Example Data Table
| Case |
Covariance |
Correlation |
Known variance |
Target |
Variance result |
| Study score model |
12 |
0.60 |
100 |
Var(X) |
4 |
| Portfolio factor check |
30 |
0.75 |
64 |
Var(Y) |
25 |
| Matrix diagonal |
120 |
0.60 |
Cov(X,X) = 400 |
Var(X) |
400 |
Formula Used
Covariance links two variables through their standard deviations and correlation.
Cov(X,Y) = r × SD(X) × SD(Y)
When Var(Y) is known, use this rearranged formula.
Var(X) = Cov(X,Y)² ÷ [r² × Var(Y)]
When Var(X) is known, use the matching form.
Var(Y) = Cov(X,Y)² ÷ [r² × Var(X)]
For a covariance matrix, the diagonal entries are the variances.
Var(X) = Cov(X,X) and Var(Y) = Cov(Y,Y)
For pasted data, sample variance uses n - 1. Population variance uses n.
How To Use This Calculator
- Select the calculation method that matches your available data.
- Choose whether you want Var(X) or Var(Y).
- Enter covariance, correlation, and known variance when using the first method.
- Enter matrix diagonal values when using matrix mode.
- Paste paired X,Y rows when using raw data mode.
- Pick sample or population denominator for raw data.
- Press Calculate to show the result below the header.
- Use CSV or PDF export for reports and records.
About the Variance From Covariance Calculator
Variance and covariance are closely linked. Variance is a special covariance. It compares a variable with itself. This calculator helps you move between those ideas. It supports three practical paths. You can use covariance, correlation, and a known companion variance. You can read a variance from a covariance matrix diagonal. You can also paste paired observations and let the tool compute the matrix.
Why Variance Matters
Variance measures spread around the mean. A larger value means the data moves more from its average. A smaller value means observations stay closer together. Analysts use variance in risk work, quality checks, forecasting, portfolios, machine learning, and classroom statistics. Covariance adds direction between two variables. A positive covariance means both variables often move together. A negative covariance means they often move in opposite directions.
Using Covariance Correctly
Covariance alone cannot always reveal one missing variance. You also need correlation and the other variable variance. The relation is direct. Covariance equals correlation times both standard deviations. Rearranging that relation gives the missing variance. This is why the correlation option asks for a nonzero correlation and a positive known variance.
Matrix and Data Options
A covariance matrix gives the fastest answer. The value on the main diagonal is the variance of that variable. Cov(X,X) is variance of X. Cov(Y,Y) is variance of Y. The off diagonal value is the covariance between X and Y. The pasted data mode checks this idea from raw pairs. It calculates means first. Then it finds deviations, products, squared deviations, covariance, variance, and correlation.
Best Practice
Choose sample mode when your rows are a sample from a larger group. Choose population mode when your rows contain every item in the group. Keep units consistent. Review the sign of covariance. Round results only after checking the detailed output. Use the export tools to save work for reports. The calculator is helpful for learning and planning. It should not replace expert statistical review when decisions carry high financial or scientific risk.
Common Checks
Check whether the covariance came from sample or population data. Mixing methods changes the answer. Negative correlation still works because it is squared. The final variance is never negative by definition in statistics.
FAQs
1. Can covariance directly give variance?
Only sometimes. In a covariance matrix, Cov(X,X) directly equals Var(X). For cross covariance Cov(X,Y), you also need correlation and the other variable variance.
2. Why is correlation required in the first method?
Correlation connects covariance to both standard deviations. Without it, many variance values could match the same covariance, so the missing variance cannot be identified.
3. Can variance be negative?
No. Variance is based on squared deviations. It is always zero or positive, even when covariance or correlation is negative.
4. What does Cov(X,X) mean?
Cov(X,X) means the covariance of X with itself. That is exactly the variance of X, so it appears on the matrix diagonal.
5. Should I use sample or population mode?
Use sample mode when your data represents part of a larger group. Use population mode when your data includes the complete group.
6. What if correlation is zero?
The rearranged covariance formula cannot divide by zero correlation. Use raw data mode or matrix mode when correlation is zero or unknown.
7. Does a negative covariance change variance?
The sign does not make variance negative. In the rearranged formula, covariance is squared, so the final variance remains non-negative.
8. What can I export?
You can export the result summary, formula, covariance matrix view, and key calculation details as CSV or PDF files.