Calculator
Example Data Table
| Observation | X1 | X2 | X3 | Y1 | Y2 |
|---|---|---|---|---|---|
| 1 | 12 | 18 | 25 | 55 | 120 |
| 2 | 15 | 20 | 28 | 58 | 128 |
| 3 | 14 | 22 | 31 | 61 | 135 |
| 4 | 18 | 26 | 35 | 67 | 148 |
| 5 | 20 | 29 | 38 | 72 | 160 |
Formula Used
For paired datasets X and Y with n observation rows, the cross covariance matrix is:
CXY = 1 / d × Σ (xk - μX)(yk - μY)
For sample covariance, d = n - 1. For population covariance, d = n. Each output cell compares one X variable with one Y variable.
The standardized correlation value is: rij = cov(Xi, Yj) / (sXi × sYj).
How to Use This Calculator
- Paste dataset X in the first box.
- Paste dataset Y in the second box.
- Keep the same number of paired observation rows.
- Select sample or population covariance.
- Choose the correct row orientation.
- Set decimal precision.
- Press the calculate button.
- Download the result as CSV or PDF.
Cross Covariance Matrix Guide
What the Matrix Means
A cross covariance matrix shows how one group of variables moves with another group. It is useful when two datasets share the same observation rows. The first matrix may contain predictors. The second matrix may contain responses. Each output cell compares one column from the first dataset with one column from the second dataset.
How the Calculation Works
This calculator keeps the workflow direct. Paste paired numeric matrices. Each row should describe the same case, time point, person, trial, or sample. The tool centers each column, multiplies matched deviations, and divides by the selected denominator. Use sample covariance when your rows represent a sample from a larger population. Use population covariance when the rows are the full group of interest.
Reading Positive and Negative Values
Positive covariance means two variables usually rise together. Negative covariance means one variable tends to rise while the other falls. A value near zero means the paired linear movement is weak. Covariance depends on measurement units, so large units can create large numbers. The optional correlation matrix helps compare relationships on a standardized scale from -1 to 1.
Where It Is Used
Cross covariance is common in multivariate statistics. It supports feature analysis, signal processing, portfolio modeling, regression diagnostics, machine learning, and experimental studies. It can show whether sensor channels move together. It can also compare marketing inputs with customer outcomes. In finance, it can connect asset returns with economic indicators.
Input Quality Tips
Clean input matters. Keep the same number of observation rows in both matrices. Do not mix dates, labels, or symbols with numeric cells. Use consistent units across columns. Outliers can strongly affect covariance, so review strange rows before trusting results. Missing values should be removed or completed before calculation.
Matrix Shape
The matrix is not limited to square output. If dataset X has three variables and dataset Y has two variables, the result has three rows and two columns. This shape makes cross covariance ideal for comparing different variable groups.
Exporting Results
Use the export buttons after calculation. CSV is best for spreadsheets and scripts. PDF is helpful for reports and classroom submissions. The example table gives a quick reference for formatting and expected interpretation. Try the sample data first, then replace it with your own paired matrices.
For best results, document column meanings and keep raw data archived for review.
FAQs
What is a cross covariance matrix?
It is a matrix that compares variables from one dataset with variables from another dataset. Each cell shows the covariance between one X column and one Y column.
Do both datasets need the same number of rows?
Yes. Each row must represent the same observation, event, subject, trial, or time point in both datasets.
Should I use sample or population covariance?
Use sample covariance when your data is a subset of a larger group. Use population covariance when your data contains the complete group being studied.
Can the output matrix be rectangular?
Yes. If X has three variables and Y has two variables, the cross covariance matrix will have three rows and two columns.
What does negative covariance mean?
Negative covariance means one variable tends to increase while the other tends to decrease. It shows opposite movement in paired observations.
Why is correlation also shown?
Covariance depends on units. Correlation standardizes the relationship, making it easier to compare strengths across different variable scales.
Can I paste spreadsheet data?
Yes. You can paste rows copied from a spreadsheet. Values may be separated by spaces, commas, tabs, or semicolons.
Why do I get undefined correlation?
Correlation is undefined when one variable has zero standard deviation. That means every value in that variable column is the same.