Multivariate statistics

Objective. Use covariance matrices and multivariate normal models.

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Learn: the key idea

Multivariate statistics studies several correlated random variables jointly, using the covariance matrix Σ to capture how each pair varies together. The multivariate normal distribution with mean vector μ and covariance Σ generalises the familiar bell curve to several dimensions, with density depending on the Mahalanobis distance (x−μ)ᵀΣ⁻¹(x−μ). Off-diagonal entries of Σ give covariances; dividing by the product of standard deviations gives correlations. For example, if two test scores have covariance 12 and standard deviations 4 and 6, their correlation is 12/(4×6) = 0.5, indicating a moderate positive linear relationship between the scores.

Worked example

For a multivariate dataset, variable X has variance 9, Y has variance 16, and their covariance is 6. Find the correlation coefficient between X and Y.

1 $Correlation formula: ρ = Cov(X,Y)/(σx σy)$
2 $σx = \sqrt9 = 3, σy = \sqrt16 = 4$
3 $ρ = 6/(3\times4) = 6/12$
4 $Simplify: ρ = 0.5$

Answer: 0.5

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Common mistake. Students assume zero covariance always implies independence; this only holds for jointly normal variables. For general distributions, uncorrelated variables can still be dependent, so independence must be checked separately, not inferred from Σ alone.

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