The sample mean and covariance estimate the first two moments of a random vector from repeated observations.
The uncentered empirical second-moment matrix averages the outer products of the observations. Its expected value is . It is an unbiased estimator of the covariance matrix when the population expected value is known to be zero. It differs from a sample covariance matrix centered at the sample mean.
For observations , the sample covariance matrix is , or the same sum divided by under the unbiased convention.
If and , then . In particular full column rank of raw does not ensure a nonsingular centered sample covariance matrix. When is singular, its Gaussian precision profile likelihood has no finite maximizer: increase precision along a nonzero null vector.
For paired scalar observations , their sample covariance is , or the same sum divided by under the unbiased convention. It is an off-diagonal entry of the sample covariance matrix of the paired vectors.
The effective rank measures the total variance of a covariance matrix relative to its largest directional variance.
For independent centered Gaussian vectors with covariance and uncentered sample covariance , with probability at least ,
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Sample mean and covariance are statistical measures that help describe the properties of a dataset. ### Sample Mean The **sample mean** is a measure of central tendency that represents the average of a set of observations. It is calculated by summing all the values in the sample and then dividing by the number of observations in that sample.