Editing Covariance (section)

== Calculating the sample covariance ==
{{further|Sample mean and sample covariance}}

The sample covariances among <math>K</math> variables based on <math>N</math> observations of each, drawn from an otherwise unobserved population, are given by the <math>K \times K</math> [[Matrix (mathematics)|matrix]] <math>\textstyle \overline{\mathbf{q}} = \left[q_{jk}\right]</math> with the entries

:<math>q_{jk} = \frac{1}{N - 1}\sum_{i=1}^N \left(X_{ij} - \bar{X}_j\right) \left(X_{ik} - \bar{X}_k\right),</math>

which is an estimate of the covariance between variable <math>j</math> and variable <math>k</math>.

The sample mean and the sample covariance matrix are [[Bias of an estimator|unbiased estimates]] of the [[mean]] and the covariance matrix of the [[random vector]] <math>\textstyle \mathbf{X}</math>, a vector whose ''j''th element <math>(j = 1,\, \ldots,\, K)</math> is one of the random variables. The reason the sample covariance matrix has <math>\textstyle N-1</math> in the denominator rather than <math>\textstyle N</math> is essentially that the population mean <math>\operatorname{E}(\mathbf{X})</math> is not known and is replaced by the sample mean <math>\mathbf{\bar{X}}</math>. If the population mean <math>\operatorname{E}(\mathbf{X})</math> is known, the analogous unbiased estimate is given by

: <math> q_{jk} = \frac{1}{N} \sum_{i=1}^N \left(X_{ij} - \operatorname{E}\left(X_j\right)\right) \left(X_{ik} - \operatorname{E}\left(X_k\right)\right)</math>.