Example Data Table
| Observation |
Height |
Width |
Score |
| 1 | 5.1 | 3.5 | 1.4 |
| 2 | 4.9 | 3.0 | 1.4 |
| 3 | 4.7 | 3.2 | 1.3 |
| 4 | 4.6 | 3.1 | 1.5 |
| 5 | 5.0 | 3.6 | 1.4 |
| 6 | 5.4 | 3.9 | 1.7 |
Formula Used
The mean vector is computed first. Each variable value is centered by subtracting its mean. The covariance entry for variables j and k is the sum of paired centered products divided by n - 1 for sample mode or n for population mode.
S[j,k] = Σ(x[i,j] - mean[j])(x[i,k] - mean[k]) / divisor
The correlation entry divides covariance by the two standard deviations. The determinant gives generalized variance. The trace gives total variance. The inverse covariance matrix is the precision matrix. Mahalanobis distance uses the centered row, the precision matrix, and a quadratic product.
For a multivariate normal model, the calculator estimates log likelihood with the determinant and Mahalanobis distances. The scatter matrix equals the covariance matrix times its divisor. In sample mode, the scatter matrix follows a Wishart form under normal assumptions.
How To Use This Calculator
- Paste observations into the data box.
- Keep each row as one observation.
- Separate each variable with commas, spaces, tabs, or semicolons.
- Enter matching variable names, or leave the field blank.
- Select sample covariance for most statistical datasets.
- Add small diagonal regularization if inversion fails.
- Press the calculate button.
- Use the CSV or PDF button for reporting.
Multivariate Covariance Matrix Distribution Guide
Why covariance matters
A multivariate dataset has many variables measured together. Each row is an observation. Each column is a variable. A covariance matrix shows how those variables move as a group. Positive covariance means two variables often rise together. Negative covariance means one variable tends to fall when the other rises. Values near zero show weak linear movement.
Matrix view
The diagonal cells are variances. They show the spread of each variable. The off diagonal cells are covariances. They describe paired movement. This layout is compact. It is also useful in regression, classification, portfolio analysis, simulation, and quality control. Many advanced models use this matrix as a core input.
Distribution diagnostics
When data is assumed to follow a multivariate normal distribution, the covariance matrix becomes part of the model shape. The determinant measures generalized variance. A larger determinant shows a wider data cloud. A smaller determinant shows tighter concentration. The inverse matrix supports Mahalanobis distance. That distance checks how far each observation sits from the center after correlation is considered.
Sample and population choices
Sample covariance divides by n minus one. It is the usual choice when the data is a sample from a larger population. Population covariance divides by n. It is useful when the dataset already represents the full group. The selected divisor changes variances, covariances, determinant, and related diagnostics.
Practical interpretation
Review the means first. Then inspect variances on the diagonal. Next compare covariance signs and sizes. Use the correlation matrix when variables have different units. Correlation rescales movement between minus one and one. Finally, inspect determinant, trace, likelihood, AIC, and BIC. These values help compare model fit and spread. If the precision matrix fails, variables may be duplicated or highly dependent. A small diagonal adjustment can stabilize inversion.
Good data practice
Use complete numeric rows. Keep units consistent inside each variable. Remove text labels from the data box before running the tool. Review extreme Mahalanobis distances because they may show unusual observations. Also compare correlation with covariance. Covariance keeps units. Correlation gives a unit free view. Both summaries support stronger multivariate decisions.
FAQs
What does a covariance matrix show?
It shows variance for each variable and covariance between variable pairs. It helps describe direction, spread, and joint movement in multivariate data.
Should I choose sample or population covariance?
Choose sample covariance when observations represent a sample. Choose population covariance when the dataset is the complete group being studied.
Why is the covariance matrix symmetric?
Covariance between variable A and variable B equals covariance between variable B and variable A. That makes matching off diagonal entries equal.
What is generalized variance?
Generalized variance is the determinant of the covariance matrix. It summarizes total multivariate spread using one value.
What is the precision matrix?
The precision matrix is the inverse of the covariance matrix. It is used in Mahalanobis distance, likelihood estimates, and dependency analysis.
Why can inversion fail?
Inversion can fail when variables are duplicated, nearly duplicated, or perfectly dependent. Too few observations can also make the matrix singular.
What does diagonal regularization do?
It adds a small value to diagonal variance entries. This can stabilize determinant and inverse calculations for difficult datasets.
Can I export my results?
Yes. Calculate the matrix first. Then use the CSV or PDF button to save the covariance report.