Calculating The Sample Covariance
The sample covariance of N observations of K variables is the K-by-K matrix with the entries
- ,
which is an estimate of the covariance between variable j and variable k.
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose jth element (j = 1, ..., K) is one of the random variables. The reason the sample covariance matrix has in the denominator rather than is essentially that the population mean is not known and is replaced by the sample mean . If the population mean is known, the analogous unbiased estimate is given by
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