Sample mean and sample covariance

The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables. The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken.

The sample mean is a vector each of whose elements is the sample mean of one of the random variables – that is, each of whose elements is the arithmetic average of the observed values of one of the variables. The sample covariance matrix is a square matrix whose i, j element is the sample covariance (an estimate of the population covariance) between the sets of observed values of two of the variables and whose i, i element is the sample variance of the observed values of one of the variables. If only one variable has had values observed, then the sample mean is a single number (the arithmetic average of the observed values of that variable) and the sample covariance matrix is also simply a single value (a 1x1 matrix containing a single number, the sample variance of the observed values of that variable).

Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics and applications to numerically represent the location and dispersion, respectively, of a distribution.

Let ${\displaystyle x_{ij}}$ be the i^th independently drawn observation (i=1,...,N) on the j^th random variable (j=1,...,K). These observations can be arranged into N column vectors, each with K entries, with the K ×1 column vector giving the i^th observations of all variables being denoted ${\displaystyle \mathbf {x} _{i}}$ (i=1,...,N).

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