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The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others.
The maximum likelihood method weights the difference between fit and data using the same weights . 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 ...
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 ().
Data can be binary, ordinal, or continuous variables. It works by normalizing the differences between each pair of variables and then computing a weighted average of these differences. The distance was defined in 1971 by Gower [1] and it takes values between 0 and 1 with smaller values indicating higher similarity.
The weighted harmonic mean is the preferable method for averaging multiples, such as the price–earnings ratio (P/E). If these ratios are averaged using a weighted arithmetic mean, high data points are given greater weights than low data points. The weighted harmonic mean, on the other hand, correctly weights each data point. [14]
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 triangular distribution has a mean equal to the average of the three parameters: μ = a + b + c 3 {\displaystyle \mu ={\frac {a+b+c}{3}}} which (unlike PERT) places equal emphasis on the extreme values which are usually less-well known than the most likely value, and is therefore less reliable.
For any q > 0 and non-negative weights summing to 1, the following inequality holds: (=) / = (=) /. The proof follows from Jensen's inequality , making use of the fact the logarithm is concave: log ∏ i = 1 n x i w i = ∑ i = 1 n w i log x i ≤ log ∑ i = 1 n w i x i . {\displaystyle \log \prod _{i=1}^{n}x_{i}^{w_{i}}=\sum _{i=1 ...