LaTeX
\Sigma=\mathbb E[(X-\mu)(X-\mu)^T]Variables
- X: random vector
- μ: mean vector
- Σ: covariance matrix
How to use this formula
Collects variances and pairwise covariances of a random vector.
Important notes
- The matrix is symmetric and positive semidefinite.
Quick example
Diagonal entries are variances; off-diagonal entries are covariances.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The matrix is symmetric and positive semidefinite.
- For the Covariance Matrix, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions.
Worked example
Diagonal entries are variances; off-diagonal entries are covariances.
Common mistakes
- Do not substitute sample and population quantities interchangeably in Covariance Matrix; map every symbol to its definition first.
- For the Covariance Matrix, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Covariance Matrix in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\Sigma=\mathbb E[(X-\mu)(X-\mu)^T]. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Covariance Matrix used for?
Collects variances and pairwise covariances of a random vector.
Can I copy this formula as LaTeX?
Yes. Copy \Sigma=\mathbb E[(X-\mu)(X-\mu)^T] or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.