the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction). Reliability weights If the weights are instead non-random (reliability weights), we can determine a correction factor to yield an unbiased estimator. Assuming each random variable is sampled from the same distribution with mean and actual variance , taking expectations we have, where and . Therefore, the bias in our estimator is , analogous to the bias in the unweighted estimator (also notice that is the effective sample size). This means that to unbias our estimator we need to pre-divide by , ensuring that the expected value of the estimated variance equals the actual variance of the sampling distribution. The final unbiased estimate of sample variance is: where . The degrees of freedom of the weighted, unbiased sample variance vary accordingly from N − 1 down to 0. The standard deviation is simply the square root of the variance above. As a side note, other approaches have been described to compute the weighted sample variance. Weighted sample covariance In a weighted sample, each row vector (each set of single observations on each of the K random variables) is assigned a weight . Then the weighted mean vector is given by And the weighted covariance matrix is given by: Similarly to weighted sample variance, there are two different unbiased estimators depending on the type of the weights. Frequency weights If the weights are frequency weights, the unbiased weighted estimate of the covariance matrix , with Bessel's correction, is given by: Note that this estimator can be unbiased only if the weights are not standardized nor normalized, these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction). Reliability weights In the case of reliability weights, the weights are normalized: (If they are not, divide the weights by their sum to normalize prior to calculating : Then the weighted mean vector can be simplified to and the unbiased weighted estimate of the covariance matrix is: The reasoning here is the same as in the previous section. Since we are assuming the weights are normalized, then and this reduces to: If all weights are the same, i.e. , then the weighted mean and covariance reduce to the unweighted sample mean and covariance