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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.
It is a measure used to evaluate the performance of regression or forecasting models. It is a variant of MAPE in which the mean absolute percent errors is treated as a weighted arithmetic mean. Most commonly the absolute percent errors are weighted by the actuals (e.g. in case of sales forecasting, errors are weighted by sales volume). [3]
A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation. [6] For example, the arithmetic mean of 3 {\displaystyle 3} and 5 {\displaystyle 5} is 3 + 5 2 = 4 {\displaystyle {\frac {3+5}{2}}=4} , or equivalently 3 ⋅ 1 2 + 5 ⋅ 1 2 = 4 ...
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 ().
The arithmetic mean (or simply mean or average) of a list of numbers, is the sum of all of the numbers divided by their count.Similarly, the mean of a sample ,, …,, usually denoted by ¯, is the sum of the sampled values divided by the number of items in the sample.
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 MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled).
In ANOVA, there is a similar usage of grand mean to calculate sum of squares (SSQ), a measurement of variation. The total variation is defined as the sum of squared differences between each score and the grand mean (designated as GM), given by the equation