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If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox .
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average .
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 ...
The inverse-variance weighted average has the least variance among all weighted averages, which can be calculated as (^) = /. If the variances of the measurements are all equal, then the inverse-variance weighted average becomes the simple average. Inverse-variance weighting is typically used in statistical meta-analysis or sensor fusion to ...
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]
In the financial field, and more specifically in the analyses of financial data, a weighted moving average (WMA) has the specific meaning of weights that decrease in arithmetical progression. [4] In an n -day WMA the latest day has weight n , the second latest n − 1 {\displaystyle n-1} , etc., down to one.
Exponential smoothing or exponential moving average (EMA) is a rule of thumb technique for smoothing time series data using the exponential window function. Whereas in the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It is an easily learned ...
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 ...