normally distributed as follows: If the population is not normally distributed, the sample mean is nonetheless approximately normally distributed if n is large and σ2/n < +∞. This is a consequence of the central limit theorem. Weighted samples In a weighted sample, each vector (each set of single observations on each of the K random variables) is assigned a weight . Without loss of generality, assume that the weights are normalized: (If they are not, divide the weights by their sum). Then the weighted mean vector is given by and the elements of the weighted covariance matrix are If all weights are the same, , the weighted mean and covariance reduce to the (biased) sample mean and covariance mentioned above. Criticism The sample mean and sample covariance are not robust statistics, meaning that they are sensitive to outliers. As robustness is often a desired trait, particularly in real-world applications, robust alternatives may prove desirable, notably quantile-based statistics such as the sample median for location, and interquartile range (IQR) for dispersion. Other alternatives include trimming and Winsorising, as in the trimmed mean and the Winsorized mean. See also Estimation of covariance matrices Scatter matrix Unbiased estimation of standard deviation References Covariance and correlation Estimation methods Summary statistics Matrices U-statistics