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The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean. [1]
The weighted mean in this case is: ¯ = ¯ (=), (where the order of the matrix–vector product is not commutative), in terms of the covariance of the weighted mean: ¯ = (=), For example, consider the weighted mean of the point [1 0] with high variance in the second component and [0 1] with high variance in the first component.
The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probabilities. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.
For normally distributed random variables inverse-variance weighted averages can also be derived as the maximum likelihood estimate for the true value. Furthermore, from a Bayesian perspective the posterior distribution for the true value given normally distributed observations and a flat prior is a normal distribution with the inverse-variance weighted average as a mean and variance ().
A weighted average is an average that has multiplying factors to give different weights to data at different positions in the sample window. Mathematically, the weighted moving average is the convolution of the data with a fixed weighting function.
The Marshall-Edgeworth index, credited to Marshall (1887) and Edgeworth (1925), [11] is a weighted relative of current period to base period sets of prices. This index uses the arithmetic average of the current and based period quantities for weighting. It is considered a pseudo-superlative formula and is symmetric. [12]
A Bayesian average is a method of estimating the mean of a population using outside information, especially a pre-existing belief, [1] which is factored into the calculation. This is a central feature of Bayesian interpretation. This is useful when the available data set is small. [2] Calculating the Bayesian average uses the prior mean m and a ...
In data analysis based on the Rasch model, the reduced chi-squared statistic is called the outfit mean-square statistic, and the information-weighted reduced chi-squared statistic is called the infit mean-square statistic. [21]
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