with a high weight contribute more to the weighted mean than do elements with a low weight. The weights cannot be negative. Some may be zero, but not all of them (since division by zero is not allowed). The formulas are simplified when the weights are normalized such that they sum up to 1, i.e., . For such normalized weights, the weighted mean is equivalently: . Note that one can always normalize the weights by making the following transformation on the original weights: . The ordinary mean is a special case of the weighted mean where all data have equal weights. If the data elements are independent and identically distributed random variables with variance , the standard error of the weighted mean, , can be shown via uncertainty propagation to be: Variance-defined weights For the weighted mean of a list of data for which each element potentially comes from a different probability distribution with known variance , all having the same mean, one possible choice for the weights is given by the reciprocal of variance: The weighted mean in this case is: and the standard error of the weighted mean (with inverse-variance weights) is: Note this reduces to when all . It is a special case of the general formula in previous section, The equations above can be combined to obtain: The significance of this choice is that this weighted mean is the maximum likelihood estimator of the mean of the probability distributions under the assumption that they are independent and normally distributed with the same mean. Statistical properties Expectancy The weighted sample mean, , is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one). If the observations have expected values then the weighted sample mean has expectation In particular, if the means are equal, , then the expectation of the weighted sample mean will be that value, Variance Simple i.i.d. case When treating the weights as constants, and having a sample of n observations from uncorrelated random variables, all with the same variance and expectation (as is the case for i.i.d random variables), then the variance of the weighted mean can be estimated as the multiplication of the variance by Kish's design effect (see proof): With , , and However, this estimation is rather