with the entries given by q_{ij}=\frac{1}{N-1}\sum_{k=1}^{N}\left( x_{ik}-\bar{x}_i \right) \left( x_{jk}-\bar{x}_j \right) . The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random variable \textstyle \mathbf{X}. The reason why the sample covariance matrix has \textstyle N-1 in the denominator rather than