
Population Covariance
Cov(X,Y) = Σ[(xi - μx)(yi - μy)] / n
Related Measures
Variance of X = Σ[(xi - center)2] / denominator
Variance of Y = Σ[(yi - center)2] / denominator
Correlation = Cov(X,Y) / (σxσy) when both standard deviations are non-zero.
How to Use This Calculator
- Enter a label for each variable.
- Paste paired numeric values into both text areas.
- Choose sample or population covariance.
- Set the decimal precision you want.
- Click the calculate button.
- Review covariance, correlation, variances, and the matrix.
- Inspect the scatter plot and regression line.
- Download your summary as CSV or PDF if needed.
Frequently Asked Questions
1. What does covariance measure?
Covariance measures how two variables move together. A positive value suggests both variables tend to rise together. A negative value suggests one often rises when the other falls. A value near zero suggests little directional co-movement.
2. When should I use sample covariance?
Use sample covariance when your paired observations represent only part of a larger population. It divides by n - 1, which corrects bias and makes the estimate better for inference and modeling work.
3. When should I use population covariance?
Use population covariance when your dataset includes every observation in the group you want to describe. It divides by n and is appropriate for complete historical or fully observed datasets.
4. Why is correlation also shown?
Correlation standardizes covariance, making the relationship easier to compare across different scales. Covariance depends on units, while correlation always stays between -1 and 1 and is often easier to interpret quickly.
5. Why must both series have equal length?
Each x value must pair with exactly one y value. Covariance is calculated from matched observations, so unequal series would break the pairing and produce invalid results.
6. What does a covariance near zero mean?
A covariance near zero means there is little overall directional movement between the variables. It does not guarantee independence, because non-linear relationships can still exist even when covariance is small.
7. Can the calculator handle negative numbers and decimals?
Yes. You can enter integers, decimals, scientific notation, and negative values. Separate entries with commas, spaces, or line breaks, and keep both variables aligned by observation order.
8. What do the CSV and PDF downloads contain?
The downloads include key summary metrics and the paired dataset used for the calculation. They are useful for reporting, audits, classroom work, dashboard documentation, and quick stakeholder sharing.