Sample mean and covariance

The sample mean vector averages observed data across observations, while the sample covariance matrix summarizes their variances and pairwise covariances.

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Arithmetic mean: The sum of numerical values divided by their count. Each coordinate of the sample mean is the arithmetic mean of that variable’s observed values.
Random vector: A vector whose components are random variables. Population location and covariance describe a random vector that the sample is used to estimate.
Multivariate normal distribution: A probability distribution for a random vector whose every linear combination of components is normally distributed. Its parameters are a mean vector and covariance matrix, often estimated from samples.
Maximum likelihood estimation: A method that selects parameter values maximizing the likelihood of observed data. For a normal model, its covariance estimate uses divisor n, unlike the common unbiased estimate’s n−1.
Sample variance: An estimate of population variance computed from a sample, commonly using deviations from the sample mean. The covariance matrix’s diagonal entries are sample variances.
Expectation (mathematics): The probability-weighted average value of a random variable or vector. The population mean is the expectation that the sample mean estimates.
Mahalanobis distance: A distance measure that scales deviations by a covariance matrix. The sample covariance can scale observations when assessing multivariate distance.
Robust statistics: Statistical methods designed to remain informative when data depart from ideal assumptions or contain outliers. Ordinary sample means and covariances can be strongly distorted by extreme observations.
Sample covariance: An estimate of how two variables vary together, calculated from paired observations. Each off-diagonal matrix entry estimates covariance between a pair of variables.
Covariance matrix: A matrix containing the variances and pairwise covariances of a random vector’s components. The sample covariance matrix estimates this population-level matrix.